Henrywood and Agarwal, Equation (12)

Average Accuracy: 59.0% → 79.3%

Time: 56.2s

Precision: binary64

Cost: 21836

Math

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

↓

\[\begin{array}{l} t_0 := 1 + 0.5 \cdot \left(\frac{\frac{M \cdot \left(0.5 \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot \left(D \cdot -0.5\right)}{d}}{\frac{1}{h}}\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;d \leq -1.85 \cdot 10^{+123}:\\ \;\;\;\;\left(\frac{t_2}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot t_0\\ \mathbf{elif}\;d \leq -1.5 \cdot 10^{-303}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \frac{t_2}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+126}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \frac{1}{\frac{\sqrt{\ell}}{\sqrt{d}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(\frac{M}{\frac{\frac{d}{0.5}}{D}}\right)}^{2}}{\ell}\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

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
         (+
          1.0
          (*
           0.5
           (*
            (/ (/ (* M (* 0.5 D)) d) l)
            (/ (/ (* M (* D -0.5)) d) (/ 1.0 h))))))
        (t_1 (sqrt (/ d h)))
        (t_2 (sqrt (- d))))
   (if (<= d -1.85e+123)
     (* (* (/ t_2 (sqrt (- h))) (sqrt (/ d l))) t_0)
     (if (<= d -1.5e-303)
       (* t_0 (* t_1 (/ t_2 (sqrt (- l)))))
       (if (<= d 1.85e+126)
         (* t_0 (* t_1 (/ 1.0 (/ (sqrt l) (sqrt d)))))
         (*
          (+ 1.0 (* -0.5 (* h (/ (pow (/ M (/ (/ d 0.5) D)) 2.0) l))))
          (/ d (* (sqrt l) (sqrt h)))))))))

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 = 1.0 + (0.5 * ((((M * (0.5 * D)) / d) / l) * (((M * (D * -0.5)) / d) / (1.0 / h))));
	double t_1 = sqrt((d / h));
	double t_2 = sqrt(-d);
	double tmp;
	if (d <= -1.85e+123) {
		tmp = ((t_2 / sqrt(-h)) * sqrt((d / l))) * t_0;
	} else if (d <= -1.5e-303) {
		tmp = t_0 * (t_1 * (t_2 / sqrt(-l)));
	} else if (d <= 1.85e+126) {
		tmp = t_0 * (t_1 * (1.0 / (sqrt(l) / sqrt(d))));
	} else {
		tmp = (1.0 + (-0.5 * (h * (pow((M / ((d / 0.5) / D)), 2.0) / l)))) * (d / (sqrt(l) * sqrt(h)));
	}
	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) :: tmp
    t_0 = 1.0d0 + (0.5d0 * ((((m * (0.5d0 * d_1)) / d) / l) * (((m * (d_1 * (-0.5d0))) / d) / (1.0d0 / h))))
    t_1 = sqrt((d / h))
    t_2 = sqrt(-d)
    if (d <= (-1.85d+123)) then
        tmp = ((t_2 / sqrt(-h)) * sqrt((d / l))) * t_0
    else if (d <= (-1.5d-303)) then
        tmp = t_0 * (t_1 * (t_2 / sqrt(-l)))
    else if (d <= 1.85d+126) then
        tmp = t_0 * (t_1 * (1.0d0 / (sqrt(l) / sqrt(d))))
    else
        tmp = (1.0d0 + ((-0.5d0) * (h * (((m / ((d / 0.5d0) / d_1)) ** 2.0d0) / l)))) * (d / (sqrt(l) * sqrt(h)))
    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 = 1.0 + (0.5 * ((((M * (0.5 * D)) / d) / l) * (((M * (D * -0.5)) / d) / (1.0 / h))));
	double t_1 = Math.sqrt((d / h));
	double t_2 = Math.sqrt(-d);
	double tmp;
	if (d <= -1.85e+123) {
		tmp = ((t_2 / Math.sqrt(-h)) * Math.sqrt((d / l))) * t_0;
	} else if (d <= -1.5e-303) {
		tmp = t_0 * (t_1 * (t_2 / Math.sqrt(-l)));
	} else if (d <= 1.85e+126) {
		tmp = t_0 * (t_1 * (1.0 / (Math.sqrt(l) / Math.sqrt(d))));
	} else {
		tmp = (1.0 + (-0.5 * (h * (Math.pow((M / ((d / 0.5) / D)), 2.0) / l)))) * (d / (Math.sqrt(l) * Math.sqrt(h)));
	}
	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 = 1.0 + (0.5 * ((((M * (0.5 * D)) / d) / l) * (((M * (D * -0.5)) / d) / (1.0 / h))))
	t_1 = math.sqrt((d / h))
	t_2 = math.sqrt(-d)
	tmp = 0
	if d <= -1.85e+123:
		tmp = ((t_2 / math.sqrt(-h)) * math.sqrt((d / l))) * t_0
	elif d <= -1.5e-303:
		tmp = t_0 * (t_1 * (t_2 / math.sqrt(-l)))
	elif d <= 1.85e+126:
		tmp = t_0 * (t_1 * (1.0 / (math.sqrt(l) / math.sqrt(d))))
	else:
		tmp = (1.0 + (-0.5 * (h * (math.pow((M / ((d / 0.5) / D)), 2.0) / l)))) * (d / (math.sqrt(l) * math.sqrt(h)))
	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 = Float64(1.0 + Float64(0.5 * Float64(Float64(Float64(Float64(M * Float64(0.5 * D)) / d) / l) * Float64(Float64(Float64(M * Float64(D * -0.5)) / d) / Float64(1.0 / h)))))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(-d))
	tmp = 0.0
	if (d <= -1.85e+123)
		tmp = Float64(Float64(Float64(t_2 / sqrt(Float64(-h))) * sqrt(Float64(d / l))) * t_0);
	elseif (d <= -1.5e-303)
		tmp = Float64(t_0 * Float64(t_1 * Float64(t_2 / sqrt(Float64(-l)))));
	elseif (d <= 1.85e+126)
		tmp = Float64(t_0 * Float64(t_1 * Float64(1.0 / Float64(sqrt(l) / sqrt(d)))));
	else
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(h * Float64((Float64(M / Float64(Float64(d / 0.5) / D)) ^ 2.0) / l)))) * Float64(d / Float64(sqrt(l) * sqrt(h))));
	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 = 1.0 + (0.5 * ((((M * (0.5 * D)) / d) / l) * (((M * (D * -0.5)) / d) / (1.0 / h))));
	t_1 = sqrt((d / h));
	t_2 = sqrt(-d);
	tmp = 0.0;
	if (d <= -1.85e+123)
		tmp = ((t_2 / sqrt(-h)) * sqrt((d / l))) * t_0;
	elseif (d <= -1.5e-303)
		tmp = t_0 * (t_1 * (t_2 / sqrt(-l)));
	elseif (d <= 1.85e+126)
		tmp = t_0 * (t_1 * (1.0 / (sqrt(l) / sqrt(d))));
	else
		tmp = (1.0 + (-0.5 * (h * (((M / ((d / 0.5) / D)) ^ 2.0) / l)))) * (d / (sqrt(l) * sqrt(h)));
	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[(1.0 + N[(0.5 * N[(N[(N[(N[(M * N[(0.5 * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(M * N[(D * -0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / N[(1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[d, -1.85e+123], N[(N[(N[(t$95$2 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[d, -1.5e-303], N[(t$95$0 * N[(t$95$1 * N[(t$95$2 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.85e+126], N[(t$95$0 * N[(t$95$1 * N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] / N[Sqrt[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(-0.5 * N[(h * N[(N[Power[N[(M / N[(N[(d / 0.5), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.84999999999999998e123

