Henrywood and Agarwal, Equation (12)

Average Error: 26.1 → 13.9

Time: 52.8s

Precision: binary64

Cost: 21836

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := 1 + 0.5 \cdot \left(h \cdot \left(\left(\frac{M}{d} \cdot \left(0.5 \cdot D\right)\right) \cdot \frac{\frac{M}{d} \cdot \left(D \cdot -0.5\right)}{\ell}\right)\right)\\ t_2 := \frac{t_0}{\sqrt{-\ell}}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{+248}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot t_2\right) \cdot \left(1 + 0.5 \cdot \left(\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{D \cdot \left(M \cdot -0.5\right)}{d}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1.1 \cdot 10^{-99}:\\ \;\;\;\;\left(\frac{t_0}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot t_1\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_1 \cdot \left(t_2 \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \frac{h}{\frac{\ell}{{\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{2}}} \cdot -0.5\right)\\ \end{array} \]

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
 (let* ((t_0 (sqrt (- d)))
        (t_1
         (+
          1.0
          (*
           0.5
           (* h (* (* (/ M d) (* 0.5 D)) (/ (* (/ M d) (* D -0.5)) l))))))
        (t_2 (/ t_0 (sqrt (- l)))))
   (if (<= l -5e+248)
     (*
      (* (sqrt (/ d h)) t_2)
      (+
       1.0
       (* 0.5 (* (/ (* D (* 0.5 M)) d) (* (/ h l) (/ (* D (* M -0.5)) d))))))
     (if (<= l -1.1e-99)
       (* (* (/ t_0 (sqrt (- h))) (sqrt (/ d l))) t_1)
       (if (<= l -5e-310)
         (* t_1 (* t_2 (/ 1.0 (sqrt (/ h d)))))
         (*
          (/ d (* (sqrt h) (sqrt l)))
          (+ 1.0 (* (/ h (/ l (pow (* 0.5 (* D (/ M d))) 2.0))) -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 t_0 = sqrt(-d);
	double t_1 = 1.0 + (0.5 * (h * (((M / d) * (0.5 * D)) * (((M / d) * (D * -0.5)) / l))));
	double t_2 = t_0 / sqrt(-l);
	double tmp;
	if (l <= -5e+248) {
		tmp = (sqrt((d / h)) * t_2) * (1.0 + (0.5 * (((D * (0.5 * M)) / d) * ((h / l) * ((D * (M * -0.5)) / d)))));
	} else if (l <= -1.1e-99) {
		tmp = ((t_0 / sqrt(-h)) * sqrt((d / l))) * t_1;
	} else if (l <= -5e-310) {
		tmp = t_1 * (t_2 * (1.0 / sqrt((h / d))));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 + ((h / (l / pow((0.5 * (D * (M / d))), 2.0))) * -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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt(-d)
    t_1 = 1.0d0 + (0.5d0 * (h * (((m / d) * (0.5d0 * d_1)) * (((m / d) * (d_1 * (-0.5d0))) / l))))
    t_2 = t_0 / sqrt(-l)
    if (l <= (-5d+248)) then
        tmp = (sqrt((d / h)) * t_2) * (1.0d0 + (0.5d0 * (((d_1 * (0.5d0 * m)) / d) * ((h / l) * ((d_1 * (m * (-0.5d0))) / d)))))
    else if (l <= (-1.1d-99)) then
        tmp = ((t_0 / sqrt(-h)) * sqrt((d / l))) * t_1
    else if (l <= (-5d-310)) then
        tmp = t_1 * (t_2 * (1.0d0 / sqrt((h / d))))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 + ((h / (l / ((0.5d0 * (d_1 * (m / d))) ** 2.0d0))) * (-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 t_0 = Math.sqrt(-d);
	double t_1 = 1.0 + (0.5 * (h * (((M / d) * (0.5 * D)) * (((M / d) * (D * -0.5)) / l))));
	double t_2 = t_0 / Math.sqrt(-l);
	double tmp;
	if (l <= -5e+248) {
