Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.7% → 82.0%

Time: 45.8s

Precision: binary64

Cost: 27396

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

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot \left(0.5 \cdot h\right)\right)}{\frac{\sqrt{-h}}{\sqrt{-d}}}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-202}:\\ \;\;\;\;-0.125 \cdot \left({\left(\frac{M \cdot D}{\sqrt{d} \cdot {\ell}^{0.75}}\right)}^{2} \cdot \sqrt{h}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \frac{{\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}}{\frac{\ell}{h \cdot -0.5}}\right)\\ \end{array} \]

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
 (if (<= d -4e-310)
   (/
    (*
     (sqrt (/ d l))
     (- 1.0 (* (/ (pow (* M (* 0.5 (/ D d))) 2.0) l) (* 0.5 h))))
    (/ (sqrt (- h)) (sqrt (- d))))
   (if (<= d 3.2e-202)
     (* -0.125 (* (pow (/ (* M D) (* (sqrt d) (pow l 0.75))) 2.0) (sqrt h)))
     (*
      (/ d (* (sqrt h) (sqrt l)))
      (+ 1.0 (/ (pow (* M (/ 0.5 (/ d D))) 2.0) (/ l (* h -0.5))))))))

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 tmp;
	if (d <= -4e-310) {
		tmp = (sqrt((d / l)) * (1.0 - ((pow((M * (0.5 * (D / d))), 2.0) / l) * (0.5 * h)))) / (sqrt(-h) / sqrt(-d));
	} else if (d <= 3.2e-202) {
		tmp = -0.125 * (pow(((M * D) / (sqrt(d) * pow(l, 0.75))), 2.0) * sqrt(h));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 + (pow((M * (0.5 / (d / D))), 2.0) / (l / (h * -0.5))));
	}
	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) :: tmp
    if (d <= (-4d-310)) then
        tmp = (sqrt((d / l)) * (1.0d0 - ((((m * (0.5d0 * (d_1 / d))) ** 2.0d0) / l) * (0.5d0 * h)))) / (sqrt(-h) / sqrt(-d))
    else if (d <= 3.2d-202) then
        tmp = (-0.125d0) * ((((m * d_1) / (sqrt(d) * (l ** 0.75d0))) ** 2.0d0) * sqrt(h))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 + (((m * (0.5d0 / (d / d_1))) ** 2.0d0) / (l / (h * (-0.5d0)))))
    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 tmp;
	if (d <= -4e-310) {
		tmp = (Math.sqrt((d / l)) * (1.0 - ((Math.pow((M * (0.5 * (D / d))), 2.0) / l) * (0.5 * h)))) / (Math.sqrt(-h) / Math.sqrt(-d));
	} else if (d <= 3.2e-202) {
		tmp = -0.125 * (Math.pow(((M * D) / (Math.sqrt(d) * Math.pow(l, 0.75))), 2.0) * Math.sqrt(h));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 + (Math.pow((M * (0.5 / (d / D))), 2.0) / (l / (h * -0.5))));
	}
	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):
	tmp = 0
	if d <= -4e-310:
		tmp = (math.sqrt((d / l)) * (1.0 - ((math.pow((M * (0.5 * (D / d))), 2.0) / l) * (0.5 * h)))) / (math.sqrt(-h) / math.sqrt(-d))
	elif d <= 3.2e-202:
		tmp = -0.125 * (math.pow(((M * D) / (math.sqrt(d) * math.pow(l, 0.75))), 2.0) * math.sqrt(h))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 + (math.pow((M * (0.5 / (d / D))), 2.0) / (l / (h * -0.5))))
	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)
	tmp = 0.0
	if (d <= -4e-310)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(Float64((Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0) / l) * Float64(0.5 * h)))) / Float64(sqrt(Float64(-h)) / sqrt(Float64(-d))));
	elseif (d <= 3.2e-202)
		tmp = Float64(-0.125 * Float64((Float64(Float64(M * D) / Float64(sqrt(d) * (l ^ 0.75))) ^ 2.0) * sqrt(h)));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 + Float64((Float64(M * Float64(0.5 / Float64(d / D))) ^ 2.0) / Float64(l / Float64(h * -0.5)))));
	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)
	tmp = 0.0;
	if (d <= -4e-310)
		tmp = (sqrt((d / l)) * (1.0 - ((((M * (0.5 * (D / d))) ^ 2.0) / l) * (0.5 * h)))) / (sqrt(-h) / sqrt(-d));
	elseif (d <= 3.2e-202)
		tmp = -0.125 * ((((M * D) / (sqrt(d) * (l ^ 0.75))) ^ 2.0) * sqrt(h));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 + (((M * (0.5 / (d / D))) ^ 2.0) / (l / (h * -0.5))));
	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_] := If[LessEqual[d, -4e-310], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[(-h)], $MachinePrecision] / N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.2e-202], N[(-0.125 * N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(N[Sqrt[d], $MachinePrecision] * N[Power[l, 0.75], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[Power[N[(M * N[(0.5 / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l / N[(h * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.999999999999988e-310

