Henrywood and Agarwal, Equation (13)

Average Error: 59.5 → 53.3

Time: 52.8s

Precision: binary64

Cost: 14412

Math

\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

↓

\[\begin{array}{l} t_0 := d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right)\\ t_1 := d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}}\\ t_2 := \frac{d}{D} \cdot \left(d \cdot \frac{\frac{c0}{w}}{D \cdot h}\right)\\ \mathbf{if}\;D \leq -2.1 \cdot 10^{-190}:\\ \;\;\;\;\frac{c0}{\frac{2}{\frac{\frac{t_0 + \sqrt{\left(t_0 + M\right) \cdot \left(t_0 - M\right)}}{1}}{w}}}\\ \mathbf{elif}\;D \leq 2.1 \cdot 10^{-108}:\\ \;\;\;\;\frac{{\left(d \cdot c0\right)}^{2}}{{\left(w \cdot D\right)}^{2}} \cdot \frac{1}{h}\\ \mathbf{elif}\;D \leq 2.35 \cdot 10^{+48}:\\ \;\;\;\;2 \cdot \left(\frac{{d}^{2}}{{D}^{2}} \cdot \left(\frac{\frac{c0}{w}}{h} \cdot \frac{c0}{w + w}\right)\right)\\ \mathbf{elif}\;D \leq 7.5 \cdot 10^{+157}:\\ \;\;\;\;\frac{t_1 + \sqrt{t_1 \cdot \left(d \cdot \frac{d}{\left(h \cdot \frac{w}{c0}\right) \cdot \left(D \cdot D\right)}\right) - M \cdot M}}{w \cdot \frac{2}{c0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{2}{\frac{t_2 + \sqrt{t_2 \cdot t_2 - M \cdot M}}{w}}}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (*
  (/ c0 (* 2.0 w))
  (+
   (/ (* c0 (* d d)) (* (* w h) (* D D)))
   (sqrt
    (-
     (*
      (/ (* c0 (* d d)) (* (* w h) (* D D)))
      (/ (* c0 (* d d)) (* (* w h) (* D D))))
     (* M M))))))

↓

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* d (* (/ d D) (/ (/ (/ c0 w) D) h))))
        (t_1 (* d (/ d (/ (* h (* D D)) (/ c0 w)))))
        (t_2 (* (/ d D) (* d (/ (/ c0 w) (* D h))))))
   (if (<= D -2.1e-190)
     (/ c0 (/ 2.0 (/ (/ (+ t_0 (sqrt (* (+ t_0 M) (- t_0 M)))) 1.0) w)))
     (if (<= D 2.1e-108)
       (* (/ (pow (* d c0) 2.0) (pow (* w D) 2.0)) (/ 1.0 h))
       (if (<= D 2.35e+48)
         (*
          2.0
          (* (/ (pow d 2.0) (pow D 2.0)) (* (/ (/ c0 w) h) (/ c0 (+ w w)))))
         (if (<= D 7.5e+157)
           (/
            (+
             t_1
             (sqrt (- (* t_1 (* d (/ d (* (* h (/ w c0)) (* D D))))) (* M M))))
            (* w (/ 2.0 c0)))
           (/ c0 (/ 2.0 (/ (+ t_2 (sqrt (- (* t_2 t_2) (* M M)))) w)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 / (2.0 * w)) * (((c0 * (d * d)) / ((w * h) * (D * D))) + sqrt(((((c0 * (d * d)) / ((w * h) * (D * D))) * ((c0 * (d * d)) / ((w * h) * (D * D)))) - (M * M))));
}

↓

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = d * ((d / D) * (((c0 / w) / D) / h));
	double t_1 = d * (d / ((h * (D * D)) / (c0 / w)));
	double t_2 = (d / D) * (d * ((c0 / w) / (D * h)));
	double tmp;
	if (D <= -2.1e-190) {
		tmp = c0 / (2.0 / (((t_0 + sqrt(((t_0 + M) * (t_0 - M)))) / 1.0) / w));
	} else if (D <= 2.1e-108) {
		tmp = (pow((d * c0), 2.0) / pow((w * D), 2.0)) * (1.0 / h);
	} else if (D <= 2.35e+48) {
		tmp = 2.0 * ((pow(d, 2.0) / pow(D, 2.0)) * (((c0 / w) / h) * (c0 / (w + w))));
	} else if (D <= 7.5e+157) {
		tmp = (t_1 + sqrt(((t_1 * (d * (d / ((h * (w / c0)) * (D * D))))) - (M * M)))) / (w * (2.0 / c0));
	} else {
		tmp = c0 / (2.0 / ((t_2 + sqrt(((t_2 * t_2) - (M * M)))) / w));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = (c0 / (2.0d0 * w)) * (((c0 * (d_1 * d_1)) / ((w * h) * (d * d))) + sqrt(((((c0 * (d_1 * d_1)) / ((w * h) * (d * d))) * ((c0 * (d_1 * d_1)) / ((w * h) * (d * d)))) - (m * m))))
end function

↓

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = d_1 * ((d_1 / d) * (((c0 / w) / d) / h))
    t_1 = d_1 * (d_1 / ((h * (d * d)) / (c0 / w)))
    t_2 = (d_1 / d) * (d_1 * ((c0 / w) / (d * h)))
    if (d <= (-2.1d-190)) then
        tmp = c0 / (2.0d0 / (((t_0 + sqrt(((t_0 + m) * (t_0 - m)))) / 1.0d0) / w))
    else if (d <= 2.1d-108) then
        tmp = (((d_1 * c0) ** 2.0d0) / ((w * d) ** 2.0d0)) * (1.0d0 / h)
    else if (d <= 2.35d+48) then
        tmp = 2.0d0 * (((d_1 ** 2.0d0) / (d ** 2.0d0)) * (((c0 / w) / h) * (c0 / (w + w))))
    else if (d <= 7.5d+157) then
        tmp = (t_1 + sqrt(((t_1 * (d_1 * (d_1 / ((h * (w / c0)) * (d * d))))) - (m * m)))) / (w * (2.0d0 / c0))
    else
        tmp = c0 / (2.0d0 / ((t_2 + sqrt(((t_2 * t_2) - (m * m)))) / w))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 / (2.0 * w)) * (((c0 * (d * d)) / ((w * h) * (D * D))) + Math.sqrt(((((c0 * (d * d)) / ((w * h) * (D * D))) * ((c0 * (d * d)) / ((w * h) * (D * D)))) - (M * M))));
}

