Henrywood and Agarwal, Equation (13)

Average Error: 59.7 → 20.8

Time: 32.8s

Precision: binary64

Cost: 7888

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c0 \leq 2.7 \cdot 10^{-266}:\\ \;\;\;\;0.25 \cdot \frac{\frac{D}{d} \cdot \left(h \cdot M\right)}{\frac{\frac{d}{D}}{M}}\\ \mathbf{elif}\;c0 \leq 9 \cdot 10^{-216}:\\ \;\;\;\;\frac{{\left(c0 \cdot d\right)}^{2}}{h \cdot \left(\left(D \cdot w\right) \cdot \left(D \cdot w\right)\right)}\\ \mathbf{elif}\;c0 \leq 3.7 \cdot 10^{-175}:\\ \;\;\;\;\mathsf{fma}\left(0.25, h \cdot \frac{\frac{D}{\frac{d}{M}}}{\frac{d}{D \cdot M}}, 0\right)\\ \mathbf{elif}\;c0 \leq 3.1 \cdot 10^{-146}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D \cdot h}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.25, \frac{\frac{D}{d} \cdot M}{\frac{\frac{d}{D}}{h \cdot M}}, 0\right)\\ \end{array} \]

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
 (if (<= c0 2.7e-266)
   (* 0.25 (/ (* (/ D d) (* h M)) (/ (/ d D) M)))
   (if (<= c0 9e-216)
     (/ (pow (* c0 d) 2.0) (* h (* (* D w) (* D w))))
     (if (<= c0 3.7e-175)
       (fma 0.25 (* h (/ (/ D (/ d M)) (/ d (* D M)))) 0.0)
       (if (<= c0 3.1e-146)
         (* (/ c0 (* 2.0 w)) (* 2.0 (* (/ c0 w) (* (/ d D) (/ d (* D h))))))
         (fma 0.25 (/ (* (/ D d) M) (/ (/ d D) (* h M))) 0.0))))))

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 tmp;
	if (c0 <= 2.7e-266) {
		tmp = 0.25 * (((D / d) * (h * M)) / ((d / D) / M));
	} else if (c0 <= 9e-216) {
		tmp = pow((c0 * d), 2.0) / (h * ((D * w) * (D * w)));
	} else if (c0 <= 3.7e-175) {
		tmp = fma(0.25, (h * ((D / (d / M)) / (d / (D * M)))), 0.0);
	} else if (c0 <= 3.1e-146) {
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / w) * ((d / D) * (d / (D * h)))));
	} else {
		tmp = fma(0.25, (((D / d) * M) / ((d / D) / (h * M))), 0.0);
	}
	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)
	tmp = 0.0
	if (c0 <= 2.7e-266)
		tmp = Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(h * M)) / Float64(Float64(d / D) / M)));
	elseif (c0 <= 9e-216)
		tmp = Float64((Float64(c0 * d) ^ 2.0) / Float64(h * Float64(Float64(D * w) * Float64(D * w))));
	elseif (c0 <= 3.7e-175)
		tmp = fma(0.25, Float64(h * Float64(Float64(D / Float64(d / M)) / Float64(d / Float64(D * M)))), 0.0);
	elseif (c0 <= 3.1e-146)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 * Float64(Float64(c0 / w) * Float64(Float64(d / D) * Float64(d / Float64(D * h))))));
	else
		tmp = fma(0.25, Float64(Float64(Float64(D / d) * M) / Float64(Float64(d / D) / Float64(h * M))), 0.0);
	end
	return 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_] := If[LessEqual[c0, 2.7e-266], N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(h * M), $MachinePrecision]), $MachinePrecision] / N[(N[(d / D), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, 9e-216], N[(N[Power[N[(c0 * d), $MachinePrecision], 2.0], $MachinePrecision] / N[(h * N[(N[(D * w), $MachinePrecision] * N[(D * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, 3.7e-175], N[(0.25 * N[(h * N[(N[(D / N[(d / M), $MachinePrecision]), $MachinePrecision] / N[(d / N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision], If[LessEqual[c0, 3.1e-146], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(c0 / w), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * N[(d / N[(D * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * M), $MachinePrecision] / N[(N[(d / D), $MachinePrecision] / N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if c0 < 2.69999999999999996e-266

   1. Initial program 59.8

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

   2. Taylor expanded in c0 around -inf 60.2

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right)} \]

