Henrywood and Agarwal, Equation (12)

Average Error: 41.65% → 26.78%

Time: 40.0s

Precision: binary64

Cost: 27528

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{if}\;d \leq -1.65 \cdot 10^{+212}:\\ \;\;\;\;t_0 \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-130}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + \left(h \cdot \frac{{\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right) \cdot -0.5\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-187}:\\ \;\;\;\;\frac{D}{\frac{\frac{d}{M}}{M \cdot D}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot t_0\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-285}:\\ \;\;\;\;\left(1 + \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \left(\frac{{\left(\frac{D}{\frac{d}{M}}\right)}^{2}}{\ell} \cdot 0.25\right)\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ (/ 1.0 l) h))))
   (if (<= d -1.65e+212)
     (* t_0 (- d))
     (if (<= d -3.5e-130)
       (*
        (* (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ d l)))
        (+ 1.0 (* (* h (/ (pow (* 0.5 (* M (/ D d))) 2.0) l)) -0.5)))
       (if (<= d -4e-187)
         (-
          (* (/ D (/ (/ d M) (* M D))) (* (sqrt (/ h (pow l 3.0))) 0.125))
          (* d t_0))
         (if (<= d 1.15e-285)
           (*
            (+ 1.0 (* (* (pow (* M (* D (/ 0.5 d))) 2.0) (/ h l)) -0.5))
            (sqrt (* (/ d l) (/ d h))))
           (*
            (/ d (* (sqrt h) (sqrt l)))
            (+
             1.0
             (* -0.5 (* h (* (/ (pow (/ D (/ d M)) 2.0) l) 0.25)))))))))))