↓

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = d * ((d / D) * (((c0 / w) / D) / h));
	double t_1 = d * (d / ((h * (D * D)) / (c0 / w)));
	double t_2 = (d / D) * (d * ((c0 / w) / (D * h)));
	double tmp;
	if (D <= -2.1e-190) {
		tmp = c0 / (2.0 / (((t_0 + Math.sqrt(((t_0 + M) * (t_0 - M)))) / 1.0) / w));
	} else if (D <= 2.1e-108) {
		tmp = (Math.pow((d * c0), 2.0) / Math.pow((w * D), 2.0)) * (1.0 / h);
	} else if (D <= 2.35e+48) {
		tmp = 2.0 * ((Math.pow(d, 2.0) / Math.pow(D, 2.0)) * (((c0 / w) / h) * (c0 / (w + w))));
	} else if (D <= 7.5e+157) {
		tmp = (t_1 + Math.sqrt(((t_1 * (d * (d / ((h * (w / c0)) * (D * D))))) - (M * M)))) / (w * (2.0 / c0));
	} else {
		tmp = c0 / (2.0 / ((t_2 + Math.sqrt(((t_2 * t_2) - (M * M)))) / w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	return (c0 / (2.0 * w)) * (((c0 * (d * d)) / ((w * h) * (D * D))) + math.sqrt(((((c0 * (d * d)) / ((w * h) * (D * D))) * ((c0 * (d * d)) / ((w * h) * (D * D)))) - (M * M))))

↓

def code(c0, w, h, D, d, M):
	t_0 = d * ((d / D) * (((c0 / w) / D) / h))
	t_1 = d * (d / ((h * (D * D)) / (c0 / w)))
	t_2 = (d / D) * (d * ((c0 / w) / (D * h)))
	tmp = 0
	if D <= -2.1e-190:
		tmp = c0 / (2.0 / (((t_0 + math.sqrt(((t_0 + M) * (t_0 - M)))) / 1.0) / w))
	elif D <= 2.1e-108:
		tmp = (math.pow((d * c0), 2.0) / math.pow((w * D), 2.0)) * (1.0 / h)
	elif D <= 2.35e+48:
		tmp = 2.0 * ((math.pow(d, 2.0) / math.pow(D, 2.0)) * (((c0 / w) / h) * (c0 / (w + w))))
	elif D <= 7.5e+157:
		tmp = (t_1 + math.sqrt(((t_1 * (d * (d / ((h * (w / c0)) * (D * D))))) - (M * M)))) / (w * (2.0 / c0))
	else:
		tmp = c0 / (2.0 / ((t_2 + math.sqrt(((t_2 * t_2) - (M * M)))) / w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D))) + sqrt(Float64(Float64(Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D))) * Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))) - Float64(M * M)))))
end

↓

function code(c0, w, h, D, d, M)
	t_0 = Float64(d * Float64(Float64(d / D) * Float64(Float64(Float64(c0 / w) / D) / h)))
	t_1 = Float64(d * Float64(d / Float64(Float64(h * Float64(D * D)) / Float64(c0 / w))))
	t_2 = Float64(Float64(d / D) * Float64(d * Float64(Float64(c0 / w) / Float64(D * h))))
	tmp = 0.0
	if (D <= -2.1e-190)
		tmp = Float64(c0 / Float64(2.0 / Float64(Float64(Float64(t_0 + sqrt(Float64(Float64(t_0 + M) * Float64(t_0 - M)))) / 1.0) / w)));
	elseif (D <= 2.1e-108)
		tmp = Float64(Float64((Float64(d * c0) ^ 2.0) / (Float64(w * D) ^ 2.0)) * Float64(1.0 / h));
	elseif (D <= 2.35e+48)
		tmp = Float64(2.0 * Float64(Float64((d ^ 2.0) / (D ^ 2.0)) * Float64(Float64(Float64(c0 / w) / h) * Float64(c0 / Float64(w + w)))));
	elseif (D <= 7.5e+157)
		tmp = Float64(Float64(t_1 + sqrt(Float64(Float64(t_1 * Float64(d * Float64(d / Float64(Float64(h * Float64(w / c0)) * Float64(D * D))))) - Float64(M * M)))) / Float64(w * Float64(2.0 / c0)));
	else
		tmp = Float64(c0 / Float64(2.0 / Float64(Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M)))) / w)));
	end
	return tmp
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = (c0 / (2.0 * w)) * (((c0 * (d * d)) / ((w * h) * (D * D))) + sqrt(((((c0 * (d * d)) / ((w * h) * (D * D))) * ((c0 * (d * d)) / ((w * h) * (D * D)))) - (M * M))));
end

↓

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = d * ((d / D) * (((c0 / w) / D) / h));
	t_1 = d * (d / ((h * (D * D)) / (c0 / w)));
	t_2 = (d / D) * (d * ((c0 / w) / (D * h)));
	tmp = 0.0;
	if (D <= -2.1e-190)
		tmp = c0 / (2.0 / (((t_0 + sqrt(((t_0 + M) * (t_0 - M)))) / 1.0) / w));
	elseif (D <= 2.1e-108)
		tmp = (((d * c0) ^ 2.0) / ((w * D) ^ 2.0)) * (1.0 / h);
	elseif (D <= 2.35e+48)
		tmp = 2.0 * (((d ^ 2.0) / (D ^ 2.0)) * (((c0 / w) / h) * (c0 / (w + w))));
	elseif (D <= 7.5e+157)
		tmp = (t_1 + sqrt(((t_1 * (d * (d / ((h * (w / c0)) * (D * D))))) - (M * M)))) / (w * (2.0 / c0));
	else
		tmp = c0 / (2.0 / ((t_2 + sqrt(((t_2 * t_2) - (M * M)))) / w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[(N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(d * N[(N[(d / D), $MachinePrecision] * N[(N[(N[(c0 / w), $MachinePrecision] / D), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d * N[(d / N[(N[(h * N[(D * D), $MachinePrecision]), $MachinePrecision] / N[(c0 / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(d / D), $MachinePrecision] * N[(d * N[(N[(c0 / w), $MachinePrecision] / N[(D * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[D, -2.1e-190], N[(c0 / N[(2.0 / N[(N[(N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 + M), $MachinePrecision] * N[(t$95$0 - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 2.1e-108], N[(N[(N[Power[N[(d * c0), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[(w * D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / h), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 2.35e+48], N[(2.0 * N[(N[(N[Power[d, 2.0], $MachinePrecision] / N[Power[D, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision] * N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 7.5e+157], N[(N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * N[(d * N[(d / N[(N[(h * N[(w / c0), $MachinePrecision]), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(w * N[(2.0 / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(2.0 / N[(N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if D < -2.09999999999999991e-190

