Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.3% → 81.9%

Time: 51.1s

Precision: binary64

Cost: 33796

• Unsound rule application detected in e-graph. Results may not be sound.

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -7.4 \cdot 10^{+53}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{t_0}{\sqrt{-\ell}} \cdot \left(1 - 0.5 \cdot {\left(\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{t_0}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \left(0.5 \cdot D\right)}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{-77}:\\ \;\;\;\;\left(1 + \frac{{\left(\frac{0.5 \cdot M}{\frac{d}{D}}\right)}^{2} \cdot \left(h \cdot -0.5\right)}{\ell}\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d \cdot \left(1 + \left(h \cdot -0.5\right) \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

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 (sqrt (- d))))
   (if (<= l -7.4e+53)
     (*
      (sqrt (/ d h))
      (*
       (/ t_0 (sqrt (- l)))
       (- 1.0 (* 0.5 (pow (* (* (* 0.5 M) (/ D d)) (sqrt (/ h l))) 2.0)))))
     (if (<= l -1e-310)
       (*
        (/ t_0 (sqrt (- h)))
        (*
         (sqrt (/ d l))
         (- 1.0 (* 0.5 (* (/ h l) (pow (/ (* M (* 0.5 D)) d) 2.0))))))
       (if (<= l 6e-77)
         (*
          (+ 1.0 (/ (* (pow (/ (* 0.5 M) (/ d D)) 2.0) (* h -0.5)) l))
          (/ d (* (sqrt h) (sqrt l))))
         (/
          (/
           (* d (+ 1.0 (* (* h -0.5) (/ (pow (* M (* 0.5 (/ D d))) 2.0) l))))
           (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 = sqrt(-d);
	double tmp;
	if (l <= -7.4e+53) {
		tmp = sqrt((d / h)) * ((t_0 / sqrt(-l)) * (1.0 - (0.5 * pow((((0.5 * M) * (D / d)) * sqrt((h / l))), 2.0))));
	} else if (l <= -1e-310) {
		tmp = (t_0 / sqrt(-h)) * (sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * pow(((M * (0.5 * D)) / d), 2.0)))));
	} else if (l <= 6e-77) {
		tmp = (1.0 + ((pow(((0.5 * M) / (d / D)), 2.0) * (h * -0.5)) / l)) * (d / (sqrt(h) * sqrt(l)));
	} else {
		tmp = ((d * (1.0 + ((h * -0.5) * (pow((M * (0.5 * (D / d))), 2.0) / l)))) / 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) :: tmp
    t_0 = sqrt(-d)
    if (l <= (-7.4d+53)) then
        tmp = sqrt((d / h)) * ((t_0 / sqrt(-l)) * (1.0d0 - (0.5d0 * ((((0.5d0 * m) * (d_1 / d)) * sqrt((h / l))) ** 2.0d0))))
    else if (l <= (-1d-310)) then
        tmp = (t_0 / sqrt(-h)) * (sqrt((d / l)) * (1.0d0 - (0.5d0 * ((h / l) * (((m * (0.5d0 * d_1)) / d) ** 2.0d0)))))
    else if (l <= 6d-77) then
        tmp = (1.0d0 + (((((0.5d0 * m) / (d / d_1)) ** 2.0d0) * (h * (-0.5d0))) / l)) * (d / (sqrt(h) * sqrt(l)))
    else
        tmp = ((d * (1.0d0 + ((h * (-0.5d0)) * (((m * (0.5d0 * (d_1 / d))) ** 2.0d0) / l)))) / 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 = Math.sqrt(-d);
	double tmp;
	if (l <= -7.4e+53) {
		tmp = Math.sqrt((d / h)) * ((t_0 / Math.sqrt(-l)) * (1.0 - (0.5 * Math.pow((((0.5 * M) * (D / d)) * Math.sqrt((h / l))), 2.0))));
	} else if (l <= -1e-310) {