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(((1.0 / l) / h));
	double tmp;
	if (d <= -1.65e+212) {
		tmp = t_0 * -d;
	} else if (d <= -3.5e-130) {
		tmp = ((sqrt(-d) / sqrt(-h)) * sqrt((d / l))) * (1.0 + ((h * (pow((0.5 * (M * (D / d))), 2.0) / l)) * -0.5));
	} else if (d <= -4e-187) {
		tmp = ((D / ((d / M) / (M * D))) * (sqrt((h / pow(l, 3.0))) * 0.125)) - (d * t_0);
	} else if (d <= 1.15e-285) {
		tmp = (1.0 + ((pow((M * (D * (0.5 / d))), 2.0) * (h / l)) * -0.5)) * sqrt(((d / l) * (d / h)));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 + (-0.5 * (h * ((pow((D / (d / M)), 2.0) / l) * 0.25))));
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((1.0d0 / l) / h))
    if (d <= (-1.65d+212)) then
        tmp = t_0 * -d
    else if (d <= (-3.5d-130)) then
        tmp = ((sqrt(-d) / sqrt(-h)) * sqrt((d / l))) * (1.0d0 + ((h * (((0.5d0 * (m * (d_1 / d))) ** 2.0d0) / l)) * (-0.5d0)))
    else if (d <= (-4d-187)) then
        tmp = ((d_1 / ((d / m) / (m * d_1))) * (sqrt((h / (l ** 3.0d0))) * 0.125d0)) - (d * t_0)
    else if (d <= 1.15d-285) then
        tmp = (1.0d0 + ((((m * (d_1 * (0.5d0 / d))) ** 2.0d0) * (h / l)) * (-0.5d0))) * sqrt(((d / l) * (d / h)))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 + ((-0.5d0) * (h * ((((d_1 / (d / m)) ** 2.0d0) / l) * 0.25d0))))
    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(((1.0 / l) / h));
	double tmp;
	if (d <= -1.65e+212) {
		tmp = t_0 * -d;
	} else if (d <= -3.5e-130) {
		tmp = ((Math.sqrt(-d) / Math.sqrt(-h)) * Math.sqrt((d / l))) * (1.0 + ((h * (Math.pow((0.5 * (M * (D / d))), 2.0) / l)) * -0.5));
	} else if (d <= -4e-187) {
		tmp = ((D / ((d / M) / (M * D))) * (Math.sqrt((h / Math.pow(l, 3.0))) * 0.125)) - (d * t_0);
	} else if (d <= 1.15e-285) {
		tmp = (1.0 + ((Math.pow((M * (D * (0.5 / d))), 2.0) * (h / l)) * -0.5)) * Math.sqrt(((d / l) * (d / h)));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 + (-0.5 * (h * ((Math.pow((D / (d / M)), 2.0) / l) * 0.25))));
	}
	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(((1.0 / l) / h))
	tmp = 0
	if d <= -1.65e+212:
		tmp = t_0 * -d
	elif d <= -3.5e-130:
		tmp = ((math.sqrt(-d) / math.sqrt(-h)) * math.sqrt((d / l))) * (1.0 + ((h * (math.pow((0.5 * (M * (D / d))), 2.0) / l)) * -0.5))
	elif d <= -4e-187:
		tmp = ((D / ((d / M) / (M * D))) * (math.sqrt((h / math.pow(l, 3.0))) * 0.125)) - (d * t_0)
	elif d <= 1.15e-285:
		tmp = (1.0 + ((math.pow((M * (D * (0.5 / d))), 2.0) * (h / l)) * -0.5)) * math.sqrt(((d / l) * (d / h)))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 + (-0.5 * (h * ((math.pow((D / (d / M)), 2.0) / l) * 0.25))))
	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(Float64(1.0 / l) / h))
	tmp = 0.0
	if (d <= -1.65e+212)
		tmp = Float64(t_0 * Float64(-d));
	elseif (d <= -3.5e-130)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * sqrt(Float64(d / l))) * Float64(1.0 + Float64(Float64(h * Float64((Float64(0.5 * Float64(M * Float64(D / d))) ^ 2.0) / l)) * -0.5)));
	elseif (d <= -4e-187)
		tmp = Float64(Float64(Float64(D / Float64(Float64(d / M) / Float64(M * D))) * Float64(sqrt(Float64(h / (l ^ 3.0))) * 0.125)) - Float64(d * t_0));
	elseif (d <= 1.15e-285)
		tmp = Float64(Float64(1.0 + Float64(Float64((Float64(M * Float64(D * Float64(0.5 / d))) ^ 2.0) * Float64(h / l)) * -0.5)) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 + Float64(-0.5 * Float64(h * Float64(Float64((Float64(D / Float64(d / M)) ^ 2.0) / l) * 0.25)))));
	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(((1.0 / l) / h));
	tmp = 0.0;
	if (d <= -1.65e+212)
		tmp = t_0 * -d;
	elseif (d <= -3.5e-130)
		tmp = ((sqrt(-d) / sqrt(-h)) * sqrt((d / l))) * (1.0 + ((h * (((0.5 * (M * (D / d))) ^ 2.0) / l)) * -0.5));
	elseif (d <= -4e-187)
		tmp = ((D / ((d / M) / (M * D))) * (sqrt((h / (l ^ 3.0))) * 0.125)) - (d * t_0);
	elseif (d <= 1.15e-285)
		tmp = (1.0 + ((((M * (D * (0.5 / d))) ^ 2.0) * (h / l)) * -0.5)) * sqrt(((d / l) * (d / h)));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 + (-0.5 * (h * ((((D / (d / M)) ^ 2.0) / l) * 0.25))));
	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[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.65e+212], N[(t$95$0 * (-d)), $MachinePrecision], If[LessEqual[d, -3.5e-130], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h * N[(N[Power[N[(0.5 * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4e-187], N[(N[(N[(D / N[(N[(d / M), $MachinePrecision] / N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] - N[(d * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.15e-285], N[(N[(1.0 + N[(N[(N[Power[N[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(h * N[(N[(N[Power[N[(D / N[(d / M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.65e212

   1. Initial program 49.35

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

   2. Simplified 47.09

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

      [Start] 49.35	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 49.35	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 49.35	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 49.35	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 49.35	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 49.35	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 49.35	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 47.09	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 45.74

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

   4. Taylor expanded in d around -inf 26.22

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

   5. Simplified 24.96

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

      [Start] 26.22	\[ -1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]
      associate-*r* [=>] 26.22	\[ \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
      neg-mul-1 [<=] 26.22	\[ \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]
      *-commutative [=>] 26.22	\[ \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]
      associate-/r* [=>] 24.96	\[ \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \cdot \left(-d\right) \]

   if -1.65e212 < d < -3.4999999999999999e-130

   1. Initial program 30.74

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

   2. Simplified 31.21

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

      [Start] 30.74	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 30.74	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 30.74	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 30.74	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 30.75	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 30.75	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 30.75	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 31.21	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 31.97