   1. Initial program 57.3

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Simplified 56.8

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \frac{c0}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)} + \sqrt{\left(\left(d \cdot d\right) \cdot \frac{c0}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)}\right) \cdot \left(\left(d \cdot d\right) \cdot \frac{c0}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)}\right) - M \cdot M}\right)} \]
      Proof

      [Start] 57.3	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      rational.json-simplify-49 [=>] 58.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(d \cdot d\right) \cdot \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      rational.json-simplify-2 [=>] 58.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \frac{c0}{\color{blue}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      rational.json-simplify-2 [=>] 58.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \frac{c0}{\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot w\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      rational.json-simplify-43 [=>] 58.7	\[ \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \frac{c0}{\color{blue}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Applied egg-rr 53.4

      \[\leadsto \color{blue}{\frac{d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}} + \sqrt{\left(d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}}\right) \cdot \left(d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}}\right) - M \cdot M}}{w \cdot \frac{2}{c0}}} \]

   4. Simplified 53.5

      \[\leadsto \color{blue}{\frac{c0}{\frac{2}{\frac{d \cdot \frac{d}{D \cdot \frac{w}{\frac{c0}{D \cdot h}}} + \sqrt{\frac{d}{D \cdot \frac{w}{\frac{c0}{D \cdot h}}} \cdot \left(\left(d \cdot \frac{d}{D \cdot \frac{w}{\frac{c0}{D \cdot h}}}\right) \cdot d\right) - M \cdot M}}{w}}}} \]
      Proof

      [Start] 53.4	\[ \frac{d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}} + \sqrt{\left(d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}}\right) \cdot \left(d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}}\right) - M \cdot M}}{w \cdot \frac{2}{c0}} \]
      rational.json-simplify-46 [=>] 53.7	\[ \color{blue}{\frac{\frac{d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}} + \sqrt{\left(d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}}\right) \cdot \left(d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}}\right) - M \cdot M}}{w}}{\frac{2}{c0}}} \]
      rational.json-simplify-61 [=>] 53.7	\[ \color{blue}{\frac{c0}{\frac{2}{\frac{d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}} + \sqrt{\left(d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}}\right) \cdot \left(d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}}\right) - M \cdot M}}{w}}}} \]

   5. Applied egg-rr 51.5

      \[\leadsto \frac{c0}{\frac{2}{\frac{\color{blue}{\frac{d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right) + \sqrt{\left(d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right) + M\right) \cdot \left(d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right) - M\right)}}{1}}}{w}}} \]

   if -2.09999999999999991e-190 < D < 2.0999999999999999e-108

   1. Initial program 63.0

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Simplified 63.0

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \frac{\frac{d \cdot d}{w \cdot h}}{D \cdot D} + \sqrt{\left(c0 \cdot \frac{\frac{d \cdot d}{w \cdot h}}{D \cdot D} + M\right) \cdot \left(c0 \cdot \frac{\frac{d \cdot d}{w \cdot h}}{D \cdot D} - M\right)}\right)} \]
      Proof

      [Start] 63.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      rational.json-simplify-2 [=>] 63.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\left(d \cdot d\right) \cdot c0}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      rational.json-simplify-49 [=>] 63.1	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{c0 \cdot \frac{d \cdot d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      rational.json-simplify-46 [=>] 63.2	\[ \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{\frac{\frac{d \cdot d}{w \cdot h}}{D \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      rational.json-simplify-34 [=>] 63.2	\[ \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \frac{\frac{d \cdot d}{w \cdot h}}{D \cdot D} + \sqrt{\color{blue}{\left(M + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M\right)}}\right) \]

   3. Taylor expanded in c0 around inf 63.0

      \[\leadsto \color{blue}{\frac{{d}^{2} \cdot {c0}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)}} \]

   4. Simplified 56.2

      \[\leadsto \color{blue}{\frac{{\left(d \cdot c0\right)}^{2}}{h \cdot {\left(w \cdot D\right)}^{2}}} \]
      Proof

      [Start] 63.0	\[ \frac{{d}^{2} \cdot {c0}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)} \]
      exponential.json-simplify-27 [=>] 62.4	\[ \frac{\color{blue}{{\left(d \cdot c0\right)}^{2}}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)} \]
      rational.json-simplify-2 [=>] 62.4	\[ \frac{{\left(d \cdot c0\right)}^{2}}{{D}^{2} \cdot \color{blue}{\left(h \cdot {w}^{2}\right)}} \]
      rational.json-simplify-43 [=>] 62.4	\[ \frac{{\left(d \cdot c0\right)}^{2}}{\color{blue}{h \cdot \left({w}^{2} \cdot {D}^{2}\right)}} \]
      exponential.json-simplify-27 [=>] 56.2	\[ \frac{{\left(d \cdot c0\right)}^{2}}{h \cdot \color{blue}{{\left(w \cdot D\right)}^{2}}} \]

   5. Applied egg-rr 55.3

      \[\leadsto \color{blue}{\frac{{\left(d \cdot c0\right)}^{2}}{{\left(w \cdot D\right)}^{2}} \cdot \frac{1}{h}} \]

   if 2.0999999999999999e-108 < D < 2.35000000000000006e48

   1. Initial program 55.3

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Simplified 55.1

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \frac{c0}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)} + \sqrt{\left(\left(d \cdot d\right) \cdot \frac{c0}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)}\right) \cdot \left(\left(d \cdot d\right) \cdot \frac{c0}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)}\right) - M \cdot M}\right)} \]
      Proof

      [Start] 55.3	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      rational.json-simplify-49 [=>] 56.2	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(d \cdot d\right) \cdot \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      rational.json-simplify-2 [=>] 56.2	\[ \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \frac{c0}{\color{blue}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      rational.json-simplify-2 [=>] 56.2	\[ \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \frac{c0}{\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot w\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      rational.json-simplify-43 [=>] 57.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \frac{c0}{\color{blue}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Applied egg-rr 52.7

      \[\leadsto \color{blue}{\frac{d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}} + \sqrt{\left(d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}}\right) \cdot \left(d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}}\right) - M \cdot M}}{w \cdot \frac{2}{c0}}} \]

   4. Taylor expanded in d around inf 55.5

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}}}{w \cdot \frac{2}{c0}} \]

   5. Simplified 55.9

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{\frac{{d}^{2} \cdot c0}{{D}^{2}}}{w \cdot h}}}{w \cdot \frac{2}{c0}} \]
      Proof

      [Start] 55.5	\[ \frac{2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}}{w \cdot \frac{2}{c0}} \]
      rational.json-simplify-46 [=>] 55.9	\[ \frac{2 \cdot \color{blue}{\frac{\frac{{d}^{2} \cdot c0}{{D}^{2}}}{w \cdot h}}}{w \cdot \frac{2}{c0}} \]