		tmp = (t_0 / Math.sqrt(-h)) * (Math.sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * Math.pow(((M * (0.5 * D)) / d), 2.0)))));
	} else if (l <= 6e-77) {
		tmp = (1.0 + ((Math.pow(((0.5 * M) / (d / D)), 2.0) * (h * -0.5)) / l)) * (d / (Math.sqrt(h) * Math.sqrt(l)));
	} else {
		tmp = ((d * (1.0 + ((h * -0.5) * (Math.pow((M * (0.5 * (D / d))), 2.0) / l)))) / 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 = math.sqrt(-d)
	tmp = 0
	if l <= -7.4e+53:
		tmp = math.sqrt((d / h)) * ((t_0 / math.sqrt(-l)) * (1.0 - (0.5 * math.pow((((0.5 * M) * (D / d)) * math.sqrt((h / l))), 2.0))))
	elif l <= -1e-310:
		tmp = (t_0 / math.sqrt(-h)) * (math.sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * math.pow(((M * (0.5 * D)) / d), 2.0)))))
	elif l <= 6e-77:
		tmp = (1.0 + ((math.pow(((0.5 * M) / (d / D)), 2.0) * (h * -0.5)) / l)) * (d / (math.sqrt(h) * math.sqrt(l)))
	else:
		tmp = ((d * (1.0 + ((h * -0.5) * (math.pow((M * (0.5 * (D / d))), 2.0) / l)))) / 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 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -7.4e+53)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(t_0 / sqrt(Float64(-l))) * Float64(1.0 - Float64(0.5 * (Float64(Float64(Float64(0.5 * M) * Float64(D / d)) * sqrt(Float64(h / l))) ^ 2.0)))));
	elseif (l <= -1e-310)
		tmp = Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M * Float64(0.5 * D)) / d) ^ 2.0))))));
	elseif (l <= 6e-77)
		tmp = Float64(Float64(1.0 + Float64(Float64((Float64(Float64(0.5 * M) / Float64(d / D)) ^ 2.0) * Float64(h * -0.5)) / l)) * Float64(d / Float64(sqrt(h) * sqrt(l))));
	else
		tmp = Float64(Float64(Float64(d * Float64(1.0 + Float64(Float64(h * -0.5) * Float64((Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0) / l)))) / 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 = sqrt(-d);
	tmp = 0.0;
	if (l <= -7.4e+53)
		tmp = sqrt((d / h)) * ((t_0 / sqrt(-l)) * (1.0 - (0.5 * ((((0.5 * M) * (D / d)) * sqrt((h / l))) ^ 2.0))));
	elseif (l <= -1e-310)
		tmp = (t_0 / sqrt(-h)) * (sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * (((M * (0.5 * D)) / d) ^ 2.0)))));
	elseif (l <= 6e-77)
		tmp = (1.0 + (((((0.5 * M) / (d / D)) ^ 2.0) * (h * -0.5)) / l)) * (d / (sqrt(h) * sqrt(l)));
	else
		tmp = ((d * (1.0 + ((h * -0.5) * (((M * (0.5 * (D / d))) ^ 2.0) / l)))) / 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[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -7.4e+53], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[Power[N[(N[(N[(0.5 * M), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-310], N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M * N[(0.5 * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6e-77], N[(N[(1.0 + N[(N[(N[Power[N[(N[(0.5 * M), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * -0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d * N[(1.0 + N[(N[(h * -0.5), $MachinePrecision] * N[(N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -7.4e53

