Henrywood and Agarwal, Equation (12)

Average Error: 42.07% → 27.1%

Time: 1.1min

Precision: binary64

Cost: 40656

Math

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

↓

\[\begin{array}{l} t_0 := 1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right) \cdot \sqrt{0.5}\right)\right)}^{2}\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;d \leq -1.35 \cdot 10^{-90}:\\ \;\;\;\;\left(\frac{t_1}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot t_0\\ \mathbf{elif}\;d \leq -7.2 \cdot 10^{-296}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{\frac{d}{h}} \cdot \frac{t_1}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-221}:\\ \;\;\;\;\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{+147}:\\ \;\;\;\;t_0 \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \left(\sqrt{\frac{1}{\ell}} \cdot \sqrt{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

FPCore

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

↓

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (-
          1.0
          (pow (* (sqrt (/ h l)) (* (* (/ 0.5 d) (* M D)) (sqrt 0.5))) 2.0)))
        (t_1 (sqrt (- d))))
   (if (<= d -1.35e-90)
     (* (* (/ t_1 (sqrt (- h))) (sqrt (/ d l))) t_0)
     (if (<= d -7.2e-296)
       (* t_0 (* (sqrt (/ d h)) (/ t_1 (sqrt (- l)))))
       (if (<= d 3.8e-221)
         (* (/ (* (* M D) (* M D)) d) (* (sqrt (/ h (pow l 3.0))) -0.125))
         (if (<= d 5.5e+147)
           (* t_0 (* (pow (/ d h) 0.5) (* (sqrt (/ 1.0 l)) (sqrt d))))
           (* d (/ (sqrt (/ 1.0 h)) (sqrt l)))))))))

C

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

↓

double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 - pow((sqrt((h / l)) * (((0.5 / d) * (M * D)) * sqrt(0.5))), 2.0);
	double t_1 = sqrt(-d);
	double tmp;
	if (d <= -1.35e-90) {
		tmp = ((t_1 / sqrt(-h)) * sqrt((d / l))) * t_0;
	} else if (d <= -7.2e-296) {
		tmp = t_0 * (sqrt((d / h)) * (t_1 / sqrt(-l)));
	} else if (d <= 3.8e-221) {
		tmp = (((M * D) * (M * D)) / d) * (sqrt((h / pow(l, 3.0))) * -0.125);
	} else if (d <= 5.5e+147) {
		tmp = t_0 * (pow((d / h), 0.5) * (sqrt((1.0 / l)) * sqrt(d)));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - ((sqrt((h / l)) * (((0.5d0 / d) * (m * d_1)) * sqrt(0.5d0))) ** 2.0d0)
    t_1 = sqrt(-d)
    if (d <= (-1.35d-90)) then
        tmp = ((t_1 / sqrt(-h)) * sqrt((d / l))) * t_0
    else if (d <= (-7.2d-296)) then
        tmp = t_0 * (sqrt((d / h)) * (t_1 / sqrt(-l)))
    else if (d <= 3.8d-221) then
        tmp = (((m * d_1) * (m * d_1)) / d) * (sqrt((h / (l ** 3.0d0))) * (-0.125d0))
    else if (d <= 5.5d+147) then
        tmp = t_0 * (((d / h) ** 0.5d0) * (sqrt((1.0d0 / l)) * sqrt(d)))
    else
        tmp = d * (sqrt((1.0d0 / h)) / sqrt(l))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 - Math.pow((Math.sqrt((h / l)) * (((0.5 / d) * (M * D)) * Math.sqrt(0.5))), 2.0);
	double t_1 = Math.sqrt(-d);
	double tmp;
	if (d <= -1.35e-90) {
		tmp = ((t_1 / Math.sqrt(-h)) * Math.sqrt((d / l))) * t_0;
	} else if (d <= -7.2e-296) {
		tmp = t_0 * (Math.sqrt((d / h)) * (t_1 / Math.sqrt(-l)));
	} else if (d <= 3.8e-221) {
		tmp = (((M * D) * (M * D)) / d) * (Math.sqrt((h / Math.pow(l, 3.0))) * -0.125);
	} else if (d <= 5.5e+147) {
		tmp = t_0 * (Math.pow((d / h), 0.5) * (Math.sqrt((1.0 / l)) * Math.sqrt(d)));
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