		tmp = (Math.sqrt((d / h)) * t_2) * (1.0 + (0.5 * (((D * (0.5 * M)) / d) * ((h / l) * ((D * (M * -0.5)) / d)))));
	} else if (l <= -1.1e-99) {
		tmp = ((t_0 / Math.sqrt(-h)) * Math.sqrt((d / l))) * t_1;
	} else if (l <= -5e-310) {
		tmp = t_1 * (t_2 * (1.0 / Math.sqrt((h / d))));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 + ((h / (l / Math.pow((0.5 * (D * (M / d))), 2.0))) * -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):
	t_0 = math.sqrt(-d)
	t_1 = 1.0 + (0.5 * (h * (((M / d) * (0.5 * D)) * (((M / d) * (D * -0.5)) / l))))
	t_2 = t_0 / math.sqrt(-l)
	tmp = 0
	if l <= -5e+248:
		tmp = (math.sqrt((d / h)) * t_2) * (1.0 + (0.5 * (((D * (0.5 * M)) / d) * ((h / l) * ((D * (M * -0.5)) / d)))))
	elif l <= -1.1e-99:
		tmp = ((t_0 / math.sqrt(-h)) * math.sqrt((d / l))) * t_1
	elif l <= -5e-310:
		tmp = t_1 * (t_2 * (1.0 / math.sqrt((h / d))))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 + ((h / (l / math.pow((0.5 * (D * (M / d))), 2.0))) * -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)
	t_0 = sqrt(Float64(-d))
	t_1 = Float64(1.0 + Float64(0.5 * Float64(h * Float64(Float64(Float64(M / d) * Float64(0.5 * D)) * Float64(Float64(Float64(M / d) * Float64(D * -0.5)) / l)))))
	t_2 = Float64(t_0 / sqrt(Float64(-l)))
	tmp = 0.0
	if (l <= -5e+248)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * t_2) * Float64(1.0 + Float64(0.5 * Float64(Float64(Float64(D * Float64(0.5 * M)) / d) * Float64(Float64(h / l) * Float64(Float64(D * Float64(M * -0.5)) / d))))));
	elseif (l <= -1.1e-99)
		tmp = Float64(Float64(Float64(t_0 / sqrt(Float64(-h))) * sqrt(Float64(d / l))) * t_1);
	elseif (l <= -5e-310)
		tmp = Float64(t_1 * Float64(t_2 * Float64(1.0 / sqrt(Float64(h / d)))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 + Float64(Float64(h / Float64(l / (Float64(0.5 * Float64(D * Float64(M / d))) ^ 2.0))) * -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)
	t_0 = sqrt(-d);
	t_1 = 1.0 + (0.5 * (h * (((M / d) * (0.5 * D)) * (((M / d) * (D * -0.5)) / l))));
	t_2 = t_0 / sqrt(-l);
	tmp = 0.0;
	if (l <= -5e+248)
		tmp = (sqrt((d / h)) * t_2) * (1.0 + (0.5 * (((D * (0.5 * M)) / d) * ((h / l) * ((D * (M * -0.5)) / d)))));
	elseif (l <= -1.1e-99)
		tmp = ((t_0 / sqrt(-h)) * sqrt((d / l))) * t_1;
	elseif (l <= -5e-310)
		tmp = t_1 * (t_2 * (1.0 / sqrt((h / d))));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 + ((h / (l / ((0.5 * (D * (M / d))) ^ 2.0))) * -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_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(0.5 * N[(h * N[(N[(N[(M / d), $MachinePrecision] * N[(0.5 * D), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M / d), $MachinePrecision] * N[(D * -0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5e+248], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision] * N[(1.0 + N[(0.5 * N[(N[(N[(D * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(N[(D * N[(M * -0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1.1e-99], N[(N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[l, -5e-310], N[(t$95$1 * N[(t$95$2 * N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / N[(l / N[Power[N[(0.5 * N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.9999999999999996e248