   1. Initial program 59.0%

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

   2. Simplified 59.7%

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

      [Start] 59.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 59.0	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 59.0	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 59.0	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 59.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 59.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 59.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 59.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 58.7%

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

      [Start] 59.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      frac-2neg [=>] 59.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \color{blue}{\frac{-h}{-\ell}}\right)\right) \]
      associate-*r/ [=>] 58.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(-h\right)}{-\ell}}\right) \]
      div-inv [=>] 58.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot \left(-h\right)}{-\ell}\right) \]
      metadata-eval [=>] 58.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(-h\right)}{-\ell}\right) \]

   4. Simplified 59.2%

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

      [Start] 58.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(-h\right)}{-\ell}\right) \]
      associate-/l* [=>] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{-\ell}{-h}}}\right) \]
      *-commutative [=>] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\color{blue}{\left(0.5 \cdot M\right)} \cdot \frac{D}{d}\right)}^{2}}{\frac{-\ell}{-h}}\right) \]
      associate-*r* [<=] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}}^{2}}{\frac{-\ell}{-h}}\right) \]
      metadata-eval [<=] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\color{blue}{\frac{1}{2}} \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{-\ell}{-h}}\right) \]
      associate-/r/ [<=] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{1}{\frac{2}{M \cdot \frac{D}{d}}}\right)}}^{2}}{\frac{-\ell}{-h}}\right) \]
      associate-*r/ [=>] 59.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\frac{1}{\frac{2}{\color{blue}{\frac{M \cdot D}{d}}}}\right)}^{2}}{\frac{-\ell}{-h}}\right) \]
      associate-/l* [<=] 59.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\frac{1}{\color{blue}{\frac{2 \cdot d}{M \cdot D}}}\right)}^{2}}{\frac{-\ell}{-h}}\right) \]
      associate-/r/ [=>] 59.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{1}{2 \cdot d} \cdot \left(M \cdot D\right)\right)}}^{2}}{\frac{-\ell}{-h}}\right) \]
      *-commutative [=>] 59.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}}^{2}}{\frac{-\ell}{-h}}\right) \]
      *-commutative [<=] 59.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2 \cdot d}\right)}^{2}}{\frac{-\ell}{-h}}\right) \]
      associate-/r* [=>] 59.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(D \cdot M\right) \cdot \color{blue}{\frac{\frac{1}{2}}{d}}\right)}^{2}}{\frac{-\ell}{-h}}\right) \]
      metadata-eval [=>] 59.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(D \cdot M\right) \cdot \frac{\color{blue}{0.5}}{d}\right)}^{2}}{\frac{-\ell}{-h}}\right) \]
      neg-mul-1 [=>] 59.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}{\frac{\color{blue}{-1 \cdot \ell}}{-h}}\right) \]
      neg-mul-1 [=>] 59.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}{\frac{-1 \cdot \ell}{\color{blue}{-1 \cdot h}}}\right) \]
      times-frac [=>] 59.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}{\color{blue}{\frac{-1}{-1} \cdot \frac{\ell}{h}}}\right) \]
      metadata-eval [=>] 59.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}{\color{blue}{1} \cdot \frac{\ell}{h}}\right) \]

   5. Applied egg-rr 59.7%

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

      [Start] 59.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}{1 \cdot \frac{\ell}{h}}\right) \]
      unpow2 [=>] 59.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}}{1 \cdot \frac{\ell}{h}}\right) \]
      *-un-lft-identity [<=] 59.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}{\color{blue}{\frac{\ell}{h}}}\right) \]
      div-inv [=>] 59.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]
      times-frac [=>] 59.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{\left(D \cdot M\right) \cdot \frac{0.5}{d}}{\ell} \cdot \frac{\left(D \cdot M\right) \cdot \frac{0.5}{d}}{\frac{1}{h}}\right)}\right) \]
      associate-*r/ [=>] 59.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\color{blue}{\frac{\left(D \cdot M\right) \cdot 0.5}{d}}}{\ell} \cdot \frac{\left(D \cdot M\right) \cdot \frac{0.5}{d}}{\frac{1}{h}}\right)\right) \]
      *-commutative [=>] 59.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\frac{\color{blue}{\left(M \cdot D\right)} \cdot 0.5}{d}}{\ell} \cdot \frac{\left(D \cdot M\right) \cdot \frac{0.5}{d}}{\frac{1}{h}}\right)\right) \]
      associate-*l* [=>] 59.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\frac{\color{blue}{M \cdot \left(D \cdot 0.5\right)}}{d}}{\ell} \cdot \frac{\left(D \cdot M\right) \cdot \frac{0.5}{d}}{\frac{1}{h}}\right)\right) \]
      associate-*r/ [=>] 59.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\ell} \cdot \frac{\color{blue}{\frac{\left(D \cdot M\right) \cdot 0.5}{d}}}{\frac{1}{h}}\right)\right) \]
      *-commutative [=>] 59.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\ell} \cdot \frac{\frac{\color{blue}{\left(M \cdot D\right)} \cdot 0.5}{d}}{\frac{1}{h}}\right)\right) \]
      associate-*l* [=>] 59.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\ell} \cdot \frac{\frac{\color{blue}{M \cdot \left(D \cdot 0.5\right)}}{d}}{\frac{1}{h}}\right)\right) \]