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

↓

def code(d, h, l, M, D):
	t_0 = 1.0 - math.pow((math.sqrt((h / l)) * (((0.5 / d) * (M * D)) * math.sqrt(0.5))), 2.0)
	t_1 = math.sqrt(-d)
	tmp = 0
	if d <= -1.35e-90:
		tmp = ((t_1 / math.sqrt(-h)) * math.sqrt((d / l))) * t_0
	elif d <= -7.2e-296:
		tmp = t_0 * (math.sqrt((d / h)) * (t_1 / math.sqrt(-l)))
	elif d <= 3.8e-221:
		tmp = (((M * D) * (M * D)) / d) * (math.sqrt((h / math.pow(l, 3.0))) * -0.125)
	elif d <= 5.5e+147:
		tmp = t_0 * (math.pow((d / h), 0.5) * (math.sqrt((1.0 / l)) * math.sqrt(d)))
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

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

↓

function code(d, h, l, M, D)
	t_0 = Float64(1.0 - (Float64(sqrt(Float64(h / l)) * Float64(Float64(Float64(0.5 / d) * Float64(M * D)) * sqrt(0.5))) ^ 2.0))
	t_1 = sqrt(Float64(-d))
	tmp = 0.0
	if (d <= -1.35e-90)
		tmp = Float64(Float64(Float64(t_1 / sqrt(Float64(-h))) * sqrt(Float64(d / l))) * t_0);
	elseif (d <= -7.2e-296)
		tmp = Float64(t_0 * Float64(sqrt(Float64(d / h)) * Float64(t_1 / sqrt(Float64(-l)))));
	elseif (d <= 3.8e-221)
		tmp = Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) / d) * Float64(sqrt(Float64(h / (l ^ 3.0))) * -0.125));
	elseif (d <= 5.5e+147)
		tmp = Float64(t_0 * Float64((Float64(d / h) ^ 0.5) * Float64(sqrt(Float64(1.0 / l)) * sqrt(d))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(d, h, l, M, D)
	t_0 = 1.0 - ((sqrt((h / l)) * (((0.5 / d) * (M * D)) * sqrt(0.5))) ^ 2.0);
	t_1 = sqrt(-d);
	tmp = 0.0;
	if (d <= -1.35e-90)
		tmp = ((t_1 / sqrt(-h)) * sqrt((d / l))) * t_0;
	elseif (d <= -7.2e-296)
		tmp = t_0 * (sqrt((d / h)) * (t_1 / sqrt(-l)));
	elseif (d <= 3.8e-221)
		tmp = (((M * D) * (M * D)) / d) * (sqrt((h / (l ^ 3.0))) * -0.125);
	elseif (d <= 5.5e+147)
		tmp = t_0 * (((d / h) ^ 0.5) * (sqrt((1.0 / l)) * sqrt(d)));
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 - N[Power[N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(0.5 / d), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[d, -1.35e-90], N[(N[(N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[d, -7.2e-296], N[(t$95$0 * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.8e-221], N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.5e+147], N[(t$95$0 * N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.34999999999999998e-90

   1. Initial program 35.1

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

   2. Applied egg-rr 33.82

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

   3. Applied egg-rr 62.05

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

   4. Simplified 33.82

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

      [Start] 62.05	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)} - 1\right)\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      expm1-def [=>] 36.11	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)\right)}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      expm1-log1p [=>] 33.82	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]

   5. Applied egg-rr 21.06

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

   if -1.34999999999999998e-90 < d < -7.19999999999999961e-296

   1. Initial program 56.12

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

   2. Applied egg-rr 51.6

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

   3. Applied egg-rr 79.13

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

   4. Simplified 51.6

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

      [Start] 79.13	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)} - 1\right)\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      expm1-def [=>] 51.95	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)\right)}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      expm1-log1p [=>] 51.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]

   5. Applied egg-rr 73.16

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

   6. Simplified 51.6

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

      [Start] 73.16	\[ \left(\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h}}\right)} - 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      expm1-def [=>] 52.3	\[ \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h}}\right)\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      expm1-log1p [=>] 51.6	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]