   1. Initial program 33.9

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

   2. Simplified 33.8

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

      [Start] 33.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 33.9	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 33.9	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 33.9	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 33.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 33.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 33.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 33.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 48.0

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

   4. Applied egg-rr 43.4

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

   5. Applied egg-rr 22.5

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

   if -4.9999999999999996e248 < l < -1.10000000000000002e-99

   1. Initial program 22.8

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

   2. Simplified 23.3

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

      [Start] 22.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 22.8	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 22.8	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 22.8	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 22.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 22.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 22.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 23.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 23.7

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

   4. Simplified 22.8

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

      [Start] 23.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(e^{\mathsf{log1p}\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)} - 1\right)\right) \]
      expm1-def [=>] 23.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]
      expm1-log1p [=>] 23.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      associate-*r/ [=>] 23.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]
      associate-*l/ [<=] 22.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell} \cdot h\right)}\right) \]
      *-commutative [=>] 22.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)}\right) \]
      associate-*r/ [=>] 22.5	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{\left(M \cdot 0.5\right) \cdot D}{d}\right)}}^{2}}{\ell}\right)\right) \]
      associate-*l/ [<=] 22.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{M \cdot 0.5}{d} \cdot D\right)}}^{2}}{\ell}\right)\right) \]
      *-commutative [=>] 22.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(D \cdot \frac{M \cdot 0.5}{d}\right)}}^{2}}{\ell}\right)\right) \]
      associate-/l* [=>] 22.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \color{blue}{\frac{M}{\frac{d}{0.5}}}\right)}^{2}}{\ell}\right)\right) \]

   5. Applied egg-rr 21.3

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

   6. Applied egg-rr 14.5

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

   if -1.10000000000000002e-99 < l < -4.999999999999985e-310

   1. Initial program 31.9

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

   2. Simplified 32.3

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

      [Start] 31.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 31.9	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 31.9	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 31.9	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 31.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 31.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 31.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 32.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 33.0

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

   4. Simplified 28.1

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

      [Start] 33.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(e^{\mathsf{log1p}\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)} - 1\right)\right) \]
      expm1-def [=>] 33.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]
      expm1-log1p [=>] 32.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      associate-*r/ [=>] 25.5	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]
      associate-*l/ [<=] 27.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell} \cdot h\right)}\right) \]
      *-commutative [=>] 27.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)}\right) \]
      associate-*r/ [=>] 27.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{\left(M \cdot 0.5\right) \cdot D}{d}\right)}}^{2}}{\ell}\right)\right) \]
      associate-*l/ [<=] 28.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{M \cdot 0.5}{d} \cdot D\right)}}^{2}}{\ell}\right)\right) \]
      *-commutative [=>] 28.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(D \cdot \frac{M \cdot 0.5}{d}\right)}}^{2}}{\ell}\right)\right) \]
      associate-/l* [=>] 28.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \color{blue}{\frac{M}{\frac{d}{0.5}}}\right)}^{2}}{\ell}\right)\right) \]

   5. Applied egg-rr 27.7

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

   6. Applied egg-rr 27.5

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

   7. Applied egg-rr 12.9

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

   if -4.999999999999985e-310 < l

   1. Initial program 26.0

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

   2. Simplified 26.5

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

      [Start] 26.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 26.0	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 26.0	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 26.0	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 26.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 26.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 26.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 26.5	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 26.9

