Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.0% → 76.8%

Time: 28.2s

Precision: binary64

Cost: 33800

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sqrt{d}}{\sqrt{\ell}}\\ \mathbf{if}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \left({\left(-\ell\right)}^{-0.5} \cdot {\left(\frac{-1}{h}\right)}^{0.5}\right)\right) \cdot \left(-1 + \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\frac{\ell \cdot 2}{h}}\right)\\ \mathbf{elif}\;h \leq 6 \cdot 10^{+84}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t_0 \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{elif}\;h \leq 10^{+297}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t_0 \cdot \left(1 - 0.5 \cdot \left(\frac{0.25}{\ell} \cdot \left(\frac{h}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{d}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \left(D \cdot \frac{M}{\frac{d}{D \cdot M}}\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (sqrt d) (sqrt l))))
   (if (<= h -2e-310)
     (*
      (* d (* (pow (- l) -0.5) (pow (/ -1.0 h) 0.5)))
      (+ -1.0 (/ (pow (* D (* 0.5 (/ M d))) 2.0) (/ (* l 2.0) h))))
     (if (<= h 6e+84)
       (*
        (/ (sqrt d) (sqrt h))
        (* t_0 (- 1.0 (* 0.5 (* (pow (* (/ M 2.0) (/ D d)) 2.0) (/ h l))))))
       (if (<= h 1e+297)
         (*
          (sqrt (/ d h))
          (*
           t_0
           (-
            1.0
            (* 0.5 (* (/ 0.25 l) (* (/ h d) (* (* M M) (/ (* D D) d))))))))
         (* (/ (sqrt h) (pow l 1.5)) (* -0.125 (* D (/ M (/ d (* D M)))))))))))

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) / sqrt(l);
	double tmp;
	if (h <= -2e-310) {
		tmp = (d * (pow(-l, -0.5) * pow((-1.0 / h), 0.5))) * (-1.0 + (pow((D * (0.5 * (M / d))), 2.0) / ((l * 2.0) / h)));
	} else if (h <= 6e+84) {
		tmp = (sqrt(d) / sqrt(h)) * (t_0 * (1.0 - (0.5 * (pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))));
	} else if (h <= 1e+297) {
		tmp = sqrt((d / h)) * (t_0 * (1.0 - (0.5 * ((0.25 / l) * ((h / d) * ((M * M) * ((D * D) / d)))))));
	} else {
		tmp = (sqrt(h) / pow(l, 1.5)) * (-0.125 * (D * (M / (d / (D * M)))));
	}
	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) / sqrt(l)
    if (h <= (-2d-310)) then
        tmp = (d * ((-l ** (-0.5d0)) * (((-1.0d0) / h) ** 0.5d0))) * ((-1.0d0) + (((d_1 * (0.5d0 * (m / d))) ** 2.0d0) / ((l * 2.0d0) / h)))
    else if (h <= 6d+84) then
        tmp = (sqrt(d) / sqrt(h)) * (t_0 * (1.0d0 - (0.5d0 * ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (h / l)))))
    else if (h <= 1d+297) then
        tmp = sqrt((d / h)) * (t_0 * (1.0d0 - (0.5d0 * ((0.25d0 / l) * ((h / d) * ((m * m) * ((d_1 * d_1) / d)))))))
    else
        tmp = (sqrt(h) / (l ** 1.5d0)) * ((-0.125d0) * (d_1 * (m / (d / (d_1 * m)))))
    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) / Math.sqrt(l);
	double tmp;
	if (h <= -2e-310) {
		tmp = (d * (Math.pow(-l, -0.5) * Math.pow((-1.0 / h), 0.5))) * (-1.0 + (Math.pow((D * (0.5 * (M / d))), 2.0) / ((l * 2.0) / h)));
	} else if (h <= 6e+84) {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * (t_0 * (1.0 - (0.5 * (Math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))));
	} else if (h <= 1e+297) {
		tmp = Math.sqrt((d / h)) * (t_0 * (1.0 - (0.5 * ((0.25 / l) * ((h / d) * ((M * M) * ((D * D) / d)))))));
	} else {
		tmp = (Math.sqrt(h) / Math.pow(l, 1.5)) * (-0.125 * (D * (M / (d / (D * M)))));
	}
	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) / math.sqrt(l)
	tmp = 0
	if h <= -2e-310:
		tmp = (d * (math.pow(-l, -0.5) * math.pow((-1.0 / h), 0.5))) * (-1.0 + (math.pow((D * (0.5 * (M / d))), 2.0) / ((l * 2.0) / h)))
	elif h <= 6e+84:
		tmp = (math.sqrt(d) / math.sqrt(h)) * (t_0 * (1.0 - (0.5 * (math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))))
	elif h <= 1e+297:
		tmp = math.sqrt((d / h)) * (t_0 * (1.0 - (0.5 * ((0.25 / l) * ((h / d) * ((M * M) * ((D * D) / d)))))))
	else:
		tmp = (math.sqrt(h) / math.pow(l, 1.5)) * (-0.125 * (D * (M / (d / (D * M)))))
	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(sqrt(d) / sqrt(l))
	tmp = 0.0
	if (h <= -2e-310)
		tmp = Float64(Float64(d * Float64((Float64(-l) ^ -0.5) * (Float64(-1.0 / h) ^ 0.5))) * Float64(-1.0 + Float64((Float64(D * Float64(0.5 * Float64(M / d))) ^ 2.0) / Float64(Float64(l * 2.0) / h))));
	elseif (h <= 6e+84)
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(t_0 * Float64(1.0 - Float64(0.5 * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * Float64(h / l))))));
	elseif (h <= 1e+297)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(t_0 * Float64(1.0 - Float64(0.5 * Float64(Float64(0.25 / l) * Float64(Float64(h / d) * Float64(Float64(M * M) * Float64(Float64(D * D) / d))))))));
	else
		tmp = Float64(Float64(sqrt(h) / (l ^ 1.5)) * Float64(-0.125 * Float64(D * Float64(M / Float64(d / Float64(D * M))))));
	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) / sqrt(l);
	tmp = 0.0;
	if (h <= -2e-310)
		tmp = (d * ((-l ^ -0.5) * ((-1.0 / h) ^ 0.5))) * (-1.0 + (((D * (0.5 * (M / d))) ^ 2.0) / ((l * 2.0) / h)));
	elseif (h <= 6e+84)
		tmp = (sqrt(d) / sqrt(h)) * (t_0 * (1.0 - (0.5 * ((((M / 2.0) * (D / d)) ^ 2.0) * (h / l)))));
	elseif (h <= 1e+297)
		tmp = sqrt((d / h)) * (t_0 * (1.0 - (0.5 * ((0.25 / l) * ((h / d) * ((M * M) * ((D * D) / d)))))));
	else
		tmp = (sqrt(h) / (l ^ 1.5)) * (-0.125 * (D * (M / (d / (D * M)))));
	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[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -2e-310], N[(N[(d * N[(N[Power[(-l), -0.5], $MachinePrecision] * N[Power[N[(-1.0 / h), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(N[Power[N[(D * N[(0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(l * 2.0), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 6e+84], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(1.0 - N[(0.5 * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 1e+297], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(1.0 - N[(0.5 * N[(N[(0.25 / l), $MachinePrecision] * N[(N[(h / d), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision] * N[(-0.125 * N[(D * N[(M / N[(d / N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -1.999999999999994e-310

