Henrywood and Agarwal, Equation (12)

Average Accuracy: 66.8% → 74.2%

Time: 54.0s

Precision: binary64

Cost: 21004

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{if}\;h \leq 5 \cdot 10^{-159}:\\ \;\;\;\;\left(1 - h \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \cdot t_0\\ \mathbf{elif}\;h \leq 1.52 \cdot 10^{-198}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;h \leq 0:\\ \;\;\;\;t_0 \cdot \left(1 - 0.125 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{D \cdot \left(h \cdot D\right)}{d \cdot \ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - h \cdot \left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

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 l)) (sqrt (/ d h)))))
   (if (<= h 5e-159)
     (* (- 1.0 (* h (/ (pow (* (* M 0.5) (/ D d)) 2.0) (/ l 0.5)))) t_0)
     (if (<= h 1.52e-198)
       (* d (- (sqrt (/ 1.0 (* h l)))))
       (if (<= h 0.0)
         (* t_0 (- 1.0 (* 0.125 (* (/ (* M M) d) (/ (* D (* h D)) (* d l))))))
         (*
          (- 1.0 (* h (* (pow (* 0.5 (* M (/ D d))) 2.0) (/ 0.5 l))))
          (/ d (* (sqrt h) (sqrt l)))))))))

C

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

↓

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l)) * sqrt((d / h));
	double tmp;
	if (h <= 5e-159) {
		tmp = (1.0 - (h * (pow(((M * 0.5) * (D / d)), 2.0) / (l / 0.5)))) * t_0;
	} else if (h <= 1.52e-198) {
		tmp = d * -sqrt((1.0 / (h * l)));
	} else if (h <= 0.0) {
		tmp = t_0 * (1.0 - (0.125 * (((M * M) / d) * ((D * (h * D)) / (d * l)))));
	} else {
		tmp = (1.0 - (h * (pow((0.5 * (M * (D / d))), 2.0) * (0.5 / l)))) * (d / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((d / l)) * sqrt((d / h))
    if (h <= 5d-159) then
        tmp = (1.0d0 - (h * ((((m * 0.5d0) * (d_1 / d)) ** 2.0d0) / (l / 0.5d0)))) * t_0
    else if (h <= 1.52d-198) then
        tmp = d * -sqrt((1.0d0 / (h * l)))
    else if (h <= 0.0d0) then
        tmp = t_0 * (1.0d0 - (0.125d0 * (((m * m) / d) * ((d_1 * (h * d_1)) / (d * l)))))
    else
        tmp = (1.0d0 - (h * (((0.5d0 * (m * (d_1 / d))) ** 2.0d0) * (0.5d0 / l)))) * (d / (sqrt(h) * sqrt(l)))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / l)) * Math.sqrt((d / h));
	double tmp;
	if (h <= 5e-159) {
		tmp = (1.0 - (h * (Math.pow(((M * 0.5) * (D / d)), 2.0) / (l / 0.5)))) * t_0;
	} else if (h <= 1.52e-198) {
		tmp = d * -Math.sqrt((1.0 / (h * l)));
	} else if (h <= 0.0) {
		tmp = t_0 * (1.0 - (0.125 * (((M * M) / d) * ((D * (h * D)) / (d * l)))));
	} else {
		tmp = (1.0 - (h * (Math.pow((0.5 * (M * (D / d))), 2.0) * (0.5 / l)))) * (d / (Math.sqrt(h) * Math.sqrt(l)));
	}
	return tmp;
}

Python

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

↓

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l)) * math.sqrt((d / h))
	tmp = 0
	if h <= 5e-159:
		tmp = (1.0 - (h * (math.pow(((M * 0.5) * (D / d)), 2.0) / (l / 0.5)))) * t_0
	elif h <= 1.52e-198:
		tmp = d * -math.sqrt((1.0 / (h * l)))
	elif h <= 0.0:
		tmp = t_0 * (1.0 - (0.125 * (((M * M) / d) * ((D * (h * D)) / (d * l)))))
	else:
		tmp = (1.0 - (h * (math.pow((0.5 * (M * (D / d))), 2.0) * (0.5 / l)))) * (d / (math.sqrt(h) * math.sqrt(l)))
	return tmp