   1. Initial program 68.0%

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

   2. Simplified 67.3%

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

      [Start] 68.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 68.0	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 68.0	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 68.0	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 68.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      *-commutative [=>] 68.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 68.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]
      times-frac [=>] 67.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 67.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 71.1%

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

      [Start] 67.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      associate-*r/ [=>] 67.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \color{blue}{\frac{0.5 \cdot h}{\ell}}\right) \]
      associate-*r/ [=>] 71.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
      div-inv [=>] 71.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]
      metadata-eval [=>] 71.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   4. Applied egg-rr 70.7%

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

      [Start] 71.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]
      clear-num [=>] 70.4	\[ \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]
      sqrt-div [=>] 70.7	\[ \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]
      metadata-eval [=>] 70.7	\[ \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   5. Applied egg-rr 70.1%

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

      [Start] 70.7	\[ \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]
      associate-*l* [=>] 70.1	\[ \color{blue}{\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)} \]
      associate-*l/ [=>] 70.1	\[ \color{blue}{\frac{1 \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}{\sqrt{\frac{h}{d}}}} \]
      *-un-lft-identity [<=] 70.1	\[ \frac{\color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)}}{\sqrt{\frac{h}{d}}} \]
      associate-/l* [=>] 66.3	\[ \frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{0.5 \cdot h}}}\right)}{\sqrt{\frac{h}{d}}} \]
      associate-/r/ [=>] 70.1	\[ \frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell} \cdot \left(0.5 \cdot h\right)}\right)}{\sqrt{\frac{h}{d}}} \]
      associate-*l* [=>] 70.1	\[ \frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2}}{\ell} \cdot \left(0.5 \cdot h\right)\right)}{\sqrt{\frac{h}{d}}} \]
      *-commutative [=>] 70.1	\[ \frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot \color{blue}{\left(h \cdot 0.5\right)}\right)}{\sqrt{\frac{h}{d}}} \]

   6. Applied egg-rr 81.1%

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

      [Start] 70.1	\[ \frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot \left(h \cdot 0.5\right)\right)}{\sqrt{\frac{h}{d}}} \]
      frac-2neg [=>] 70.1	\[ \frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot \left(h \cdot 0.5\right)\right)}{\sqrt{\color{blue}{\frac{-h}{-d}}}} \]
      sqrt-div [=>] 81.1	\[ \frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot \left(h \cdot 0.5\right)\right)}{\color{blue}{\frac{\sqrt{-h}}{\sqrt{-d}}}} \]

   if -3.999999999999988e-310 < d < 3.2000000000000001e-202

   1. Initial program 34.3%

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

   2. Simplified 34.3%

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

      [Start] 34.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 34.3	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 34.3	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 34.3	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 34.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      *-commutative [=>] 34.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 34.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]
      times-frac [=>] 34.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 34.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Taylor expanded in d around 0 42.8%

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

   4. Simplified 43.2%

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

      [Start] 42.8	\[ -0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]
      associate-*l/ [=>] 43.0	\[ -0.125 \cdot \color{blue}{\frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}} \]
      unpow2 [=>] 43.0	\[ -0.125 \cdot \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d} \]
      associate-*l* [=>] 43.2	\[ -0.125 \cdot \frac{\color{blue}{\left(D \cdot \left(D \cdot {M}^{2}\right)\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d} \]
      unpow2 [=>] 43.2	\[ -0.125 \cdot \frac{\left(D \cdot \left(D \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d} \]

   5. Applied egg-rr 59.6%

      \[\leadsto -0.125 \cdot \color{blue}{\left(0 + \frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right)} \]
      Step-by-step derivation