   1. Initial program 66.2%

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

   2. Applied egg-rr 21.1%

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

      [Start] 66.2%	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      expm1-log1p-u [=>] 35.8%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]
      expm1-udef [=>] 27.9%	\[ \color{blue}{e^{\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} - 1} \]

   3. Simplified 54.5%

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

      [Start] 21.1%	\[ e^{\mathsf{log1p}\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1 \]
      expm1-def [=>] 26.7%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]
      expm1-log1p [=>] 54.5%	\[ \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]
      unpow1/2 [=>] 54.5%	\[ \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]
      associate-/l/ [=>] 54.5%	\[ \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(D \cdot \color{blue}{\frac{M}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]
      *-commutative [=>] 54.5%	\[ \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \frac{\color{blue}{h \cdot 0.5}}{\ell}\right) \]
      associate-/l* [=>] 54.5%	\[ \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{h}{\frac{\ell}{0.5}}}\right) \]

   4. Taylor expanded in d around -inf 71.0%

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

   5. Simplified 71.6%

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

      [Start] 71.0%	\[ \left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right) \cdot \left(1 - {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\frac{\ell}{0.5}}\right) \]
      *-commutative [<=] 71.0%	\[ \left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right)\right) \cdot \left(1 - {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\frac{\ell}{0.5}}\right) \]
      neg-mul-1 [<=] 71.0%	\[ \color{blue}{\left(-d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\frac{\ell}{0.5}}\right) \]
      distribute-rgt-neg-in [=>] 71.0%	\[ \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\frac{\ell}{0.5}}\right) \]
      unpow-1 [<=] 71.0%	\[ \left(d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right)\right) \cdot \left(1 - {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\frac{\ell}{0.5}}\right) \]
      sqr-pow [=>] 71.1%	\[ \left(d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}}\right)\right) \cdot \left(1 - {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\frac{\ell}{0.5}}\right) \]
      rem-sqrt-square [=>] 71.6%	\[ \left(d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|}\right)\right) \cdot \left(1 - {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\frac{\ell}{0.5}}\right) \]
      metadata-eval [=>] 71.6%	\[ \left(d \cdot \left(-\left|{\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right|\right)\right) \cdot \left(1 - {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\frac{\ell}{0.5}}\right) \]
      sqr-pow [=>] 71.4%	\[ \left(d \cdot \left(-\left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}}\right|\right)\right) \cdot \left(1 - {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\frac{\ell}{0.5}}\right) \]
      fabs-sqr [=>] 71.4%	\[ \left(d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}}\right)\right) \cdot \left(1 - {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\frac{\ell}{0.5}}\right) \]
      sqr-pow [<=] 71.6%	\[ \left(d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right)\right) \cdot \left(1 - {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\frac{\ell}{0.5}}\right) \]