   3. Simplified 58.5

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{D}{d \cdot \frac{d}{D}} \cdot \frac{w}{\frac{c0}{h \cdot \left(M \cdot M\right)}}, c0 \cdot \left(-\left(\frac{\frac{d}{h}}{D \cdot D} \cdot \frac{d}{w}\right) \cdot 0\right)\right)} \]
      Proof

      [Start] 60.2	\[ \frac{c0}{2 \cdot w} \cdot \left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right) \]
      fma-def [=>] 60.2	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0}, -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right)} \]
      times-frac [=>] 60.0	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \color{blue}{\frac{{D}^{2}}{{d}^{2}} \cdot \frac{w \cdot \left({M}^{2} \cdot h\right)}{c0}}, -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right) \]
      unpow2 [=>] 60.0	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{w \cdot \left({M}^{2} \cdot h\right)}{c0}, -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right) \]
      unpow2 [=>] 60.0	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \frac{w \cdot \left({M}^{2} \cdot h\right)}{c0}, -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right) \]
      associate-/l* [=>] 59.8	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \color{blue}{\frac{D}{\frac{d \cdot d}{D}}} \cdot \frac{w \cdot \left({M}^{2} \cdot h\right)}{c0}, -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right) \]
      associate-*r/ [<=] 59.7	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \frac{D}{\color{blue}{d \cdot \frac{d}{D}}} \cdot \frac{w \cdot \left({M}^{2} \cdot h\right)}{c0}, -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right) \]
      *-commutative [<=] 59.7	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \frac{D}{d \cdot \frac{d}{D}} \cdot \frac{w \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{c0}, -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right) \]
      associate-/l* [=>] 59.8	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \frac{D}{d \cdot \frac{d}{D}} \cdot \color{blue}{\frac{w}{\frac{c0}{h \cdot {M}^{2}}}}, -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right) \]
      unpow2 [=>] 59.8	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \frac{D}{d \cdot \frac{d}{D}} \cdot \frac{w}{\frac{c0}{h \cdot \color{blue}{\left(M \cdot M\right)}}}, -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right) \]
      mul-1-neg [=>] 59.8	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \frac{D}{d \cdot \frac{d}{D}} \cdot \frac{w}{\frac{c0}{h \cdot \left(M \cdot M\right)}}, \color{blue}{-\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0}\right) \]
      *-commutative [=>] 59.8	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \frac{D}{d \cdot \frac{d}{D}} \cdot \frac{w}{\frac{c0}{h \cdot \left(M \cdot M\right)}}, -\color{blue}{c0 \cdot \left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}\right) \]
      distribute-rgt-neg-in [=>] 59.8	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \frac{D}{d \cdot \frac{d}{D}} \cdot \frac{w}{\frac{c0}{h \cdot \left(M \cdot M\right)}}, \color{blue}{c0 \cdot \left(-\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)\right)}\right) \]
      distribute-rgt1-in [=>] 59.8	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \frac{D}{d \cdot \frac{d}{D}} \cdot \frac{w}{\frac{c0}{h \cdot \left(M \cdot M\right)}}, c0 \cdot \left(-\color{blue}{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}\right)\right) \]
      metadata-eval [=>] 59.8	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \frac{D}{d \cdot \frac{d}{D}} \cdot \frac{w}{\frac{c0}{h \cdot \left(M \cdot M\right)}}, c0 \cdot \left(-\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)\right) \]
      +-inverses [<=] 61.5	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \frac{D}{d \cdot \frac{d}{D}} \cdot \frac{w}{\frac{c0}{h \cdot \left(M \cdot M\right)}}, c0 \cdot \left(-\color{blue}{\left({\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - {\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2}\right)} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)\right) \]
      *-commutative [=>] 61.5	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \frac{D}{d \cdot \frac{d}{D}} \cdot \frac{w}{\frac{c0}{h \cdot \left(M \cdot M\right)}}, c0 \cdot \left(-\color{blue}{\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} \cdot \left({\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - {\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2}\right)}\right)\right) \]
      unpow2 [=>] 61.5	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \frac{D}{d \cdot \frac{d}{D}} \cdot \frac{w}{\frac{c0}{h \cdot \left(M \cdot M\right)}}, c0 \cdot \left(-\frac{\color{blue}{d \cdot d}}{{D}^{2} \cdot \left(w \cdot h\right)} \cdot \left({\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - {\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2}\right)\right)\right) \]
      *-commutative [=>] 61.5	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \frac{D}{d \cdot \frac{d}{D}} \cdot \frac{w}{\frac{c0}{h \cdot \left(M \cdot M\right)}}, c0 \cdot \left(-\frac{d \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot {D}^{2}}} \cdot \left({\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - {\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2}\right)\right)\right) \]
      unpow2 [=>] 61.5	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \frac{D}{d \cdot \frac{d}{D}} \cdot \frac{w}{\frac{c0}{h \cdot \left(M \cdot M\right)}}, c0 \cdot \left(-\frac{d \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} \cdot \left({\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - {\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2}\right)\right)\right) \]
      associate-*r* [<=] 61.6	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \frac{D}{d \cdot \frac{d}{D}} \cdot \frac{w}{\frac{c0}{h \cdot \left(M \cdot M\right)}}, c0 \cdot \left(-\frac{d \cdot d}{\color{blue}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}} \cdot \left({\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - {\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2}\right)\right)\right) \]
      times-frac [=>] 60.5	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \frac{D}{d \cdot \frac{d}{D}} \cdot \frac{w}{\frac{c0}{h \cdot \left(M \cdot M\right)}}, c0 \cdot \left(-\color{blue}{\left(\frac{d}{w} \cdot \frac{d}{h \cdot \left(D \cdot D\right)}\right)} \cdot \left({\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - {\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2}\right)\right)\right) \]
      *-commutative [=>] 60.5	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \frac{D}{d \cdot \frac{d}{D}} \cdot \frac{w}{\frac{c0}{h \cdot \left(M \cdot M\right)}}, c0 \cdot \left(-\color{blue}{\left(\frac{d}{h \cdot \left(D \cdot D\right)} \cdot \frac{d}{w}\right)} \cdot \left({\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - {\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2}\right)\right)\right) \]
      associate-/r* [=>] 60.9	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \frac{D}{d \cdot \frac{d}{D}} \cdot \frac{w}{\frac{c0}{h \cdot \left(M \cdot M\right)}}, c0 \cdot \left(-\left(\color{blue}{\frac{\frac{d}{h}}{D \cdot D}} \cdot \frac{d}{w}\right) \cdot \left({\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - {\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2}\right)\right)\right) \]
      +-inverses [=>] 58.5	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \frac{D}{d \cdot \frac{d}{D}} \cdot \frac{w}{\frac{c0}{h \cdot \left(M \cdot M\right)}}, c0 \cdot \left(-\left(\frac{\frac{d}{h}}{D \cdot D} \cdot \frac{d}{w}\right) \cdot \color{blue}{0}\right)\right) \]