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

   4. Simplified 25.2

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

      [Start] 26.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(e^{\mathsf{log1p}\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)} - 1\right)\right) \]
      expm1-def [=>] 26.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]
      expm1-log1p [=>] 26.5	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      associate-*r/ [=>] 25.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]
      associate-*l/ [<=] 25.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell} \cdot h\right)}\right) \]
      *-commutative [=>] 25.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)}\right) \]
      associate-*r/ [=>] 25.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{\left(M \cdot 0.5\right) \cdot D}{d}\right)}}^{2}}{\ell}\right)\right) \]
      associate-*l/ [<=] 25.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{M \cdot 0.5}{d} \cdot D\right)}}^{2}}{\ell}\right)\right) \]
      *-commutative [=>] 25.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(D \cdot \frac{M \cdot 0.5}{d}\right)}}^{2}}{\ell}\right)\right) \]
      associate-/l* [=>] 25.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \color{blue}{\frac{M}{\frac{d}{0.5}}}\right)}^{2}}{\ell}\right)\right) \]

   5. Applied egg-rr 24.1

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

   6. Applied egg-rr 12.9

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

   7. Simplified 12.5

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

      [Start] 12.9	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(\frac{h \cdot {\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{2}}{\ell} \cdot -0.5\right) \]
      *-rgt-identity [<=] 12.9	\[ \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot 1} + \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(\frac{h \cdot {\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{2}}{\ell} \cdot -0.5\right) \]
      distribute-lft-in [<=] 12.9	\[ \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \frac{h \cdot {\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{2}}{\ell} \cdot -0.5\right)} \]
      associate-/l* [=>] 12.5	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \color{blue}{\frac{h}{\frac{\ell}{{\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{2}}}} \cdot -0.5\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 13.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{+248}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{D \cdot \left(M \cdot -0.5\right)}{d}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1.1 \cdot 10^{-99}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(h \cdot \left(\left(\frac{M}{d} \cdot \left(0.5 \cdot D\right)\right) \cdot \frac{\frac{M}{d} \cdot \left(D \cdot -0.5\right)}{\ell}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(h \cdot \left(\left(\frac{M}{d} \cdot \left(0.5 \cdot D\right)\right) \cdot \frac{\frac{M}{d} \cdot \left(D \cdot -0.5\right)}{\ell}\right)\right)\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \frac{h}{\frac{\ell}{{\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{2}}} \cdot -0.5\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	15.7
Cost	21840

\[\begin{array}{l} t_0 := D \cdot \left(0.5 \cdot M\right)\\ t_1 := D \cdot \left(M \cdot -0.5\right)\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := t_2 \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{if}\;d \leq -1.3 \cdot 10^{+256}:\\ \;\;\;\;\left(t_2 \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -2.35 \cdot 10^{+175}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{+43}:\\ \;\;\;\;t_3 \cdot \left(1 + 0.5 \cdot \left(t_0 \cdot \left(\frac{h}{d \cdot d} \cdot \frac{t_1}{\ell}\right)\right)\right)\\ \mathbf{elif}\;d \leq -4.4 \cdot 10^{-299}:\\ \;\;\;\;t_3 \cdot \left(1 + 0.5 \cdot \left(\frac{t_0}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{t_1}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \frac{h}{\frac{\ell}{{\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{2}}} \cdot -0.5\right)\\ \end{array} \]

Alternative 2
Error	18.0
Cost	21712

\[\begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ t_1 := 0.5 \cdot t_0\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := t_2 \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{if}\;d \leq -5 \cdot 10^{+254}:\\ \;\;\;\;t_3 \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -1.12 \cdot 10^{+166}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -2.35 \cdot 10^{-122}:\\ \;\;\;\;t_3 \cdot \left(1 + 0.5 \cdot \left(\frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot d} \cdot \left(h \cdot \frac{D \cdot \left(M \cdot -0.5\right)}{\ell}\right)\right)\right)\\ \mathbf{elif}\;d \leq -6.2 \cdot 10^{-182}:\\ \;\;\;\;\left(t_2 \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(\frac{M}{\frac{\ell \cdot d}{h \cdot \frac{D}{\frac{d}{D}}}} \cdot \left(M \cdot -0.25\right)\right)\right)\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{-212}:\\ \;\;\;\;\left(t_2 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + 0.5 \cdot \frac{t_1 \cdot \left(h \cdot \left(t_0 \cdot -0.5\right)\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \frac{h}{\frac{\ell}{{t_1}^{2}}} \cdot -0.5\right)\\ \end{array} \]