   1. Initial program 49.3%

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

   2. Simplified 49.3%

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

      [Start] 49.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 49.3	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 49.3	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 49.3	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 49.3	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 49.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      associate-*l* [=>] 49.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)\right) \]
      metadata-eval [=>] 49.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      times-frac [=>] 49.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   3. Applied egg-rr 53.8%

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

      [Start] 49.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      add-sqr-sqrt [=>] 49.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right)\right) \]
      pow2 [=>] 49.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right)\right) \]
      sqrt-prod [=>] 49.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right)\right) \]
      unpow2 [=>] 49.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot {\left(\sqrt{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right) \]
      sqrt-prod [=>] 30.8	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\sqrt{\frac{M}{2} \cdot \frac{D}{d}} \cdot \sqrt{\frac{M}{2} \cdot \frac{D}{d}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right) \]
      add-sqr-sqrt [<=] 53.8	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right) \]
      div-inv [=>] 53.8	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right) \]
      metadata-eval [=>] 53.8	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot {\left(\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right) \]

   4. Applied egg-rr 70.1%

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

      [Start] 53.8	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot {\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right) \]
      frac-2neg [=>] 53.8	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\color{blue}{\frac{-d}{-\ell}}} \cdot \left(1 - 0.5 \cdot {\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right) \]
      sqrt-div [=>] 70.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}} \cdot \left(1 - 0.5 \cdot {\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right) \]

   if -7.4e53 < l < -9.999999999999969e-311

   1. Initial program 72.7%

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

   2. Simplified 71.5%

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

      [Start] 72.7	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 72.7	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 72.7	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 72.7	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 72.7	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 72.7	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      associate-*l* [=>] 72.7	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)\right) \]
      metadata-eval [=>] 72.7	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      times-frac [=>] 71.5	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   3. Applied egg-rr 72.7%

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

      [Start] 71.5	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      associate-*r/ [=>] 72.7	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{\frac{M}{2} \cdot D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      div-inv [=>] 72.7	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      metadata-eval [=>] 72.7	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\left(M \cdot \color{blue}{0.5}\right) \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      associate-*l* [=>] 72.7	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\color{blue}{M \cdot \left(0.5 \cdot D\right)}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   4. Applied egg-rr 84.8%

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

      [Start] 72.7	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M \cdot \left(0.5 \cdot D\right)}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      frac-2neg [=>] 72.7	\[ \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M \cdot \left(0.5 \cdot D\right)}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      sqrt-div [=>] 84.8	\[ \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M \cdot \left(0.5 \cdot D\right)}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   if -9.999999999999969e-311 < l < 6.00000000000000033e-77

   1. Initial program 66.0%

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

   2. Simplified 64.1%

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

      [Start] 66.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 66.0	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 66.0	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 66.0	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 66.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      *-commutative [=>] 66.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 66.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]
      times-frac [=>] 64.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 64.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 73.7%

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

      [Start] 64.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      sub-neg [=>] 64.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\left(1 + \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} \]
      distribute-lft-in [=>] 49.1	\[ \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot 1 + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
      *-commutative [<=] 49.1	\[ \color{blue}{1 \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      *-un-lft-identity [<=] 49.1	\[ \color{blue}{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      sqrt-div [=>] 50.9	\[ \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      sqrt-div [=>] 57.1	\[ \frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      frac-times [=>] 57.2	\[ \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      add-sqr-sqrt [<=] 57.2	\[ \frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      sqrt-div [=>] 67.6	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      sqrt-div [=>] 73.7	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      frac-times [=>] 73.7	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}} \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      add-sqr-sqrt [<=] 73.7	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      distribute-rgt-neg-in [=>] 73.7	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Simplified 81.9%

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

      [Start] 73.7	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right) \]
      *-commutative [<=] 73.7	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \color{blue}{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]
      distribute-rgt1-in [=>] 81.9	\[ \color{blue}{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right) + 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]
      distribute-lft-neg-in [=>] 81.9	\[ \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \color{blue}{\left(\left(-0.5\right) \cdot \frac{h}{\ell}\right)} + 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]
      metadata-eval [=>] 81.9	\[ \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{-0.5} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]

   5. Applied egg-rr 95.8%

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

      [Start] 81.9	\[ \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right) + 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]
      associate-*r/ [=>] 81.9	\[ \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \color{blue}{\frac{-0.5 \cdot h}{\ell}} + 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]
      associate-*r/ [=>] 95.8	\[ \left(\color{blue}{\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(-0.5 \cdot h\right)}{\ell}} + 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]
      associate-*l* [=>] 95.8	\[ \left(\frac{{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \left(-0.5 \cdot h\right)}{\ell} + 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]
      clear-num [=>] 95.8	\[ \left(\frac{{\left(M \cdot \left(0.5 \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)\right)}^{2} \cdot \left(-0.5 \cdot h\right)}{\ell} + 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]
      un-div-inv [=>] 95.8	\[ \left(\frac{{\left(M \cdot \color{blue}{\frac{0.5}{\frac{d}{D}}}\right)}^{2} \cdot \left(-0.5 \cdot h\right)}{\ell} + 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]