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

   4. Simplified 27.98

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

      [Start] 31.97	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(e^{\mathsf{log1p}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)} - 1\right)\right) \]
      expm1-def [=>] 31.97	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]
      expm1-log1p [=>] 31.21	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      associate-*r/ [=>] 29.48	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}}\right) \]
      associate-*l/ [<=] 27.98	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot h\right)}\right) \]
      *-commutative [=>] 27.98	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)}\right) \]
      *-commutative [=>] 27.98	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(M \cdot \color{blue}{\left(\frac{D}{d} \cdot 0.5\right)}\right)}^{2}}{\ell}\right)\right) \]
      associate-*r* [=>] 27.98	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot 0.5\right)}}^{2}}{\ell}\right)\right) \]

   5. Applied egg-rr 18.21

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

   if -3.4999999999999999e-130 < d < -4.0000000000000001e-187

   1. Initial program 51.82

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

   2. Simplified 54.63

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

      [Start] 51.82	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 51.82	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 51.82	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 51.82	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 51.82	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 51.82	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 51.82	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 54.63	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 55.14

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

   4. Taylor expanded in d around -inf 62.17

      \[\leadsto \color{blue}{0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + -1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]

   5. Simplified 54.92

      \[\leadsto \color{blue}{\frac{D}{\frac{\frac{d}{M}}{D \cdot M}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
      Proof

      [Start] 62.17	\[ 0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + -1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]
      mul-1-neg [=>] 62.17	\[ 0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
      unsub-neg [=>] 62.17	\[ \color{blue}{0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
      *-commutative [=>] 62.17	\[ \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot 0.125} - d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]
      associate-*l* [=>] 62.17	\[ \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right)} - d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]
      associate-/l* [=>] 63.99	\[ \color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]
      unpow2 [=>] 63.99	\[ \frac{\color{blue}{D \cdot D}}{\frac{d}{{M}^{2}}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]
      associate-/l* [=>] 62.06	\[ \color{blue}{\frac{D}{\frac{\frac{d}{{M}^{2}}}{D}}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]
      unpow2 [=>] 62.06	\[ \frac{D}{\frac{\frac{d}{\color{blue}{M \cdot M}}}{D}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]
      associate-/r* [=>] 60.54	\[ \frac{D}{\frac{\color{blue}{\frac{\frac{d}{M}}{M}}}{D}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]
      associate-/l/ [=>] 55.24	\[ \frac{D}{\color{blue}{\frac{\frac{d}{M}}{D \cdot M}}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]
      associate-/r* [=>] 54.92	\[ \frac{D}{\frac{\frac{d}{M}}{D \cdot M}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   if -4.0000000000000001e-187 < d < 1.14999999999999998e-285

   1. Initial program 65.58

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

   2. Applied egg-rr 74.01

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

   3. Simplified 74.01

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

      [Start] 74.01	\[ \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} + \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(-0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      *-lft-identity [<=] 74.01	\[ \color{blue}{1 \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} + \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(-0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      *-commutative [<=] 74.01	\[ 1 \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} + \color{blue}{\left(-0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]
      distribute-rgt-in [<=] 74.01	\[ \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
      *-commutative [=>] 74.01	\[ \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \color{blue}{\left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5}\right) \]

   if 1.14999999999999998e-285 < d

   1. Initial program 41.2

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

   2. Simplified 41.57

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

      [Start] 41.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 41.2	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 41.19	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 41.19	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 41.19	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 41.19	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 41.19	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 41.57	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 42.22