   6. Applied egg-rr 54.3

      \[\leadsto \color{blue}{\frac{{d}^{2} \cdot \frac{c0}{w \cdot h}}{{D}^{2} \cdot \frac{w + w}{c0}} + \frac{{d}^{2} \cdot \frac{c0}{w \cdot h}}{{D}^{2} \cdot \frac{w + w}{c0}}} \]

   7. Simplified 55.6

      \[\leadsto \color{blue}{2 \cdot \left(\frac{{d}^{2}}{{D}^{2}} \cdot \left(\frac{\frac{c0}{w}}{h} \cdot \frac{c0}{w + w}\right)\right)} \]
      Proof

      [Start] 54.3	\[ \frac{{d}^{2} \cdot \frac{c0}{w \cdot h}}{{D}^{2} \cdot \frac{w + w}{c0}} + \frac{{d}^{2} \cdot \frac{c0}{w \cdot h}}{{D}^{2} \cdot \frac{w + w}{c0}} \]
      rational.json-simplify-49 [=>] 54.7	\[ \color{blue}{\frac{c0}{w \cdot h} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \frac{w + w}{c0}}} + \frac{{d}^{2} \cdot \frac{c0}{w \cdot h}}{{D}^{2} \cdot \frac{w + w}{c0}} \]
      rational.json-simplify-2 [=>] 54.7	\[ \color{blue}{\frac{{d}^{2}}{{D}^{2} \cdot \frac{w + w}{c0}} \cdot \frac{c0}{w \cdot h}} + \frac{{d}^{2} \cdot \frac{c0}{w \cdot h}}{{D}^{2} \cdot \frac{w + w}{c0}} \]
      rational.json-simplify-49 [=>] 54.5	\[ \frac{{d}^{2}}{{D}^{2} \cdot \frac{w + w}{c0}} \cdot \frac{c0}{w \cdot h} + \color{blue}{\frac{c0}{w \cdot h} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \frac{w + w}{c0}}} \]
      rational.json-simplify-51 [=>] 54.5	\[ \color{blue}{\frac{{d}^{2}}{{D}^{2} \cdot \frac{w + w}{c0}} \cdot \left(\frac{c0}{w \cdot h} + \frac{c0}{w \cdot h}\right)} \]
      rational.json-simplify-7 [<=] 54.5	\[ \frac{{d}^{2}}{{D}^{2} \cdot \frac{w + w}{c0}} \cdot \left(\frac{c0}{w \cdot h} + \color{blue}{\frac{\frac{c0}{w \cdot h}}{1}}\right) \]
      rational.json-simplify-30 [<=] 54.5	\[ \frac{{d}^{2}}{{D}^{2} \cdot \frac{w + w}{c0}} \cdot \color{blue}{\left(\left(1 + 1\right) \cdot \frac{\frac{c0}{w \cdot h}}{1}\right)} \]
      metadata-eval [=>] 54.5	\[ \frac{{d}^{2}}{{D}^{2} \cdot \frac{w + w}{c0}} \cdot \left(\color{blue}{2} \cdot \frac{\frac{c0}{w \cdot h}}{1}\right) \]
      rational.json-simplify-7 [=>] 54.5	\[ \frac{{d}^{2}}{{D}^{2} \cdot \frac{w + w}{c0}} \cdot \left(2 \cdot \color{blue}{\frac{c0}{w \cdot h}}\right) \]
      rational.json-simplify-43 [<=] 54.5	\[ \color{blue}{\frac{c0}{w \cdot h} \cdot \left(\frac{{d}^{2}}{{D}^{2} \cdot \frac{w + w}{c0}} \cdot 2\right)} \]
      rational.json-simplify-43 [<=] 54.5	\[ \color{blue}{2 \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \frac{w + w}{c0}}\right)} \]
      rational.json-simplify-49 [<=] 54.3	\[ 2 \cdot \color{blue}{\frac{{d}^{2} \cdot \frac{c0}{w \cdot h}}{{D}^{2} \cdot \frac{w + w}{c0}}} \]

   if 2.35000000000000006e48 < D < 7.5e157

   1. Initial program 56.7

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Simplified 55.9

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \frac{c0}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)} + \sqrt{\left(\left(d \cdot d\right) \cdot \frac{c0}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)}\right) \cdot \left(\left(d \cdot d\right) \cdot \frac{c0}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)}\right) - M \cdot M}\right)} \]
      Proof

      [Start] 56.7	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      rational.json-simplify-49 [=>] 57.5	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(d \cdot d\right) \cdot \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      rational.json-simplify-2 [=>] 57.5	\[ \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \frac{c0}{\color{blue}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      rational.json-simplify-2 [=>] 57.5	\[ \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \frac{c0}{\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot w\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      rational.json-simplify-43 [=>] 58.5	\[ \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \frac{c0}{\color{blue}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Applied egg-rr 50.5

      \[\leadsto \color{blue}{\frac{d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}} + \sqrt{\left(d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}}\right) \cdot \left(d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}}\right) - M \cdot M}}{w \cdot \frac{2}{c0}}} \]

   4. Applied egg-rr 52.4

      \[\leadsto \frac{d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}} + \sqrt{\left(d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}}\right) \cdot \left(d \cdot \frac{d}{\color{blue}{\left(h \cdot \frac{w}{c0}\right) \cdot \left(D \cdot D\right)}}\right) - M \cdot M}}{w \cdot \frac{2}{c0}} \]

   if 7.5e157 < D

   1. Initial program 63.0

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Simplified 61.4

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \frac{c0}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)} + \sqrt{\left(\left(d \cdot d\right) \cdot \frac{c0}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)}\right) \cdot \left(\left(d \cdot d\right) \cdot \frac{c0}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)}\right) - M \cdot M}\right)} \]
      Proof

      [Start] 63.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      rational.json-simplify-49 [=>] 63.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(d \cdot d\right) \cdot \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      rational.json-simplify-2 [=>] 63.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \frac{c0}{\color{blue}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      rational.json-simplify-2 [=>] 63.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \frac{c0}{\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot w\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      rational.json-simplify-43 [=>] 63.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \frac{c0}{\color{blue}{h \cdot \left(w \cdot \left(D \cdot D\right)\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Applied egg-rr 52.3

      \[\leadsto \color{blue}{\frac{d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}} + \sqrt{\left(d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}}\right) \cdot \left(d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}}\right) - M \cdot M}}{w \cdot \frac{2}{c0}}} \]

