Henrywood and Agarwal, Equation (12)

Average Error: 41.47% → 24.29%

Time: 52.6s

Precision: binary64

Cost: 27460

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

Math

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

↓

\[\begin{array}{l} t_0 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_1 := \sqrt{-d}\\ t_2 := \frac{t_1}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\\ t_3 := 1 + h \cdot \frac{\left(0.5 \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{0.5}{\ell} \cdot \left(M \cdot \left(D \cdot \frac{-0.5}{d}\right)\right)\right)}{d}\\ \mathbf{if}\;h \leq -5.8 \cdot 10^{+180}:\\ \;\;\;\;t_2 \cdot \left(1 + h \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{-0.5}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\left(t_0 \cdot \frac{t_1}{\sqrt{-\ell}}\right) \cdot t_3\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_2 \cdot \left(1 + \left(\frac{M}{\frac{\ell \cdot \frac{d}{D}}{h}} \cdot \frac{M}{\frac{d}{D}}\right) \cdot -0.125\right)\\ \mathbf{elif}\;h \leq 4.1 \cdot 10^{-149}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot \frac{1}{\sqrt{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \]

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 (pow (/ d h) 0.5))
        (t_1 (sqrt (- d)))
        (t_2 (* (/ t_1 (sqrt (- h))) (pow (/ d l) 0.5)))
        (t_3
         (+
          1.0
          (*
           h
           (/ (* (* 0.5 (* D M)) (* (/ 0.5 l) (* M (* D (/ -0.5 d))))) d)))))
   (if (<= h -5.8e+180)
     (* t_2 (+ 1.0 (* h (* (pow (* D (* M (/ 0.5 d))) 2.0) (/ -0.5 l)))))
     (if (<= h -2e-16)
       (* (* t_0 (/ t_1 (sqrt (- l)))) t_3)
       (if (<= h -2e-310)
         (* t_2 (+ 1.0 (* (* (/ M (/ (* l (/ d D)) h)) (/ M (/ d D))) -0.125)))
         (if (<= h 4.1e-149)
           (* d (* (pow l -0.5) (/ 1.0 (sqrt h))))
           (* t_3 (* t_0 (/ (sqrt d) (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 = pow((d / h), 0.5);
	double t_1 = sqrt(-d);
	double t_2 = (t_1 / sqrt(-h)) * pow((d / l), 0.5);
	double t_3 = 1.0 + (h * (((0.5 * (D * M)) * ((0.5 / l) * (M * (D * (-0.5 / d))))) / d));
	double tmp;
	if (h <= -5.8e+180) {
		tmp = t_2 * (1.0 + (h * (pow((D * (M * (0.5 / d))), 2.0) * (-0.5 / l))));
	} else if (h <= -2e-16) {
		tmp = (t_0 * (t_1 / sqrt(-l))) * t_3;
	} else if (h <= -2e-310) {
		tmp = t_2 * (1.0 + (((M / ((l * (d / D)) / h)) * (M / (d / D))) * -0.125));
	} else if (h <= 4.1e-149) {
		tmp = d * (pow(l, -0.5) * (1.0 / sqrt(h)));
	} else {
		tmp = t_3 * (t_0 * (sqrt(d) / 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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (d / h) ** 0.5d0
    t_1 = sqrt(-d)
    t_2 = (t_1 / sqrt(-h)) * ((d / l) ** 0.5d0)
    t_3 = 1.0d0 + (h * (((0.5d0 * (d_1 * m)) * ((0.5d0 / l) * (m * (d_1 * ((-0.5d0) / d))))) / d))
    if (h <= (-5.8d+180)) then
        tmp = t_2 * (1.0d0 + (h * (((d_1 * (m * (0.5d0 / d))) ** 2.0d0) * ((-0.5d0) / l))))
    else if (h <= (-2d-16)) then
        tmp = (t_0 * (t_1 / sqrt(-l))) * t_3
    else if (h <= (-2d-310)) then
        tmp = t_2 * (1.0d0 + (((m / ((l * (d / d_1)) / h)) * (m / (d / d_1))) * (-0.125d0)))
    else if (h <= 4.1d-149) then
        tmp = d * ((l ** (-0.5d0)) * (1.0d0 / sqrt(h)))
    else
        tmp = t_3 * (t_0 * (sqrt(d) / sqrt(l)))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.pow((d / h), 0.5);
	double t_1 = Math.sqrt(-d);
	double t_2 = (t_1 / Math.sqrt(-h)) * Math.pow((d / l), 0.5);
	double t_3 = 1.0 + (h * (((0.5 * (D * M)) * ((0.5 / l) * (M * (D * (-0.5 / d))))) / d));
	double tmp;
	if (h <= -5.8e+180) {
		tmp = t_2 * (1.0 + (h * (Math.pow((D * (M * (0.5 / d))), 2.0) * (-0.5 / l))));
	} else if (h <= -2e-16) {
		tmp = (t_0 * (t_1 / Math.sqrt(-l))) * t_3;
	} else if (h <= -2e-310) {
		tmp = t_2 * (1.0 + (((M / ((l * (d / D)) / h)) * (M / (d / D))) * -0.125));
	} else if (h <= 4.1e-149) {
		tmp = d * (Math.pow(l, -0.5) * (1.0 / Math.sqrt(h)));
	} else {
		tmp = t_3 * (t_0 * (Math.sqrt(d) / Math.sqrt(l)));
	}
	return tmp;
}