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

   4. Simplified 39.65

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

      [Start] 42.22	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(e^{\mathsf{log1p}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)} - 1\right)\right) \]
      expm1-def [=>] 42.22	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]
      expm1-log1p [=>] 41.57	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      associate-*r/ [=>] 40.26	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}}\right) \]
      associate-*l/ [<=] 39.65	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot h\right)}\right) \]
      *-commutative [=>] 39.65	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)}\right) \]
      *-commutative [=>] 39.65	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(M \cdot \color{blue}{\left(\frac{D}{d} \cdot 0.5\right)}\right)}^{2}}{\ell}\right)\right) \]
      associate-*r* [=>] 39.65	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot 0.5\right)}}^{2}}{\ell}\right)\right) \]

   5. Applied egg-rr 19.29

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

   6. Simplified 19.4

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

      [Start] 19.29	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(\left(-0.5 \cdot h\right) \cdot \frac{{\left(\frac{M}{\frac{d}{D}}\right)}^{2}}{\frac{\ell}{0.25}}\right) \]
      *-lft-identity [<=] 19.29	\[ \color{blue}{1 \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} + \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(\left(-0.5 \cdot h\right) \cdot \frac{{\left(\frac{M}{\frac{d}{D}}\right)}^{2}}{\frac{\ell}{0.25}}\right) \]
      *-commutative [<=] 19.29	\[ 1 \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \color{blue}{\left(\left(-0.5 \cdot h\right) \cdot \frac{{\left(\frac{M}{\frac{d}{D}}\right)}^{2}}{\frac{\ell}{0.25}}\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]
      distribute-rgt-in [<=] 19.29	\[ \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(-0.5 \cdot h\right) \cdot \frac{{\left(\frac{M}{\frac{d}{D}}\right)}^{2}}{\frac{\ell}{0.25}}\right)} \]
      associate-*l* [=>] 19.29	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \color{blue}{-0.5 \cdot \left(h \cdot \frac{{\left(\frac{M}{\frac{d}{D}}\right)}^{2}}{\frac{\ell}{0.25}}\right)}\right) \]
      associate-/r/ [=>] 19.29	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \color{blue}{\left(\frac{{\left(\frac{M}{\frac{d}{D}}\right)}^{2}}{\ell} \cdot 0.25\right)}\right)\right) \]
      associate-/l* [<=] 19.23	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \left(\frac{{\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}}{\ell} \cdot 0.25\right)\right)\right) \]
      *-commutative [<=] 19.23	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \left(\frac{{\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2}}{\ell} \cdot 0.25\right)\right)\right) \]
      associate-/l* [=>] 19.4	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \left(\frac{{\color{blue}{\left(\frac{D}{\frac{d}{M}}\right)}}^{2}}{\ell} \cdot 0.25\right)\right)\right) \]

3. Recombined 5 regimes into one program.

4. Final simplification 26.78

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.65 \cdot 10^{+212}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-130}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + \left(h \cdot \frac{{\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right) \cdot -0.5\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-187}:\\ \;\;\;\;\frac{D}{\frac{\frac{d}{M}}{M \cdot D}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-285}:\\ \;\;\;\;\left(1 + \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \left(\frac{{\left(\frac{D}{\frac{d}{M}}\right)}^{2}}{\ell} \cdot 0.25\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	25.19%
Cost	27268

\[\begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -3.8 \cdot 10^{-49}:\\ \;\;\;\;\frac{t_0}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{h}}\right)\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{t_0}{\sqrt{-\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{M}{\left(d \cdot \frac{d}{h}\right) \cdot \frac{\ell}{D \cdot \left(M \cdot D\right)}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \left(\frac{{\left(\frac{D}{\frac{d}{M}}\right)}^{2}}{\ell} \cdot 0.25\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	27.34%
Cost	21580

\[\begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ t_1 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;\ell \leq -2.25 \cdot 10^{+104}:\\ \;\;\;\;t_1 \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{t_0}{\frac{\ell}{h}} \cdot \frac{t_0}{4}, 1\right)\right)\\ \mathbf{elif}\;\ell \leq -4.3 \cdot 10^{-70}:\\ \;\;\;\;\frac{D}{\frac{\frac{d}{M}}{M \cdot D}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(t_1 \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{M}{\left(d \cdot \frac{d}{h}\right) \cdot \frac{\ell}{D \cdot \left(M \cdot D\right)}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \left(\frac{{\left(\frac{D}{\frac{d}{M}}\right)}^{2}}{\ell} \cdot 0.25\right)\right)\right)\\ \end{array} \]