      [Start] 43.2	\[ -0.125 \cdot \frac{\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d} \]
      add-log-exp [=>] 43.2	\[ -0.125 \cdot \color{blue}{\log \left(e^{\frac{\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}}\right)} \]
      *-un-lft-identity [=>] 43.2	\[ -0.125 \cdot \log \color{blue}{\left(1 \cdot e^{\frac{\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}}\right)} \]
      log-prod [=>] 43.2	\[ -0.125 \cdot \color{blue}{\left(\log 1 + \log \left(e^{\frac{\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}}\right)\right)} \]
      metadata-eval [=>] 43.2	\[ -0.125 \cdot \left(\color{blue}{0} + \log \left(e^{\frac{\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}}\right)\right) \]
      add-log-exp [<=] 43.2	\[ -0.125 \cdot \left(0 + \color{blue}{\frac{\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}}\right) \]
      associate-/l* [=>] 43.2	\[ -0.125 \cdot \left(0 + \color{blue}{\frac{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}}\right) \]
      associate-/r/ [=>] 43.1	\[ -0.125 \cdot \left(0 + \color{blue}{\frac{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}}\right) \]

   6. Simplified 59.4%

      \[\leadsto -0.125 \cdot \color{blue}{\left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot {\ell}^{1.5}} \cdot \sqrt{h}\right)} \]
      Step-by-step derivation

      [Start] 59.6	\[ -0.125 \cdot \left(0 + \frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right) \]
      +-lft-identity [=>] 59.6	\[ -0.125 \cdot \color{blue}{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right)} \]
      associate-*l/ [=>] 55.8	\[ -0.125 \cdot \color{blue}{\frac{{\left(D \cdot M\right)}^{2} \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}}{d}} \]
      associate-/l* [=>] 59.6	\[ -0.125 \cdot \color{blue}{\frac{{\left(D \cdot M\right)}^{2}}{\frac{d}{\frac{\sqrt{h}}{{\ell}^{1.5}}}}} \]
      associate-/l* [<=] 59.4	\[ -0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{\color{blue}{\frac{d \cdot {\ell}^{1.5}}{\sqrt{h}}}} \]
      associate-/r/ [=>] 59.4	\[ -0.125 \cdot \color{blue}{\left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot {\ell}^{1.5}} \cdot \sqrt{h}\right)} \]

   7. Applied egg-rr 84.3%

      \[\leadsto -0.125 \cdot \left(\color{blue}{{\left(\frac{D \cdot M}{\sqrt{d} \cdot {\ell}^{0.75}}\right)}^{2}} \cdot \sqrt{h}\right) \]
      Step-by-step derivation

      [Start] 59.4	\[ -0.125 \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot {\ell}^{1.5}} \cdot \sqrt{h}\right) \]
      add-sqr-sqrt [=>] 59.4	\[ -0.125 \cdot \left(\color{blue}{\left(\sqrt{\frac{{\left(D \cdot M\right)}^{2}}{d \cdot {\ell}^{1.5}}} \cdot \sqrt{\frac{{\left(D \cdot M\right)}^{2}}{d \cdot {\ell}^{1.5}}}\right)} \cdot \sqrt{h}\right) \]
      pow2 [=>] 59.4	\[ -0.125 \cdot \left(\color{blue}{{\left(\sqrt{\frac{{\left(D \cdot M\right)}^{2}}{d \cdot {\ell}^{1.5}}}\right)}^{2}} \cdot \sqrt{h}\right) \]
      sqrt-div [=>] 59.4	\[ -0.125 \cdot \left({\color{blue}{\left(\frac{\sqrt{{\left(D \cdot M\right)}^{2}}}{\sqrt{d \cdot {\ell}^{1.5}}}\right)}}^{2} \cdot \sqrt{h}\right) \]
      unpow2 [=>] 59.4	\[ -0.125 \cdot \left({\left(\frac{\sqrt{\color{blue}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}}{\sqrt{d \cdot {\ell}^{1.5}}}\right)}^{2} \cdot \sqrt{h}\right) \]
      sqrt-prod [=>] 26.6	\[ -0.125 \cdot \left({\left(\frac{\color{blue}{\sqrt{D \cdot M} \cdot \sqrt{D \cdot M}}}{\sqrt{d \cdot {\ell}^{1.5}}}\right)}^{2} \cdot \sqrt{h}\right) \]
      add-sqr-sqrt [<=] 60.2	\[ -0.125 \cdot \left({\left(\frac{\color{blue}{D \cdot M}}{\sqrt{d \cdot {\ell}^{1.5}}}\right)}^{2} \cdot \sqrt{h}\right) \]
      sqrt-prod [=>] 68.2	\[ -0.125 \cdot \left({\left(\frac{D \cdot M}{\color{blue}{\sqrt{d} \cdot \sqrt{{\ell}^{1.5}}}}\right)}^{2} \cdot \sqrt{h}\right) \]
      sqrt-pow1 [=>] 84.3	\[ -0.125 \cdot \left({\left(\frac{D \cdot M}{\sqrt{d} \cdot \color{blue}{{\ell}^{\left(\frac{1.5}{2}\right)}}}\right)}^{2} \cdot \sqrt{h}\right) \]
      metadata-eval [=>] 84.3	\[ -0.125 \cdot \left({\left(\frac{D \cdot M}{\sqrt{d} \cdot {\ell}^{\color{blue}{0.75}}}\right)}^{2} \cdot \sqrt{h}\right) \]