Julia

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

↓

function code(d, h, l, M, D)
	t_0 = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)))
	tmp = 0.0
	if (h <= 5e-159)
		tmp = Float64(Float64(1.0 - Float64(h * Float64((Float64(Float64(M * 0.5) * Float64(D / d)) ^ 2.0) / Float64(l / 0.5)))) * t_0);
	elseif (h <= 1.52e-198)
		tmp = Float64(d * Float64(-sqrt(Float64(1.0 / Float64(h * l)))));
	elseif (h <= 0.0)
		tmp = Float64(t_0 * Float64(1.0 - Float64(0.125 * Float64(Float64(Float64(M * M) / d) * Float64(Float64(D * Float64(h * D)) / Float64(d * l))))));
	else
		tmp = Float64(Float64(1.0 - Float64(h * Float64((Float64(0.5 * Float64(M * Float64(D / d))) ^ 2.0) * Float64(0.5 / l)))) * Float64(d / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l)) * sqrt((d / h));
	tmp = 0.0;
	if (h <= 5e-159)
		tmp = (1.0 - (h * ((((M * 0.5) * (D / d)) ^ 2.0) / (l / 0.5)))) * t_0;
	elseif (h <= 1.52e-198)
		tmp = d * -sqrt((1.0 / (h * l)));
	elseif (h <= 0.0)
		tmp = t_0 * (1.0 - (0.125 * (((M * M) / d) * ((D * (h * D)) / (d * l)))));
	else
		tmp = (1.0 - (h * (((0.5 * (M * (D / d))) ^ 2.0) * (0.5 / l)))) * (d / (sqrt(h) * sqrt(l)));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, 5e-159], N[(N[(1.0 - N[(h * N[(N[Power[N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[h, 1.52e-198], N[(d * (-N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[h, 0.0], N[(t$95$0 * N[(1.0 - N[(0.125 * N[(N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision] * N[(N[(D * N[(h * D), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(h * N[(N[Power[N[(0.5 * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if h < 5.00000000000000032e-159

   1. Initial program 63.7%

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

   2. Simplified 62.4%

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

      [Start] 63.7	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 63.7	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 63.7	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 63.7	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 63.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      *-commutative [=>] 63.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 63.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]
      times-frac [=>] 62.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 62.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 62.4%

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

      [Start] 62.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      expm1-log1p-u [=>] 62.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)}\right) \]
      expm1-udef [=>] 62.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} - 1\right)}\right) \]
      log1p-udef [=>] 62.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(e^{\color{blue}{\log \left(1 + {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}} - 1\right)\right) \]
      add-exp-log [<=] 62.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\left(1 + {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} - 1\right)\right) \]
      +-commutative [=>] 62.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right) + 1\right)} - 1\right)\right) \]
      div-inv [=>] 62.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right) + 1\right) - 1\right)\right) \]
      metadata-eval [=>] 62.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right) + 1\right) - 1\right)\right) \]

   4. Simplified 64.3%

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

      [Start] 62.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right) + 1\right) - 1\right)\right) \]
      associate--l+ [=>] 62.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right) + \left(1 - 1\right)\right)}\right) \]
      metadata-eval [=>] 62.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right) + \color{blue}{0}\right)\right) \]
      +-rgt-identity [=>] 62.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)}\right) \]
      associate-*r* [=>] 62.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]
      associate-*r/ [=>] 64.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
      associate-*l/ [<=] 64.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5}{\ell} \cdot h}\right) \]
      *-commutative [=>] 64.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{h \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5}{\ell}}\right) \]
      associate-/l* [=>] 64.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \color{blue}{\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{0.5}}}\right) \]

   if 5.00000000000000032e-159 < h < 1.5199999999999999e-198

   1. Initial program 69.1%

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

   2. Simplified 68.1%

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

      [Start] 69.1	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 69.1	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 69.1	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 69.1	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 69.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      *-commutative [=>] 69.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 69.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]
      times-frac [=>] 68.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 68.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 70.3%