   6. Applied egg-rr 95.8%

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

      [Start] 95.8	\[ \left(\frac{{\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2} \cdot \left(-0.5 \cdot h\right)}{\ell} + 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]
      associate-*r/ [=>] 95.8	\[ \left(\frac{{\color{blue}{\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}}^{2} \cdot \left(-0.5 \cdot h\right)}{\ell} + 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]

   if 6.00000000000000033e-77 < l

   1. Initial program 62.6%

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

   2. Simplified 62.6%

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

      [Start] 62.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 62.6	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 62.6	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 62.6	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 62.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      *-commutative [=>] 62.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 62.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]
      times-frac [=>] 62.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 62.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 79.7%

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

      [Start] 62.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      sub-neg [=>] 62.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\left(1 + \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} \]
      distribute-lft-in [=>] 57.9	\[ \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot 1 + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
      *-commutative [<=] 57.9	\[ \color{blue}{1 \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      *-un-lft-identity [<=] 57.9	\[ \color{blue}{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      sqrt-div [=>] 58.5	\[ \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      sqrt-div [=>] 62.4	\[ \frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      frac-times [=>] 62.3	\[ \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      add-sqr-sqrt [<=] 62.4	\[ \frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      sqrt-div [=>] 78.8	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      sqrt-div [=>] 79.8	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      frac-times [=>] 79.7	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}} \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      add-sqr-sqrt [<=] 79.7	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      distribute-rgt-neg-in [=>] 79.7	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Simplified 83.5%

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

      [Start] 79.7	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right) \]
      *-commutative [<=] 79.7	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \color{blue}{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]
      distribute-rgt1-in [=>] 83.5	\[ \color{blue}{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right) + 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]
      distribute-lft-neg-in [=>] 83.5	\[ \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \color{blue}{\left(\left(-0.5\right) \cdot \frac{h}{\ell}\right)} + 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]
      metadata-eval [=>] 83.5	\[ \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{-0.5} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]

   5. Applied egg-rr 79.9%

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

      [Start] 83.5	\[ \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right) + 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]
      associate-*r/ [=>] 83.5	\[ \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \color{blue}{\frac{-0.5 \cdot h}{\ell}} + 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]
      associate-*r/ [=>] 80.0	\[ \left(\color{blue}{\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(-0.5 \cdot h\right)}{\ell}} + 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]
      associate-*l* [=>] 80.0	\[ \left(\frac{{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \left(-0.5 \cdot h\right)}{\ell} + 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]
      clear-num [=>] 79.9	\[ \left(\frac{{\left(M \cdot \left(0.5 \cdot \color{blue}{\frac{1}{\frac{d}{D}}}\right)\right)}^{2} \cdot \left(-0.5 \cdot h\right)}{\ell} + 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]
      un-div-inv [=>] 79.9	\[ \left(\frac{{\left(M \cdot \color{blue}{\frac{0.5}{\frac{d}{D}}}\right)}^{2} \cdot \left(-0.5 \cdot h\right)}{\ell} + 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]