Python

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

↓

def code(d, h, l, M, D):
	t_0 = math.pow((d / h), 0.5)
	t_1 = math.sqrt(-d)
	t_2 = (t_1 / math.sqrt(-h)) * math.pow((d / l), 0.5)
	t_3 = 1.0 + (h * (((0.5 * (D * M)) * ((0.5 / l) * (M * (D * (-0.5 / d))))) / d))
	tmp = 0
	if h <= -5.8e+180:
		tmp = t_2 * (1.0 + (h * (math.pow((D * (M * (0.5 / d))), 2.0) * (-0.5 / l))))
	elif h <= -2e-16:
		tmp = (t_0 * (t_1 / math.sqrt(-l))) * t_3
	elif h <= -2e-310:
		tmp = t_2 * (1.0 + (((M / ((l * (d / D)) / h)) * (M / (d / D))) * -0.125))
	elif h <= 4.1e-149:
		tmp = d * (math.pow(l, -0.5) * (1.0 / math.sqrt(h)))
	else:
		tmp = t_3 * (t_0 * (math.sqrt(d) / 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(d / h) ^ 0.5
	t_1 = sqrt(Float64(-d))
	t_2 = Float64(Float64(t_1 / sqrt(Float64(-h))) * (Float64(d / l) ^ 0.5))
	t_3 = Float64(1.0 + Float64(h * Float64(Float64(Float64(0.5 * Float64(D * M)) * Float64(Float64(0.5 / l) * Float64(M * Float64(D * Float64(-0.5 / d))))) / d)))
	tmp = 0.0
	if (h <= -5.8e+180)
		tmp = Float64(t_2 * Float64(1.0 + Float64(h * Float64((Float64(D * Float64(M * Float64(0.5 / d))) ^ 2.0) * Float64(-0.5 / l)))));
	elseif (h <= -2e-16)
		tmp = Float64(Float64(t_0 * Float64(t_1 / sqrt(Float64(-l)))) * t_3);
	elseif (h <= -2e-310)
		tmp = Float64(t_2 * Float64(1.0 + Float64(Float64(Float64(M / Float64(Float64(l * Float64(d / D)) / h)) * Float64(M / Float64(d / D))) * -0.125)));
	elseif (h <= 4.1e-149)
		tmp = Float64(d * Float64((l ^ -0.5) * Float64(1.0 / sqrt(h))));
	else
		tmp = Float64(t_3 * Float64(t_0 * Float64(sqrt(d) / 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 = (d / h) ^ 0.5;
	t_1 = sqrt(-d);
	t_2 = (t_1 / sqrt(-h)) * ((d / l) ^ 0.5);
	t_3 = 1.0 + (h * (((0.5 * (D * M)) * ((0.5 / l) * (M * (D * (-0.5 / d))))) / d));
	tmp = 0.0;
	if (h <= -5.8e+180)
		tmp = t_2 * (1.0 + (h * (((D * (M * (0.5 / d))) ^ 2.0) * (-0.5 / l))));
	elseif (h <= -2e-16)
		tmp = (t_0 * (t_1 / sqrt(-l))) * t_3;
	elseif (h <= -2e-310)
		tmp = t_2 * (1.0 + (((M / ((l * (d / D)) / h)) * (M / (d / D))) * -0.125));
	elseif (h <= 4.1e-149)
		tmp = d * ((l ^ -0.5) * (1.0 / sqrt(h)));
	else
		tmp = t_3 * (t_0 * (sqrt(d) / 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[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + N[(h * N[(N[(N[(0.5 * N[(D * M), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 / l), $MachinePrecision] * N[(M * N[(D * N[(-0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -5.8e+180], N[(t$95$2 * N[(1.0 + N[(h * N[(N[Power[N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -2e-16], N[(N[(t$95$0 * N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[h, -2e-310], N[(t$95$2 * N[(1.0 + N[(N[(N[(M / N[(N[(l * N[(d / D), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision] * N[(M / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 4.1e-149], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[(1.0 / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 * N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if h < -5.80000000000000015e180