   4. Taylor expanded in c0 around 0 36.1

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

   5. Simplified 29.7

      \[\leadsto \color{blue}{0.25 \cdot \left(\frac{D}{d \cdot \frac{d}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)} \]
      Proof

      [Start] 36.1	\[ 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}} \]
      *-commutative [=>] 36.1	\[ 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{{d}^{2}} \]
      associate-/l* [=>] 36.0	\[ 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{{M}^{2} \cdot h}}} \]
      unpow2 [=>] 36.0	\[ 0.25 \cdot \frac{{D}^{2}}{\frac{{d}^{2}}{\color{blue}{\left(M \cdot M\right)} \cdot h}} \]
      *-commutative [<=] 36.0	\[ 0.25 \cdot \frac{{D}^{2}}{\frac{{d}^{2}}{\color{blue}{h \cdot \left(M \cdot M\right)}}} \]
      associate-/r/ [=>] 35.9	\[ 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)} \]
      unpow2 [=>] 35.9	\[ 0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]
      associate-/l* [=>] 33.1	\[ 0.25 \cdot \left(\color{blue}{\frac{D}{\frac{{d}^{2}}{D}}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]
      unpow2 [=>] 33.1	\[ 0.25 \cdot \left(\frac{D}{\frac{\color{blue}{d \cdot d}}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]
      associate-*r/ [<=] 29.7	\[ 0.25 \cdot \left(\frac{D}{\color{blue}{d \cdot \frac{d}{D}}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