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

      [Start] 68.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      add-sqr-sqrt [=>] 68.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)}}\right) \]
      pow2 [=>] 68.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)}\right)}^{2}}\right) \]
      sqrt-prod [=>] 68.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{0.5 \cdot \frac{h}{\ell}}\right)}}^{2}\right) \]
      unpow2 [=>] 68.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\sqrt{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}} \cdot \sqrt{0.5 \cdot \frac{h}{\ell}}\right)}^{2}\right) \]
      sqrt-prod [=>] 44.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\color{blue}{\left(\sqrt{\frac{M}{2} \cdot \frac{D}{d}} \cdot \sqrt{\frac{M}{2} \cdot \frac{D}{d}}\right)} \cdot \sqrt{0.5 \cdot \frac{h}{\ell}}\right)}^{2}\right) \]
      add-sqr-sqrt [<=] 70.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{0.5 \cdot \frac{h}{\ell}}\right)}^{2}\right) \]
      div-inv [=>] 70.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right) \cdot \sqrt{0.5 \cdot \frac{h}{\ell}}\right)}^{2}\right) \]
      metadata-eval [=>] 70.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5 \cdot \frac{h}{\ell}}\right)}^{2}\right) \]

   4. Applied egg-rr 70.3%

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

      [Start] 70.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5 \cdot \frac{h}{\ell}}\right)}^{2}\right) \]
      clear-num [=>] 70.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - {\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5 \cdot \frac{h}{\ell}}\right)}^{2}\right) \]
      sqrt-div [=>] 70.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - {\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5 \cdot \frac{h}{\ell}}\right)}^{2}\right) \]
      metadata-eval [=>] 70.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - {\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5 \cdot \frac{h}{\ell}}\right)}^{2}\right) \]

   5. Taylor expanded in d around -inf 26.4%

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

   6. Simplified 26.4%

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

      [Start] 26.4	\[ -1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]
      mul-1-neg [=>] 26.4	\[ \color{blue}{-d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
      *-commutative [=>] 26.4	\[ -\color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
      distribute-rgt-neg-in [=>] 26.4	\[ \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]
      *-commutative [=>] 26.4	\[ \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-d\right) \]

   if 1.5199999999999999e-198 < h < 0.0

   1. Initial program 69.1%

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

   2. Simplified 68.1%

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

      [Start] 69.1	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 69.1	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 69.1	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 69.1	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 69.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      *-commutative [=>] 69.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 69.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]
      times-frac [=>] 68.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 68.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Taylor expanded in M around 0 43.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]

   4. Simplified 55.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{D \cdot \left(D \cdot h\right)}{d \cdot \ell}\right)}\right) \]
      Step-by-step derivation

      [Start] 43.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right) \]
      associate-*r/ [=>] 43.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right) \]
      *-commutative [=>] 43.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right) \]
      associate-*r/ [<=] 43.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
      associate-*r* [=>] 44.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}\right) \]
      *-commutative [=>] 44.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{\left({M}^{2} \cdot {D}^{2}\right)} \cdot h}{{d}^{2} \cdot \ell}\right) \]
      associate-*l* [=>] 44.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{{M}^{2} \cdot \left({D}^{2} \cdot h\right)}}{{d}^{2} \cdot \ell}\right) \]
      unpow2 [=>] 44.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{{M}^{2} \cdot \left({D}^{2} \cdot h\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]
      associate-*l* [=>] 49.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{{M}^{2} \cdot \left({D}^{2} \cdot h\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]
      times-frac [=>] 54.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\left(\frac{{M}^{2}}{d} \cdot \frac{{D}^{2} \cdot h}{d \cdot \ell}\right)}\right) \]
      unpow2 [=>] 54.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{\color{blue}{M \cdot M}}{d} \cdot \frac{{D}^{2} \cdot h}{d \cdot \ell}\right)\right) \]
      unpow2 [=>] 54.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot h}{d \cdot \ell}\right)\right) \]
      associate-*l* [=>] 55.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{\color{blue}{D \cdot \left(D \cdot h\right)}}{d \cdot \ell}\right)\right) \]

   if 0.0 < h

   1. Initial program 76.8%

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

   2. Simplified 76.1%

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

      [Start] 76.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 76.8	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 76.8	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 76.8	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 76.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      *-commutative [=>] 76.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 76.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]
      times-frac [=>] 76.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 76.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 76.1%