   7. Applied egg-rr 38.42

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

   if -7.19999999999999961e-296 < d < 3.8000000000000001e-221

   1. Initial program 67.29

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

   2. Simplified 69.12

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

      [Start] 67.29	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 67.29	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 67.29	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 67.29	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 67.29	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 67.29	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 67.29	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 69.12	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 69.33

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

   4. Taylor expanded in d around 0 80.04

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

   5. Simplified 74.48

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

      [Start] 80.04	\[ -0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]
      *-commutative [=>] 80.04	\[ \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot -0.125} \]
      associate-*l* [=>] 80.04	\[ \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)} \]
      unpow2 [=>] 80.04	\[ \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right) \]
      unpow2 [=>] 80.04	\[ \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right) \]
      unswap-sqr [=>] 74.48	\[ \frac{\color{blue}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right) \]

   if 3.8000000000000001e-221 < d < 5.4999999999999997e147

   1. Initial program 35.34

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

   2. Applied egg-rr 32.33

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

   3. Applied egg-rr 25.37

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

   4. Simplified 25.37

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

      [Start] 25.37	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      *-commutative [=>] 25.37	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\frac{1}{\ell}} \cdot \sqrt{d}\right)}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]

   if 5.4999999999999997e147 < d

   1. Initial program 46.5

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

   2. Simplified 45.83

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

      [Start] 46.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 46.5	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 46.5	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 46.5	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 46.49	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 46.49	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 46.49	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 45.83	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Taylor expanded in d around inf 25.18

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

   4. Simplified 24.1

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
      Proof

      [Start] 25.18	\[ \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]
      *-commutative [=>] 25.18	\[ \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
      associate-/r* [=>] 24.1	\[ d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   5. Applied egg-rr 7.89

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

   6. Simplified 7.89

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
      Proof

      [Start] 7.89	\[ d \cdot \frac{1 \cdot \sqrt{\frac{1}{h}}}{\sqrt{\ell}} \]
      *-lft-identity [=>] 7.89	\[ d \cdot \frac{\color{blue}{\sqrt{\frac{1}{h}}}}{\sqrt{\ell}} \]

3. Recombined 5 regimes into one program.

4. Final simplification 27.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.35 \cdot 10^{-90}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right)\\ \mathbf{elif}\;d \leq -7.2 \cdot 10^{-296}:\\ \;\;\;\;\left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-221}:\\ \;\;\;\;\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{+147}:\\ \;\;\;\;\left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \left(\sqrt{\frac{1}{\ell}} \cdot \sqrt{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternatives

Alternative 1
Error	27.85%
Cost	40592

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{-d}\\ t_2 := \sqrt{\frac{h}{\ell}}\\ \mathbf{if}\;d \leq -2.1 \cdot 10^{-61}:\\ \;\;\;\;\left(\frac{t_1}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) \cdot -0.5\right)\\ \mathbf{elif}\;d \leq -7.2 \cdot 10^{-296}:\\ \;\;\;\;\left(t_0 \cdot \frac{t_1}{\sqrt{-\ell}}\right) \cdot \left(1 + {\left(t_2 \cdot \left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)\right)}^{2} \cdot -0.5\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-221}:\\ \;\;\;\;\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{+147}:\\ \;\;\;\;\left(1 - {\left(t_2 \cdot \left(\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \cdot \left(t_0 \cdot \left(\sqrt{d} \cdot \frac{1}{\sqrt{\ell}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 2
Error	26.75%
Cost	40592

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := 1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right) \cdot \sqrt{0.5}\right)\right)}^{2}\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;d \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\left(\frac{t_2}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) \cdot -0.5\right)\\ \mathbf{elif}\;d \leq -7.2 \cdot 10^{-296}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \frac{t_2}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-221}:\\ \;\;\;\;\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)\\ \mathbf{elif}\;d \leq 1.12 \cdot 10^{+148}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \left(\sqrt{d} \cdot \frac{1}{\sqrt{\ell}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 3
Error	27.11%
Cost	40592