   6. Applied egg-rr 85.2%

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

      [Start] 79.9	\[ \left(\frac{{\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2} \cdot \left(-0.5 \cdot h\right)}{\ell} + 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]
      associate-*r/ [=>] 83.8	\[ \color{blue}{\frac{\left(\frac{{\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2} \cdot \left(-0.5 \cdot h\right)}{\ell} + 1\right) \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}} \]
      *-commutative [=>] 83.8	\[ \frac{\left(\frac{{\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2} \cdot \left(-0.5 \cdot h\right)}{\ell} + 1\right) \cdot d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]
      associate-/r* [=>] 82.7	\[ \color{blue}{\frac{\frac{\left(\frac{{\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2} \cdot \left(-0.5 \cdot h\right)}{\ell} + 1\right) \cdot d}{\sqrt{\ell}}}{\sqrt{h}}} \]
      *-commutative [=>] 82.7	\[ \frac{\frac{\color{blue}{d \cdot \left(\frac{{\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2} \cdot \left(-0.5 \cdot h\right)}{\ell} + 1\right)}}{\sqrt{\ell}}}{\sqrt{h}} \]
      associate-/l* [=>] 85.1	\[ \frac{\frac{d \cdot \left(\color{blue}{\frac{{\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}}{\frac{\ell}{-0.5 \cdot h}}} + 1\right)}{\sqrt{\ell}}}{\sqrt{h}} \]
      associate-/r/ [=>] 85.2	\[ \frac{\frac{d \cdot \left(\color{blue}{\frac{{\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}}{\ell} \cdot \left(-0.5 \cdot h\right)} + 1\right)}{\sqrt{\ell}}}{\sqrt{h}} \]
      div-inv [=>] 85.2	\[ \frac{\frac{d \cdot \left(\frac{{\left(M \cdot \color{blue}{\left(0.5 \cdot \frac{1}{\frac{d}{D}}\right)}\right)}^{2}}{\ell} \cdot \left(-0.5 \cdot h\right) + 1\right)}{\sqrt{\ell}}}{\sqrt{h}} \]
      clear-num [<=] 85.2	\[ \frac{\frac{d \cdot \left(\frac{{\left(M \cdot \left(0.5 \cdot \color{blue}{\frac{D}{d}}\right)\right)}^{2}}{\ell} \cdot \left(-0.5 \cdot h\right) + 1\right)}{\sqrt{\ell}}}{\sqrt{h}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.4 \cdot 10^{+53}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 - 0.5 \cdot {\left(\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \left(0.5 \cdot D\right)}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{-77}:\\ \;\;\;\;\left(1 + \frac{{\left(\frac{0.5 \cdot M}{\frac{d}{D}}\right)}^{2} \cdot \left(h \cdot -0.5\right)}{\ell}\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d \cdot \left(1 + \left(h \cdot -0.5\right) \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	77.2%
Cost	27528

\[\begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;h \leq -4 \cdot 10^{+230}:\\ \;\;\;\;\frac{t_1}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\frac{h}{\ell} \cdot \left(t_0 \cdot t_0\right)\right)\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-294}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{t_1}{\sqrt{-\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{{\left(\frac{0.5 \cdot M}{\frac{d}{D}}\right)}^{2} \cdot \left(h \cdot -0.5\right)}{\ell}\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 2
Accuracy	77.5%
Cost	27528

\[\begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;h \leq -6 \cdot 10^{+238}:\\ \;\;\;\;\frac{t_1}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\frac{h}{\ell} \cdot \left(t_0 \cdot t_0\right)\right)\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-294}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{t_1}{\sqrt{-\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \left(0.5 \cdot D\right)}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{{\left(\frac{0.5 \cdot M}{\frac{d}{D}}\right)}^{2} \cdot \left(h \cdot -0.5\right)}{\ell}\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 3
Accuracy	81.8%
Cost	27396

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-78}:\\ \;\;\;\;\left(1 + \frac{{\left(\frac{0.5 \cdot M}{\frac{d}{D}}\right)}^{2} \cdot \left(h \cdot -0.5\right)}{\ell}\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d \cdot \left(1 + \left(h \cdot -0.5\right) \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 4
Accuracy	81.8%
Cost	27396

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \left(0.5 \cdot D\right)}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{-75}:\\ \;\;\;\;\left(1 + \frac{{\left(\frac{0.5 \cdot M}{\frac{d}{D}}\right)}^{2} \cdot \left(h \cdot -0.5\right)}{\ell}\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d \cdot \left(1 + \left(h \cdot -0.5\right) \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 5
Accuracy	75.5%
Cost	21128

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.8 \cdot 10^{+93}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\ell \leq -3.6 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \left(0.5 \cdot D\right)}{d}\right)}^{2}\right)\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)\\ \mathbf{elif}\;\ell \leq 10^{-77}:\\ \;\;\;\;\left(1 + \frac{{\left(\frac{0.5 \cdot M}{\frac{d}{D}}\right)}^{2} \cdot \left(h \cdot -0.5\right)}{\ell}\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d \cdot \left(1 + \left(h \cdot -0.5\right) \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 6
Accuracy	71.6%
Cost	21004