   6. Applied egg-rr 28.9

      \[\leadsto 0.25 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

   7. Applied egg-rr 19.4

      \[\leadsto 0.25 \cdot \color{blue}{\frac{\frac{D}{d} \cdot \left(h \cdot M\right)}{\frac{\frac{d}{D}}{M}}} \]

   if 2.69999999999999996e-266 < c0 < 8.9999999999999997e-216

   1. Initial program 59.8

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

   2. Applied egg-rr 60.1

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

   3. Simplified 43.5

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

      [Start] 60.1	\[ \frac{\left(\left({\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - {\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2}\right) + M \cdot M\right) \cdot \left(-c0\right)}{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right) \cdot \left(w \cdot -2\right)} \]
      *-commutative [=>] 60.1	\[ \frac{\color{blue}{\left(-c0\right) \cdot \left(\left({\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - {\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2}\right) + M \cdot M\right)}}{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right) \cdot \left(w \cdot -2\right)} \]
      *-commutative [=>] 60.1	\[ \frac{\left(-c0\right) \cdot \left(\left({\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - {\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2}\right) + M \cdot M\right)}{\color{blue}{\left(w \cdot -2\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right)}} \]
      distribute-lft-neg-out [=>] 60.1	\[ \frac{\color{blue}{-c0 \cdot \left(\left({\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - {\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2}\right) + M \cdot M\right)}}{\left(w \cdot -2\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right)} \]
      distribute-rgt-neg-in [=>] 60.1	\[ \frac{\color{blue}{c0 \cdot \left(-\left(\left({\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - {\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2}\right) + M \cdot M\right)\right)}}{\left(w \cdot -2\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right)} \]
      neg-sub0 [=>] 60.1	\[ \frac{c0 \cdot \color{blue}{\left(0 - \left(\left({\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - {\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2}\right) + M \cdot M\right)\right)}}{\left(w \cdot -2\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right)} \]
      +-inverses [<=] 60.1	\[ \frac{c0 \cdot \left(\color{blue}{\left({\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - {\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2}\right)} - \left(\left({\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - {\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2}\right) + M \cdot M\right)\right)}{\left(w \cdot -2\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right)} \]
      associate--r+ [=>] 60.1	\[ \frac{c0 \cdot \color{blue}{\left(\left(\left({\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - {\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2}\right) - \left({\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - {\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2}\right)\right) - M \cdot M\right)}}{\left(w \cdot -2\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right)} \]
      +-inverses [=>] 60.1	\[ \frac{c0 \cdot \left(\left(\color{blue}{0} - \left({\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - {\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2}\right)\right) - M \cdot M\right)}{\left(w \cdot -2\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right)} \]
      +-inverses [=>] 48.0	\[ \frac{c0 \cdot \left(\left(0 - \color{blue}{0}\right) - M \cdot M\right)}{\left(w \cdot -2\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right)} \]
      metadata-eval [=>] 48.0	\[ \frac{c0 \cdot \left(\color{blue}{0} - M \cdot M\right)}{\left(w \cdot -2\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right)} \]
      associate-*l* [=>] 48.0	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{\color{blue}{w \cdot \left(-2 \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right)\right)}} \]
      *-commutative [=>] 48.0	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \color{blue}{\left(\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right) \cdot -2\right)}} \]
      *-commutative [=>] 48.0	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}} - \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      associate-*l/ [=>] 47.8	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\color{blue}{\frac{d \cdot \frac{d}{D}}{D}} \cdot \frac{\frac{c0}{h}}{w} - \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      associate-*r/ [=>] 48.4	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\frac{\color{blue}{\frac{d \cdot d}{D}}}{D} \cdot \frac{\frac{c0}{h}}{w} - \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      associate-/r* [<=] 49.7	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\color{blue}{\frac{d \cdot d}{D \cdot D}} \cdot \frac{\frac{c0}{h}}{w} - \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      unpow2 [<=] 49.7	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\frac{\color{blue}{{d}^{2}}}{D \cdot D} \cdot \frac{\frac{c0}{h}}{w} - \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      unpow2 [<=] 49.7	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\frac{{d}^{2}}{\color{blue}{{D}^{2}}} \cdot \frac{\frac{c0}{h}}{w} - \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      associate-/l/ [=>] 50.1	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\frac{{d}^{2}}{{D}^{2}} \cdot \color{blue}{\frac{c0}{w \cdot h}} - \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      times-frac [<=] 52.0	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\color{blue}{\frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}} - \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      associate-/r* [=>] 51.9	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\color{blue}{\frac{\frac{{d}^{2} \cdot c0}{{D}^{2}}}{w \cdot h}} - \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      *-commutative [=>] 51.9	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\frac{\frac{\color{blue}{c0 \cdot {d}^{2}}}{{D}^{2}}}{w \cdot h} - \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      unpow2 [=>] 51.9	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\frac{\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{{D}^{2}}}{w \cdot h} - \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      associate-*r* [=>] 51.6	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\frac{\frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{{D}^{2}}}{w \cdot h} - \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      unpow2 [=>] 