\[\begin{array}{l} t_0 := 1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right) \cdot \sqrt{0.5}\right)\right)}^{2}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;d \leq -1.35 \cdot 10^{-90}:\\ \;\;\;\;\left(\frac{t_2}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot t_0\\ \mathbf{elif}\;d \leq -7.2 \cdot 10^{-296}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \frac{t_2}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-221}:\\ \;\;\;\;\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{+147}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(\sqrt{d} \cdot \frac{1}{\sqrt{\ell}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 4
Error	27.09%
Cost	40528

\[\begin{array}{l} t_0 := 1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right) \cdot \sqrt{0.5}\right)\right)}^{2}\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;d \leq -1.35 \cdot 10^{-90}:\\ \;\;\;\;\left(\frac{t_1}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot t_0\\ \mathbf{elif}\;d \leq -7.2 \cdot 10^{-296}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{\frac{d}{h}} \cdot \frac{t_1}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-221}:\\ \;\;\;\;\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{+147}:\\ \;\;\;\;t_0 \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 5
Error	29.27%
Cost	34196

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_2 := \sqrt{-d}\\ t_3 := \left(\frac{t_2}{\sqrt{-\ell}} \cdot t_1\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ \mathbf{if}\;\ell \leq -1.55 \cdot 10^{+180}:\\ \;\;\;\;\left(\frac{t_2}{\sqrt{-h}} \cdot t_0\right) \cdot \left(1 + \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) \cdot -0.5\right)\\ \mathbf{elif}\;\ell \leq -1.6 \cdot 10^{+111}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq -1.8 \cdot 10^{+94}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -7.2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(-d, \sqrt{\frac{1}{h \cdot \ell}}, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot 0.125}{d}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-70}:\\ \;\;\;\;\left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \cdot \left(t_0 \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;\ell \leq -1.16 \cdot 10^{-222}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq 6.4 \cdot 10^{-288}:\\ \;\;\;\;\left(t_1 \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + h \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{-0.5}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 6
Error	30.07%
Cost	34064

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{h}{\ell}}\\ \mathbf{if}\;d \leq -1.95 \cdot 10^{+49}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t_1\right) \cdot \left(1 + \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) \cdot -0.5\right)\\ \mathbf{elif}\;d \leq -8 \cdot 10^{-282}:\\ \;\;\;\;\left(1 - {\left(t_2 \cdot \left(\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \cdot \left(t_1 \cdot t_0\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-221}:\\ \;\;\;\;\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)\\ \mathbf{elif}\;d \leq 5.6 \cdot 10^{+147}:\\ \;\;\;\;\left(1 + {\left(t_2 \cdot \left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)\right)}^{2} \cdot -0.5\right) \cdot \left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 7
Error	28.41%
Cost	34064

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := 1 + {\left(\sqrt{\frac{h}{\ell}} \cdot \left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)\right)}^{2} \cdot -0.5\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;d \leq -7.8 \cdot 10^{-60}:\\ \;\;\;\;\left(\frac{t_2}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) \cdot -0.5\right)\\ \mathbf{elif}\;d \leq -7.2 \cdot 10^{-296}:\\ \;\;\;\;\left(t_0 \cdot \frac{t_2}{\sqrt{-\ell}}\right) \cdot t_1\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-221}:\\ \;\;\;\;\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{+147}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 8
Error	29.5%
Cost	28120

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := M \cdot \left(0.5 \cdot \frac{D}{d}\right)\\ t_2 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_3 := \sqrt{-d}\\ t_4 := \left(\frac{t_3}{\sqrt{-\ell}} \cdot t_2\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ \mathbf{if}\;\ell \leq -3.45 \cdot 10^{+179}:\\ \;\;\;\;\left(\frac{t_3}{\sqrt{-h}} \cdot t_0\right) \cdot \left(1 + \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) \cdot -0.5\right)\\ \mathbf{elif}\;\ell \leq -1.6 \cdot 10^{+111}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\ell \leq -1.28 \cdot 10^{+92}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -0.0053:\\ \;\;\;\;\mathsf{fma}\left(-d, \sqrt{\frac{1}{h \cdot \ell}}, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot 0.125}{d}\right)\\ \mathbf{elif}\;\ell \leq -7.2 \cdot 10^{-69}:\\ \;\;\;\;\left(1 + {\left(\sqrt{\frac{h}{\ell}} \cdot t_1\right)}^{2} \cdot -0.5\right) \cdot \left(t_0 \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;\ell \leq -1.6 \cdot 10^{-222}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{-275}:\\ \;\;\;\;\left(t_2 \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + h \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{-0.5}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + {t_1}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 9
Error	30.02%
Cost	27528