   6. Applied egg-rr 80.7%

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

      [Start] 59.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\frac{1}{h}}\right)\right) \]
      frac-2neg [=>] 59.7	\[ \left(\sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\frac{1}{h}}\right)\right) \]
      sqrt-div [=>] 80.7	\[ \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\frac{1}{h}}\right)\right) \]

   if -1.84999999999999998e123 < d < -1.50000000000000014e-303

   1. Initial program 59.5%

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

   2. Simplified 58.3%

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

      [Start] 59.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 59.5	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 59.5	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 59.5	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 59.5	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 59.5	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 59.5	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 58.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 60.6%

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

      [Start] 58.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      frac-2neg [=>] 58.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \color{blue}{\frac{-h}{-\ell}}\right)\right) \]
      associate-*r/ [=>] 60.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(-h\right)}{-\ell}}\right) \]
      div-inv [=>] 60.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot \left(-h\right)}{-\ell}\right) \]
      metadata-eval [=>] 60.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(-h\right)}{-\ell}\right) \]

   4. Simplified 59.9%

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

      [Start] 60.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(-h\right)}{-\ell}\right) \]
      associate-/l* [=>] 58.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{-\ell}{-h}}}\right) \]
      *-commutative [=>] 58.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\color{blue}{\left(0.5 \cdot M\right)} \cdot \frac{D}{d}\right)}^{2}}{\frac{-\ell}{-h}}\right) \]
      associate-*r* [<=] 58.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}}^{2}}{\frac{-\ell}{-h}}\right) \]
      metadata-eval [<=] 58.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\color{blue}{\frac{1}{2}} \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{-\ell}{-h}}\right) \]
      associate-/r/ [<=] 58.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{1}{\frac{2}{M \cdot \frac{D}{d}}}\right)}}^{2}}{\frac{-\ell}{-h}}\right) \]
      associate-*r/ [=>] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\frac{1}{\frac{2}{\color{blue}{\frac{M \cdot D}{d}}}}\right)}^{2}}{\frac{-\ell}{-h}}\right) \]
      associate-/l* [<=] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\frac{1}{\color{blue}{\frac{2 \cdot d}{M \cdot D}}}\right)}^{2}}{\frac{-\ell}{-h}}\right) \]
      associate-/r/ [=>] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{1}{2 \cdot d} \cdot \left(M \cdot D\right)\right)}}^{2}}{\frac{-\ell}{-h}}\right) \]
      *-commutative [=>] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}}^{2}}{\frac{-\ell}{-h}}\right) \]
      *-commutative [<=] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2 \cdot d}\right)}^{2}}{\frac{-\ell}{-h}}\right) \]
      associate-/r* [=>] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(D \cdot M\right) \cdot \color{blue}{\frac{\frac{1}{2}}{d}}\right)}^{2}}{\frac{-\ell}{-h}}\right) \]
      metadata-eval [=>] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(D \cdot M\right) \cdot \frac{\color{blue}{0.5}}{d}\right)}^{2}}{\frac{-\ell}{-h}}\right) \]
      neg-mul-1 [=>] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}{\frac{\color{blue}{-1 \cdot \ell}}{-h}}\right) \]
      neg-mul-1 [=>] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}{\frac{-1 \cdot \ell}{\color{blue}{-1 \cdot h}}}\right) \]
      times-frac [=>] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}{\color{blue}{\frac{-1}{-1} \cdot \frac{\ell}{h}}}\right) \]
      metadata-eval [=>] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}{\color{blue}{1} \cdot \frac{\ell}{h}}\right) \]

   5. Applied egg-rr 67.4%

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

      [Start] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}{1 \cdot \frac{\ell}{h}}\right) \]
      unpow2 [=>] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}}{1 \cdot \frac{\ell}{h}}\right) \]
      *-un-lft-identity [<=] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}{\color{blue}{\frac{\ell}{h}}}\right) \]
      div-inv [=>] 59.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]
      times-frac [=>] 67.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{\left(D \cdot M\right) \cdot \frac{0.5}{d}}{\ell} \cdot \frac{\left(D \cdot M\right) \cdot \frac{0.5}{d}}{\frac{1}{h}}\right)}\right) \]
      associate-*r/ [=>] 67.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\color{blue}{\frac{\left(D \cdot M\right) \cdot 0.5}{d}}}{\ell} \cdot \frac{\left(D \cdot M\right) \cdot \frac{0.5}{d}}{\frac{1}{h}}\right)\right) \]
      *-commutative [=>] 67.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\frac{\color{blue}{\left(M \cdot D\right)} \cdot 0.5}{d}}{\ell} \cdot \frac{\left(D \cdot M\right) \cdot \frac{0.5}{d}}{\frac{1}{h}}\right)\right) \]
      associate-*l* [=>] 67.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\frac{\color{blue}{M \cdot \left(D \cdot 0.5\right)}}{d}}{\ell} \cdot \frac{\left(D \cdot M\right) \cdot \frac{0.5}{d}}{\frac{1}{h}}\right)\right) \]
      associate-*r/ [=>] 67.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\ell} \cdot \frac{\color{blue}{\frac{\left(D \cdot M\right) \cdot 0.5}{d}}}{\frac{1}{h}}\right)\right) \]
      *-commutative [=>] 67.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\ell} \cdot \frac{\frac{\color{blue}{\left(M \cdot D\right)} \cdot 0.5}{d}}{\frac{1}{h}}\right)\right) \]
      associate-*l* [=>] 67.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\ell} \cdot \frac{\frac{\color{blue}{M \cdot \left(D \cdot 0.5\right)}}{d}}{\frac{1}{h}}\right)\right) \]