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

      [Start] 76.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      expm1-log1p-u [=>] 75.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)}\right) \]
      expm1-udef [=>] 75.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} - 1\right)}\right) \]
      log1p-udef [=>] 75.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(e^{\color{blue}{\log \left(1 + {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)}} - 1\right)\right) \]
      add-exp-log [<=] 76.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\left(1 + {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} - 1\right)\right) \]
      +-commutative [=>] 76.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right) + 1\right)} - 1\right)\right) \]
      div-inv [=>] 76.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right) + 1\right) - 1\right)\right) \]
      metadata-eval [=>] 76.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right) + 1\right) - 1\right)\right) \]

   4. Simplified 77.8%

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

      [Start] 76.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right) + 1\right) - 1\right)\right) \]
      associate--l+ [=>] 76.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right) + \left(1 - 1\right)\right)}\right) \]
      metadata-eval [=>] 76.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right) + \color{blue}{0}\right)\right) \]
      +-rgt-identity [=>] 76.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)}\right) \]
      associate-*r* [=>] 76.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]
      associate-*r/ [=>] 76.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
      associate-*l/ [<=] 77.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5}{\ell} \cdot h}\right) \]
      *-commutative [=>] 77.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{h \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5}{\ell}}\right) \]
      associate-/l* [=>] 77.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \color{blue}{\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{0.5}}}\right) \]

   5. Applied egg-rr 79.9%

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

      [Start] 77.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]
      sub-neg [=>] 77.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\left(1 + \left(-h \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right)\right)} \]
      distribute-lft-in [=>] 71.1	\[ \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot 1 + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-h \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right)} \]
      *-commutative [<=] 71.1	\[ \color{blue}{1 \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-h \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]
      *-un-lft-identity [<=] 71.1	\[ \color{blue}{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-h \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]
      sqrt-div [=>] 71.9	\[ \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-h \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]
      sqrt-div [=>] 74.7	\[ \frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-h \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]
      frac-times [=>] 74.8	\[ \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-h \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]
      add-sqr-sqrt [<=] 74.8	\[ \frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-h \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]
      sqrt-div [=>] 80.4	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-h \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]
      sqrt-div [=>] 81.9	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(-h \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]
      frac-times [=>] 81.9	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}} \cdot \left(-h \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]
      add-sqr-sqrt [<=] 81.9	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(-h \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \]

   6. Simplified 86.6%

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

      [Start] 79.9	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(\left(\left(-h\right) \cdot {\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.5\right)}^{2}\right) \cdot \frac{0.5}{\ell}\right) \]
      *-commutative [<=] 79.9	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \color{blue}{\left(\left(\left(-h\right) \cdot {\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.5\right)}^{2}\right) \cdot \frac{0.5}{\ell}\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]
      associate-*l* [=>] 81.9	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \color{blue}{\left(\left(-h\right) \cdot \left({\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.5\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right)} \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]
      associate-*r/ [=>] 81.9	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \left(\left(-h\right) \cdot \color{blue}{\frac{{\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.5\right)}^{2} \cdot 0.5}{\ell}}\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]
      associate-*l/ [<=] 81.9	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \left(\left(-h\right) \cdot \color{blue}{\left(\frac{{\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.5\right)}^{2}}{\ell} \cdot 0.5\right)}\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]
      *-commutative [<=] 81.9	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \left(\left(-h\right) \cdot \color{blue}{\left(0.5 \cdot \frac{{\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.5\right)}^{2}}{\ell}\right)}\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]
      distribute-lft-neg-in [<=] 81.9	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \color{blue}{\left(-h \cdot \left(0.5 \cdot \frac{{\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.5\right)}^{2}}{\ell}\right)\right)} \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]
      distribute-rgt1-in [=>] 86.6	\[ \color{blue}{\left(\left(-h \cdot \left(0.5 \cdot \frac{{\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.5\right)}^{2}}{\ell}\right)\right) + 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 5 \cdot 10^{-159}:\\ \;\;\;\;\left(1 - h \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;h \leq 1.52 \cdot 10^{-198}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;h \leq 0:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{D \cdot \left(h \cdot D\right)}{d \cdot \ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - h \cdot \left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	56.1%
Cost	34060