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;\ell \leq -3.5 \cdot 10^{+227}:\\ \;\;\;\;d \cdot \left(-t_1\right)\\ \mathbf{elif}\;\ell \leq -1.1 \cdot 10^{+134}:\\ \;\;\;\;\left(t_0 \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 + \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) \cdot -0.5\right)\\ \mathbf{elif}\;\ell \leq -3.9 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(-d, t_1, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot 0.125}{d}\right)\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{-245}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{-0.5 \cdot \left(h \cdot {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}\right)}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{-156}:\\ \;\;\;\;d \cdot \frac{1}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{-147}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, 0.25 \cdot \left(\left(h \cdot \frac{\frac{D}{d}}{\ell}\right) \cdot \left(\frac{D}{d} \cdot \left(M \cdot M\right)\right)\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 10
Error	31.25%
Cost	27468

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;\ell \leq -9.5 \cdot 10^{+226}:\\ \;\;\;\;d \cdot \left(-t_2\right)\\ \mathbf{elif}\;\ell \leq -1.95 \cdot 10^{+129}:\\ \;\;\;\;t_0 \cdot t_1\\ \mathbf{elif}\;\ell \leq -4.5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(-d, t_2, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot 0.125}{d}\right)\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{-245}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{-0.5 \cdot \left(h \cdot {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}\right)}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 2.15 \cdot 10^{-156}:\\ \;\;\;\;d \cdot \frac{1}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{-146}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \mathsf{fma}\left(-0.5, 0.25 \cdot \left(\left(h \cdot \frac{\frac{D}{d}}{\ell}\right) \cdot \left(\frac{D}{d} \cdot \left(M \cdot M\right)\right)\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 11
Error	29.84%
Cost	27396

\[\begin{array}{l} t_0 := \sqrt{\frac{h}{{\ell}^{3}}}\\ t_1 := \left(M \cdot D\right) \cdot \left(M \cdot D\right)\\ \mathbf{if}\;d \leq -8.8 \cdot 10^{-177}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) \cdot -0.5\right)\\ \mathbf{elif}\;d \leq -8 \cdot 10^{-282}:\\ \;\;\;\;\mathsf{fma}\left(-d, \sqrt{\frac{1}{h \cdot \ell}}, t_0 \cdot \frac{t_1 \cdot 0.125}{d}\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-221}:\\ \;\;\;\;\frac{t_1}{d} \cdot \left(t_0 \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 12
Error	33.54%
Cost	21720

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := t_0 \cdot t_1\\ \mathbf{if}\;\ell \leq -9.5 \cdot 10^{+226}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq -1.5 \cdot 10^{+129}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -3.9 \cdot 10^{-7}:\\ \;\;\;\;\frac{D}{\frac{\frac{d}{M}}{M \cdot D}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{-292}:\\ \;\;\;\;t_2 \cdot \left(1 + 0.5 \cdot \left(\frac{D}{\frac{\frac{d}{M} \cdot \left(d \cdot \frac{\ell}{h \cdot M}\right)}{D}} \cdot -0.25\right)\right)\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{-156}:\\ \;\;\;\;\frac{d}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \mathbf{elif}\;\ell \leq 4.4 \cdot 10^{-147}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \mathsf{fma}\left(-0.5, 0.25 \cdot \left(\left(h \cdot \frac{\frac{D}{d}}{\ell}\right) \cdot \left(\frac{D}{d} \cdot \left(M \cdot M\right)\right)\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 13
Error	32.02%
Cost	21720