51.6	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\frac{\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{D \cdot D}}}{w \cdot h} - \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      associate-*l/ [<=] 51.2	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\frac{\color{blue}{\frac{c0 \cdot d}{D \cdot D} \cdot d}}{w \cdot h} - \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      associate-*r/ [<=] 51.2	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\color{blue}{\frac{c0 \cdot d}{D \cdot D} \cdot \frac{d}{w \cdot h}} - \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      *-commutative [=>] 51.2	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\color{blue}{\frac{d}{w \cdot h} \cdot \frac{c0 \cdot d}{D \cdot D}} - \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      associate-/r* [=>] 51.3	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\color{blue}{\frac{\frac{d}{w}}{h}} \cdot \frac{c0 \cdot d}{D \cdot D} - \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      associate-*l/ [=>] 51.0	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\color{blue}{\frac{\frac{d}{w} \cdot \frac{c0 \cdot d}{D \cdot D}}{h}} - \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      *-commutative [=>] 51.0	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\frac{\frac{d}{w} \cdot \frac{\color{blue}{d \cdot c0}}{D \cdot D}}{h} - \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      *-commutative [=>] 51.0	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\frac{\frac{d}{w} \cdot \frac{d \cdot c0}{D \cdot D}}{h} - \sqrt{{\color{blue}{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right)}}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      associate-*l/ [=>] 51.0	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\frac{\frac{d}{w} \cdot \frac{d \cdot c0}{D \cdot D}}{h} - \sqrt{{\left(\color{blue}{\frac{d \cdot \frac{d}{D}}{D}} \cdot \frac{\frac{c0}{h}}{w}\right)}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      associate-*r/ [=>] 51.0	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\frac{\frac{d}{w} \cdot \frac{d \cdot c0}{D \cdot D}}{h} - \sqrt{{\left(\frac{\color{blue}{\frac{d \cdot d}{D}}}{D} \cdot \frac{\frac{c0}{h}}{w}\right)}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      associate-/r* [<=] 51.5	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\frac{\frac{d}{w} \cdot \frac{d \cdot c0}{D \cdot D}}{h} - \sqrt{{\left(\color{blue}{\frac{d \cdot d}{D \cdot D}} \cdot \frac{\frac{c0}{h}}{w}\right)}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      unpow2 [<=] 51.5	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\frac{\frac{d}{w} \cdot \frac{d \cdot c0}{D \cdot D}}{h} - \sqrt{{\left(\frac{\color{blue}{{d}^{2}}}{D \cdot D} \cdot \frac{\frac{c0}{h}}{w}\right)}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      unpow2 [<=] 51.5	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\frac{\frac{d}{w} \cdot \frac{d \cdot c0}{D \cdot D}}{h} - \sqrt{{\left(\frac{{d}^{2}}{\color{blue}{{D}^{2}}} \cdot \frac{\frac{c0}{h}}{w}\right)}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      associate-/l/ [=>] 49.6	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\frac{\frac{d}{w} \cdot \frac{d \cdot c0}{D \cdot D}}{h} - \sqrt{{\left(\frac{{d}^{2}}{{D}^{2}} \cdot \color{blue}{\frac{c0}{w \cdot h}}\right)}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      times-frac [<=] 45.9	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\frac{\frac{d}{w} \cdot \frac{d \cdot c0}{D \cdot D}}{h} - \sqrt{{\color{blue}{\left(\frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      associate-/r* [=>] 46.4	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\frac{\frac{d}{w} \cdot \frac{d \cdot c0}{D \cdot D}}{h} - \sqrt{{\color{blue}{\left(\frac{\frac{{d}^{2} \cdot c0}{{D}^{2}}}{w \cdot h}\right)}}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      *-commutative [=>] 46.4	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\frac{\frac{d}{w} \cdot \frac{d \cdot c0}{D \cdot D}}{h} - \sqrt{{\left(\frac{\frac{\color{blue}{c0 \cdot {d}^{2}}}{{D}^{2}}}{w \cdot h}\right)}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      unpow2 [=>] 46.4	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\frac{\frac{d}{w} \cdot \frac{d \cdot c0}{D \cdot D}}{h} - \sqrt{{\left(\frac{\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{{D}^{2}}}{w \cdot h}\right)}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      associate-*r* [=>] 45.9	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\frac{\frac{d}{w} \cdot \frac{d \cdot c0}{D \cdot D}}{h} - \sqrt{{\left(\frac{\frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{{D}^{2}}}{w \cdot h}\right)}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      unpow2 [=>] 45.9	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\frac{\frac{d}{w} \cdot \frac{d \cdot c0}{D \cdot D}}{h} - \sqrt{{\left(\frac{\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{D \cdot D}}}{w \cdot h}\right)}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      associate-*l/ [<=] 45.7	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\frac{\frac{d}{w} \cdot \frac{d \cdot c0}{D \cdot D}}{h} - \sqrt{{\left(\frac{\color{blue}{\frac{c0 \cdot d}{D \cdot D} \cdot d}}{w \cdot h}\right)}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      associate-*r/ [<=] 45.5	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\frac{\frac{d}{w} \cdot \frac{d \cdot c0}{D \cdot D}}{h} - \sqrt{{\color{blue}{\left(\frac{c0 \cdot d}{D \cdot D} \cdot \frac{d}{w \cdot h}\right)}}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      *-commutative [=>] 45.5	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\frac{\frac{d}{w} \cdot \frac{d \cdot c0}{D \cdot D}}{h} - \sqrt{{\color{blue}{\left(\frac{d}{w \cdot h} \cdot \frac{c0 \cdot d}{D \cdot D}\right)}}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      associate-/r* [=>] 43.4	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\frac{\frac{d}{w} \cdot \frac{d \cdot c0}{D \cdot D}}{h} - \sqrt{{\left(\color{blue}{\frac{\frac{d}{w}}{h}} \cdot \frac{c0 \cdot d}{D \cdot D}\right)}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      associate-*l/ [=>] 43.5	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\frac{\frac{d}{w} \cdot \frac{d \cdot c0}{D \cdot D}}{h} - \sqrt{{\color{blue}{\left(\frac{\frac{d}{w} \cdot \frac{c0 \cdot d}{D \cdot D}}{h}\right)}}^{2} - M \cdot M}\right) \cdot -2\right)} \]
      *-commutative [=>] 43.5	\[ \frac{c0 \cdot \left(0 - M \cdot M\right)}{w \cdot \left(\left(\frac{\frac{d}{w} \cdot \frac{d \cdot c0}{D \cdot D}}{h} - \sqrt{{\left(\frac{\frac{d}{w} \cdot \frac{\color{blue}{d \cdot c0}}{D \cdot D}}{h}\right)}^{2} - M \cdot M}\right) \cdot -2\right)} \]