   4. Simplified 49.5

      \[\leadsto \color{blue}{\frac{c0}{\frac{2}{\frac{d \cdot \frac{d}{D \cdot \frac{w}{\frac{c0}{D \cdot h}}} + \sqrt{\frac{d}{D \cdot \frac{w}{\frac{c0}{D \cdot h}}} \cdot \left(\left(d \cdot \frac{d}{D \cdot \frac{w}{\frac{c0}{D \cdot h}}}\right) \cdot d\right) - M \cdot M}}{w}}}} \]
      Proof

      [Start] 52.3	\[ \frac{d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}} + \sqrt{\left(d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}}\right) \cdot \left(d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}}\right) - M \cdot M}}{w \cdot \frac{2}{c0}} \]
      rational.json-simplify-46 [=>] 52.3	\[ \color{blue}{\frac{\frac{d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}} + \sqrt{\left(d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}}\right) \cdot \left(d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}}\right) - M \cdot M}}{w}}{\frac{2}{c0}}} \]
      rational.json-simplify-61 [=>] 52.3	\[ \color{blue}{\frac{c0}{\frac{2}{\frac{d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}} + \sqrt{\left(d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}}\right) \cdot \left(d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}}\right) - M \cdot M}}{w}}}} \]

   5. Applied egg-rr 46.6

      \[\leadsto \frac{c0}{\frac{2}{\frac{\color{blue}{\left(d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right) + \sqrt{\left(d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right) + M\right) \cdot \left(d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right) - M\right)}\right) - 0}}{w}}} \]

   6. Simplified 45.9

      \[\leadsto \frac{c0}{\frac{2}{\frac{\color{blue}{\frac{d}{D} \cdot \left(d \cdot \frac{\frac{c0}{w}}{D \cdot h}\right) + \sqrt{\left(\frac{d}{D} \cdot \left(d \cdot \frac{\frac{c0}{w}}{D \cdot h}\right)\right) \cdot \left(\frac{d}{D} \cdot \left(d \cdot \frac{\frac{c0}{w}}{D \cdot h}\right)\right) - M \cdot M}}}{w}}} \]
      Proof

      [Start] 46.6	\[ \frac{c0}{\frac{2}{\frac{\left(d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right) + \sqrt{\left(d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right) + M\right) \cdot \left(d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right) - M\right)}\right) - 0}{w}}} \]
      rational.json-simplify-5 [=>] 46.6	\[ \frac{c0}{\frac{2}{\frac{\color{blue}{d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right) + \sqrt{\left(d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right) + M\right) \cdot \left(d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right) - M\right)}}}{w}}} \]
      rational.json-simplify-6 [<=] 46.6	\[ \frac{c0}{\frac{2}{\frac{d \cdot \color{blue}{\left(1 \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right)\right)} + \sqrt{\left(d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right) + M\right) \cdot \left(d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right) - M\right)}}{w}}} \]
      rational.json-simplify-43 [=>] 46.6	\[ \frac{c0}{\frac{2}{\frac{d \cdot \color{blue}{\left(\frac{d}{D} \cdot \left(\frac{\frac{\frac{c0}{w}}{D}}{h} \cdot 1\right)\right)} + \sqrt{\left(d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right) + M\right) \cdot \left(d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right) - M\right)}}{w}}} \]
      rational.json-simplify-43 [=>] 47.7	\[ \frac{c0}{\frac{2}{\frac{\color{blue}{\frac{d}{D} \cdot \left(\left(\frac{\frac{\frac{c0}{w}}{D}}{h} \cdot 1\right) \cdot d\right)} + \sqrt{\left(d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right) + M\right) \cdot \left(d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right) - M\right)}}{w}}} \]
      rational.json-simplify-2 [=>] 47.7	\[ \frac{c0}{\frac{2}{\frac{\frac{d}{D} \cdot \color{blue}{\left(d \cdot \left(\frac{\frac{\frac{c0}{w}}{D}}{h} \cdot 1\right)\right)} + \sqrt{\left(d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right) + M\right) \cdot \left(d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right) - M\right)}}{w}}} \]
      rational.json-simplify-2 [=>] 47.7	\[ \frac{c0}{\frac{2}{\frac{\frac{d}{D} \cdot \left(d \cdot \color{blue}{\left(1 \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right)}\right) + \sqrt{\left(d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right) + M\right) \cdot \left(d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right) - M\right)}}{w}}} \]
      rational.json-simplify-6 [=>] 47.7	\[ \frac{c0}{\frac{2}{\frac{\frac{d}{D} \cdot \left(d \cdot \color{blue}{\frac{\frac{\frac{c0}{w}}{D}}{h}}\right) + \sqrt{\left(d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right) + M\right) \cdot \left(d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right) - M\right)}}{w}}} \]
      rational.json-simplify-47 [=>] 52.1	\[ \frac{c0}{\frac{2}{\frac{\frac{d}{D} \cdot \left(d \cdot \color{blue}{\frac{\frac{c0}{w}}{D \cdot h}}\right) + \sqrt{\left(d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right) + M\right) \cdot \left(d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right) - M\right)}}{w}}} \]
      rational.json-simplify-1 [=>] 52.1	\[ \frac{c0}{\frac{2}{\frac{\frac{d}{D} \cdot \left(d \cdot \frac{\frac{c0}{w}}{D \cdot h}\right) + \sqrt{\color{blue}{\left(M + d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right)\right)} \cdot \left(d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right) - M\right)}}{w}}} \]

3. Recombined 5 regimes into one program.

4. Final simplification 53.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq -2.1 \cdot 10^{-190}:\\ \;\;\;\;\frac{c0}{\frac{2}{\frac{\frac{d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right) + \sqrt{\left(d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right) + M\right) \cdot \left(d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right) - M\right)}}{1}}{w}}}\\ \mathbf{elif}\;D \leq 2.1 \cdot 10^{-108}:\\ \;\;\;\;\frac{{\left(d \cdot c0\right)}^{2}}{{\left(w \cdot D\right)}^{2}} \cdot \frac{1}{h}\\ \mathbf{elif}\;D \leq 2.35 \cdot 10^{+48}:\\ \;\;\;\;2 \cdot \left(\frac{{d}^{2}}{{D}^{2}} \cdot \left(\frac{\frac{c0}{w}}{h} \cdot \frac{c0}{w + w}\right)\right)\\ \mathbf{elif}\;D \leq 7.5 \cdot 10^{+157}:\\ \;\;\;\;\frac{d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}} + \sqrt{\left(d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}}\right) \cdot \left(d \cdot \frac{d}{\left(h \cdot \frac{w}{c0}\right) \cdot \left(D \cdot D\right)}\right) - M \cdot M}}{w \cdot \frac{2}{c0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{2}{\frac{\frac{d}{D} \cdot \left(d \cdot \frac{\frac{c0}{w}}{D \cdot h}\right) + \sqrt{\left(\frac{d}{D} \cdot \left(d \cdot \frac{\frac{c0}{w}}{D \cdot h}\right)\right) \cdot \left(\frac{d}{D} \cdot \left(d \cdot \frac{\frac{c0}{w}}{D \cdot h}\right)\right) - M \cdot M}}{w}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	53.2
Cost	14348