   6. Applied egg-rr 75.1%

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

      [Start] 71.6%	\[ \left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right) \cdot \left(1 - {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\frac{\ell}{0.5}}\right) \]
      associate-*r/ [=>] 75.1%	\[ \left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right) \cdot \left(1 - \color{blue}{\frac{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot h}{\frac{\ell}{0.5}}}\right) \]
      associate-/l/ [<=] 75.1%	\[ \left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right) \cdot \left(1 - \frac{{\left(D \cdot \color{blue}{\frac{\frac{M}{d}}{2}}\right)}^{2} \cdot h}{\frac{\ell}{0.5}}\right) \]
      div-inv [=>] 75.1%	\[ \left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right) \cdot \left(1 - \frac{{\left(D \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{1}{2}\right)}\right)}^{2} \cdot h}{\frac{\ell}{0.5}}\right) \]
      metadata-eval [=>] 75.1%	\[ \left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right) \cdot \left(1 - \frac{{\left(D \cdot \left(\frac{M}{d} \cdot \color{blue}{0.5}\right)\right)}^{2} \cdot h}{\frac{\ell}{0.5}}\right) \]
      div-inv [=>] 75.1%	\[ \left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right) \cdot \left(1 - \frac{{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2} \cdot h}{\color{blue}{\ell \cdot \frac{1}{0.5}}}\right) \]
      metadata-eval [=>] 75.1%	\[ \left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right) \cdot \left(1 - \frac{{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2} \cdot h}{\ell \cdot \color{blue}{2}}\right) \]

   7. Simplified 72.4%

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

      [Start] 75.1%	\[ \left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right) \cdot \left(1 - \frac{{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2} \cdot h}{\ell \cdot 2}\right) \]
      associate-/l* [=>] 72.4%	\[ \left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right) \cdot \left(1 - \color{blue}{\frac{{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2}}{\frac{\ell \cdot 2}{h}}}\right) \]
      *-commutative [=>] 72.4%	\[ \left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right) \cdot \left(1 - \frac{{\left(D \cdot \color{blue}{\left(0.5 \cdot \frac{M}{d}\right)}\right)}^{2}}{\frac{\ell \cdot 2}{h}}\right) \]
      *-commutative [=>] 72.4%	\[ \left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right) \cdot \left(1 - \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\frac{\color{blue}{2 \cdot \ell}}{h}}\right) \]

   8. Taylor expanded in h around -inf 79.8%

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

   9. Simplified 83.4%

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

      [Start] 79.8%	\[ \left(d \cdot \left(-e^{-0.5 \cdot \left(\log \left(-1 \cdot \ell\right) + -1 \cdot \log \left(\frac{-1}{h}\right)\right)}\right)\right) \cdot \left(1 - \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\frac{2 \cdot \ell}{h}}\right) \]
      distribute-lft-in [=>] 79.8%	\[ \left(d \cdot \left(-e^{\color{blue}{-0.5 \cdot \log \left(-1 \cdot \ell\right) + -0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{h}\right)\right)}}\right)\right) \cdot \left(1 - \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\frac{2 \cdot \ell}{h}}\right) \]
      exp-sum [=>] 79.9%	\[ \left(d \cdot \left(-\color{blue}{e^{-0.5 \cdot \log \left(-1 \cdot \ell\right)} \cdot e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{h}\right)\right)}}\right)\right) \cdot \left(1 - \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\frac{2 \cdot \ell}{h}}\right) \]
      *-commutative [<=] 79.9%	\[ \left(d \cdot \left(-e^{\color{blue}{\log \left(-1 \cdot \ell\right) \cdot -0.5}} \cdot e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{h}\right)\right)}\right)\right) \cdot \left(1 - \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\frac{2 \cdot \ell}{h}}\right) \]
      exp-to-pow [=>] 80.6%	\[ \left(d \cdot \left(-\color{blue}{{\left(-1 \cdot \ell\right)}^{-0.5}} \cdot e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{h}\right)\right)}\right)\right) \cdot \left(1 - \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\frac{2 \cdot \ell}{h}}\right) \]
      mul-1-neg [=>] 80.6%	\[ \left(d \cdot \left(-{\color{blue}{\left(-\ell\right)}}^{-0.5} \cdot e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{h}\right)\right)}\right)\right) \cdot \left(1 - \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\frac{2 \cdot \ell}{h}}\right) \]
      *-commutative [<=] 80.6%	\[ \left(d \cdot \left(-{\left(-\ell\right)}^{-0.5} \cdot e^{\color{blue}{\left(-1 \cdot \log \left(\frac{-1}{h}\right)\right) \cdot -0.5}}\right)\right) \cdot \left(1 - \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\frac{2 \cdot \ell}{h}}\right) \]
      *-commutative [=>] 80.6%	\[ \left(d \cdot \left(-{\left(-\ell\right)}^{-0.5} \cdot e^{\color{blue}{\left(\log \left(\frac{-1}{h}\right) \cdot -1\right)} \cdot -0.5}\right)\right) \cdot \left(1 - \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\frac{2 \cdot \ell}{h}}\right) \]
      associate-*l* [=>] 80.6%	\[ \left(d \cdot \left(-{\left(-\ell\right)}^{-0.5} \cdot e^{\color{blue}{\log \left(\frac{-1}{h}\right) \cdot \left(-1 \cdot -0.5\right)}}\right)\right) \cdot \left(1 - \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\frac{2 \cdot \ell}{h}}\right) \]
      metadata-eval [=>] 80.6%	\[ \left(d \cdot \left(-{\left(-\ell\right)}^{-0.5} \cdot e^{\log \left(\frac{-1}{h}\right) \cdot \color{blue}{0.5}}\right)\right) \cdot \left(1 - \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\frac{2 \cdot \ell}{h}}\right) \]
      exp-to-pow [=>] 83.4%	\[ \left(d \cdot \left(-{\left(-\ell\right)}^{-0.5} \cdot \color{blue}{{\left(\frac{-1}{h}\right)}^{0.5}}\right)\right) \cdot \left(1 - \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\frac{2 \cdot \ell}{h}}\right) \]