Alternative 3
Error	16.3
Cost	21708

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -3.8 \cdot 10^{+256}:\\ \;\;\;\;\left(t_0 \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -3 \cdot 10^{+179}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -4.4 \cdot 10^{-299}:\\ \;\;\;\;\left(t_0 \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{D \cdot \left(M \cdot -0.5\right)}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \frac{h}{\frac{\ell}{{\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{2}}} \cdot -0.5\right)\\ \end{array} \]

Alternative 4
Error	14.0
Cost	21576

\[\begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;d \leq -3.1 \cdot 10^{+35}:\\ \;\;\;\;\left(\frac{t_0}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(h \cdot \left(\left(\frac{M}{d} \cdot \left(0.5 \cdot D\right)\right) \cdot \frac{\frac{M}{d} \cdot \left(D \cdot -0.5\right)}{\ell}\right)\right)\right)\\ \mathbf{elif}\;d \leq -4.4 \cdot 10^{-299}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{t_0}{\sqrt{-\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{D \cdot \left(M \cdot -0.5\right)}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \frac{h}{\frac{\ell}{{\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{2}}} \cdot -0.5\right)\\ \end{array} \]

Alternative 5
Error	18.0
Cost	21268

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := t_0 \cdot \sqrt{\frac{d}{\ell}}\\ t_2 := D \cdot \frac{M}{d}\\ t_3 := 0.5 \cdot t_2\\ \mathbf{if}\;d \leq -3.8 \cdot 10^{+256}:\\ \;\;\;\;t_1 \cdot \left(1 + 0.5 \cdot \frac{\left(M \cdot \frac{D}{d}\right) \cdot \left(\frac{D}{d} \cdot \left(M \cdot -0.5\right)\right)}{2 \cdot \frac{\ell}{h}}\right)\\ \mathbf{elif}\;d \leq -1.65 \cdot 10^{+168}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -2.2 \cdot 10^{-116}:\\ \;\;\;\;\left(t_0 \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 + 0.5 \cdot \left(\frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot d} \cdot \left(h \cdot \frac{D \cdot \left(M \cdot -0.5\right)}{\ell}\right)\right)\right)\\ \mathbf{elif}\;d \leq -8 \cdot 10^{-182}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{-213}:\\ \;\;\;\;t_1 \cdot \left(1 + 0.5 \cdot \frac{t_3 \cdot \left(h \cdot \left(t_2 \cdot -0.5\right)\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \frac{h}{\frac{\ell}{{t_3}^{2}}} \cdot -0.5\right)\\ \end{array} \]

Alternative 6
Error	17.9
Cost	21268

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := t_0 \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\\ t_2 := D \cdot \frac{M}{d}\\ t_3 := 0.5 \cdot t_2\\ \mathbf{if}\;d \leq -4.4 \cdot 10^{+255}:\\ \;\;\;\;t_1 \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -2.7 \cdot 10^{+167}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -1.26 \cdot 10^{-117}:\\ \;\;\;\;t_1 \cdot \left(1 + 0.5 \cdot \left(\frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot d} \cdot \left(h \cdot \frac{D \cdot \left(M \cdot -0.5\right)}{\ell}\right)\right)\right)\\ \mathbf{elif}\;d \leq -6.4 \cdot 10^{-183}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq 3.7 \cdot 10^{-212}:\\ \;\;\;\;\left(t_0 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + 0.5 \cdot \frac{t_3 \cdot \left(h \cdot \left(t_2 \cdot -0.5\right)\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \frac{h}{\frac{\ell}{{t_3}^{2}}} \cdot -0.5\right)\\ \end{array} \]