   4. Taylor expanded in c0 around inf 61.6

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

   5. Simplified 46.0

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

      [Start] 61.6	\[ \frac{{d}^{2} \cdot {c0}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)} \]
      *-commutative [=>] 61.6	\[ \frac{\color{blue}{{c0}^{2} \cdot {d}^{2}}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)} \]
      unpow2 [=>] 61.6	\[ \frac{\color{blue}{\left(c0 \cdot c0\right)} \cdot {d}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)} \]
      unpow2 [=>] 61.6	\[ \frac{\left(c0 \cdot c0\right) \cdot \color{blue}{\left(d \cdot d\right)}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)} \]
      swap-sqr [<=] 54.2	\[ \frac{\color{blue}{\left(c0 \cdot d\right) \cdot \left(c0 \cdot d\right)}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)} \]
      unpow2 [<=] 54.2	\[ \frac{\color{blue}{{\left(c0 \cdot d\right)}^{2}}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)} \]
      associate-*r* [=>] 54.9	\[ \frac{{\left(c0 \cdot d\right)}^{2}}{\color{blue}{\left({D}^{2} \cdot {w}^{2}\right) \cdot h}} \]
      unpow2 [=>] 54.9	\[ \frac{{\left(c0 \cdot d\right)}^{2}}{\left(\color{blue}{\left(D \cdot D\right)} \cdot {w}^{2}\right) \cdot h} \]
      unpow2 [=>] 54.9	\[ \frac{{\left(c0 \cdot d\right)}^{2}}{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(w \cdot w\right)}\right) \cdot h} \]
      unswap-sqr [=>] 46.0	\[ \frac{{\left(c0 \cdot d\right)}^{2}}{\color{blue}{\left(\left(D \cdot w\right) \cdot \left(D \cdot w\right)\right)} \cdot h} \]

   if 8.9999999999999997e-216 < c0 < 3.69999999999999998e-175

   1. Initial program 57.0

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

   2. Simplified 61.9

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot {\left(\frac{d}{D}\right)}^{3}\right), -M \cdot M\right)}\right)} \]
      Proof