\[\begin{array}{l} t_0 := {\left(w \cdot D\right)}^{2}\\ t_1 := h \cdot t_0\\ t_2 := {\left(d \cdot c0\right)}^{2}\\ t_3 := \frac{d}{D} \cdot \left(d \cdot \frac{\frac{c0}{w}}{D \cdot h}\right)\\ \mathbf{if}\;w \leq -4.2 \cdot 10^{+123}:\\ \;\;\;\;\frac{t_2}{t_1}\\ \mathbf{elif}\;w \leq 6.2 \cdot 10^{+51}:\\ \;\;\;\;\frac{c0}{\frac{2}{\frac{t_3 + \sqrt{t_3 \cdot t_3 - M \cdot M}}{w}}}\\ \mathbf{elif}\;w \leq 4.5 \cdot 10^{+145}:\\ \;\;\;\;\left({d}^{2} \cdot \frac{c0}{t_1}\right) \cdot \left(2 \cdot \frac{c0}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \left(1 - \left(-t_2\right)\right)}{h} \cdot \frac{1}{t_0}\\ \end{array} \]

Alternative 2
Error	52.8
Cost	14284

\[\begin{array}{l} t_0 := {\left(w \cdot D\right)}^{2}\\ t_1 := d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right)\\ t_2 := {\left(d \cdot c0\right)}^{2}\\ t_3 := \frac{d}{D} \cdot \left(d \cdot \frac{\frac{c0}{w}}{D \cdot h}\right)\\ \mathbf{if}\;w \leq -3.8 \cdot 10^{+129}:\\ \;\;\;\;\frac{t_2}{h \cdot t_0}\\ \mathbf{elif}\;w \leq 7.8 \cdot 10^{+51}:\\ \;\;\;\;\frac{c0}{\frac{2}{\frac{t_3 + \sqrt{t_3 \cdot t_3 - M \cdot M}}{w}}}\\ \mathbf{elif}\;w \leq 2.5 \cdot 10^{+214}:\\ \;\;\;\;\frac{\frac{\frac{1}{t_0}}{2}}{\frac{\frac{h}{t_2}}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{2}{\frac{\frac{t_1 + \sqrt{\left(t_1 + M\right) \cdot \left(t_1 - M\right)}}{1}}{w}}}\\ \end{array} \]

Alternative 3
Error	52.7
Cost	14028

\[\begin{array}{l} t_0 := {\left(w \cdot D\right)}^{2}\\ t_1 := d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right)\\ t_2 := {\left(d \cdot c0\right)}^{2}\\ t_3 := \frac{d}{D} \cdot \left(d \cdot \frac{\frac{c0}{w}}{D \cdot h}\right)\\ \mathbf{if}\;w \leq -1.15 \cdot 10^{+124}:\\ \;\;\;\;\frac{t_2}{h \cdot t_0}\\ \mathbf{elif}\;w \leq 9 \cdot 10^{+81}:\\ \;\;\;\;\frac{c0}{\frac{2}{\frac{t_3 + \sqrt{t_3 \cdot t_3 - M \cdot M}}{w}}}\\ \mathbf{elif}\;w \leq 2.3 \cdot 10^{+216}:\\ \;\;\;\;\frac{\frac{1}{h}}{\frac{t_0}{t_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{2}{\frac{\frac{t_1 + \sqrt{\left(t_1 + M\right) \cdot \left(t_1 - M\right)}}{1}}{w}}}\\ \end{array} \]

Alternative 4
Error	52.8
Cost	14028

\[\begin{array}{l} t_0 := {\left(w \cdot D\right)}^{2}\\ t_1 := d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right)\\ t_2 := {\left(d \cdot c0\right)}^{2}\\ t_3 := \frac{d}{D} \cdot \left(d \cdot \frac{\frac{c0}{w}}{D \cdot h}\right)\\ \mathbf{if}\;w \leq -4.4 \cdot 10^{+123}:\\ \;\;\;\;\frac{t_2}{h \cdot t_0}\\ \mathbf{elif}\;w \leq 7.5 \cdot 10^{+51}:\\ \;\;\;\;\frac{c0}{\frac{2}{\frac{t_3 + \sqrt{t_3 \cdot t_3 - M \cdot M}}{w}}}\\ \mathbf{elif}\;w \leq 7.5 \cdot 10^{+213}:\\ \;\;\;\;\frac{\frac{1}{t_0}}{\frac{h}{t_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{2}{\frac{\frac{t_1 + \sqrt{\left(t_1 + M\right) \cdot \left(t_1 - M\right)}}{1}}{w}}}\\ \end{array} \]

Alternative 5
Error	53.0
Cost	13900

\[\begin{array}{l} t_0 := \frac{\frac{{\left(d \cdot c0\right)}^{2}}{h}}{{\left(w \cdot D\right)}^{2}}\\ t_1 := \frac{d}{D} \cdot \left(d \cdot \frac{\frac{c0}{w}}{D \cdot h}\right)\\ t_2 := d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right)\\ \mathbf{if}\;w \leq -9.5 \cdot 10^{+129}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;w \leq 4.4 \cdot 10^{+80}:\\ \;\;\;\;\frac{c0}{\frac{2}{\frac{t_1 + \sqrt{t_1 \cdot t_1 - M \cdot M}}{w}}}\\ \mathbf{elif}\;w \leq 2.4 \cdot 10^{+216}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{2}{\frac{\frac{t_2 + \sqrt{\left(t_2 + M\right) \cdot \left(t_2 - M\right)}}{1}}{w}}}\\ \end{array} \]