Alternative 7
Error	21.9
Cost	15444

\[\begin{array}{l} t_0 := \left(1 + 0.5 \cdot \left(\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{D \cdot \left(M \cdot -0.5\right)}{d}\right)\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{if}\;h \leq -3.45 \cdot 10^{-124}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq -2.7 \cdot 10^{-187}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;h \leq 6 \cdot 10^{-309}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq 2.4 \cdot 10^{+63}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \mathbf{elif}\;h \leq 5.6 \cdot 10^{+230}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 8
Error	20.4
Cost	15444

\[\begin{array}{l} t_0 := \left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ t_1 := D \cdot \left(M \cdot -0.5\right)\\ t_2 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ t_3 := D \cdot \left(0.5 \cdot M\right)\\ \mathbf{if}\;d \leq -3.2 \cdot 10^{+256}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -1.26 \cdot 10^{+138}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -2.05 \cdot 10^{-120}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(t_3 \cdot \left(\frac{h}{d \cdot d} \cdot \frac{t_1}{\ell}\right)\right)\right) \cdot t_2\\ \mathbf{elif}\;d \leq -1.95 \cdot 10^{-182}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 6200000000:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(\frac{t_3}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{t_1}{d}\right)\right)\right) \cdot t_2\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 9
Error	20.0
Cost	15444

\[\begin{array}{l} t_0 := D \cdot \left(0.5 \cdot M\right)\\ t_1 := \left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ t_2 := D \cdot \left(M \cdot -0.5\right)\\ t_3 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -3.7 \cdot 10^{+255}:\\ \;\;\;\;t_3 \cdot \left(1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot \left(D \cdot \frac{D \cdot \left(\frac{M}{d} \cdot -0.5\right)}{2 \cdot \frac{d}{M}}\right)\right)\right)\\ \mathbf{elif}\;d \leq -5.5 \cdot 10^{+137}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -1.7 \cdot 10^{-117}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(t_0 \cdot \left(\frac{h}{d \cdot d} \cdot \frac{t_2}{\ell}\right)\right)\right) \cdot t_3\\ \mathbf{elif}\;d \leq -6.2 \cdot 10^{-182}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 100000000000:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(\frac{t_0}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{t_2}{d}\right)\right)\right) \cdot t_3\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 10
Error	19.5
Cost	15444

\[\begin{array}{l} t_0 := D \cdot \left(0.5 \cdot M\right)\\ t_1 := D \cdot \left(M \cdot -0.5\right)\\ t_2 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -3.8 \cdot 10^{+256}:\\ \;\;\;\;t_2 \cdot \left(1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot \left(D \cdot \frac{D \cdot \left(\frac{M}{d} \cdot -0.5\right)}{2 \cdot \frac{d}{M}}\right)\right)\right)\\ \mathbf{elif}\;d \leq -3.7 \cdot 10^{+137}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq -2.05 \cdot 10^{-132}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(\frac{t_0}{d \cdot d} \cdot \left(h \cdot \frac{t_1}{\ell}\right)\right)\right) \cdot t_2\\ \mathbf{elif}\;d \leq -5.9 \cdot 10^{-182}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;d \leq 110000000000:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(\frac{t_0}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{t_1}{d}\right)\right)\right) \cdot t_2\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 11
Error	19.8
Cost	15444

\[\begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ t_1 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -3.8 \cdot 10^{+256}:\\ \;\;\;\;t_1 \cdot \left(1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot \left(D \cdot \frac{D \cdot \left(\frac{M}{d} \cdot -0.5\right)}{2 \cdot \frac{d}{M}}\right)\right)\right)\\ \mathbf{elif}\;d \leq -8 \cdot 10^{+137}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq -2.7 \cdot 10^{-132}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(\frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot d} \cdot \left(h \cdot \frac{D \cdot \left(M \cdot -0.5\right)}{\ell}\right)\right)\right) \cdot t_1\\ \mathbf{elif}\;d \leq -5.1 \cdot 10^{-183}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;d \leq 18000000000000:\\ \;\;\;\;t_1 \cdot \left(1 + 0.5 \cdot \frac{\left(0.5 \cdot t_0\right) \cdot \left(h \cdot \left(t_0 \cdot -0.5\right)\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 12
Error	19.8
Cost	15444