   6. Applied egg-rr 76.9%

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

      [Start] 67.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\frac{1}{h}}\right)\right) \]
      frac-2neg [=>] 67.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{-d}{-\ell}}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\frac{1}{h}}\right)\right) \]
      sqrt-div [=>] 76.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\frac{1}{h}}\right)\right) \]

   if -1.50000000000000014e-303 < d < 1.8499999999999999e126

   1. Initial program 59.6%

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

   2. Simplified 58.4%

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

      [Start] 59.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 59.6	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 59.6	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 59.6	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 59.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 59.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 59.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 58.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 59.8%

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

      [Start] 58.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      frac-2neg [=>] 58.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \color{blue}{\frac{-h}{-\ell}}\right)\right) \]
      associate-*r/ [=>] 59.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(-h\right)}{-\ell}}\right) \]
      div-inv [=>] 59.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot \left(-h\right)}{-\ell}\right) \]
      metadata-eval [=>] 59.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(-h\right)}{-\ell}\right) \]

   4. Simplified 59.9%

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

      [Start] 59.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(-h\right)}{-\ell}\right) \]
      associate-/l* [=>] 58.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{-\ell}{-h}}}\right) \]
      *-commutative [=>] 58.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\color{blue}{\left(0.5 \cdot M\right)} \cdot \frac{D}{d}\right)}^{2}}{\frac{-\ell}{-h}}\right) \]
      associate-*r* [<=] 58.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}}^{2}}{\frac{-\ell}{-h}}\right) \]
      metadata-eval [<=] 58.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\color{blue}{\frac{1}{2}} \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{-\ell}{-h}}\right) \]
      associate-/r/ [<=] 58.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{1}{\frac{2}{M \cdot \frac{D}{d}}}\right)}}^{2}}{\frac{-\ell}{-h}}\right) \]
      associate-*r/ [=>] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\frac{1}{\frac{2}{\color{blue}{\frac{M \cdot D}{d}}}}\right)}^{2}}{\frac{-\ell}{-h}}\right) \]
      associate-/l* [<=] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\frac{1}{\color{blue}{\frac{2 \cdot d}{M \cdot D}}}\right)}^{2}}{\frac{-\ell}{-h}}\right) \]
      associate-/r/ [=>] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{1}{2 \cdot d} \cdot \left(M \cdot D\right)\right)}}^{2}}{\frac{-\ell}{-h}}\right) \]
      *-commutative [=>] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}}^{2}}{\frac{-\ell}{-h}}\right) \]
      *-commutative [<=] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2 \cdot d}\right)}^{2}}{\frac{-\ell}{-h}}\right) \]
      associate-/r* [=>] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(D \cdot M\right) \cdot \color{blue}{\frac{\frac{1}{2}}{d}}\right)}^{2}}{\frac{-\ell}{-h}}\right) \]
      metadata-eval [=>] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(D \cdot M\right) \cdot \frac{\color{blue}{0.5}}{d}\right)}^{2}}{\frac{-\ell}{-h}}\right) \]
      neg-mul-1 [=>] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}{\frac{\color{blue}{-1 \cdot \ell}}{-h}}\right) \]
      neg-mul-1 [=>] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}{\frac{-1 \cdot \ell}{\color{blue}{-1 \cdot h}}}\right) \]
      times-frac [=>] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}{\color{blue}{\frac{-1}{-1} \cdot \frac{\ell}{h}}}\right) \]
      metadata-eval [=>] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}{\color{blue}{1} \cdot \frac{\ell}{h}}\right) \]

   5. Applied egg-rr 66.2%

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

      [Start] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}{1 \cdot \frac{\ell}{h}}\right) \]
      unpow2 [=>] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}}{1 \cdot \frac{\ell}{h}}\right) \]
      *-un-lft-identity [<=] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}{\color{blue}{\frac{\ell}{h}}}\right) \]
      div-inv [=>] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]
      times-frac [=>] 66.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{\left(D \cdot M\right) \cdot \frac{0.5}{d}}{\ell} \cdot \frac{\left(D \cdot M\right) \cdot \frac{0.5}{d}}{\frac{1}{h}}\right)}\right) \]
      associate-*r/ [=>] 66.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\color{blue}{\frac{\left(D \cdot M\right) \cdot 0.5}{d}}}{\ell} \cdot \frac{\left(D \cdot M\right) \cdot \frac{0.5}{d}}{\frac{1}{h}}\right)\right) \]
      *-commutative [=>] 66.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\frac{\color{blue}{\left(M \cdot D\right)} \cdot 0.5}{d}}{\ell} \cdot \frac{\left(D \cdot M\right) \cdot \frac{0.5}{d}}{\frac{1}{h}}\right)\right) \]
      associate-*l* [=>] 66.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\frac{\color{blue}{M \cdot \left(D \cdot 0.5\right)}}{d}}{\ell} \cdot \frac{\left(D \cdot M\right) \cdot \frac{0.5}{d}}{\frac{1}{h}}\right)\right) \]
      associate-*r/ [=>] 66.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\ell} \cdot \frac{\color{blue}{\frac{\left(D \cdot M\right) \cdot 0.5}{d}}}{\frac{1}{h}}\right)\right) \]
      *-commutative [=>] 66.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\ell} \cdot \frac{\frac{\color{blue}{\left(M \cdot D\right)} \cdot 0.5}{d}}{\frac{1}{h}}\right)\right) \]
      associate-*l* [=>] 66.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\ell} \cdot \frac{\frac{\color{blue}{M \cdot \left(D \cdot 0.5\right)}}{d}}{\frac{1}{h}}\right)\right) \]

   6. Applied egg-rr 74.4%

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

      [Start] 66.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\frac{1}{h}}\right)\right) \]
      sqrt-div [=>] 74.5	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\frac{1}{h}}\right)\right) \]
      clear-num [=>] 74.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\sqrt{d}}}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\frac{1}{h}}\right)\right) \]

   if 1.8499999999999999e126 < d

   1. Initial program 56.2%

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

   2. Simplified 56.9%

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

      [Start] 56.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 56.2	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 56.2	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 56.2	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 56.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 56.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 56.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 56.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 56.3%