\[\begin{array}{l} t_0 := \left(M \cdot 0.5\right) \cdot \frac{D}{d}\\ t_1 := \sqrt{-d}\\ t_2 := \frac{t_1}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq 6.6 \cdot 10^{+59}:\\ \;\;\;\;t_2 \cdot \left(1 - h \cdot \frac{{t_0}^{2}}{\frac{\ell}{0.5}}\right)\\ \mathbf{elif}\;h \leq 1.55 \cdot 10^{-170}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{t_1}{\sqrt{-\ell}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;h \leq 0:\\ \;\;\;\;t_2 \cdot \left(1 - {\left(t_0 \cdot \sqrt{0.5 \cdot \frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - h \cdot \left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 2
Accuracy	48.7%
Cost	27660

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := 1 - h \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{0.5}}\\ t_2 := \sqrt{-d}\\ t_3 := \frac{t_2}{\sqrt{-h}}\\ \mathbf{if}\;h \leq 1.2 \cdot 10^{+163}:\\ \;\;\;\;\left(t_3 \cdot t_0\right) \cdot t_1\\ \mathbf{elif}\;h \leq 1.5 \cdot 10^{-195}:\\ \;\;\;\;t_1 \cdot \left(\frac{t_2}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;h \leq 0:\\ \;\;\;\;t_3 \cdot \left(t_0 \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - h \cdot \left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 3
Accuracy	44.2%
Cost	27528

\[\begin{array}{l} t_0 := 1 - h \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{0.5}}\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq 1.5 \cdot 10^{+202}:\\ \;\;\;\;t_0 \cdot \left(\frac{t_1}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;\ell \leq 0:\\ \;\;\;\;\left(\frac{t_1}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 - h \cdot \left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 4
Accuracy	40.4%
Cost	27528

\[\begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq 1.02 \cdot 10^{+242}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{t_0}{\sqrt{-\ell}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 0:\\ \;\;\;\;\left(\frac{t_0}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - h \cdot \left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 5
Accuracy	50.5%
Cost	27396

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;\ell \leq 1.05 \cdot 10^{+102}:\\ \;\;\;\;\left(1 - h \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot t_0\right)\\ \mathbf{elif}\;\ell \leq 4.65 \cdot 10^{-33}:\\ \;\;\;\;\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq 1.36 \cdot 10^{-307}:\\ \;\;\;\;\frac{1 + \frac{\frac{0.25}{{\left(\frac{d}{M \cdot D}\right)}^{2}} \cdot \left(h \cdot -0.5\right)}{\ell}}{\frac{\sqrt{\frac{\ell}{d}}}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - h \cdot \left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 6
Accuracy	36.1%
Cost	21004

\[\begin{array}{l} \mathbf{if}\;\ell \leq 3.5 \cdot 10^{+121}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 1.08 \cdot 10^{-35}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}{\ell} \cdot \frac{\frac{h}{d}}{d}\right)\right)\\ \mathbf{elif}\;\ell \leq 0:\\ \;\;\;\;\left(1 + \frac{h}{\ell} \cdot \frac{-0.125}{{\left(\frac{d}{M \cdot D}\right)}^{2}}\right) \cdot \sqrt{\frac{d}{h \cdot \frac{\ell}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	36.6%
Cost	21004

\[\begin{array}{l} \mathbf{if}\;\ell \leq 5.6 \cdot 10^{+119}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{-36}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}{\ell} \cdot \frac{\frac{h}{d}}{d}\right)\right)\\ \mathbf{elif}\;\ell \leq 0:\\ \;\;\;\;\left(1 + \frac{h}{\ell} \cdot \frac{-0.125}{{\left(\frac{d}{M \cdot D}\right)}^{2}}\right) \cdot \sqrt{\frac{d}{h \cdot \frac{\ell}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right)}^{2}}{\ell} \cdot \left(h \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	36.7%
Cost	21004