   if 3.2000000000000001e-202 < d

   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. Simplified 66.1%

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

      [Start] 66.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 66.2	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 66.2	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 66.2	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 66.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      *-commutative [=>] 66.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 66.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]
      times-frac [=>] 66.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 66.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 68.4%

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

      [Start] 66.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      associate-*r/ [=>] 66.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \color{blue}{\frac{0.5 \cdot h}{\ell}}\right) \]
      associate-*r/ [=>] 68.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
      div-inv [=>] 68.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]
      metadata-eval [=>] 68.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   4. Applied egg-rr 80.0%

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

      [Start] 68.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]
      sub-neg [=>] 68.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\left(1 + \left(-\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)} \]
      distribute-lft-in [=>] 56.5	\[ \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot 1 + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)} \]
      *-commutative [<=] 56.5	\[ \color{blue}{1 \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]
      *-un-lft-identity [<=] 56.5	\[ \color{blue}{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]
      sqrt-div [=>] 56.6	\[ \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]
      sqrt-div [=>] 62.9	\[ \frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]
      frac-times [=>] 62.9	\[ \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]
      add-sqr-sqrt [<=] 63.0	\[ \frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]
      sqrt-div [=>] 75.7	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]
      sqrt-div [=>] 80.0	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(-\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]
      frac-times [=>] 80.0	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}} \cdot \left(-\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]
      add-sqr-sqrt [<=] 80.0	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(-\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   5. Simplified 84.9%

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

      [Start] 80.0	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-h \cdot 0.5\right)}{\ell} \]
      *-commutative [<=] 80.0	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-h \cdot 0.5\right)}{\ell} \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]
      distribute-rgt1-in [=>] 88.5	\[ \color{blue}{\left(\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-h \cdot 0.5\right)}{\ell} + 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]
      *-commutative [=>] 88.5	\[ \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-h \cdot 0.5\right)}{\ell} + 1\right)} \]
      associate-*l/ [<=] 88.6	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(\color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot \left(-h \cdot 0.5\right)} + 1\right) \]
      distribute-rgt-neg-in [<=] 88.6	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(\color{blue}{\left(-\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot \left(h \cdot 0.5\right)\right)} + 1\right) \]
      +-commutative [=>] 88.6	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \color{blue}{\left(1 + \left(-\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot \left(h \cdot 0.5\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 88.6	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot \left(-h \cdot 0.5\right)}\right) \]
      associate-/r/ [<=] 84.9	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{-h \cdot 0.5}}}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot \left(0.5 \cdot h\right)\right)}{\frac{\sqrt{-h}}{\sqrt{-d}}}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-202}:\\ \;\;\;\;-0.125 \cdot \left({\left(\frac{M \cdot D}{\sqrt{d} \cdot {\ell}^{0.75}}\right)}^{2} \cdot \sqrt{h}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \frac{{\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}}{\frac{\ell}{h \cdot -0.5}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	74.7%
Cost	27396

\[\begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;h \leq -1.8 \cdot 10^{-255}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{t_0}{\sqrt{-\ell}}\right) \cdot \left(1 - {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-311}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \frac{t_0}{\sqrt{-h}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{D \cdot \left(D \cdot h\right)}{d \cdot \ell}\right)\right)\\ \mathbf{elif}\;h \leq 5.6 \cdot 10^{+125}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\sqrt{h} \cdot {\left(\frac{D}{\sqrt{d}} \cdot \frac{M}{{\ell}^{0.75}}\right)}^{2}\right)\\ \end{array} \]