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

      [Start] 56.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      frac-2neg [=>] 56.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \color{blue}{\frac{-h}{-\ell}}\right)\right) \]
      associate-*r/ [=>] 56.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(-h\right)}{-\ell}}\right) \]
      div-inv [=>] 56.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot \left(-h\right)}{-\ell}\right) \]
      metadata-eval [=>] 56.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(-h\right)}{-\ell}\right) \]

   4. Simplified 56.7%

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

      [Start] 56.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(-h\right)}{-\ell}\right) \]
      associate-/l* [=>] 57.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{-\ell}{-h}}}\right) \]
      *-commutative [=>] 57.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\color{blue}{\left(0.5 \cdot M\right)} \cdot \frac{D}{d}\right)}^{2}}{\frac{-\ell}{-h}}\right) \]
      associate-*r* [<=] 57.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}}^{2}}{\frac{-\ell}{-h}}\right) \]
      metadata-eval [<=] 57.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\color{blue}{\frac{1}{2}} \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{-\ell}{-h}}\right) \]
      associate-/r/ [<=] 57.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{1}{\frac{2}{M \cdot \frac{D}{d}}}\right)}}^{2}}{\frac{-\ell}{-h}}\right) \]
      associate-*r/ [=>] 56.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\frac{1}{\frac{2}{\color{blue}{\frac{M \cdot D}{d}}}}\right)}^{2}}{\frac{-\ell}{-h}}\right) \]
      associate-/l* [<=] 56.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\frac{1}{\color{blue}{\frac{2 \cdot d}{M \cdot D}}}\right)}^{2}}{\frac{-\ell}{-h}}\right) \]
      associate-/r/ [=>] 56.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{1}{2 \cdot d} \cdot \left(M \cdot D\right)\right)}}^{2}}{\frac{-\ell}{-h}}\right) \]
      *-commutative [=>] 56.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}}^{2}}{\frac{-\ell}{-h}}\right) \]
      *-commutative [<=] 56.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2 \cdot d}\right)}^{2}}{\frac{-\ell}{-h}}\right) \]
      associate-/r* [=>] 56.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(D \cdot M\right) \cdot \color{blue}{\frac{\frac{1}{2}}{d}}\right)}^{2}}{\frac{-\ell}{-h}}\right) \]
      metadata-eval [=>] 56.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(D \cdot M\right) \cdot \frac{\color{blue}{0.5}}{d}\right)}^{2}}{\frac{-\ell}{-h}}\right) \]
      neg-mul-1 [=>] 56.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}{\frac{\color{blue}{-1 \cdot \ell}}{-h}}\right) \]
      neg-mul-1 [=>] 56.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}{\frac{-1 \cdot \ell}{\color{blue}{-1 \cdot h}}}\right) \]
      times-frac [=>] 56.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}{\color{blue}{\frac{-1}{-1} \cdot \frac{\ell}{h}}}\right) \]
      metadata-eval [=>] 56.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}{\color{blue}{1} \cdot \frac{\ell}{h}}\right) \]

   5. Applied egg-rr 56.7%

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

      [Start] 56.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}{1 \cdot \frac{\ell}{h}}\right) \]
      unpow2 [=>] 56.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}}{1 \cdot \frac{\ell}{h}}\right) \]
      *-un-lft-identity [<=] 56.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}{\color{blue}{\frac{\ell}{h}}}\right) \]
      div-inv [=>] 56.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]
      times-frac [=>] 56.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{\left(D \cdot M\right) \cdot \frac{0.5}{d}}{\ell} \cdot \frac{\left(D \cdot M\right) \cdot \frac{0.5}{d}}{\frac{1}{h}}\right)}\right) \]
      associate-*r/ [=>] 56.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\color{blue}{\frac{\left(D \cdot M\right) \cdot 0.5}{d}}}{\ell} \cdot \frac{\left(D \cdot M\right) \cdot \frac{0.5}{d}}{\frac{1}{h}}\right)\right) \]
      *-commutative [=>] 56.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\frac{\color{blue}{\left(M \cdot D\right)} \cdot 0.5}{d}}{\ell} \cdot \frac{\left(D \cdot M\right) \cdot \frac{0.5}{d}}{\frac{1}{h}}\right)\right) \]
      associate-*l* [=>] 56.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\frac{\color{blue}{M \cdot \left(D \cdot 0.5\right)}}{d}}{\ell} \cdot \frac{\left(D \cdot M\right) \cdot \frac{0.5}{d}}{\frac{1}{h}}\right)\right) \]
      associate-*r/ [=>] 56.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\ell} \cdot \frac{\color{blue}{\frac{\left(D \cdot M\right) \cdot 0.5}{d}}}{\frac{1}{h}}\right)\right) \]
      *-commutative [=>] 56.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\ell} \cdot \frac{\frac{\color{blue}{\left(M \cdot D\right)} \cdot 0.5}{d}}{\frac{1}{h}}\right)\right) \]
      associate-*l* [=>] 56.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\ell} \cdot \frac{\frac{\color{blue}{M \cdot \left(D \cdot 0.5\right)}}{d}}{\frac{1}{h}}\right)\right) \]

   6. Applied egg-rr 96.1%

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

      [Start] 56.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\frac{1}{h}}\right)\right) \]
      cancel-sign-sub-inv [=>] 56.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\left(1 + \left(-0.5\right) \cdot \left(\frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\frac{1}{h}}\right)\right)} \]
      distribute-lft-in [=>] 56.7	\[ \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot 1 + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(\left(-0.5\right) \cdot \left(\frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\frac{1}{h}}\right)\right)} \]
      *-commutative [<=] 56.7	\[ \color{blue}{1 \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(\left(-0.5\right) \cdot \left(\frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\frac{1}{h}}\right)\right) \]
      *-un-lft-identity [<=] 56.7	\[ \color{blue}{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(\left(-0.5\right) \cdot \left(\frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\frac{1}{h}}\right)\right) \]
      sqrt-div [=>] 56.6	\[ \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(\left(-0.5\right) \cdot \left(\frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\frac{1}{h}}\right)\right) \]
      sqrt-div [=>] 56.5	\[ \frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(\left(-0.5\right) \cdot \left(\frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\frac{1}{h}}\right)\right) \]
      frac-times [=>] 56.4	\[ \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(\left(-0.5\right) \cdot \left(\frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\frac{1}{h}}\right)\right) \]
      add-sqr-sqrt [<=] 56.6	\[ \frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(\left(-0.5\right) \cdot \left(\frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\frac{1}{h}}\right)\right) \]