   if -1.999999999999994e-310 < h < 5.99999999999999992e84

   1. Initial program 65.5%

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

   2. Simplified 65.5%

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

      [Start] 65.5%	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 65.5%	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 65.5%	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 65.5%	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 65.5%	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 65.5%	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      associate-*l* [=>] 65.5%	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)\right) \]
      metadata-eval [=>] 65.5%	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      times-frac [=>] 65.5%	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   3. Applied egg-rr 70.0%

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

      [Start] 65.5%	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      sqrt-div [=>] 70.0%	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   4. Applied egg-rr 79.7%

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

      [Start] 70.0%	\[ \sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      sqrt-div [=>] 79.7%	\[ \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   if 5.99999999999999992e84 < h < 1e297

   1. Initial program 59.1%

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

   2. Simplified 59.1%

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

      [Start] 59.1%	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 59.1%	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 59.1%	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 59.1%	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 59.1%	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 59.1%	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      associate-*l* [=>] 59.1%	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)\right) \]
      metadata-eval [=>] 59.1%	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      times-frac [=>] 59.1%	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   3. Applied egg-rr 62.6%

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

      [Start] 59.1%	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      sqrt-div [=>] 62.6%	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   4. Taylor expanded in M around 0 60.5%

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

   5. Simplified 78.2%

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

      [Start] 60.5%	\[ \sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)\right)\right) \]
      associate-*r/ [=>] 60.5%	\[ \sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right)\right) \]
      *-commutative [<=] 60.5%	\[ \sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \frac{0.25 \cdot \left({D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{\ell \cdot {d}^{2}}\right)\right) \]
      times-frac [=>] 66.4%	\[ \sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{0.25}{\ell} \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}\right)}\right)\right) \]
      *-commutative [=>] 66.4%	\[ \sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{0.25}{\ell} \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{{d}^{2}}\right)\right)\right) \]
      associate-*r* [=>] 74.2%	\[ \sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{0.25}{\ell} \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2}}\right)\right)\right) \]
      unpow2 [=>] 74.2%	\[ \sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{0.25}{\ell} \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}{\color{blue}{d \cdot d}}\right)\right)\right) \]
      times-frac [=>] 80.1%	\[ \sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{0.25}{\ell} \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \frac{h}{d}\right)}\right)\right)\right) \]
      associate-/l* [=>] 80.1%	\[ \sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{0.25}{\ell} \cdot \left(\color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \frac{h}{d}\right)\right)\right)\right) \]
      associate-/r/ [=>] 78.2%	\[ \sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{0.25}{\ell} \cdot \left(\color{blue}{\left(\frac{{D}^{2}}{d} \cdot {M}^{2}\right)} \cdot \frac{h}{d}\right)\right)\right)\right) \]
      unpow2 [=>] 78.2%	\[ \sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{0.25}{\ell} \cdot \left(\left(\frac{\color{blue}{D \cdot D}}{d} \cdot {M}^{2}\right) \cdot \frac{h}{d}\right)\right)\right)\right) \]
      unpow2 [=>] 78.2%	\[ \sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{0.25}{\ell} \cdot \left(\left(\frac{D \cdot D}{d} \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \frac{h}{d}\right)\right)\right)\right) \]

   if 1e297 < h

   1. Initial program 1.8%

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

   2. Taylor expanded in d around 0 33.3%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]

   3. Simplified 34.9%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{\frac{d}{D}}\right)\right)} \]
      Step-by-step derivation