\[\begin{array}{l} \mathbf{if}\;\ell \leq 2.2 \cdot 10^{+119}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{-36}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}{\ell} \cdot \frac{\frac{h}{d}}{d}\right)\right)\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{-309}:\\ \;\;\;\;\left(1 + \frac{h}{\ell} \cdot \frac{-0.125}{{\left(\frac{d}{M \cdot D}\right)}^{2}}\right) \cdot \sqrt{\frac{d}{h \cdot \frac{\ell}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - h \cdot \left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 9
Accuracy	46.1%
Cost	21000

\[\begin{array}{l} \mathbf{if}\;\ell \leq 8.2 \cdot 10^{+142}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - h \cdot \left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 10
Accuracy	72.4%
Cost	21000

\[\begin{array}{l} \mathbf{if}\;d \leq 2.7 \cdot 10^{-52}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - h \cdot \left(\frac{0.5}{\ell} \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 8.6 \cdot 10^{-299}:\\ \;\;\;\;\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - h \cdot \left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 11
Accuracy	72.4%
Cost	21000

\[\begin{array}{l} \mathbf{if}\;d \leq 1.9 \cdot 10^{-51}:\\ \;\;\;\;\left(1 - h \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{0.5}}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;d \leq 8.6 \cdot 10^{-299}:\\ \;\;\;\;\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - h \cdot \left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 12
Accuracy	72.4%
Cost	21000

\[\begin{array}{l} \mathbf{if}\;d \leq 2.8 \cdot 10^{-52}:\\ \;\;\;\;\frac{1 + \frac{\frac{0.25}{{\left(\frac{d}{M \cdot D}\right)}^{2}} \cdot \left(h \cdot -0.5\right)}{\ell}}{\frac{\sqrt{\frac{\ell}{d}}}{\sqrt{\frac{d}{h}}}}\\ \mathbf{elif}\;d \leq 8.6 \cdot 10^{-299}:\\ \;\;\;\;\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - h \cdot \left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 13
Accuracy	34.7%
Cost	15056

\[\begin{array}{l} t_0 := \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{D \cdot \left(h \cdot D\right)}{d \cdot \ell}\right)\right)\\ \mathbf{if}\;\ell \leq 6.3 \cdot 10^{+112}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{-36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-110}:\\ \;\;\;\;\left(1 + \frac{h}{\ell} \cdot \frac{-0.125}{{\left(\frac{d}{M \cdot D}\right)}^{2}}\right) \cdot \sqrt{\frac{d}{h \cdot \frac{\ell}{d}}}\\ \mathbf{elif}\;\ell \leq 1.26 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 14
Accuracy	33.7%
Cost	15056

\[\begin{array}{l} \mathbf{if}\;\ell \leq 8 \cdot 10^{+117}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{-36}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}{\ell} \cdot \frac{\frac{h}{d}}{d}\right)\right)\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{-108}:\\ \;\;\;\;\left(1 + \frac{h}{\ell} \cdot \frac{-0.125}{{\left(\frac{d}{M \cdot D}\right)}^{2}}\right) \cdot \sqrt{\frac{d}{h \cdot \frac{\ell}{d}}}\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{+154}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{D \cdot \left(h \cdot D\right)}{d \cdot \ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{+203}:\\ \;\;\;\;\left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 15
Accuracy	29.9%
Cost	14864

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{if}\;d \leq 4.4 \cdot 10^{+214}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq 6.4 \cdot 10^{-49}:\\ \;\;\;\;t_0 \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \frac{\left(M \cdot D\right) \cdot \left(\frac{h}{\ell} \cdot \left(D \cdot \frac{M}{d}\right)\right)}{d}\right)\right)\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{-183}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \cdot t_0\\ \end{array} \]

Alternative 16
Accuracy	29.4%
Cost	14864

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{if}\;d \leq 1.5 \cdot 10^{+227}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq 8.6 \cdot 10^{-49}:\\ \;\;\;\;t_0 \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \frac{\left(M \cdot D\right) \cdot \frac{h}{\frac{\ell}{D \cdot \frac{M}{d}}}}{d}\right)\right)\\ \mathbf{elif}\;d \leq 6.6 \cdot 10^{-183}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \cdot t_0\\ \end{array} \]