      [Start] 57.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*l/ [<=] 57.7	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \left(d \cdot d\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      *-commutative [=>] 57.7	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(d \cdot d\right) \cdot \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-def [=>] 59.0	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]
      times-frac [=>] 60.5	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\color{blue}{\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 60.5	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right) \cdot \color{blue}{\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right)} - M \cdot M}\right) \]
      swap-sqr [=>] 62.2	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\color{blue}{\left(\frac{c0}{w \cdot h} \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\frac{d \cdot d}{D \cdot D} \cdot \frac{d \cdot d}{D \cdot D}\right)} - M \cdot M}\right) \]
      associate-*l* [=>] 61.9	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\color{blue}{\frac{c0}{w \cdot h} \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d \cdot d}{D \cdot D} \cdot \frac{d \cdot d}{D \cdot D}\right)\right)} - M \cdot M}\right) \]
      fma-neg [=>] 61.9	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\color{blue}{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{c0}{w \cdot h} \cdot \left(\frac{d \cdot d}{D \cdot D} \cdot \frac{d \cdot d}{D \cdot D}\right), -M \cdot M\right)}}\right) \]
      times-frac [=>] 61.9	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{c0}{w \cdot h} \cdot \left(\color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)} \cdot \frac{d \cdot d}{D \cdot D}\right), -M \cdot M\right)}\right) \]
      associate-*l* [=>] 61.9	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{c0}{w \cdot h} \cdot \color{blue}{\left(\frac{d}{D} \cdot \left(\frac{d}{D} \cdot \frac{d \cdot d}{D \cdot D}\right)\right)}, -M \cdot M\right)}\right) \]
      times-frac [=>] 61.9	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \left(\frac{d}{D} \cdot \color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)}\right)\right), -M \cdot M\right)}\right) \]
      cube-unmult [=>] 61.9	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \color{blue}{{\left(\frac{d}{D}\right)}^{3}}\right), -M \cdot M\right)}\right) \]

   3. Taylor expanded in c0 around -inf 59.5

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w} + 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]

   4. Simplified 27.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{0}{w} \cdot \left(c0 \cdot c0\right)\right)} \]
      Proof