Alternative 6
Error	52.6
Cost	13900

\[\begin{array}{l} t_0 := {\left(w \cdot D\right)}^{2}\\ t_1 := d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right)\\ t_2 := {\left(d \cdot c0\right)}^{2}\\ t_3 := \frac{d}{D} \cdot \left(d \cdot \frac{\frac{c0}{w}}{D \cdot h}\right)\\ \mathbf{if}\;w \leq -2.3 \cdot 10^{+56}:\\ \;\;\;\;\frac{\frac{t_2}{t_0}}{h}\\ \mathbf{elif}\;w \leq 3.6 \cdot 10^{+84}:\\ \;\;\;\;\frac{c0}{\frac{2}{\frac{t_3 + \sqrt{t_3 \cdot t_3 - M \cdot M}}{w}}}\\ \mathbf{elif}\;w \leq 6.5 \cdot 10^{+213}:\\ \;\;\;\;\frac{\frac{t_2}{h}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{2}{\frac{\frac{t_1 + \sqrt{\left(t_1 + M\right) \cdot \left(t_1 - M\right)}}{1}}{w}}}\\ \end{array} \]

Alternative 7
Error	52.9
Cost	13900

\[\begin{array}{l} t_0 := d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right)\\ t_1 := {\left(w \cdot D\right)}^{2}\\ t_2 := {\left(d \cdot c0\right)}^{2}\\ t_3 := \frac{d}{D} \cdot \left(d \cdot \frac{\frac{c0}{w}}{D \cdot h}\right)\\ \mathbf{if}\;w \leq -1.1 \cdot 10^{+130}:\\ \;\;\;\;\frac{t_2}{h \cdot t_1}\\ \mathbf{elif}\;w \leq 1.5 \cdot 10^{+82}:\\ \;\;\;\;\frac{c0}{\frac{2}{\frac{t_3 + \sqrt{t_3 \cdot t_3 - M \cdot M}}{w}}}\\ \mathbf{elif}\;w \leq 1.15 \cdot 10^{+212}:\\ \;\;\;\;\frac{\frac{t_2}{h}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{2}{\frac{\frac{t_0 + \sqrt{\left(t_0 + M\right) \cdot \left(t_0 - M\right)}}{1}}{w}}}\\ \end{array} \]

Alternative 8
Error	54.4
Cost	10964

\[\begin{array}{l} t_0 := \frac{d}{D} \cdot \left(d \cdot \frac{\frac{c0}{w}}{D \cdot h}\right)\\ t_1 := c0 \cdot \frac{\frac{d \cdot d}{w \cdot h}}{D \cdot D}\\ t_2 := \frac{c0}{D \cdot D} \cdot \frac{d}{w \cdot \frac{h}{d}}\\ t_3 := \frac{c0}{\frac{2}{\frac{t_0 + \sqrt{t_0 \cdot t_0 - M \cdot M}}{w}}}\\ t_4 := \frac{d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \left(d \cdot c0\right)\\ \mathbf{if}\;D \cdot D \leq 5 \cdot 10^{-264}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;D \cdot D \leq 10^{-108}:\\ \;\;\;\;0.5 \cdot \left(c0 \cdot \frac{t_2 + \sqrt{t_2 \cdot t_2 - M \cdot M}}{w}\right)\\ \mathbf{elif}\;D \cdot D \leq 2000:\\ \;\;\;\;\frac{d \cdot \frac{d}{D \cdot \left(\frac{D}{\frac{c0}{w}} \cdot h\right)} + \sqrt{\left(d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}}\right) \cdot \left(d \cdot \frac{d}{\left(h \cdot \frac{w}{c0}\right) \cdot \left(D \cdot D\right)}\right) - M \cdot M}}{w \cdot \frac{2}{c0}}\\ \mathbf{elif}\;D \cdot D \leq 2 \cdot 10^{+22}:\\ \;\;\;\;\frac{c0}{\frac{w}{\left(t_4 + \sqrt{t_4 \cdot t_4 - M \cdot M}\right) \cdot 0.5}}\\ \mathbf{elif}\;D \cdot D \leq 5 \cdot 10^{+96}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \left(\frac{\frac{d}{w \cdot h}}{D} \cdot \frac{d}{D}\right) + \sqrt{\left(t_1 + M\right) \cdot \left(t_1 - M\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 9
Error	54.0
Cost	10964

\[\begin{array}{l} t_0 := \frac{d}{D} \cdot \left(d \cdot \frac{\frac{c0}{w}}{D \cdot h}\right)\\ t_1 := w \cdot \frac{2}{c0}\\ t_2 := \frac{c0}{D \cdot D} \cdot \frac{d}{w \cdot \frac{h}{d}}\\ t_3 := \frac{c0}{\frac{2}{\frac{t_0 + \sqrt{t_0 \cdot t_0 - M \cdot M}}{w}}}\\ t_4 := d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}}\\ t_5 := \frac{d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \left(d \cdot c0\right)\\ \mathbf{if}\;D \cdot D \leq 5 \cdot 10^{-264}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;D \cdot D \leq 10^{-108}:\\ \;\;\;\;0.5 \cdot \left(c0 \cdot \frac{t_2 + \sqrt{t_2 \cdot t_2 - M \cdot M}}{w}\right)\\ \mathbf{elif}\;D \cdot D \leq 2000:\\ \;\;\;\;\frac{d \cdot \frac{d}{D \cdot \left(\frac{D}{\frac{c0}{w}} \cdot h\right)} + \sqrt{t_4 \cdot \left(d \cdot \frac{d}{\left(h \cdot \frac{w}{c0}\right) \cdot \left(D \cdot D\right)}\right) - M \cdot M}}{t_1}\\ \mathbf{elif}\;D \cdot D \leq 2 \cdot 10^{+22}:\\ \;\;\;\;\frac{c0}{\frac{w}{\left(t_5 + \sqrt{t_5 \cdot t_5 - M \cdot M}\right) \cdot 0.5}}\\ \mathbf{elif}\;D \cdot D \leq 5 \cdot 10^{+141}:\\ \;\;\;\;\frac{t_4 + \sqrt{t_4 \cdot t_4 - M \cdot M}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 10
Error	53.6
Cost	10964