\[\begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ t_1 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -3.8 \cdot 10^{+256}:\\ \;\;\;\;t_1 \cdot \left(1 + 0.5 \cdot \frac{\left(M \cdot \frac{D}{d}\right) \cdot \left(\frac{D}{d} \cdot \left(M \cdot -0.5\right)\right)}{2 \cdot \frac{\ell}{h}}\right)\\ \mathbf{elif}\;d \leq -5.2 \cdot 10^{+137}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq -1.55 \cdot 10^{-132}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(\frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot d} \cdot \left(h \cdot \frac{D \cdot \left(M \cdot -0.5\right)}{\ell}\right)\right)\right) \cdot t_1\\ \mathbf{elif}\;d \leq -5.9 \cdot 10^{-182}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;d \leq 52000000000000:\\ \;\;\;\;t_1 \cdot \left(1 + 0.5 \cdot \frac{\left(0.5 \cdot t_0\right) \cdot \left(h \cdot \left(t_0 \cdot -0.5\right)\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 13
Error	20.2
Cost	15444

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := t_0 \cdot \sqrt{\frac{d}{\ell}}\\ t_2 := D \cdot \frac{M}{d}\\ \mathbf{if}\;d \leq -1.22 \cdot 10^{+256}:\\ \;\;\;\;t_1 \cdot \left(1 + 0.5 \cdot \frac{\left(M \cdot \frac{D}{d}\right) \cdot \left(\frac{D}{d} \cdot \left(M \cdot -0.5\right)\right)}{2 \cdot \frac{\ell}{h}}\right)\\ \mathbf{elif}\;d \leq -1.16 \cdot 10^{+168}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -2.1 \cdot 10^{-119}:\\ \;\;\;\;\left(t_0 \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 + 0.5 \cdot \left(\frac{D \cdot \left(0.5 \cdot M\right)}{d \cdot d} \cdot \left(h \cdot \frac{D \cdot \left(M \cdot -0.5\right)}{\ell}\right)\right)\right)\\ \mathbf{elif}\;d \leq -6 \cdot 10^{-183}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq 48000000000000:\\ \;\;\;\;t_1 \cdot \left(1 + 0.5 \cdot \frac{\left(0.5 \cdot t_2\right) \cdot \left(h \cdot \left(t_2 \cdot -0.5\right)\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 14
Error	20.6
Cost	15440

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \left(1 + 0.5 \cdot \left(h \cdot \left(\left(\frac{M}{d} \cdot \left(0.5 \cdot D\right)\right) \cdot \frac{\frac{M}{d} \cdot \left(D \cdot -0.5\right)}{\ell}\right)\right)\right) \cdot \left(t_0 \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right)\\ \mathbf{if}\;h \leq -3.6 \cdot 10^{-124}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;h \leq -4.2 \cdot 10^{-190}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;h \leq 6 \cdot 10^{-309}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(\frac{D \cdot \left(0.5 \cdot M\right)}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{D \cdot \left(M \cdot -0.5\right)}{d}\right)\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot t_0\right)\\ \mathbf{elif}\;h \leq 6.1 \cdot 10^{-112}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	25.3
Cost	15184