   7. Simplified 96.1%

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

      [Start] 96.1	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(\left(\frac{{\left(\frac{M}{\frac{\frac{d}{0.5}}{D}}\right)}^{2}}{\ell} \cdot h\right) \cdot -0.5\right) \]
      *-rgt-identity [<=] 96.1	\[ \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot 1} + \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(\left(\frac{{\left(\frac{M}{\frac{\frac{d}{0.5}}{D}}\right)}^{2}}{\ell} \cdot h\right) \cdot -0.5\right) \]
      distribute-lft-in [<=] 96.1	\[ \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(\frac{{\left(\frac{M}{\frac{\frac{d}{0.5}}{D}}\right)}^{2}}{\ell} \cdot h\right) \cdot -0.5\right)} \]
      *-commutative [=>] 96.1	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \color{blue}{-0.5 \cdot \left(\frac{{\left(\frac{M}{\frac{\frac{d}{0.5}}{D}}\right)}^{2}}{\ell} \cdot h\right)}\right) \]
      *-commutative [=>] 96.1	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(\frac{M}{\frac{\frac{d}{0.5}}{D}}\right)}^{2}}{\ell}\right)}\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.85 \cdot 10^{+123}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(\frac{\frac{M \cdot \left(0.5 \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot \left(D \cdot -0.5\right)}{d}}{\frac{1}{h}}\right)\right)\\ \mathbf{elif}\;d \leq -1.5 \cdot 10^{-303}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(\frac{\frac{M \cdot \left(0.5 \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot \left(D \cdot -0.5\right)}{d}}{\frac{1}{h}}\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+126}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(\frac{\frac{M \cdot \left(0.5 \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot \left(D \cdot -0.5\right)}{d}}{\frac{1}{h}}\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \frac{1}{\frac{\sqrt{\ell}}{\sqrt{d}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(\frac{M}{\frac{\frac{d}{0.5}}{D}}\right)}^{2}}{\ell}\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	73.0%
Cost	21972

\[\begin{array}{l} t_0 := \frac{M \cdot \left(D \cdot -0.5\right)}{d}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}} \cdot t_1\\ t_3 := \frac{M \cdot \left(0.5 \cdot D\right)}{d}\\ \mathbf{if}\;d \leq -5 \cdot 10^{+122}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq -3 \cdot 10^{-112}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(t_3 \cdot \left(\frac{h}{\ell} \cdot t_0\right)\right)\right) \cdot t_2\\ \mathbf{elif}\;d \leq -1.05 \cdot 10^{-244}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot 0.125\right) - d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -1.5 \cdot 10^{-303}:\\ \;\;\;\;t_2 \cdot \left(1 + 0.5 \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{D \cdot \left(M \cdot -0.5\right)}{d \cdot \frac{\ell}{h}}\right)\right)\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{+126}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(\frac{t_3}{\ell} \cdot \frac{t_0}{\frac{1}{h}}\right)\right) \cdot \left(t_1 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(\frac{M}{\frac{\frac{d}{0.5}}{D}}\right)}^{2}}{\ell}\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 2
Accuracy	75.7%
Cost	21836

\[\begin{array}{l} t_0 := \frac{M \cdot \left(D \cdot -0.5\right)}{d}\\ t_1 := \frac{M \cdot \left(0.5 \cdot D\right)}{d}\\ \mathbf{if}\;d \leq -3.7 \cdot 10^{-143}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(t_1 \cdot \left(\frac{h}{\ell} \cdot t_0\right)\right)\right)\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{-284}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot 0.125\right) - d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq 2.35 \cdot 10^{+126}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(\frac{t_1}{\ell} \cdot \frac{t_0}{\frac{1}{h}}\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \frac{1}{\frac{\sqrt{\ell}}{\sqrt{d}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(\frac{M}{\frac{\frac{d}{0.5}}{D}}\right)}^{2}}{\ell}\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 3
Accuracy	79.2%
Cost	21836

\[\begin{array}{l} t_0 := \frac{M \cdot \left(D \cdot -0.5\right)}{d}\\ t_1 := \frac{M \cdot \left(0.5 \cdot D\right)}{d}\\ t_2 := 1 + 0.5 \cdot \left(\frac{t_1}{\ell} \cdot \frac{t_0}{\frac{1}{h}}\right)\\ t_3 := \sqrt{-d}\\ t_4 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -2.15 \cdot 10^{+123}:\\ \;\;\;\;\left(\frac{t_3}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(t_1 \cdot \left(\frac{h}{\ell} \cdot t_0\right)\right)\right)\\ \mathbf{elif}\;d \leq -1.5 \cdot 10^{-303}:\\ \;\;\;\;t_2 \cdot \left(t_4 \cdot \frac{t_3}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+126}:\\ \;\;\;\;t_2 \cdot \left(t_4 \cdot \frac{1}{\frac{\sqrt{\ell}}{\sqrt{d}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(\frac{M}{\frac{\frac{d}{0.5}}{D}}\right)}^{2}}{\ell}\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 4
Accuracy	75.7%
Cost	21708

\[\begin{array}{l} t_0 := \frac{M \cdot \left(D \cdot -0.5\right)}{d}\\ t_1 := \frac{M \cdot \left(0.5 \cdot D\right)}{d}\\ \mathbf{if}\;d \leq -9.5 \cdot 10^{-143}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(t_1 \cdot \left(\frac{h}{\ell} \cdot t_0\right)\right)\right)\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{-284}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot 0.125\right) - d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+126}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(\frac{t_1}{\ell} \cdot \frac{t_0}{\frac{1}{h}}\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(\frac{M}{\frac{\frac{d}{0.5}}{D}}\right)}^{2}}{\ell}\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 5
Accuracy	70.3%
Cost	21401

\[\begin{array}{l} t_0 := \frac{M \cdot \left(D \cdot -0.5\right)}{d}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}} \cdot t_1\\ t_3 := d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ t_4 := \frac{M \cdot \left(0.5 \cdot D\right)}{d}\\ \mathbf{if}\;d \leq -1.15 \cdot 10^{+123}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq -7.5 \cdot 10^{-142}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(t_4 \cdot \left(\frac{h}{\ell} \cdot t_0\right)\right)\right) \cdot t_2\\ \mathbf{elif}\;d \leq -8.5 \cdot 10^{-245}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-283}:\\ \;\;\;\;t_2 \cdot \left(1 + 0.5 \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{D \cdot \left(M \cdot -0.5\right)}{d \cdot \frac{\ell}{h}}\right)\right)\\ \mathbf{elif}\;d \leq 1.46 \cdot 10^{-96} \lor \neg \left(d \leq 1.85 \cdot 10^{+126}\right):\\ \;\;\;\;\left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(\frac{M}{\frac{\frac{d}{0.5}}{D}}\right)}^{2}}{\ell}\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(\frac{t_4}{\ell} \cdot \frac{t_0}{\frac{1}{h}}\right)\right) \cdot \left(t_1 \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)\\ \end{array} \]