\[\begin{array}{l} t_0 := \frac{d}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \left(d \cdot c0\right)\\ t_1 := d \cdot \left(\frac{d}{D} \cdot \frac{\frac{\frac{c0}{w}}{D}}{h}\right)\\ t_2 := \frac{d}{D} \cdot \left(d \cdot \frac{\frac{c0}{w}}{D \cdot h}\right)\\ t_3 := w \cdot \frac{2}{c0}\\ t_4 := \frac{c0}{D \cdot D} \cdot \frac{d}{w \cdot \frac{h}{d}}\\ t_5 := d \cdot \frac{d}{\frac{h \cdot \left(D \cdot D\right)}{\frac{c0}{w}}}\\ \mathbf{if}\;D \cdot D \leq 5 \cdot 10^{-264}:\\ \;\;\;\;\frac{c0}{\frac{2}{\frac{\frac{t_1 + \sqrt{\left(t_1 + M\right) \cdot \left(t_1 - M\right)}}{1}}{w}}}\\ \mathbf{elif}\;D \cdot D \leq 10^{-108}:\\ \;\;\;\;0.5 \cdot \left(c0 \cdot \frac{t_4 + \sqrt{t_4 \cdot t_4 - M \cdot M}}{w}\right)\\ \mathbf{elif}\;D \cdot D \leq 2000:\\ \;\;\;\;\frac{d \cdot \frac{d}{D \cdot \left(\frac{D}{\frac{c0}{w}} \cdot h\right)} + \sqrt{t_5 \cdot \left(d \cdot \frac{d}{\left(h \cdot \frac{w}{c0}\right) \cdot \left(D \cdot D\right)}\right) - M \cdot M}}{t_3}\\ \mathbf{elif}\;D \cdot D \leq 2 \cdot 10^{+22}:\\ \;\;\;\;\frac{c0}{\frac{w}{\left(t_0 + \sqrt{t_0 \cdot t_0 - M \cdot M}\right) \cdot 0.5}}\\ \mathbf{elif}\;D \cdot D \leq 5 \cdot 10^{+141}:\\ \;\;\;\;\frac{t_5 + \sqrt{t_5 \cdot t_5 - M \cdot M}}{t_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{2}{\frac{t_2 + \sqrt{t_2 \cdot t_2 - M \cdot M}}{w}}}\\ \end{array} \]

Alternative 11
Error	55.5
Cost	10184

\[\begin{array}{l} t_0 := d \cdot \frac{\frac{c0}{D \cdot \left(w \cdot h\right)}}{D}\\ t_1 := \frac{c0}{D \cdot D} \cdot \frac{d}{w \cdot \frac{h}{d}}\\ t_2 := \frac{c0}{\frac{2}{\frac{d \cdot t_0 + \sqrt{d \cdot \left(d \cdot \left(t_0 \cdot t_0\right)\right) - M \cdot M}}{w}}}\\ \mathbf{if}\;D \cdot D \leq 0:\\ \;\;\;\;t_2\\ \mathbf{elif}\;D \cdot D \leq 5 \cdot 10^{+141}:\\ \;\;\;\;0.5 \cdot \left(c0 \cdot \frac{t_1 + \sqrt{t_1 \cdot t_1 - M \cdot M}}{w}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 12
Error	53.7
Cost	10184

\[\begin{array}{l} t_0 := d \cdot \left(\frac{d}{D} \cdot \frac{c0}{h \cdot \left(D \cdot w\right)}\right)\\ t_1 := \frac{c0}{D \cdot D} \cdot \frac{d}{w \cdot \frac{h}{d}}\\ t_2 := \frac{c0}{\frac{2}{\frac{t_0 + \sqrt{t_0 \cdot t_0 - M \cdot M}}{w}}}\\ \mathbf{if}\;D \cdot D \leq 5 \cdot 10^{-264}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;D \cdot D \leq 5 \cdot 10^{+141}:\\ \;\;\;\;0.5 \cdot \left(c0 \cdot \frac{t_1 + \sqrt{t_1 \cdot t_1 - M \cdot M}}{w}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 13
Error	54.5
Cost	10184

\[\begin{array}{l} t_0 := d \cdot \left(\frac{d}{D} \cdot \frac{c0}{h \cdot \left(D \cdot w\right)}\right)\\ t_1 := \frac{d}{D \cdot \frac{w}{\frac{c0}{D \cdot h}}}\\ t_2 := d \cdot t_1\\ t_3 := \frac{c0}{D \cdot D} \cdot \frac{d}{w \cdot \frac{h}{d}}\\ \mathbf{if}\;D \cdot D \leq 5 \cdot 10^{-264}:\\ \;\;\;\;\frac{c0}{\frac{2}{\frac{t_0 + \sqrt{t_0 \cdot t_0 - M \cdot M}}{w}}}\\ \mathbf{elif}\;D \cdot D \leq 10^{-75}:\\ \;\;\;\;0.5 \cdot \left(c0 \cdot \frac{t_3 + \sqrt{t_3 \cdot t_3 - M \cdot M}}{w}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{2}{\frac{t_2 + \sqrt{t_1 \cdot \left(t_2 \cdot d\right) - M \cdot M}}{w}}}\\ \end{array} \]

Alternative 14
Error	53.8
Cost	10184

\[\begin{array}{l} t_0 := \frac{d}{D} \cdot \left(d \cdot \frac{\frac{c0}{w}}{D \cdot h}\right)\\ t_1 := \frac{c0}{D \cdot D} \cdot \frac{d}{w \cdot \frac{h}{d}}\\ t_2 := \frac{c0}{\frac{2}{\frac{t_0 + \sqrt{t_0 \cdot t_0 - M \cdot M}}{w}}}\\ \mathbf{if}\;D \cdot D \leq 5 \cdot 10^{-264}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;D \cdot D \leq 10^{-108}:\\ \;\;\;\;0.5 \cdot \left(c0 \cdot \frac{t_1 + \sqrt{t_1 \cdot t_1 - M \cdot M}}{w}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 15
Error	57.9
Cost	9924

\[\begin{array}{l} t_0 := \frac{d}{w \cdot h}\\ t_1 := \frac{c0}{D \cdot D}\\ t_2 := t_1 \cdot \frac{d}{w \cdot \frac{h}{d}}\\ t_3 := t_0 \cdot \left(d \cdot t_1\right)\\ \mathbf{if}\;M \cdot M \leq 2 \cdot 10^{-264}:\\ \;\;\;\;0.5 \cdot \frac{t_3 + \sqrt{t_0 \cdot \left(t_1 \cdot \left(d \cdot t_3\right)\right) - M \cdot M}}{\frac{w}{c0}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(c0 \cdot \frac{t_2 + \sqrt{t_2 \cdot t_2 - M \cdot M}}{w}\right)\\ \end{array} \]

Alternative 16
Error	57.8
Cost	9664

\[\begin{array}{l} t_0 := \frac{c0}{D \cdot D} \cdot \frac{d}{w \cdot \frac{h}{d}}\\ 0.5 \cdot \left(c0 \cdot \frac{t_0 + \sqrt{t_0 \cdot t_0 - M \cdot M}}{w}\right) \end{array} \]

Alternative 17
Error	64.0
Cost	6976

\[\frac{c0}{2 \cdot w} \cdot \left(\sqrt{-1} \cdot M\right) \]

Reproduce

herbie shell --seed 2023075 
(FPCore (c0 w h D d M)
  :name "Henrywood and Agarwal, Equation (13)"
  :precision binary64
  (* (/ c0 (* 2.0 w)) (+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))))