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ t_1 := \left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ t_2 := \left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{if}\;\ell \leq -2.8 \cdot 10^{+230}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -2.8 \cdot 10^{+144}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right) \cdot t_0\\ \mathbf{elif}\;\ell \leq -5.5 \cdot 10^{+135}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -4.6 \cdot 10^{+76}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(\frac{M}{\frac{\ell}{h} \cdot \frac{d \cdot d}{D \cdot D}} \cdot \left(M \cdot -0.25\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -7.2 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -8 \cdot 10^{-207}:\\ \;\;\;\;\left(1 + -0.5 \cdot \frac{h \cdot {\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}\right) \cdot t_0\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{-275}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 16
Error	24.1
Cost	14996

\[\begin{array}{l} t_0 := \left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ t_1 := \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ t_2 := \left(1 + -0.5 \cdot \frac{h \cdot {\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}\right) \cdot t_1\\ \mathbf{if}\;d \leq -3.8 \cdot 10^{+256}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right) \cdot t_1\\ \mathbf{elif}\;d \leq -1.28 \cdot 10^{+132}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -2.6 \cdot 10^{-116}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-278}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -4.4 \cdot 10^{-299}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 17
Error	23.9
Cost	14732

\[\begin{array}{l} t_0 := \left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ t_1 := \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{if}\;d \leq -3.8 \cdot 10^{+256}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -3.9 \cdot 10^{+43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.95 \cdot 10^{-119}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-291}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 18
Error	23.1
Cost	13776

\[\begin{array}{l} t_0 := \left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{if}\;d \leq -3.8 \cdot 10^{+256}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -9.2 \cdot 10^{+131}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -2.7 \cdot 10^{-113}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{-\frac{\ell}{d}}}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 1.12 \cdot 10^{-288}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 19
Error	24.1
Cost	13512

\[\begin{array}{l} \mathbf{if}\;d \leq -3.8 \cdot 10^{+256}:\\ \;\;\;\;\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \mathbf{elif}\;d \leq 10^{-288}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 20
Error	23.7
Cost	13512

\[\begin{array}{l} \mathbf{if}\;d \leq -3.2 \cdot 10^{+256}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-290}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 21
Error	24.1
Cost	13384

\[\begin{array}{l} \mathbf{if}\;d \leq -3.8 \cdot 10^{+256}:\\ \;\;\;\;\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \mathbf{elif}\;d \leq 2.45 \cdot 10^{-291}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 22
Error	27.9
Cost	7244

\[\begin{array}{l} \mathbf{if}\;h \leq -6.2 \cdot 10^{+165}:\\ \;\;\;\;\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \mathbf{elif}\;h \leq 3 \cdot 10^{-295}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;h \leq 9.2 \cdot 10^{+122}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \end{array} \]

Alternative 23
Error	27.8
Cost	7244

\[\begin{array}{l} \mathbf{if}\;h \leq -1.9 \cdot 10^{+168}:\\ \;\;\;\;\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \mathbf{elif}\;h \leq 3 \cdot 10^{-295}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;h \leq 5.1 \cdot 10^{+122}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \end{array} \]

Alternative 24
Error	32.9
Cost	7113

\[\begin{array}{l} \mathbf{if}\;h \leq -3.2 \cdot 10^{-304} \lor \neg \left(h \leq 1.8 \cdot 10^{+118}\right):\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]

Alternative 25
Error	32.9
Cost	7112

\[\begin{array}{l} \mathbf{if}\;h \leq -2.4 \cdot 10^{-306}:\\ \;\;\;\;\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \mathbf{elif}\;h \leq 1.65 \cdot 10^{+118}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \end{array} \]

Alternative 26
Error	33.2
Cost	7112

\[\begin{array}{l} \mathbf{if}\;h \leq 3 \cdot 10^{-292}:\\ \;\;\;\;\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \mathbf{elif}\;h \leq 2.2 \cdot 10^{+122}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \end{array} \]

Alternative 27
Error	33.2
Cost	7112

\[\begin{array}{l} \mathbf{if}\;h \leq 1.15 \cdot 10^{-291}:\\ \;\;\;\;\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \mathbf{elif}\;h \leq 1.58 \cdot 10^{+119}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \end{array} \]

Alternative 28
Error	34.7
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq -1.3 \cdot 10^{-274}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]

Alternative 29
Error	43.8
Cost	6720

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

Reproduce

herbie shell --seed 2023045 
(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)))))