      [Start] 33.3%	\[ -0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]
      associate-*r* [=>] 33.3%	\[ \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]
      *-commutative [=>] 33.3%	\[ \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]
      associate-/l* [=>] 33.3%	\[ \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}}\right) \]
      associate-/r/ [=>] 33.3%	\[ \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{d} \cdot {M}^{2}\right)}\right) \]
      *-commutative [=>] 33.3%	\[ \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{d}\right)}\right) \]
      unpow2 [=>] 33.3%	\[ \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{d}\right)\right) \]
      unpow2 [=>] 33.3%	\[ \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{d}\right)\right) \]
      associate-/l* [=>] 34.9%	\[ \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(\left(M \cdot M\right) \cdot \color{blue}{\frac{D}{\frac{d}{D}}}\right)\right) \]

   4. Taylor expanded in M around 0 33.3%

      \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d}}\right) \]

   5. Simplified 36.6%

      \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(D \cdot \frac{M}{\frac{d}{D \cdot M}}\right)}\right) \]
      Step-by-step derivation

      [Start] 33.3%	\[ \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \]
      *-commutative [=>] 33.3%	\[ \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{d}\right) \]
      associate-/l* [=>] 33.3%	\[ \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\frac{{M}^{2}}{\frac{d}{{D}^{2}}}}\right) \]
      unpow2 [=>] 33.3%	\[ \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{M}^{2}}{\frac{d}{\color{blue}{D \cdot D}}}\right) \]
      associate-/l/ [<=] 34.9%	\[ \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{M}^{2}}{\color{blue}{\frac{\frac{d}{D}}{D}}}\right) \]
      associate-/l* [<=] 34.9%	\[ \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\frac{{M}^{2} \cdot D}{\frac{d}{D}}}\right) \]
      unpow2 [=>] 34.9%	\[ \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot D}{\frac{d}{D}}\right) \]
      associate-/r/ [=>] 34.9%	\[ \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(\frac{\left(M \cdot M\right) \cdot D}{d} \cdot D\right)}\right) \]
      *-commutative [=>] 34.9%	\[ \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(D \cdot \frac{\left(M \cdot M\right) \cdot D}{d}\right)}\right) \]
      associate-*l* [=>] 36.6%	\[ \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \frac{\color{blue}{M \cdot \left(M \cdot D\right)}}{d}\right)\right) \]
      associate-/l* [=>] 36.6%	\[ \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \color{blue}{\frac{M}{\frac{d}{M \cdot D}}}\right)\right) \]
      *-commutative [<=] 36.6%	\[ \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \frac{M}{\frac{d}{\color{blue}{D \cdot M}}}\right)\right) \]

   6. Applied egg-rr 36.6%

      \[\leadsto \color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}} \cdot \left(-0.125 \cdot \left(D \cdot \frac{M}{\frac{d}{D \cdot M}}\right)\right) \]
      Step-by-step derivation

      [Start] 36.6%	\[ \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \frac{M}{\frac{d}{D \cdot M}}\right)\right) \]
      sqrt-div [=>] 36.6%	\[ \color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}} \cdot \left(-0.125 \cdot \left(D \cdot \frac{M}{\frac{d}{D \cdot M}}\right)\right) \]

   7. Simplified 68.2%

      \[\leadsto \color{blue}{\frac{\sqrt{h}}{{\ell}^{1.5}}} \cdot \left(-0.125 \cdot \left(D \cdot \frac{M}{\frac{d}{D \cdot M}}\right)\right) \]
      Step-by-step derivation