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.7 \cdot 10^{+93}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\ell \leq -3.6 \cdot 10^{-308}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.125 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{-77}:\\ \;\;\;\;\left(1 + \frac{{\left(\frac{0.5 \cdot M}{\frac{d}{D}}\right)}^{2} \cdot \left(h \cdot -0.5\right)}{\ell}\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d \cdot \left(1 + \left(h \cdot -0.5\right) \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 7
Accuracy	75.4%
Cost	21004

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.8 \cdot 10^{+93}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\ell \leq -3.6 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{-77}:\\ \;\;\;\;\left(1 + \frac{{\left(\frac{0.5 \cdot M}{\frac{d}{D}}\right)}^{2} \cdot \left(h \cdot -0.5\right)}{\ell}\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d \cdot \left(1 + \left(h \cdot -0.5\right) \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 8
Accuracy	75.4%
Cost	21004

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.2 \cdot 10^{+93}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\ell \leq -3.6 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \left(0.5 \cdot D\right)}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{-78}:\\ \;\;\;\;\left(1 + \frac{{\left(\frac{0.5 \cdot M}{\frac{d}{D}}\right)}^{2} \cdot \left(h \cdot -0.5\right)}{\ell}\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d \cdot \left(1 + \left(h \cdot -0.5\right) \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 9
Accuracy	66.0%
Cost	20872

\[\begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ t_1 := 1 + \left(h \cdot -0.5\right) \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\\ \mathbf{if}\;d \leq -1.4 \cdot 10^{+185}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2}, 1\right)\\ \mathbf{elif}\;d \leq -2.1 \cdot 10^{-298}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.125 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\\ \mathbf{elif}\;d \leq 10^{-44}:\\ \;\;\;\;\frac{d \cdot t_1}{t_0}\\ \mathbf{elif}\;d \leq 3.7 \cdot 10^{+122}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.125 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{h}{d} \cdot \frac{M \cdot M}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\frac{t_0}{d}}\\ \end{array} \]

Alternative 10
Accuracy	67.2%
Cost	20872

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.02 \cdot 10^{+93}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\ell \leq 2.45 \cdot 10^{-284}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.125 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 11
Accuracy	67.9%
Cost	20872

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.1 \cdot 10^{+93}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\ell \leq -3.6 \cdot 10^{-308}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.125 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \frac{-0.5 \cdot {\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}}{\frac{\ell}{h}}\right)\\ \end{array} \]

Alternative 12
Accuracy	70.1%
Cost	20872

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+93}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\ell \leq -3.6 \cdot 10^{-308}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.125 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \frac{\left(h \cdot -0.5\right) \cdot {\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}}{\ell}\right)\\ \end{array} \]

Alternative 13
Accuracy	70.2%
Cost	20872

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.4 \cdot 10^{+93}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\ell \leq -3.6 \cdot 10^{-308}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.125 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{{\left(\frac{0.5 \cdot M}{\frac{d}{D}}\right)}^{2} \cdot \left(h \cdot -0.5\right)}{\ell}\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 14
Accuracy	65.7%
Cost	14928

\[\begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ t_1 := 1 + \left(h \cdot -0.5\right) \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\\ \mathbf{if}\;d \leq -1.3 \cdot 10^{+185}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -4.2 \cdot 10^{-296}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{-43}:\\ \;\;\;\;\frac{d \cdot t_1}{t_0}\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{+123}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.125 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{h}{d} \cdot \frac{M \cdot M}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\frac{t_0}{d}}\\ \end{array} \]