   1. Initial program 52.27

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

   2. Applied egg-rr 52.23

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

   3. Simplified 44.97

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

      [Start] 52.23	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + 0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]
      +-commutative [=>] 52.23	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) + 1\right)} - 1\right)\right) \]
      associate--l+ [=>] 52.23	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) + \left(1 - 1\right)\right)}\right) \]
      metadata-eval [=>] 52.23	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) + \color{blue}{0}\right)\right) \]
      +-commutative [=>] 52.23	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(0 + 0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]
      metadata-eval [<=] 52.23	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(0 + \color{blue}{\left(--0.5\right)} \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      cancel-sign-sub-inv [<=] 52.23	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(0 - -0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]
      metadata-eval [<=] 52.23	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(0 - \color{blue}{\frac{0.5}{-1}} \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      metadata-eval [<=] 52.23	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(0 - \frac{0.5}{\color{blue}{-1}} \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      associate-*r/ [=>] 47.98	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(0 - \frac{0.5}{-1} \cdot \color{blue}{\frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot h}{\ell}}\right)\right) \]
      times-frac [<=] 47.98	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(0 - \color{blue}{\frac{0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot h\right)}{\left(-1\right) \cdot \ell}}\right)\right) \]
      metadata-eval [=>] 47.98	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(0 - \frac{0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot h\right)}{\color{blue}{-1} \cdot \ell}\right)\right) \]
      neg-mul-1 [<=] 47.98	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(0 - \frac{0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot h\right)}{\color{blue}{-\ell}}\right)\right) \]
      neg-sub0 [<=] 47.98	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(-\frac{0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot h\right)}{-\ell}\right)}\right) \]
      distribute-frac-neg [<=] 47.98	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{-0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot h\right)}{-\ell}}\right) \]

   4. Applied egg-rr 38.42

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

   if -5.80000000000000015e180 < h < -2e-16

   1. Initial program 31.21

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

   2. Applied egg-rr 31.3

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

   3. Simplified 27.8

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

      [Start] 31.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + 0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]
      +-commutative [=>] 31.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) + 1\right)} - 1\right)\right) \]
      associate--l+ [=>] 31.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) + \left(1 - 1\right)\right)}\right) \]
      metadata-eval [=>] 31.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) + \color{blue}{0}\right)\right) \]
      +-commutative [=>] 31.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(0 + 0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]
      metadata-eval [<=] 31.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(0 + \color{blue}{\left(--0.5\right)} \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      cancel-sign-sub-inv [<=] 31.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(0 - -0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]
      metadata-eval [<=] 31.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(0 - \color{blue}{\frac{0.5}{-1}} \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      metadata-eval [<=] 31.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(0 - \frac{0.5}{\color{blue}{-1}} \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      associate-*r/ [=>] 30.03	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(0 - \frac{0.5}{-1} \cdot \color{blue}{\frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot h}{\ell}}\right)\right) \]
      times-frac [<=] 30.03	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(0 - \color{blue}{\frac{0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot h\right)}{\left(-1\right) \cdot \ell}}\right)\right) \]
      metadata-eval [=>] 30.03	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(0 - \frac{0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot h\right)}{\color{blue}{-1} \cdot \ell}\right)\right) \]
      neg-mul-1 [<=] 30.03	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(0 - \frac{0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot h\right)}{\color{blue}{-\ell}}\right)\right) \]
      neg-sub0 [<=] 30.03	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(-\frac{0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot h\right)}{-\ell}\right)}\right) \]
      distribute-frac-neg [<=] 30.03	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{-0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot h\right)}{-\ell}}\right) \]