Alternative 17
Accuracy	29.9%
Cost	14864

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{if}\;d \leq 2.7 \cdot 10^{+216}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq 6.4 \cdot 10^{-49}:\\ \;\;\;\;t_0 \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \frac{\left(M \cdot D\right) \cdot \frac{\frac{M \cdot \left(h \cdot D\right)}{\ell}}{d}}{d}\right)\right)\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-182}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \cdot t_0\\ \end{array} \]

Alternative 18
Accuracy	36.5%
Cost	14472

\[\begin{array}{l} \mathbf{if}\;\ell \leq 2.35 \cdot 10^{+64}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+65}:\\ \;\;\;\;\left(1 + \frac{h}{\ell} \cdot \frac{-0.125}{{\left(\frac{d}{M \cdot D}\right)}^{2}}\right) \cdot \sqrt{\frac{d}{h \cdot \frac{\ell}{d}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 19
Accuracy	40.4%
Cost	14220

\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.35 \cdot 10^{-44}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{-306}:\\ \;\;\;\;d \cdot {\left({\left(h \cdot \ell\right)}^{-3}\right)}^{0.16666666666666666}\\ \mathbf{elif}\;\ell \leq 1.52 \cdot 10^{-44}:\\ \;\;\;\;-0.125 \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \end{array} \]

Alternative 20
Accuracy	40.4%
Cost	14220

\[\begin{array}{l} \mathbf{if}\;\ell \leq 3.6 \cdot 10^{-45}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{-306}:\\ \;\;\;\;d \cdot {\left({\left(h \cdot \ell\right)}^{-3}\right)}^{0.16666666666666666}\\ \mathbf{elif}\;\ell \leq 1.42 \cdot 10^{-44}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \end{array} \]

Alternative 21
Accuracy	39.7%
Cost	13972

\[\begin{array}{l} t_0 := d \cdot {\left({\left(h \cdot \ell\right)}^{-3}\right)}^{0.16666666666666666}\\ \mathbf{if}\;h \leq 10^{+120}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;h \leq 4 \cdot 10^{+62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq 2.05 \cdot 10^{-78}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;h \leq 1.05 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq 1.2 \cdot 10^{-292}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \end{array} \]

Alternative 22
Accuracy	40.5%
Cost	13576

\[\begin{array}{l} \mathbf{if}\;\ell \leq 3.6 \cdot 10^{-45}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 0:\\ \;\;\;\;d \cdot {\left({\left(h \cdot \ell\right)}^{-3}\right)}^{0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \end{array} \]

Alternative 23
Accuracy	43.1%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.22 \cdot 10^{-127}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 0:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(h \cdot \ell\right)}^{-1.5}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 24
Accuracy	40.5%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;\ell \leq 3.6 \cdot 10^{-45}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 0:\\ \;\;\;\;d \cdot {\left({\left(h \cdot \ell\right)}^{-3}\right)}^{0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 25
Accuracy	40.3%
Cost	13448

\[\begin{array}{l} \mathbf{if}\;\ell \leq 8.2 \cdot 10^{-129}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{-306}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(h \cdot \ell\right)}^{-1.5}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]

Alternative 26
Accuracy	43.1%
Cost	13448

\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.65 \cdot 10^{-128}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 0:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(h \cdot \ell\right)}^{-1.5}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]

Alternative 27
Accuracy	40.3%
Cost	13384

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;\ell \leq 1.25 \cdot 10^{-129}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{-303}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left|t_0\right|\\ \end{array} \]

Alternative 28
Accuracy	40.3%
Cost	7308

\[\begin{array}{l} t_0 := d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{if}\;\ell \leq 8.5 \cdot 10^{-128}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{-173}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq 1.62 \cdot 10^{-127}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 29
Accuracy	34.4%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.26 \cdot 10^{-129}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 30
Accuracy	36.2%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\ell \leq 3.5 \cdot 10^{-129}:\\ \;\;\;\;\sqrt{\frac{d}{\frac{\ell}{\frac{d}{h}}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 31
Accuracy	26.0%
Cost	6784

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

Alternative 32
Accuracy	26.0%
Cost	6720

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

Reproduce

herbie shell --seed 2023159 
(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)))))