Alternative 15
Accuracy	56.7%
Cost	14869

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;d \leq -6.5 \cdot 10^{-138}:\\ \;\;\;\;\left|t_0\right|\\ \mathbf{elif}\;d \leq -7.5 \cdot 10^{-274}:\\ \;\;\;\;d \cdot {\left({\left(\ell \cdot h\right)}^{-3}\right)}^{0.16666666666666666}\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{+160} \lor \neg \left(d \leq 2.5 \cdot 10^{+305}\right):\\ \;\;\;\;\left(1 + {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 16
Accuracy	66.6%
Cost	14864

\[\begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;\ell \leq -6.1 \cdot 10^{+113}:\\ \;\;\;\;\left|\frac{d}{t_0}\right|\\ \mathbf{elif}\;\ell \leq -4.2 \cdot 10^{-70}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{D \cdot \left(h \cdot D\right)}{\ell \cdot d}\right)\right)\\ \mathbf{elif}\;\ell \leq -2.6 \cdot 10^{-85}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq -3.6 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \left(1 + \left(h \cdot -0.5\right) \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)}{t_0}\\ \end{array} \]

Alternative 17
Accuracy	66.1%
Cost	14792

\[\begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+93}:\\ \;\;\;\;\left|\frac{d}{t_0}\right|\\ \mathbf{elif}\;\ell \leq -3.6 \cdot 10^{-308}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.125 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \left(1 + \left(h \cdot -0.5\right) \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)}{t_0}\\ \end{array} \]

Alternative 18
Accuracy	60.0%
Cost	14604

\[\begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;d \leq -8.5 \cdot 10^{-135}:\\ \;\;\;\;\left|\frac{d}{t_0}\right|\\ \mathbf{elif}\;d \leq -1.75 \cdot 10^{-273}:\\ \;\;\;\;d \cdot {\left({\left(\ell \cdot h\right)}^{-3}\right)}^{0.16666666666666666}\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \left(1 + \left(h \cdot -0.5\right) \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)}{t_0}\\ \end{array} \]

Alternative 19
Accuracy	65.8%
Cost	14600

\[\begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;\ell \leq -2.1 \cdot 10^{-20}:\\ \;\;\;\;\left|\frac{d}{t_0}\right|\\ \mathbf{elif}\;\ell \leq -3.6 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \left(1 + \left(h \cdot -0.5\right) \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)}{t_0}\\ \end{array} \]

Alternative 20
Accuracy	49.2%
Cost	14352

\[\begin{array}{l} \mathbf{if}\;d \leq -6.4 \cdot 10^{-136}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;d \leq -9 \cdot 10^{-274}:\\ \;\;\;\;d \cdot {\left({\left(\ell \cdot h\right)}^{-3}\right)}^{0.16666666666666666}\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{-21}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{D}{\frac{\frac{\frac{d}{M}}{M}}{D}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 21
Accuracy	47.2%
Cost	13580

\[\begin{array}{l} \mathbf{if}\;d \leq -4.8 \cdot 10^{-138}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;d \leq -7.5 \cdot 10^{-274}:\\ \;\;\;\;d \cdot {\left({\left(\ell \cdot h\right)}^{-3}\right)}^{0.16666666666666666}\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-283}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\frac{\sqrt{\ell}}{{h}^{-0.5}}}\\ \end{array} \]

Alternative 22
Accuracy	45.5%
Cost	13316

\[\begin{array}{l} \mathbf{if}\;\ell \leq 6.2 \cdot 10^{-220}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 23
Accuracy	45.5%
Cost	13316

\[\begin{array}{l} \mathbf{if}\;\ell \leq 6.2 \cdot 10^{-220}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\frac{\sqrt{\ell}}{{h}^{-0.5}}}\\ \end{array} \]

Alternative 24
Accuracy	42.6%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;\ell \leq 6.2 \cdot 10^{-220}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]

Alternative 25
Accuracy	42.6%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{-219}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]

Alternative 26
Accuracy	36.8%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq -4.7 \cdot 10^{-132}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]

Alternative 27
Accuracy	38.1%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq -7.5 \cdot 10^{-139}:\\ \;\;\;\;\sqrt{\frac{d}{\frac{\ell}{\frac{d}{h}}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]

Alternative 28
Accuracy	27.4%
Cost	6784

\[d \cdot {\left(\ell \cdot h\right)}^{-0.5} \]

Alternative 29
Accuracy	27.4%
Cost	6720

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

Reproduce

herbie shell --seed 2023160 
(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)))))