      [Start] 36.6%	\[ \frac{\sqrt{h}}{\sqrt{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \frac{M}{\frac{d}{D \cdot M}}\right)\right) \]
      sqr-pow [=>] 36.6%	\[ \frac{\sqrt{h}}{\sqrt{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)} \cdot {\ell}^{\left(\frac{3}{2}\right)}}}} \cdot \left(-0.125 \cdot \left(D \cdot \frac{M}{\frac{d}{D \cdot M}}\right)\right) \]
      rem-sqrt-square [=>] 68.2%	\[ \frac{\sqrt{h}}{\color{blue}{\left|{\ell}^{\left(\frac{3}{2}\right)}\right|}} \cdot \left(-0.125 \cdot \left(D \cdot \frac{M}{\frac{d}{D \cdot M}}\right)\right) \]
      sqr-pow [=>] 68.2%	\[ \frac{\sqrt{h}}{\left|\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}\right|} \cdot \left(-0.125 \cdot \left(D \cdot \frac{M}{\frac{d}{D \cdot M}}\right)\right) \]
      fabs-sqr [=>] 68.2%	\[ \frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}} \cdot \left(-0.125 \cdot \left(D \cdot \frac{M}{\frac{d}{D \cdot M}}\right)\right) \]
      sqr-pow [<=] 68.2%	\[ \frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)}}} \cdot \left(-0.125 \cdot \left(D \cdot \frac{M}{\frac{d}{D \cdot M}}\right)\right) \]
      metadata-eval [=>] 68.2%	\[ \frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}} \cdot \left(-0.125 \cdot \left(D \cdot \frac{M}{\frac{d}{D \cdot M}}\right)\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \left({\left(-\ell\right)}^{-0.5} \cdot {\left(\frac{-1}{h}\right)}^{0.5}\right)\right) \cdot \left(-1 + \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\frac{\ell \cdot 2}{h}}\right)\\ \mathbf{elif}\;h \leq 6 \cdot 10^{+84}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{elif}\;h \leq 10^{+297}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{0.25}{\ell} \cdot \left(\frac{h}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{d}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \left(D \cdot \frac{M}{\frac{d}{D \cdot M}}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	76.8%
Cost	33800

\[\begin{array}{l} t_0 := \frac{\sqrt{d}}{\sqrt{\ell}}\\ \mathbf{if}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \left({\left(-\ell\right)}^{-0.5} \cdot {\left(\frac{-1}{h}\right)}^{0.5}\right)\right) \cdot \left(-1 + \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\frac{\ell \cdot 2}{h}}\right)\\ \mathbf{elif}\;h \leq 6 \cdot 10^{+84}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t_0 \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{elif}\;h \leq 10^{+297}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t_0 \cdot \left(1 - 0.5 \cdot \left(\frac{0.25}{\ell} \cdot \left(\frac{h}{d} \cdot \left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{d}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \left(D \cdot \frac{M}{\frac{d}{D \cdot M}}\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	77.3%
Cost	41288

\[\begin{array}{l} t_0 := \frac{\sqrt{d}}{\sqrt{\ell}}\\ t_1 := \sqrt[3]{D \cdot \frac{D}{d}}\\ \mathbf{if}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \left({\left(-\ell\right)}^{-0.5} \cdot {\left(\frac{-1}{h}\right)}^{0.5}\right)\right) \cdot \left(-1 + \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\frac{\ell \cdot 2}{h}}\right)\\ \mathbf{elif}\;h \leq 4.1 \cdot 10^{+83}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t_0 \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t_0 \cdot \left(1 - 0.5 \cdot \left(\frac{0.25}{\ell} \cdot \left(\left(\left(t_1 \cdot \left(t_1 \cdot t_1\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d}\right)\right)\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	73.1%
Cost	21396

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right)}^{2}}{\ell}\right)\right)\right)\\ t_1 := D \cdot \frac{M}{\frac{d}{D \cdot M}}\\ \mathbf{if}\;d \leq -1.16 \cdot 10^{+148}:\\ \;\;\;\;\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(-1 + \frac{h \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell \cdot 2}\right)\\ \mathbf{elif}\;d \leq -5.6 \cdot 10^{-133}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.35 \cdot 10^{-306}:\\ \;\;\;\;0.125 \cdot \left(t_1 \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\\ \mathbf{elif}\;d \leq 2 \cdot 10^{-153}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot t_1\right)\\ \mathbf{elif}\;d \leq 3 \cdot 10^{+225}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 4
Accuracy	76.9%
Cost	21132

\[\begin{array}{l} \mathbf{if}\;d \leq -1.35 \cdot 10^{-306}:\\ \;\;\;\;\left(d \cdot \left({\left(-\ell\right)}^{-0.5} \cdot {\left(\frac{-1}{h}\right)}^{0.5}\right)\right) \cdot \left(-1 + {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\frac{\ell}{0.5}}\right)\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-152}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \left(D \cdot \frac{M}{\frac{d}{D \cdot M}}\right)\right)\\ \mathbf{elif}\;d \leq 1.02 \cdot 10^{+225}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 5
Accuracy	77.0%
Cost	21132

\[\begin{array}{l} \mathbf{if}\;d \leq -1.35 \cdot 10^{-306}:\\ \;\;\;\;\left(d \cdot \left({\left(-\ell\right)}^{-0.5} \cdot {\left(\frac{-1}{h}\right)}^{0.5}\right)\right) \cdot \left(-1 + \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\frac{\ell \cdot 2}{h}}\right)\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-152}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \left(D \cdot \frac{M}{\frac{d}{D \cdot M}}\right)\right)\\ \mathbf{elif}\;d \leq 8 \cdot 10^{+224}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 6
Accuracy	65.7%
Cost	14996