Alternative 2
Accuracy	76.5%
Cost	27396

\[\begin{array}{l} t_0 := {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\\ \mathbf{if}\;h \leq -4 \cdot 10^{-311}:\\ \;\;\;\;\left(1 - \frac{\left(0.5 \cdot h\right) \cdot t_0}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;h \leq 1.22 \cdot 10^{+117}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + t_0 \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\sqrt{h} \cdot {\left(\frac{D}{\sqrt{d}} \cdot \frac{M}{{\ell}^{0.75}}\right)}^{2}\right)\\ \end{array} \]

Alternative 3
Accuracy	78.1%
Cost	27396

\[\begin{array}{l} t_0 := {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\\ \mathbf{if}\;h \leq -4 \cdot 10^{-311}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right) \cdot \left(1 - \frac{\left(0.5 \cdot h\right) \cdot t_0}{\ell}\right)\\ \mathbf{elif}\;h \leq 8.5 \cdot 10^{+130}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + t_0 \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\sqrt{h} \cdot {\left(\frac{D}{\sqrt{d}} \cdot \frac{M}{{\ell}^{0.75}}\right)}^{2}\right)\\ \end{array} \]

Alternative 4
Accuracy	74.7%
Cost	27092

\[\begin{array}{l} t_0 := {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -3.6 \cdot 10^{+90}:\\ \;\;\;\;\left(1 - \frac{\left(0.5 \cdot h\right) \cdot t_0}{\ell}\right) \cdot \left(t_1 \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;h \leq -4.8 \cdot 10^{-194}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left({\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2} \cdot \left(h \cdot \frac{0.5}{\ell}\right) + -1\right)\\ \mathbf{elif}\;h \leq -4.4 \cdot 10^{-241}:\\ \;\;\;\;\frac{t_1 \cdot \left(1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot \left(0.5 \cdot h\right)\right)}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-311}:\\ \;\;\;\;\left(t_1 \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{D \cdot \left(D \cdot h\right)}{d \cdot \ell}\right)\right)\\ \mathbf{elif}\;h \leq 10^{+117}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + t_0 \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\sqrt{h} \cdot {\left(\frac{D}{\sqrt{d}} \cdot \frac{M}{{\ell}^{0.75}}\right)}^{2}\right)\\ \end{array} \]

Alternative 5
Accuracy	77.3%
Cost	21136

\[\begin{array}{l} t_0 := \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left({\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2} \cdot \left(h \cdot \frac{0.5}{\ell}\right) + -1\right)\\ \mathbf{if}\;d \leq -2.1 \cdot 10^{+128}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -2.8 \cdot 10^{-115}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\frac{h}{d} \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d}\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq -1.1 \cdot 10^{-308}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.32 \cdot 10^{-229}:\\ \;\;\;\;-0.125 \cdot \left(\sqrt{h} \cdot \left(\frac{M \cdot D}{{\ell}^{1.5}} \cdot \frac{M \cdot D}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	77.4%
Cost	21136

\[\begin{array}{l} t_0 := {\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}\\ t_1 := \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(t_0 \cdot \left(h \cdot \frac{0.5}{\ell}\right) + -1\right)\\ \mathbf{if}\;d \leq -4.5 \cdot 10^{+126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -6 \cdot 10^{-116}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\frac{h}{d} \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d}\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq -1.1 \cdot 10^{-308}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{-228}:\\ \;\;\;\;-0.125 \cdot \left(\sqrt{h} \cdot \left(\frac{M \cdot D}{{\ell}^{1.5}} \cdot \frac{M \cdot D}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \frac{t_0}{\frac{\ell}{h \cdot -0.5}}\right)\\ \end{array} \]