   4. Applied egg-rr 27.57

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

   5. Applied egg-rr 16.82

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

   if -2e-16 < h < -1.999999999999994e-310

   1. Initial program 43.04

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

   2. Taylor expanded in M around 0 68.42

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

   3. Simplified 60.34

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

      [Start] 68.42	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right) \]
      associate-/r* [=>] 65.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}}\right) \]
      associate-/l/ [=>] 68.42	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
      associate-*r* [=>] 69.18	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}\right) \]
      times-frac [=>] 66.24	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right)}\right) \]
      *-commutative [=>] 66.24	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)}\right) \]
      *-commutative [=>] 66.24	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}}\right)\right) \]
      associate-/l* [=>] 66.59	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot \color{blue}{\frac{{M}^{2}}{\frac{{d}^{2}}{{D}^{2}}}}\right)\right) \]
      unpow2 [=>] 66.59	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot \frac{\color{blue}{M \cdot M}}{\frac{{d}^{2}}{{D}^{2}}}\right)\right) \]
      unpow2 [=>] 66.59	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot \frac{M \cdot M}{\frac{\color{blue}{d \cdot d}}{{D}^{2}}}\right)\right) \]
      associate-/l* [=>] 60.34	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot \frac{M \cdot M}{\color{blue}{\frac{d}{\frac{{D}^{2}}{d}}}}\right)\right) \]
      unpow2 [=>] 60.34	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot \frac{M \cdot M}{\frac{d}{\frac{\color{blue}{D \cdot D}}{d}}}\right)\right) \]

   4. Applied egg-rr 50.23

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

   5. Applied egg-rr 43.43

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

   6. Applied egg-rr 26.18

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

   if -1.999999999999994e-310 < h < 4.10000000000000007e-149

   1. Initial program 54.6

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

   2. Simplified 56.18

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

      [Start] 54.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 54.6	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 54.6	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 54.6	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 54.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 54.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 54.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 56.18	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Taylor expanded in d around inf 37.17

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

   4. Simplified 37.19

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

      [Start] 37.17	\[ \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]
      *-commutative [=>] 37.17	\[ \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
      associate-/r* [=>] 37.19	\[ d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   5. Applied egg-rr 22.03

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

   if 4.10000000000000007e-149 < h

   1. Initial program 37.85

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

   2. Applied egg-rr 38.04

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

   3. Simplified 34.76

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

      [Start] 38.04	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + 0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]
      +-commutative [=>] 38.04	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) + 1\right)} - 1\right)\right) \]
      associate--l+ [=>] 38.04	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) + \left(1 - 1\right)\right)}\right) \]
      metadata-eval [=>] 38.04	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) + \color{blue}{0}\right)\right) \]
      +-commutative [=>] 38.04	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(0 + 0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]
      metadata-eval [<=] 38.04	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(0 + \color{blue}{\left(--0.5\right)} \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      cancel-sign-sub-inv [<=] 38.04	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(0 - -0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]
      metadata-eval [<=] 38.04	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(0 - \color{blue}{\frac{0.5}{-1}} \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      metadata-eval [<=] 38.04	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(0 - \frac{0.5}{\color{blue}{-1}} \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      associate-*r/ [=>] 36.38	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(0 - \frac{0.5}{-1} \cdot \color{blue}{\frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot h}{\ell}}\right)\right) \]
      times-frac [<=] 36.38	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(0 - \color{blue}{\frac{0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot h\right)}{\left(-1\right) \cdot \ell}}\right)\right) \]
      metadata-eval [=>] 36.38	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(0 - \frac{0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot h\right)}{\color{blue}{-1} \cdot \ell}\right)\right) \]
      neg-mul-1 [<=] 36.38	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(0 - \frac{0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot h\right)}{\color{blue}{-\ell}}\right)\right) \]
      neg-sub0 [<=] 36.38	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(-\frac{0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot h\right)}{-\ell}\right)}\right) \]
      distribute-frac-neg [<=] 36.38	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{-0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot h\right)}{-\ell}}\right) \]

   4. Applied egg-rr 34.03

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

   5. Applied egg-rr 23.29

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - h \cdot \frac{\left(\left(D \cdot M\right) \cdot 0.5\right) \cdot \left(\left(M \cdot \left(\frac{0.5}{d} \cdot D\right)\right) \cdot \frac{0.5}{\ell}\right)}{d}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 24.29