Alternative 3
Error	28.68%
Cost	21524

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := t_0 \cdot \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -3.9 \cdot 10^{+176}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -7 \cdot 10^{+101}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t_0\right) \cdot \left(1 + 0.5 \cdot \left(\frac{M}{\frac{\frac{\frac{\ell}{M \cdot D}}{D} \cdot \left(d \cdot d\right)}{h}} \cdot -0.25\right)\right)\\ \mathbf{elif}\;d \leq -2.1 \cdot 10^{+41}:\\ \;\;\;\;t_1 \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{M \cdot M}{\frac{\ell}{h}}}}{D}}\right)\right)\\ \mathbf{elif}\;d \leq -9 \cdot 10^{-133}:\\ \;\;\;\;t_1 \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{h}{\frac{\frac{\ell \cdot \left(d \cdot d\right)}{M \cdot D}}{M \cdot D}}\right)\right)\\ \mathbf{elif}\;d \leq 5 \cdot 10^{-294}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \left(\frac{{\left(\frac{D}{\frac{d}{M}}\right)}^{2}}{\ell} \cdot 0.25\right)\right)\right)\\ \end{array} \]

Alternative 4
Error	29.27%
Cost	21448

\[\begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := t_1 \cdot \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -5.2 \cdot 10^{+188}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -2.4 \cdot 10^{+101}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t_1\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{M}{\left(d \cdot \frac{d}{h}\right) \cdot \frac{\ell}{D \cdot \left(M \cdot D\right)}}\right)\right)\\ \mathbf{elif}\;d \leq -2.1 \cdot 10^{+41}:\\ \;\;\;\;t_2 \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{M \cdot M}{\frac{\ell}{h}}}}{D}}\right)\right)\\ \mathbf{elif}\;d \leq -3.1 \cdot 10^{-125}:\\ \;\;\;\;t_2 \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{h}{\frac{\frac{\ell \cdot \left(d \cdot d\right)}{M \cdot D}}{M \cdot D}}\right)\right)\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{-277}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \left(\frac{{\left(\frac{D}{\frac{d}{M}}\right)}^{2}}{\ell} \cdot 0.25\right)\right)\right)\\ \end{array} \]

Alternative 5
Error	28.96%
Cost	21448

\[\begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := t_1 \cdot \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -1.1 \cdot 10^{+173}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -2.4 \cdot 10^{+101}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t_1\right) \cdot \left(1 + 0.5 \cdot \left(\frac{M}{\frac{\frac{\frac{\ell}{M \cdot D}}{D} \cdot \left(d \cdot d\right)}{h}} \cdot -0.25\right)\right)\\ \mathbf{elif}\;d \leq -2.1 \cdot 10^{+41}:\\ \;\;\;\;t_2 \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{M \cdot M}{\frac{\ell}{h}}}}{D}}\right)\right)\\ \mathbf{elif}\;d \leq -6.8 \cdot 10^{-125}:\\ \;\;\;\;t_2 \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{h}{\frac{\frac{\ell \cdot \left(d \cdot d\right)}{M \cdot D}}{M \cdot D}}\right)\right)\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{-277}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \left(\frac{{\left(\frac{D}{\frac{d}{M}}\right)}^{2}}{\ell} \cdot 0.25\right)\right)\right)\\ \end{array} \]

Alternative 6
Error	32.43%
Cost	21136

\[\begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)\\ \mathbf{if}\;d \leq -1.8 \cdot 10^{+103}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -6.1 \cdot 10^{-125}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{h}{\frac{\frac{\ell \cdot \left(d \cdot d\right)}{M \cdot D}}{M \cdot D}}\right)\right)\\ \mathbf{elif}\;d \leq -5.8 \cdot 10^{-231}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -2.85 \cdot 10^{-267}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}, -0.5, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.125 \cdot \left(M \cdot \left(D \cdot \left(\left(M \cdot \frac{D}{\ell}\right) \cdot \frac{h}{d \cdot d}\right)\right)\right)\right)\\ \end{array} \]