\[\begin{array}{l} t_0 := \left(1 - \frac{h \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell \cdot 2}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ t_1 := D \cdot \frac{M}{\frac{d}{D \cdot M}}\\ \mathbf{if}\;d \leq -1.55 \cdot 10^{+143}:\\ \;\;\;\;{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -8.2 \cdot 10^{-133}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.35 \cdot 10^{-306}:\\ \;\;\;\;0.125 \cdot \left(t_1 \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{-135}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot t_1\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 7
Accuracy	64.8%
Cost	14932

\[\begin{array}{l} t_0 := D \cdot \frac{M}{\frac{d}{D \cdot M}}\\ t_1 := {\left(h \cdot \ell\right)}^{-0.5}\\ t_2 := d \cdot t_1\\ \mathbf{if}\;d \leq -5.4 \cdot 10^{+205}:\\ \;\;\;\;t_1 \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -9 \cdot 10^{-171}:\\ \;\;\;\;t_2 \cdot \left(-1 + 0.125 \cdot \left(\frac{h}{\ell} \cdot \left(\left(M \cdot M\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq -1.35 \cdot 10^{-306}:\\ \;\;\;\;0.125 \cdot \left(t_0 \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{-137}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot t_0\right)\\ \mathbf{elif}\;d \leq 2.25 \cdot 10^{+212}:\\ \;\;\;\;\left(1 - {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\frac{\ell}{0.5}}\right) \cdot t_2\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 8
Accuracy	65.8%
Cost	14932

\[\begin{array}{l} t_0 := 1 - {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\frac{\ell}{0.5}}\\ t_1 := {\left(h \cdot \ell\right)}^{-0.5}\\ t_2 := D \cdot \frac{M}{\frac{d}{D \cdot M}}\\ \mathbf{if}\;d \leq -2.1 \cdot 10^{+181}:\\ \;\;\;\;t_1 \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -8.2 \cdot 10^{-133}:\\ \;\;\;\;t_0 \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -1.35 \cdot 10^{-306}:\\ \;\;\;\;0.125 \cdot \left(t_2 \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\\ \mathbf{elif}\;d \leq 8.6 \cdot 10^{-140}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot t_2\right)\\ \mathbf{elif}\;d \leq 1.62 \cdot 10^{+212}:\\ \;\;\;\;t_0 \cdot \left(d \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 9
Accuracy	65.9%
Cost	14932

\[\begin{array}{l} t_0 := {\left(h \cdot \ell\right)}^{-0.5}\\ t_1 := \frac{h}{\frac{\ell}{0.5}}\\ t_2 := D \cdot \frac{M}{\frac{d}{D \cdot M}}\\ \mathbf{if}\;d \leq -9.8 \cdot 10^{+179}:\\ \;\;\;\;t_0 \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -6.6 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - t_1 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\\ \mathbf{elif}\;d \leq -1.35 \cdot 10^{-306}:\\ \;\;\;\;0.125 \cdot \left(t_2 \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\\ \mathbf{elif}\;d \leq 4.8 \cdot 10^{-140}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot t_2\right)\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{+211}:\\ \;\;\;\;\left(1 - {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2} \cdot t_1\right) \cdot \left(d \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 10
Accuracy	69.2%
Cost	14732

\[\begin{array}{l} \mathbf{if}\;d \leq -1.35 \cdot 10^{-306}:\\ \;\;\;\;\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(-1 + {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\frac{\ell}{0.5}}\right)\\ \mathbf{elif}\;d \leq 3 \cdot 10^{-138}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \left(D \cdot \frac{M}{\frac{d}{D \cdot M}}\right)\right)\\ \mathbf{elif}\;d \leq 5.6 \cdot 10^{+153}:\\ \;\;\;\;\left(1 - \frac{h \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell \cdot 2}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 11
Accuracy	71.6%
Cost	14732

\[\begin{array}{l} t_0 := \frac{h \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell \cdot 2}\\ \mathbf{if}\;d \leq -1.35 \cdot 10^{-306}:\\ \;\;\;\;\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(-1 + t_0\right)\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-137}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \left(D \cdot \frac{M}{\frac{d}{D \cdot M}}\right)\right)\\ \mathbf{elif}\;d \leq 2.85 \cdot 10^{+153}:\\ \;\;\;\;\left(1 - t_0\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 12
Accuracy	56.9%
Cost	14352

\[\begin{array}{l} t_0 := {\left(h \cdot \ell\right)}^{-0.5}\\ t_1 := \sqrt{\frac{h}{{\ell}^{3}}}\\ \mathbf{if}\;d \leq -2 \cdot 10^{+206}:\\ \;\;\;\;t_0 \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -3.3 \cdot 10^{-170}:\\ \;\;\;\;\left(d \cdot t_0\right) \cdot \left(-1 + 0.125 \cdot \left(\frac{h}{\ell} \cdot \left(\left(M \cdot M\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq -1.35 \cdot 10^{-306}:\\ \;\;\;\;0.125 \cdot \left(\left(D \cdot \frac{M}{\frac{d}{D \cdot M}}\right) \cdot t_1\right)\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-160}:\\ \;\;\;\;-0.125 \cdot \left(t_1 \cdot \frac{D \cdot D}{\frac{d}{M \cdot M}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 13
Accuracy	58.7%
Cost	14352