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -5.8 \cdot 10^{+180}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + h \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{-0.5}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 + h \cdot \frac{\left(0.5 \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{0.5}{\ell} \cdot \left(M \cdot \left(D \cdot \frac{-0.5}{d}\right)\right)\right)}{d}\right)\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \left(\frac{M}{\frac{\ell \cdot \frac{d}{D}}{h}} \cdot \frac{M}{\frac{d}{D}}\right) \cdot -0.125\right)\\ \mathbf{elif}\;h \leq 4.1 \cdot 10^{-149}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot \frac{1}{\sqrt{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + h \cdot \frac{\left(0.5 \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{0.5}{\ell} \cdot \left(M \cdot \left(D \cdot \frac{-0.5}{d}\right)\right)\right)}{d}\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	27.21%
Cost	21776

\[\begin{array}{l} t_0 := {\left(\frac{d}{\ell}\right)}^{0.5}\\ t_1 := 1 + \left(\frac{M}{\frac{\ell \cdot \frac{d}{D}}{h}} \cdot \frac{M}{\frac{d}{D}}\right) \cdot -0.125\\ t_2 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_3 := \sqrt{-d}\\ \mathbf{if}\;d \leq -2.7 \cdot 10^{+82}:\\ \;\;\;\;\left(\frac{t_3}{\sqrt{-h}} \cdot t_0\right) \cdot t_1\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-101}:\\ \;\;\;\;\left(t_0 \cdot t_2\right) \cdot \left(1 + \frac{h \cdot \left(D \cdot \left(M \cdot -0.5\right)\right)}{\ell \cdot \frac{d}{M \cdot \left(0.5 \cdot \frac{D}{d \cdot 2}\right)}}\right)\\ \mathbf{elif}\;d \leq -2.75 \cdot 10^{-272}:\\ \;\;\;\;\left(t_2 \cdot \frac{t_3}{\sqrt{-\ell}}\right) \cdot t_1\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{+67}:\\ \;\;\;\;\left(1 + h \cdot \frac{\left(0.5 \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{0.5}{\ell} \cdot \left(M \cdot \left(D \cdot \frac{-0.5}{d}\right)\right)\right)}{d}\right) \cdot \left(t_2 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \end{array} \]

Alternative 2
Error	23.48%
Cost	21644

\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := 1 + h \cdot \frac{\left(0.5 \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{0.5}{\ell} \cdot \left(M \cdot \left(D \cdot \frac{-0.5}{d}\right)\right)\right)}{d}\\ t_2 := {\left(\frac{d}{h}\right)}^{0.5}\\ \mathbf{if}\;h \leq -5.2 \cdot 10^{-20}:\\ \;\;\;\;\left(t_2 \cdot \frac{t_0}{\sqrt{-\ell}}\right) \cdot t_1\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{t_0}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \left(\frac{M}{\frac{\ell \cdot \frac{d}{D}}{h}} \cdot \frac{M}{\frac{d}{D}}\right) \cdot -0.125\right)\\ \mathbf{elif}\;h \leq 9.6 \cdot 10^{-149}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot \frac{1}{\sqrt{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(t_2 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \]

Alternative 3
Error	27.87%
Cost	21588

\[\begin{array}{l} t_0 := {\left(\frac{d}{\ell}\right)}^{0.5}\\ t_1 := 1 + \left(\frac{M}{\frac{\ell \cdot \frac{d}{D}}{h}} \cdot \frac{M}{\frac{d}{D}}\right) \cdot -0.125\\ t_2 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_3 := \sqrt{-d}\\ \mathbf{if}\;d \leq -3.6 \cdot 10^{+82}:\\ \;\;\;\;\left(\frac{t_3}{\sqrt{-h}} \cdot t_0\right) \cdot t_1\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-101}:\\ \;\;\;\;\left(t_0 \cdot t_2\right) \cdot \left(1 + \frac{h \cdot \left(D \cdot \left(M \cdot -0.5\right)\right)}{\ell \cdot \frac{d}{M \cdot \left(0.5 \cdot \frac{D}{d \cdot 2}\right)}}\right)\\ \mathbf{elif}\;d \leq -2.75 \cdot 10^{-272}:\\ \;\;\;\;\left(t_2 \cdot \frac{t_3}{\sqrt{-\ell}}\right) \cdot t_1\\ \mathbf{elif}\;d \leq 4.3 \cdot 10^{-86}:\\ \;\;\;\;t_1 \cdot \left(t_2 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, 0.25 \cdot \left(\frac{h \cdot \left(D \cdot M\right)}{d \cdot \ell} \cdot \frac{D \cdot M}{d}\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \end{array} \]