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;\ell \leq -9.5 \cdot 10^{+226}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq -2.2 \cdot 10^{+131}:\\ \;\;\;\;t_0 \cdot t_1\\ \mathbf{elif}\;\ell \leq -7.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{D}{\frac{\frac{d}{M}}{M \cdot D}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -2.15 \cdot 10^{-305}:\\ \;\;\;\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \cdot \left(t_0 \cdot {\left(\frac{d}{h}\right)}^{0.5}\right)\\ \mathbf{elif}\;\ell \leq 2.05 \cdot 10^{-156}:\\ \;\;\;\;\frac{d}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{-147}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \mathsf{fma}\left(-0.5, 0.25 \cdot \left(\left(h \cdot \frac{\frac{D}{d}}{\ell}\right) \cdot \left(\frac{D}{d} \cdot \left(M \cdot M\right)\right)\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 14
Error	30.92%
Cost	21720

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+227}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq -6.2 \cdot 10^{+130}:\\ \;\;\;\;t_0 \cdot t_1\\ \mathbf{elif}\;\ell \leq -7.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{D}{\frac{\frac{d}{M}}{M \cdot D}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{-244}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{-0.5 \cdot \left(h \cdot {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}\right)}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{-156}:\\ \;\;\;\;d \cdot \frac{1}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{-147}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \mathsf{fma}\left(-0.5, 0.25 \cdot \left(\left(h \cdot \frac{\frac{D}{d}}{\ell}\right) \cdot \left(\frac{D}{d} \cdot \left(M \cdot M\right)\right)\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 15
Error	34.67%
Cost	21400

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -1.6 \cdot 10^{+98}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -9.5 \cdot 10^{+76}:\\ \;\;\;\;t_0 \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -2.9 \cdot 10^{+54}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-104}:\\ \;\;\;\;t_0 \cdot \left(1 + 0.5 \cdot \left(\frac{D}{\frac{\frac{d \cdot d}{\left(h \cdot M\right) \cdot \frac{M}{\ell}}}{D}} \cdot -0.25\right)\right)\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-281}:\\ \;\;\;\;t_0 \cdot \left(1 + 0.5 \cdot \left(\frac{D}{\frac{\frac{d}{M} \cdot \left(d \cdot \frac{\ell}{h \cdot M}\right)}{D}} \cdot -0.25\right)\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-221}:\\ \;\;\;\;\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 16
Error	33%
Cost	21400

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{h}{{\ell}^{3}}}\\ t_2 := \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{if}\;d \leq -2.2 \cdot 10^{+98}:\\ \;\;\;\;\left(-d\right) \cdot t_2\\ \mathbf{elif}\;d \leq -6.5 \cdot 10^{+76}:\\ \;\;\;\;t_0 \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -4.3 \cdot 10^{+51}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq -2.65 \cdot 10^{-137}:\\ \;\;\;\;t_0 \cdot \left(1 + 0.5 \cdot \left(\frac{D}{\frac{\frac{d \cdot d}{\left(h \cdot M\right) \cdot \frac{M}{\ell}}}{D}} \cdot -0.25\right)\right)\\ \mathbf{elif}\;d \leq -9.5 \cdot 10^{-282}:\\ \;\;\;\;\frac{D}{\frac{\frac{d}{M}}{M \cdot D}} \cdot \left(t_1 \cdot 0.125\right) - d \cdot t_2\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{-221}:\\ \;\;\;\;\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot \left(t_1 \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 17
Error	36.81%
Cost	21145

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{if}\;\ell \leq -2.9 \cdot 10^{+227}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq -8.5 \cdot 10^{+133}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -115:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{-292}:\\ \;\;\;\;t_0 \cdot \left(1 + 0.5 \cdot \left(\frac{D}{\frac{\frac{d}{M} \cdot \left(d \cdot \frac{\ell}{h \cdot M}\right)}{D}} \cdot -0.25\right)\right)\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{-128} \lor \neg \left(\ell \leq 1.12 \cdot 10^{+30}\right):\\ \;\;\;\;\frac{d}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\left(D \cdot \frac{M}{d}\right)}^{2}, \frac{h}{\ell} \cdot -0.125, 1\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 18
Error	33.88%
Cost	21136