\[\begin{array}{l} t_0 := D \cdot \frac{M}{\frac{d}{D \cdot M}}\\ t_1 := {\left(h \cdot \ell\right)}^{-0.5}\\ t_2 := \sqrt{\frac{h}{{\ell}^{3}}}\\ \mathbf{if}\;d \leq -4.2 \cdot 10^{+202}:\\ \;\;\;\;t_1 \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -6.5 \cdot 10^{-170}:\\ \;\;\;\;\left(d \cdot t_1\right) \cdot \left(-1 + 0.125 \cdot \left(\frac{h}{\ell} \cdot \left(\left(M \cdot M\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq -1.35 \cdot 10^{-306}:\\ \;\;\;\;0.125 \cdot \left(t_0 \cdot t_2\right)\\ \mathbf{elif}\;d \leq 4.05 \cdot 10^{-110}:\\ \;\;\;\;\left(-0.125 \cdot t_0\right) \cdot t_2\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 14
Accuracy	60.5%
Cost	14352

\[\begin{array}{l} t_0 := D \cdot \frac{M}{\frac{d}{D \cdot M}}\\ t_1 := {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{if}\;d \leq -6.5 \cdot 10^{+205}:\\ \;\;\;\;t_1 \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-170}:\\ \;\;\;\;\left(d \cdot t_1\right) \cdot \left(-1 + 0.125 \cdot \left(\frac{h}{\ell} \cdot \left(\left(M \cdot M\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq -1.35 \cdot 10^{-306}:\\ \;\;\;\;0.125 \cdot \left(t_0 \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\\ \mathbf{elif}\;d \leq 5.3 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 15
Accuracy	56.7%
Cost	14220

\[\begin{array}{l} t_0 := {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{if}\;d \leq -1.8 \cdot 10^{+204}:\\ \;\;\;\;t_0 \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -1.65 \cdot 10^{-305}:\\ \;\;\;\;\left(d \cdot t_0\right) \cdot \left(-1 + 0.125 \cdot \left(\frac{h}{\ell} \cdot \left(\left(M \cdot M\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 4.1 \cdot 10^{-160}:\\ \;\;\;\;-0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{D \cdot D}{\frac{d}{M \cdot M}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 16
Accuracy	55.4%
Cost	13512

\[\begin{array}{l} t_0 := {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{if}\;d \leq -4.7 \cdot 10^{+206}:\\ \;\;\;\;t_0 \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -1.65 \cdot 10^{-305}:\\ \;\;\;\;\left(d \cdot t_0\right) \cdot \left(-1 + 0.125 \cdot \left(\frac{h}{\ell} \cdot \left(\left(M \cdot M\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 17
Accuracy	55.4%
Cost	13512

\[\begin{array}{l} t_0 := {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{if}\;d \leq -2.05 \cdot 10^{+207}:\\ \;\;\;\;t_0 \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -1.65 \cdot 10^{-305}:\\ \;\;\;\;\left(d \cdot t_0\right) \cdot \left(-1 + 0.125 \cdot \left(\frac{h}{\ell} \cdot \left(\left(M \cdot M\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 18
Accuracy	55.3%
Cost	8524

\[\begin{array}{l} t_0 := {\left(h \cdot \ell\right)}^{-0.5}\\ t_1 := d \cdot t_0\\ \mathbf{if}\;d \leq -6 \cdot 10^{+204}:\\ \;\;\;\;t_0 \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq 5.2 \cdot 10^{-264}:\\ \;\;\;\;t_1 \cdot \left(-1 + 0.125 \cdot \left(\frac{h}{\ell} \cdot \left(\left(M \cdot M\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{+125}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{h}{d} \cdot \frac{M \cdot M}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 19
Accuracy	51.3%
Cost	8392

\[\begin{array}{l} t_0 := {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{if}\;d \leq -2.6 \cdot 10^{+77}:\\ \;\;\;\;t_0 \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{+125}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{h}{d} \cdot \frac{M \cdot M}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot t_0\\ \end{array} \]

Alternative 20
Accuracy	43.0%
Cost	6980

\[\begin{array}{l} t_0 := {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{if}\;d \leq 4.5 \cdot 10^{-239}:\\ \;\;\;\;t_0 \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot t_0\\ \end{array} \]

Alternative 21
Accuracy	26.3%
Cost	6848

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

Alternative 22
Accuracy	26.2%
Cost	6784

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

Reproduce

herbie shell --seed 2023165 
(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)))))