Alternative 7
Accuracy	76.4%
Cost	21136

\[\begin{array}{l} t_0 := {\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}\\ t_1 := d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ t_2 := \left(M \cdot D\right) \cdot \left(M \cdot D\right)\\ \mathbf{if}\;d \leq -5.6 \cdot 10^{+127}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \left(h \cdot \frac{0.5}{\ell}\right) + -1\right)\\ \mathbf{elif}\;d \leq -1.86 \cdot 10^{-150}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\frac{h}{d} \cdot \frac{t_2}{d}\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{t_2 \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{\frac{d}{0.125}} - t_1\\ \mathbf{elif}\;d \leq 1.95 \cdot 10^{-229}:\\ \;\;\;\;-0.125 \cdot \left(\sqrt{h} \cdot \left(\frac{M \cdot D}{{\ell}^{1.5}} \cdot \frac{M \cdot D}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \frac{t_0}{\frac{\ell}{h \cdot -0.5}}\right)\\ \end{array} \]

Alternative 8
Accuracy	70.8%
Cost	15180

\[\begin{array}{l} t_0 := d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ t_1 := \frac{D}{d} \cdot \left(M \cdot 0.5\right)\\ t_2 := {\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2} \cdot \left(h \cdot \frac{0.5}{\ell}\right)\\ t_3 := \frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 - \left(0.5 \cdot h\right) \cdot \left(t_1 \cdot \frac{t_1}{\ell}\right)\right)}{\sqrt{\frac{h}{d}}}\\ \mathbf{if}\;h \leq -1.35 \cdot 10^{+53}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;h \leq -2.65 \cdot 10^{-194}:\\ \;\;\;\;t_0 \cdot \left(t_2 + -1\right)\\ \mathbf{elif}\;h \leq -1.25 \cdot 10^{-238}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-311}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;h \leq 3.9 \cdot 10^{+102}:\\ \;\;\;\;\left(1 - t_2\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\sqrt{h} \cdot \left(\frac{M \cdot D}{{\ell}^{1.5}} \cdot \frac{M \cdot D}{d}\right)\right)\\ \end{array} \]

Alternative 9
Accuracy	63.2%
Cost	14864

\[\begin{array}{l} \mathbf{if}\;d \leq -9.5 \cdot 10^{+156}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -1.1 \cdot 10^{-308}:\\ \;\;\;\;\left(1 - h \cdot \left(\frac{0.5}{\ell} \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}\right)\right) \cdot \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{-178}:\\ \;\;\;\;-0.125 \cdot \left(\sqrt{h} \cdot \left(\frac{M \cdot D}{{\ell}^{1.5}} \cdot \frac{M \cdot D}{d}\right)\right)\\ \mathbf{elif}\;d \leq 2.25 \cdot 10^{+162}:\\ \;\;\;\;\left(1 + \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot -0.5}}\right) \cdot \sqrt{d \cdot \frac{\frac{d}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 10
Accuracy	62.3%
Cost	14732

\[\begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{d \cdot \frac{\frac{d}{h}}{\ell}} \cdot \left(1 - h \cdot \left({\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{-178}:\\ \;\;\;\;-0.125 \cdot \left(\sqrt{h} \cdot \left(\frac{M \cdot D}{{\ell}^{1.5}} \cdot \frac{M \cdot D}{d}\right)\right)\\ \mathbf{elif}\;d \leq 9.2 \cdot 10^{+160}:\\ \;\;\;\;\left(1 + \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot -0.5}}\right) \cdot \sqrt{d \cdot \frac{\frac{d}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 11
Accuracy	70.6%
Cost	14732

\[\begin{array}{l} t_0 := d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ t_1 := {\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}\\ t_2 := t_1 \cdot \left(h \cdot \frac{0.5}{\ell}\right)\\ \mathbf{if}\;h \leq -4.2 \cdot 10^{+218}:\\ \;\;\;\;\sqrt{d \cdot \frac{\frac{d}{h}}{\ell}} \cdot \left(1 - h \cdot \left(t_1 \cdot \frac{0.5}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-311}:\\ \;\;\;\;t_0 \cdot \left(t_2 + -1\right)\\ \mathbf{elif}\;h \leq 1.75 \cdot 10^{+103}:\\ \;\;\;\;\left(1 - t_2\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\sqrt{h} \cdot \left(\frac{M \cdot D}{{\ell}^{1.5}} \cdot \frac{M \cdot D}{d}\right)\right)\\ \end{array} \]