Alternative 7
Error	28.38%
Cost	21136

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)\\ \mathbf{if}\;d \leq -1.65 \cdot 10^{+108}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -3.1 \cdot 10^{-125}:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{h}{\frac{\frac{\ell \cdot \left(d \cdot d\right)}{M \cdot D}}{M \cdot D}}\right)\right)\\ \mathbf{elif}\;d \leq -4.9 \cdot 10^{-231}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-293}:\\ \;\;\;\;\left(t_0 \cdot \left(1 + -0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{h}}\right)\right) \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \left(\frac{{\left(\frac{D}{\frac{d}{M}}\right)}^{2}}{\ell} \cdot 0.25\right)\right)\right)\\ \end{array} \]

Alternative 8
Error	29.69%
Cost	21060

\[\begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ \mathbf{if}\;\ell \leq -3.1 \cdot 10^{+105}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{t_0}{\frac{\ell}{h}} \cdot \frac{t_0}{4}, 1\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-67}:\\ \;\;\;\;\frac{D}{\frac{\frac{d}{M}}{M \cdot D}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -1.1 \cdot 10^{-286}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \left(\frac{{\left(\frac{D}{\frac{d}{M}}\right)}^{2}}{\ell} \cdot 0.25\right)\right)\right)\\ \end{array} \]

Alternative 9
Error	28.08%
Cost	21000

\[\begin{array}{l} \mathbf{if}\;d \leq -3.7 \cdot 10^{+117}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-292}:\\ \;\;\;\;\left(1 + \left(h \cdot \frac{{\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right) \cdot -0.5\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \left(\frac{{\left(\frac{D}{\frac{d}{M}}\right)}^{2}}{\ell} \cdot 0.25\right)\right)\right)\\ \end{array} \]

Alternative 10
Error	38.79%
Cost	14928

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{if}\;h \leq -3.2 \cdot 10^{+268}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)\\ \mathbf{elif}\;h \leq 5.4 \cdot 10^{+14}:\\ \;\;\;\;\frac{d \cdot \frac{1}{\sqrt{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;h \leq 1.9 \cdot 10^{+225}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.125 \cdot \left(M \cdot \left(D \cdot \left(\left(M \cdot \frac{D}{\ell}\right) \cdot \frac{h}{d \cdot d}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(1 + {\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 11
Error	30.79%
Cost	14928

\[\begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)\\ \mathbf{if}\;d \leq -3.4 \cdot 10^{+116}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -9.5 \cdot 10^{-115}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \left(\frac{M \cdot D}{d \cdot \ell} \cdot \left(h \cdot \left(M \cdot \frac{D}{d}\right)\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{-277}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.125 \cdot \left(M \cdot \left(D \cdot \left(\left(M \cdot \frac{D}{\ell}\right) \cdot \frac{h}{d \cdot d}\right)\right)\right)\right)\\ \end{array} \]

Alternative 12
Error	31.54%
Cost	14928

\[\begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)\\ \mathbf{if}\;d \leq -1.8 \cdot 10^{+104}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -6.8 \cdot 10^{-125}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{h}{\frac{\frac{\ell \cdot \left(d \cdot d\right)}{M \cdot D}}{M \cdot D}}\right)\right)\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{-277}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.125 \cdot \left(M \cdot \left(D \cdot \left(\left(M \cdot \frac{D}{\ell}\right) \cdot \frac{h}{d \cdot d}\right)\right)\right)\right)\\ \end{array} \]

Alternative 13
Error	38.3%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;h \leq -5.8 \cdot 10^{+268}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;h \leq 1.35 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 14
Error	42.72%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;h \leq -3.4 \cdot 10^{+268}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]

Alternative 15
Error	42.5%
Cost	7176

\[\begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{if}\;h \leq -2.1 \cdot 10^{+268}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_0 \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot t_0\\ \end{array} \]

Alternative 16
Error	51.62%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.25 \cdot 10^{-304}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]

Alternative 17
Error	51.57%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\ell \leq 2.9 \cdot 10^{-306}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]

Alternative 18
Error	51.57%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\ell \leq 2.45 \cdot 10^{-306}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]

Alternative 19
Error	68.39%
Cost	6720

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

Reproduce

herbie shell --seed 2023102 
(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)))))