Alternative 4
Error	33.99%
Cost	21456

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -1.1 \cdot 10^{+165}:\\ \;\;\;\;\left(1 + \left(\frac{h}{\ell} \cdot \frac{M \cdot M}{\frac{d}{\frac{D \cdot D}{d}}}\right) \cdot -0.125\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t_1\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\left(1 + h \cdot \frac{\left(0.5 \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{0.5}{\ell} \cdot \left(M \cdot \left(D \cdot \frac{-0.5}{d}\right)\right)\right)}{d}\right) \cdot \left(t_0 \cdot t_1\right)\\ \mathbf{elif}\;d \leq 1.95 \cdot 10^{-67}:\\ \;\;\;\;\frac{{\ell}^{-0.5}}{\frac{\sqrt{h}}{d}}\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+66}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \mathsf{fma}\left(-0.5, 0.25 \cdot \left(\frac{h \cdot \left(D \cdot M\right)}{d \cdot \ell} \cdot \frac{D \cdot M}{d}\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \end{array} \]

Alternative 5
Error	32.69%
Cost	21456

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -3.2 \cdot 10^{+165}:\\ \;\;\;\;\left(1 + \left(\frac{h}{\ell} \cdot \frac{M \cdot M}{\frac{d}{\frac{D \cdot D}{d}}}\right) \cdot -0.125\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t_1\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\left(1 + h \cdot \frac{\left(0.5 \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{0.5}{\ell} \cdot \left(M \cdot \left(D \cdot \frac{-0.5}{d}\right)\right)\right)}{d}\right) \cdot \left(t_0 \cdot t_1\right)\\ \mathbf{elif}\;d \leq 2.75 \cdot 10^{-86}:\\ \;\;\;\;\left(1 + \left(\frac{M}{\frac{\ell \cdot \frac{d}{D}}{h}} \cdot \frac{M}{\frac{d}{D}}\right) \cdot -0.125\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{elif}\;d \leq 7.4 \cdot 10^{+66}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \mathsf{fma}\left(-0.5, 0.25 \cdot \left(\frac{h \cdot \left(D \cdot M\right)}{d \cdot \ell} \cdot \frac{D \cdot M}{d}\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \end{array} \]

Alternative 6
Error	32.69%
Cost	21456

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -7.5 \cdot 10^{+157}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 + \left(\frac{h}{\ell} \cdot \frac{M \cdot M}{\frac{d}{\frac{D \cdot D}{d}}}\right) \cdot -0.125\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\left(1 + h \cdot \frac{\left(0.5 \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{0.5}{\ell} \cdot \left(M \cdot \left(D \cdot \frac{-0.5}{d}\right)\right)\right)}{d}\right) \cdot \left(t_0 \cdot t_1\right)\\ \mathbf{elif}\;d \leq 1.36 \cdot 10^{-88}:\\ \;\;\;\;\left(1 + \left(\frac{M}{\frac{\ell \cdot \frac{d}{D}}{h}} \cdot \frac{M}{\frac{d}{D}}\right) \cdot -0.125\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{+67}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \mathsf{fma}\left(-0.5, 0.25 \cdot \left(\frac{h \cdot \left(D \cdot M\right)}{d \cdot \ell} \cdot \frac{D \cdot M}{d}\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \end{array} \]

Alternative 7
Error	28.78%
Cost	21456

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := 1 + \left(\frac{M}{\frac{\ell \cdot \frac{d}{D}}{h}} \cdot \frac{M}{\frac{d}{D}}\right) \cdot -0.125\\ \mathbf{if}\;d \leq -3.6 \cdot 10^{+82}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot t_2\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\left(1 + h \cdot \frac{\left(0.5 \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{0.5}{\ell} \cdot \left(M \cdot \left(D \cdot \frac{-0.5}{d}\right)\right)\right)}{d}\right) \cdot \left(t_0 \cdot t_1\right)\\ \mathbf{elif}\;d \leq 6 \cdot 10^{-87}:\\ \;\;\;\;t_2 \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{elif}\;d \leq 7 \cdot 10^{+66}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \mathsf{fma}\left(-0.5, 0.25 \cdot \left(\frac{h \cdot \left(D \cdot M\right)}{d \cdot \ell} \cdot \frac{D \cdot M}{d}\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \end{array} \]