Alternative 6
Accuracy	72.3%
Cost	21132

\[\begin{array}{l} \mathbf{if}\;d \leq -4.2 \cdot 10^{+122}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq -3.15 \cdot 10^{-112}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(\frac{M \cdot \left(0.5 \cdot D\right)}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{M \cdot \left(D \cdot -0.5\right)}{d}\right)\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-283}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot 0.125\right) - d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(\frac{M}{\frac{\frac{d}{0.5}}{D}}\right)}^{2}}{\ell}\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 7
Accuracy	68.1%
Cost	15832

\[\begin{array}{l} t_0 := \frac{M \cdot \left(D \cdot -0.5\right)}{d}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}} \cdot t_1\\ t_3 := \sqrt{\frac{1}{h \cdot \ell}}\\ t_4 := d \cdot \left(-t_3\right)\\ t_5 := \frac{M \cdot \left(0.5 \cdot D\right)}{d}\\ \mathbf{if}\;d \leq -8.5 \cdot 10^{+122}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;d \leq -2.7 \cdot 10^{-142}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(t_5 \cdot \left(\frac{h}{\ell} \cdot t_0\right)\right)\right) \cdot t_2\\ \mathbf{elif}\;d \leq -8.5 \cdot 10^{-245}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-283}:\\ \;\;\;\;t_2 \cdot \left(1 + 0.5 \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{D \cdot \left(M \cdot -0.5\right)}{d \cdot \frac{\ell}{h}}\right)\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-123}:\\ \;\;\;\;d \cdot t_3\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{+126}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(\frac{t_5}{\ell} \cdot \frac{t_0}{\frac{1}{h}}\right)\right) \cdot \left(t_1 \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 8
Accuracy	66.9%
Cost	15576

\[\begin{array}{l} t_0 := \sqrt{\frac{1}{h \cdot \ell}}\\ t_1 := d \cdot \left(-t_0\right)\\ t_2 := \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ t_3 := D \cdot \left(M \cdot -0.5\right)\\ t_4 := t_2 \cdot \left(1 + 0.5 \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{t_3}{d \cdot \frac{\ell}{h}}\right)\right)\\ \mathbf{if}\;d \leq -4.2 \cdot 10^{+122}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -1.05 \cdot 10^{-143}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;d \leq -8.5 \cdot 10^{-245}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-283}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-119}:\\ \;\;\;\;d \cdot t_0\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+126}:\\ \;\;\;\;t_2 \cdot \left(1 + 0.5 \cdot \left(\frac{M \cdot D}{\ell} \cdot \frac{t_3}{\frac{d}{\frac{h}{d \cdot 2}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 9
Accuracy	67.2%
Cost	15576

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ t_1 := D \cdot \left(M \cdot -0.5\right)\\ t_2 := \sqrt{\frac{1}{h \cdot \ell}}\\ t_3 := d \cdot \left(-t_2\right)\\ \mathbf{if}\;d \leq -1 \cdot 10^{+123}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq -3.6 \cdot 10^{-142}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(\frac{M \cdot \left(0.5 \cdot D\right)}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{M \cdot \left(D \cdot -0.5\right)}{d}\right)\right)\right) \cdot t_0\\ \mathbf{elif}\;d \leq -1.1 \cdot 10^{-244}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-283}:\\ \;\;\;\;t_0 \cdot \left(1 + 0.5 \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{t_1}{d \cdot \frac{\ell}{h}}\right)\right)\\ \mathbf{elif}\;d \leq 3 \cdot 10^{-119}:\\ \;\;\;\;d \cdot t_2\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+126}:\\ \;\;\;\;t_0 \cdot \left(1 + 0.5 \cdot \left(\frac{M \cdot D}{\ell} \cdot \frac{t_1}{\frac{d}{\frac{h}{d \cdot 2}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 10
Accuracy	68.1%
Cost	15576

\[\begin{array}{l} t_0 := \frac{M \cdot \left(D \cdot -0.5\right)}{d}\\ t_1 := \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ t_2 := M \cdot \left(0.5 \cdot D\right)\\ t_3 := \sqrt{\frac{1}{h \cdot \ell}}\\ t_4 := d \cdot \left(-t_3\right)\\ \mathbf{if}\;d \leq -4.6 \cdot 10^{+122}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;d \leq -1.1 \cdot 10^{-143}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(\frac{t_2}{d} \cdot \left(\frac{h}{\ell} \cdot t_0\right)\right)\right) \cdot t_1\\ \mathbf{elif}\;d \leq -9 \cdot 10^{-245}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-283}:\\ \;\;\;\;t_1 \cdot \left(1 + 0.5 \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{D \cdot \left(M \cdot -0.5\right)}{d \cdot \frac{\ell}{h}}\right)\right)\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{-122}:\\ \;\;\;\;d \cdot t_3\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{+126}:\\ \;\;\;\;t_1 \cdot \left(1 + 0.5 \cdot \left(\frac{\frac{h}{\frac{d}{t_2}}}{\ell} \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 11
Accuracy	67.8%
Cost	15576