Alternative 12
Accuracy	70.7%
Cost	14732

\[\begin{array}{l} t_0 := d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ t_1 := {\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2} \cdot \left(h \cdot \frac{0.5}{\ell}\right)\\ \mathbf{if}\;h \leq -4.4 \cdot 10^{+201}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.125 \cdot \frac{D \cdot \left(M \cdot \frac{M}{d}\right)}{\frac{d}{h} \cdot \frac{\ell}{D}}\right)\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-311}:\\ \;\;\;\;t_0 \cdot \left(t_1 + -1\right)\\ \mathbf{elif}\;h \leq 1.72 \cdot 10^{+103}:\\ \;\;\;\;\left(1 - t_1\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\sqrt{h} \cdot \left(\frac{M \cdot D}{{\ell}^{1.5}} \cdot \frac{M \cdot D}{d}\right)\right)\\ \end{array} \]

Alternative 13
Accuracy	69.9%
Cost	14732

\[\begin{array}{l} t_0 := d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ t_1 := {\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2} \cdot \left(h \cdot \frac{0.5}{\ell}\right)\\ \mathbf{if}\;h \leq -1.35 \cdot 10^{+185}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(0.125 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-311}:\\ \;\;\;\;t_0 \cdot \left(t_1 + -1\right)\\ \mathbf{elif}\;h \leq 8 \cdot 10^{+102}:\\ \;\;\;\;\left(1 - t_1\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\sqrt{h} \cdot \left(\frac{M \cdot D}{{\ell}^{1.5}} \cdot \frac{M \cdot D}{d}\right)\right)\\ \end{array} \]

Alternative 14
Accuracy	70.8%
Cost	14732

\[\begin{array}{l} t_0 := d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ t_1 := {\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2} \cdot \left(h \cdot \frac{0.5}{\ell}\right)\\ \mathbf{if}\;h \leq -1.35 \cdot 10^{+185}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{0.125 \cdot \left(\frac{h}{d} \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d}\right)}{\ell}\right)\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-311}:\\ \;\;\;\;t_0 \cdot \left(t_1 + -1\right)\\ \mathbf{elif}\;h \leq 1.75 \cdot 10^{+103}:\\ \;\;\;\;\left(1 - t_1\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\sqrt{h} \cdot \left(\frac{M \cdot D}{{\ell}^{1.5}} \cdot \frac{M \cdot D}{d}\right)\right)\\ \end{array} \]

Alternative 15
Accuracy	63.2%
Cost	14600

\[\begin{array}{l} \mathbf{if}\;h \leq 2.9 \cdot 10^{-208}:\\ \;\;\;\;\left(1 - h \cdot \left(\frac{0.5}{\ell} \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}\right)\right) \cdot \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}\\ \mathbf{elif}\;h \leq 3.9 \cdot 10^{+102}:\\ \;\;\;\;\left(1 - {\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2} \cdot \left(h \cdot \frac{0.5}{\ell}\right)\right) \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\sqrt{h} \cdot \left(\frac{M \cdot D}{{\ell}^{1.5}} \cdot \frac{M \cdot D}{d}\right)\right)\\ \end{array} \]

Alternative 16
Accuracy	62.7%
Cost	14468

\[\begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{d \cdot \frac{\frac{d}{h}}{\ell}} \cdot \left(1 - h \cdot \left({\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 2 \cdot 10^{-118}:\\ \;\;\;\;-0.125 \cdot \left(\sqrt{h} \cdot \left(\frac{M \cdot D}{{\ell}^{1.5}} \cdot \frac{M \cdot D}{d}\right)\right)\\ \mathbf{elif}\;d \leq 1.3 \cdot 10^{+103}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - 0.125 \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\ell} \cdot \frac{h}{d \cdot d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 17
Accuracy	57.8%
Cost	14352

\[\begin{array}{l} t_0 := \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}\\ \mathbf{if}\;d \leq -2.25 \cdot 10^{+117}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -1.65 \cdot 10^{-219}:\\ \;\;\;\;t_0 \cdot \left(1 - 0.125 \cdot \left(\frac{h}{d} \cdot \frac{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}{d \cdot \ell}\right)\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-119}:\\ \;\;\;\;-0.125 \cdot \left(\sqrt{h} \cdot \left(\frac{D \cdot D}{d} \cdot \frac{M \cdot M}{{\ell}^{1.5}}\right)\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{+100}:\\ \;\;\;\;t_0 \cdot \left(1 - 0.125 \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\ell} \cdot \frac{h}{d \cdot d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 18
Accuracy	60.3%
Cost	14352