Alternative 8
Error	33.56%
Cost	20872

\[\begin{array}{l} \mathbf{if}\;h \leq -4.3 \cdot 10^{-226}:\\ \;\;\;\;\left(1 + h \cdot \frac{\left(0.5 \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{0.5}{\ell} \cdot \left(M \cdot \left(D \cdot \frac{-0.5}{d}\right)\right)\right)}{d}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;h \leq 1.4 \cdot 10^{-283}:\\ \;\;\;\;\left(1 + \left(\frac{h}{d} \cdot \left(D \cdot \left(\frac{0.5}{\ell} \cdot M\right)\right)\right) \cdot \left(M \cdot \left(\frac{D}{d} \cdot -0.25\right)\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \end{array} \]

Alternative 9
Error	41.01%
Cost	15320

\[\begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ t_1 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := \frac{0.5}{\ell} \cdot M\\ t_4 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -1.95 \cdot 10^{+129}:\\ \;\;\;\;t_2 \cdot t_4\\ \mathbf{elif}\;d \leq -6 \cdot 10^{-73}:\\ \;\;\;\;\left(1 + \left(\frac{h}{d} \cdot \left(D \cdot t_3\right)\right) \cdot \left(M \cdot \left(\frac{D}{d} \cdot -0.25\right)\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(1 + h \cdot \left(\frac{0.5}{\ell} \cdot \left(\left(t_0 \cdot t_0\right) \cdot -0.25\right)\right)\right) \cdot t_1\\ \mathbf{elif}\;d \leq 7.6 \cdot 10^{-67}:\\ \;\;\;\;\frac{{\ell}^{-0.5}}{\frac{\sqrt{h}}{d}}\\ \mathbf{elif}\;d \leq 2.75 \cdot 10^{+16}:\\ \;\;\;\;\left(1 - \left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \left(h \cdot \left(t_3 \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)\right)\right) \cdot t_1\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{+28}:\\ \;\;\;\;t_2 \cdot \left(t_4 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot \frac{1}{\sqrt{h}}\right)\\ \end{array} \]

Alternative 10
Error	40.91%
Cost	15316

\[\begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ t_1 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -2.75 \cdot 10^{+129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -5.9 \cdot 10^{-73}:\\ \;\;\;\;\left(1 + \left(\frac{h}{d} \cdot \left(D \cdot \left(\frac{0.5}{\ell} \cdot M\right)\right)\right) \cdot \left(M \cdot \left(\frac{D}{d} \cdot -0.25\right)\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(1 + h \cdot \left(\frac{0.5}{\ell} \cdot \left(\left(t_0 \cdot t_0\right) \cdot -0.25\right)\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;d \leq 5.7 \cdot 10^{-36}:\\ \;\;\;\;\frac{{\ell}^{-0.5}}{\frac{\sqrt{h}}{d}}\\ \mathbf{elif}\;d \leq 9 \cdot 10^{+30}:\\ \;\;\;\;t_1 \cdot \left(1 + 0.5 \cdot \left(\left(\frac{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}{\ell} \cdot \frac{\frac{h}{d}}{d}\right) \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot \frac{1}{\sqrt{h}}\right)\\ \end{array} \]

Alternative 11
Error	34.4%
Cost	15180

\[\begin{array}{l} t_0 := \left(1 + h \cdot \frac{\left(0.5 \cdot \left(D \cdot M\right)\right) \cdot \left(\frac{0.5}{\ell} \cdot \left(M \cdot \left(D \cdot \frac{-0.5}{d}\right)\right)\right)}{d}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{if}\;h \leq -1 \cdot 10^{-226}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(1 + \left(\frac{h}{d} \cdot \left(D \cdot \left(\frac{0.5}{\ell} \cdot M\right)\right)\right) \cdot \left(M \cdot \left(\frac{D}{d} \cdot -0.25\right)\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq 5.6 \cdot 10^{-65}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	37.12%
Cost	15120

\[\begin{array}{l} t_0 := \left(1 + \left(\frac{M}{\frac{\ell \cdot \frac{d}{D}}{h}} \cdot \frac{M}{\frac{d}{D}}\right) \cdot -0.125\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{if}\;h \leq -2.4 \cdot 10^{-226}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(1 + \left(\frac{h}{d} \cdot \left(D \cdot \left(\frac{0.5}{\ell} \cdot M\right)\right)\right) \cdot \left(M \cdot \left(\frac{D}{d} \cdot -0.25\right)\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq 1.2 \cdot 10^{+88}:\\ \;\;\;\;\frac{d}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \mathbf{elif}\;h \leq 2.8 \cdot 10^{+281}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(h \cdot \ell\right)}^{-1.5}}\\ \end{array} \]