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -2.9 \cdot 10^{+104}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -3.8 \cdot 10^{-8}:\\ \;\;\;\;\left(1 + \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) \cdot -0.5\right) \cdot \left(t_0 \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-281}:\\ \;\;\;\;\left(t_0 \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + 0.5 \cdot \left(\frac{D}{\frac{\frac{d}{M} \cdot \left(d \cdot \frac{\ell}{h \cdot M}\right)}{D}} \cdot -0.25\right)\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-221}:\\ \;\;\;\;\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 19
Error	34.44%
Cost	21004

\[\begin{array}{l} \mathbf{if}\;d \leq -4.3 \cdot 10^{+110}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -8 \cdot 10^{-282}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + 0.5 \cdot \left(\frac{D}{\frac{\frac{d}{M} \cdot \left(d \cdot \frac{\ell}{h \cdot M}\right)}{D}} \cdot -0.25\right)\right)\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{-220}:\\ \;\;\;\;\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 20
Error	37.5%
Cost	15184

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ t_1 := d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{if}\;\ell \leq -9.5 \cdot 10^{+226}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -8 \cdot 10^{+133}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -7 \cdot 10^{-22}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -1.2 \cdot 10^{-70}:\\ \;\;\;\;t_0 \cdot \left(1 + 0.5 \cdot \left(\left(\frac{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}{\ell} \cdot \frac{\frac{h}{d}}{d}\right) \cdot -0.25\right)\right)\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{-292}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \end{array} \]

Alternative 21
Error	37.51%
Cost	15056

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{if}\;\ell \leq -1.45 \cdot 10^{+227}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -9.8 \cdot 10^{+133}:\\ \;\;\;\;t_0 \cdot t_1\\ \mathbf{elif}\;\ell \leq -7 \cdot 10^{-43}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -6 \cdot 10^{-71}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{-292}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \end{array} \]

Alternative 22
Error	36.03%
Cost	14920

\[\begin{array}{l} \mathbf{if}\;d \leq -1.82 \cdot 10^{+109}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -1.12 \cdot 10^{-281}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + 0.5 \cdot \left(\frac{D}{\frac{\frac{d}{M} \cdot \left(d \cdot \frac{\ell}{h \cdot M}\right)}{D}} \cdot -0.25\right)\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-221}:\\ \;\;\;\;\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \end{array} \]

Alternative 23
Error	37.2%
Cost	14600

\[\begin{array}{l} t_0 := \left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{if}\;d \leq -5 \cdot 10^{+101}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.1 \cdot 10^{-140}:\\ \;\;\;\;\left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;d \leq -6.5 \cdot 10^{-295}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-221}:\\ \;\;\;\;\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \end{array} \]

Alternative 24
Error	36.71%
Cost	14352

\[\begin{array}{l} \mathbf{if}\;d \leq -2.15 \cdot 10^{+42}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -1.15 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -6.5 \cdot 10^{-295}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq 6 \cdot 10^{-221}:\\ \;\;\;\;\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \end{array} \]

Alternative 25
Error	35.85%
Cost	13448

\[\begin{array}{l} \mathbf{if}\;h \leq -4.2 \cdot 10^{+185}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;h \leq -5.5 \cdot 10^{-307}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \end{array} \]

Alternative 26
Error	34.96%
Cost	13448

\[\begin{array}{l} \mathbf{if}\;h \leq -2.05 \cdot 10^{+102}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \end{array} \]

Alternative 27
Error	35.83%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;h \leq -1.5 \cdot 10^{+182}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;h \leq -5.5 \cdot 10^{-307}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 28
Error	42.44%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;h \leq -6.5 \cdot 10^{+179}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;h \leq -5.5 \cdot 10^{-307}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]

Alternative 29
Error	42.2%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;h \leq -6 \cdot 10^{+177}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;h \leq -5.5 \cdot 10^{-307}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]

Alternative 30
Error	53.45%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;\ell \leq -9 \cdot 10^{+231}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 9.8 \cdot 10^{-213}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 31
Error	55.45%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\ell \leq -6.8 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 32
Error	52.94%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq 3.8 \cdot 10^{-221}:\\ \;\;\;\;\sqrt{\frac{d}{h \cdot \frac{\ell}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 33
Error	52.9%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq 9.5 \cdot 10^{-221}:\\ \;\;\;\;\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 34
Error	52.75%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq 4.2 \cdot 10^{-221}:\\ \;\;\;\;\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]

Alternative 35
Error	68.32%
Cost	6720

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

Alternative 36
Error	95.98%
Cost	64

\[d \]

Reproduce

herbie shell --seed 2023103 
(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)))))