\[\begin{array}{l} t_0 := \frac{M \cdot \left(D \cdot -0.5\right)}{d}\\ t_1 := \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ t_2 := M \cdot \left(0.5 \cdot D\right)\\ t_3 := \sqrt{\frac{1}{h \cdot \ell}}\\ t_4 := d \cdot \left(-t_3\right)\\ \mathbf{if}\;d \leq -1.12 \cdot 10^{+123}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;d \leq -2.2 \cdot 10^{-142}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(\frac{t_2}{d} \cdot \left(\frac{h}{\ell} \cdot t_0\right)\right)\right) \cdot t_1\\ \mathbf{elif}\;d \leq -1.05 \cdot 10^{-244}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-283}:\\ \;\;\;\;t_1 \cdot \left(1 + 0.5 \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{D \cdot \left(M \cdot -0.5\right)}{d \cdot \frac{\ell}{h}}\right)\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-123}:\\ \;\;\;\;d \cdot t_3\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+126}:\\ \;\;\;\;t_1 \cdot \left(1 + 0.5 \cdot \left(\left(t_2 \cdot \frac{h}{d}\right) \cdot \frac{t_0}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 12
Accuracy	65.6%
Cost	15448

\[\begin{array}{l} t_0 := \frac{M \cdot D}{d}\\ t_1 := \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(t_0 \cdot t_0\right)\right) \cdot -0.25\right)\right)\\ t_2 := \sqrt{\frac{1}{h \cdot \ell}}\\ t_3 := d \cdot \left(-t_2\right)\\ \mathbf{if}\;d \leq -4.5 \cdot 10^{+122}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-143}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -8.5 \cdot 10^{-245}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq -1.2 \cdot 10^{-284}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-94}:\\ \;\;\;\;d \cdot t_2\\ \mathbf{elif}\;d \leq 4.8 \cdot 10^{+42}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 13
Accuracy	65.6%
Cost	15448

\[\begin{array}{l} t_0 := \frac{M \cdot D}{d}\\ t_1 := \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ t_2 := t_1 \cdot \left(1 + 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(t_0 \cdot t_0\right)\right) \cdot -0.25\right)\right)\\ t_3 := \sqrt{\frac{1}{h \cdot \ell}}\\ t_4 := d \cdot \left(-t_3\right)\\ \mathbf{if}\;d \leq -5.8 \cdot 10^{+122}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;d \leq -1.05 \cdot 10^{-142}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -1.05 \cdot 10^{-244}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;d \leq -3.4 \cdot 10^{-284}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{-96}:\\ \;\;\;\;d \cdot t_3\\ \mathbf{elif}\;d \leq 9.2 \cdot 10^{+42}:\\ \;\;\;\;t_1 \cdot \left(1 + 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(\left(M \cdot D\right) \cdot \frac{M \cdot D}{d \cdot d}\right)\right) \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 14
Accuracy	66.6%
Cost	15448

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ t_1 := t_0 \cdot \left(1 + 0.5 \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{D \cdot \left(M \cdot -0.5\right)}{d \cdot \frac{\ell}{h}}\right)\right)\\ t_2 := \sqrt{\frac{1}{h \cdot \ell}}\\ t_3 := d \cdot \left(-t_2\right)\\ \mathbf{if}\;d \leq -4.6 \cdot 10^{+122}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq -6.5 \cdot 10^{-143}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -1.2 \cdot 10^{-243}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-283}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{-121}:\\ \;\;\;\;d \cdot t_2\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{+126}:\\ \;\;\;\;t_0 \cdot \left(1 + 0.5 \cdot \left(\frac{M \cdot D}{\frac{\ell}{h} \cdot \frac{d \cdot d}{M \cdot D}} \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 15
Accuracy	67.2%
Cost	15316

\[\begin{array}{l} t_0 := \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + 0.5 \cdot \left(\frac{M \cdot D}{\left(d \cdot \frac{\ell}{h}\right) \cdot \frac{d}{M \cdot D}} \cdot -0.25\right)\right)\\ t_1 := \sqrt{\frac{1}{h \cdot \ell}}\\ t_2 := d \cdot \left(-t_1\right)\\ \mathbf{if}\;d \leq -4.2 \cdot 10^{+122}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -7.2 \cdot 10^{-143}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -2.9 \cdot 10^{-273}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-123}:\\ \;\;\;\;d \cdot t_1\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{+42}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 16
Accuracy	66.8%
Cost	15316

\[\begin{array}{l} t_0 := \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + 0.5 \cdot \left(\frac{M \cdot D}{\frac{\ell}{h} \cdot \frac{d \cdot d}{M \cdot D}} \cdot -0.25\right)\right)\\ t_1 := \sqrt{\frac{1}{h \cdot \ell}}\\ t_2 := d \cdot \left(-t_1\right)\\ \mathbf{if}\;d \leq -6 \cdot 10^{+122}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -4.5 \cdot 10^{-142}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -2.5 \cdot 10^{-273}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{-117}:\\ \;\;\;\;d \cdot t_1\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+126}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 17
Accuracy	61.2%
Cost	14997

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.02 \cdot 10^{-54}:\\ \;\;\;\;\frac{-d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq -1.1 \cdot 10^{-78}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(\frac{D \cdot D}{d} \cdot \frac{M \cdot M}{d}\right)\right) \cdot -0.25\right)\right)\\ \mathbf{elif}\;\ell \leq -6 \cdot 10^{-262}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+45} \lor \neg \left(\ell \leq 4.5 \cdot 10^{+110}\right):\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(\left(0.5 \cdot D\right) \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \end{array} \]

Alternative 18
Accuracy	62.9%
Cost	14600

\[\begin{array}{l} t_0 := d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{if}\;d \leq -1.82 \cdot 10^{+121}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.7 \cdot 10^{-141}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-310}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 19
Accuracy	64.1%
Cost	13644

\[\begin{array}{l} t_0 := d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{if}\;d \leq -4.5 \cdot 10^{+122}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -6.9 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-310}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 20
Accuracy	63.0%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 21
Accuracy	63.0%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 22
Accuracy	55.9%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.55 \cdot 10^{-240}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 23
Accuracy	56.1%
Cost	6916

\[\begin{array}{l} t_0 := \sqrt{h \cdot \ell}\\ \mathbf{if}\;\ell \leq -4.55 \cdot 10^{-240}:\\ \;\;\;\;\frac{-d}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t_0}\\ \end{array} \]

Alternative 24
Accuracy	31.6%
Cost	6720

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

Reproduce

herbie shell --seed 2023135 
(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)))))