Alternative 13
Error	41.02%
Cost	13580

\[\begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ \mathbf{if}\;d \leq -6.2 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -5.1 \cdot 10^{-73}:\\ \;\;\;\;\left(1 + \left(\frac{h}{d} \cdot \left(D \cdot \left(\frac{0.5}{\ell} \cdot M\right)\right)\right) \cdot \left(M \cdot \left(\frac{D}{d} \cdot -0.25\right)\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq 1.65 \cdot 10^{-308}:\\ \;\;\;\;\left(1 + h \cdot \left(\frac{0.5}{\ell} \cdot \left(\left(t_0 \cdot t_0\right) \cdot -0.25\right)\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \end{array} \]

Alternative 14
Error	41.72%
Cost	13448

\[\begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ \mathbf{if}\;h \leq -6.4 \cdot 10^{-116}:\\ \;\;\;\;\left(1 + h \cdot \left(\frac{0.5}{\ell} \cdot \left(\left(t_0 \cdot t_0\right) \cdot -0.25\right)\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(1 + \left(\frac{h}{d} \cdot \left(D \cdot \left(\frac{0.5}{\ell} \cdot M\right)\right)\right) \cdot \left(M \cdot \left(\frac{D}{d} \cdot -0.25\right)\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \end{array} \]

Alternative 15
Error	41.71%
Cost	13384

\[\begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ \mathbf{if}\;h \leq -8.5 \cdot 10^{-111}:\\ \;\;\;\;\left(1 + h \cdot \left(\frac{0.5}{\ell} \cdot \left(\left(t_0 \cdot t_0\right) \cdot -0.25\right)\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(1 + \left(\frac{h}{d} \cdot \left(D \cdot \left(\frac{0.5}{\ell} \cdot M\right)\right)\right) \cdot \left(M \cdot \left(\frac{D}{d} \cdot -0.25\right)\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 16
Error	46.08%
Cost	8652

\[\begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ t_1 := \left(1 + h \cdot \left(\frac{0.5}{\ell} \cdot \left(\left(t_0 \cdot t_0\right) \cdot -0.25\right)\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{if}\;h \leq -3.5 \cdot 10^{-116}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;h \leq 1.4 \cdot 10^{-283}:\\ \;\;\;\;\left(1 + \left(\frac{h}{d} \cdot \left(D \cdot \left(\frac{0.5}{\ell} \cdot M\right)\right)\right) \cdot \left(M \cdot \left(\frac{D}{d} \cdot -0.25\right)\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq 1.9 \cdot 10^{+121}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 17
Error	50.56%
Cost	8520

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{if}\;h \leq -5.6 \cdot 10^{+146}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq 1.4 \cdot 10^{-283}:\\ \;\;\;\;\left(1 + \left(\frac{h}{d} \cdot \left(D \cdot \left(\frac{0.5}{\ell} \cdot M\right)\right)\right) \cdot \left(M \cdot \left(\frac{D}{d} \cdot -0.25\right)\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq 1.1 \cdot 10^{+116}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;h \leq 2.2 \cdot 10^{+217}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 18
Error	51.89%
Cost	7508

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{if}\;h \leq -2.35 \cdot 10^{-226}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq 3.7 \cdot 10^{-304}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq 2.5 \cdot 10^{-303}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq 3.2 \cdot 10^{+118}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;h \leq 6 \cdot 10^{+216}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 19
Error	52.1%
Cost	7244

\[\begin{array}{l} \mathbf{if}\;h \leq 2.5 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{\frac{d}{h \cdot \frac{\ell}{d}}}\\ \mathbf{elif}\;h \leq 5.7 \cdot 10^{+125}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;h \leq 2.5 \cdot 10^{+217}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 20
Error	52.1%
Cost	7244

\[\begin{array}{l} \mathbf{if}\;h \leq 2.5 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{\frac{d}{h \cdot \frac{\ell}{d}}}\\ \mathbf{elif}\;h \leq 9.5 \cdot 10^{+118}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;h \leq 1.7 \cdot 10^{+217}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 21
Error	54.4%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;h \leq 5 \cdot 10^{-293}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 22
Error	68.68%
Cost	6784

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

Alternative 23
Error	68.68%
Cost	6720

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

Reproduce

herbie shell --seed 2023089 
(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)))))