\[\begin{array}{l} t_0 := \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}\\ \mathbf{if}\;d \leq -5 \cdot 10^{+117}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -7.6 \cdot 10^{-222}:\\ \;\;\;\;t_0 \cdot \left(1 - 0.125 \cdot \left(\frac{h}{d} \cdot \frac{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}{d \cdot \ell}\right)\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{-119}:\\ \;\;\;\;-0.125 \cdot \left(\sqrt{h} \cdot \left(\frac{M \cdot D}{{\ell}^{1.5}} \cdot \frac{M \cdot D}{d}\right)\right)\\ \mathbf{elif}\;d \leq 3.4 \cdot 10^{+102}:\\ \;\;\;\;t_0 \cdot \left(1 - 0.125 \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\ell} \cdot \frac{h}{d \cdot d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 19
Accuracy	55.7%
Cost	14041

\[\begin{array}{l} t_0 := \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}\\ t_1 := t_0 \cdot \left(1 - 0.125 \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\ell} \cdot \frac{h}{d \cdot d}\right)\right)\\ \mathbf{if}\;d \leq -7 \cdot 10^{+153}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -1.1 \cdot 10^{-171}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{-250}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;d \leq 2.15 \cdot 10^{-223}:\\ \;\;\;\;t_0 \cdot \left(1 - 0.125 \cdot \left(\frac{h}{d} \cdot \frac{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}{d \cdot \ell}\right)\right)\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-145} \lor \neg \left(d \leq 1.65 \cdot 10^{+100}\right):\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 20
Accuracy	54.2%
Cost	14041

\[\begin{array}{l} t_0 := \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}\\ t_1 := t_0 \cdot \left(1 - 0.125 \cdot \left(\frac{h}{d} \cdot \frac{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}{d \cdot \ell}\right)\right)\\ \mathbf{if}\;d \leq -2 \cdot 10^{+117}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -8.5 \cdot 10^{-219}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 6 \cdot 10^{-273}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{-223}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-145} \lor \neg \left(d \leq 9.6 \cdot 10^{+103}\right):\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(1 - 0.125 \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\ell} \cdot \frac{h}{d \cdot d}\right)\right)\\ \end{array} \]

Alternative 21
Accuracy	55.1%
Cost	13913

\[\begin{array}{l} t_0 := \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}\\ t_1 := t_0 \cdot \left(1 - 0.125 \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\ell} \cdot \frac{h}{d \cdot d}\right)\right)\\ \mathbf{if}\;d \leq -2.9 \cdot 10^{+153}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -2.5 \cdot 10^{-172}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 9 \cdot 10^{-251}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{-223}:\\ \;\;\;\;t_0 \cdot \left(1 - 0.125 \cdot \left(\frac{h}{d} \cdot \frac{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}{d \cdot \ell}\right)\right)\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{-145} \lor \neg \left(d \leq 8.7 \cdot 10^{+99}\right):\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 22
Accuracy	54.0%
Cost	8788

\[\begin{array}{l} t_0 := \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - 0.125 \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\ell} \cdot \frac{h}{d \cdot d}\right)\right)\\ \mathbf{if}\;d \leq -9.5 \cdot 10^{+153}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -4.8 \cdot 10^{-172}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{-248}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;d \leq 1.12 \cdot 10^{-144}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{+103}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]

Alternative 23
Accuracy	52.3%
Cost	8392

\[\begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{if}\;\ell \leq -1.1 \cdot 10^{-55}:\\ \;\;\;\;d \cdot \left(-t_0\right)\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{-22}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - 0.125 \cdot \left(\frac{h}{d} \cdot \frac{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}{d \cdot \ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot t_0\\ \end{array} \]

Alternative 24
Accuracy	42.2%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;\ell \leq 8.8 \cdot 10^{-243}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]

Alternative 25
Accuracy	42.2%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;\ell \leq 2.1 \cdot 10^{-244}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]

Alternative 26
Accuracy	35.8%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]

Alternative 27
Accuracy	37.1%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.18 \cdot 10^{-222}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]

Alternative 28
Accuracy	26.3%
Cost	6784

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

Alternative 29
Accuracy	26.3%
Cost	6720

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

Reproduce

herbie shell --seed 2023161 
(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)))))