      [Start] 59.5	\[ -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w} + 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]
      +-commutative [=>] 59.5	\[ \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} + -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}} \]
      fma-def [=>] 59.5	\[ \color{blue}{\mathsf{fma}\left(0.25, \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}, -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right)} \]
      *-commutative [<=] 59.5	\[ \mathsf{fma}\left(0.25, \frac{{D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2}}, -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right) \]
      associate-/l* [=>] 59.5	\[ \mathsf{fma}\left(0.25, \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}}, -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right) \]
      associate-/r/ [=>] 59.5	\[ \mathsf{fma}\left(0.25, \color{blue}{\frac{{D}^{2}}{{d}^{2}} \cdot \left(h \cdot {M}^{2}\right)}, -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right) \]
      unpow2 [=>] 59.5	\[ \mathsf{fma}\left(0.25, \frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \left(h \cdot {M}^{2}\right), -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right) \]
      unpow2 [=>] 59.5	\[ \mathsf{fma}\left(0.25, \frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \left(h \cdot {M}^{2}\right), -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right) \]
      times-frac [=>] 59.5	\[ \mathsf{fma}\left(0.25, \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(h \cdot {M}^{2}\right), -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right) \]
      *-commutative [=>] 59.5	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\left({M}^{2} \cdot h\right)}, -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right) \]
      unpow2 [=>] 59.5	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right), -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right) \]
      associate-*l* [=>] 59.2	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(M \cdot \left(M \cdot h\right)\right)}, -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right) \]
      associate-/l* [=>] 59.2	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), -0.5 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{\frac{w}{{c0}^{2}}}}\right) \]
      associate-*r/ [=>] 59.2	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \color{blue}{\frac{-0.5 \cdot \left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}{\frac{w}{{c0}^{2}}}}\right) \]
      distribute-rgt1-in [=>] 59.2	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{-0.5 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}}{\frac{w}{{c0}^{2}}}\right) \]
      metadata-eval [=>] 59.2	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{-0.5 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}{\frac{w}{{c0}^{2}}}\right) \]
      mul0-lft [=>] 27.2	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{-0.5 \cdot \color{blue}{0}}{\frac{w}{{c0}^{2}}}\right) \]
      metadata-eval [=>] 27.2	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{\color{blue}{0}}{\frac{w}{{c0}^{2}}}\right) \]
      mul0-lft [<=] 59.2	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{\color{blue}{0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}}{\frac{w}{{c0}^{2}}}\right) \]
      metadata-eval [<=] 59.2	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{\color{blue}{\left(-1 + 1\right)} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{\frac{w}{{c0}^{2}}}\right) \]
      distribute-rgt1-in [<=] 59.2	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{\color{blue}{\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}}{\frac{w}{{c0}^{2}}}\right) \]
      associate-/r/ [=>] 59.2	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{w} \cdot {c0}^{2}}\right) \]
      distribute-rgt1-in [=>] 59.2	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{\color{blue}{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}}{w} \cdot {c0}^{2}\right) \]
      metadata-eval [=>] 59.2	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{w} \cdot {c0}^{2}\right) \]
      mul0-lft [=>] 27.2	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{\color{blue}{0}}{w} \cdot {c0}^{2}\right) \]
      unpow2 [=>] 27.2	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{0}{w} \cdot \color{blue}{\left(c0 \cdot c0\right)}\right) \]

   5. Taylor expanded in w around 0 27.2

      \[\leadsto \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \color{blue}{0}\right) \]

   6. Applied egg-rr 23.4

      \[\leadsto \mathsf{fma}\left(0.25, \color{blue}{\frac{\frac{D}{d} \cdot M}{\frac{\frac{d}{D}}{M \cdot h}}}, 0\right) \]

   7. Applied egg-rr 23.0

      \[\leadsto \mathsf{fma}\left(0.25, \color{blue}{\frac{\frac{D}{\frac{d}{M}}}{\frac{d}{D \cdot M}} \cdot h}, 0\right) \]

   if 3.69999999999999998e-175 < c0 < 3.0999999999999998e-146

   1. Initial program 58.4

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

   2. Simplified 54.7

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

      [Start] 58.4	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 58.8	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-def [=>] 59.2	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{D \cdot D}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]
      associate-/r* [=>] 59.2	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{c0}{w}}{h}}, \frac{d \cdot d}{D \cdot D}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 59.2	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \color{blue}{\frac{d}{D} \cdot \frac{d}{D}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      difference-of-squares [=>] 59.2	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M\right)}}\right) \]
      sub-neg [=>] 59.2	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + M\right) \cdot \color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \left(-M\right)\right)}}\right) \]
      times-frac [=>] 59.2	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}} + M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \left(-M\right)\right)}\right) \]
      fma-def [=>] 59.2	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\color{blue}{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{D \cdot D}, M\right)} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \left(-M\right)\right)}\right) \]
      associate-/r* [=>] 59.2	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\mathsf{fma}\left(\color{blue}{\frac{\frac{c0}{w}}{h}}, \frac{d \cdot d}{D \cdot D}, M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \left(-M\right)\right)}\right) \]
      times-frac [=>] 59.2	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \color{blue}{\frac{d}{D} \cdot \frac{d}{D}}, M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \left(-M\right)\right)}\right) \]
      sub-neg [<=] 59.2	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, M\right) \cdot \color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M\right)}}\right) \]
      times-frac [=>] 59.2	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, M\right) \cdot \left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}} - M\right)}\right) \]
      associate-/r* [=>] 58.7	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, M\right) \cdot \left(\color{blue}{\frac{\frac{c0}{w}}{h}} \cdot \frac{d \cdot d}{D \cdot D} - M\right)}\right) \]
      times-frac [=>] 54.7	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, M\right) \cdot \left(\frac{\frac{c0}{w}}{h} \cdot \color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)} - M\right)}\right) \]

   3. Taylor expanded in c0 around inf 57.8

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

   4. Simplified 57.4

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

      [Start] 57.8	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]
      times-frac [=>] 58.5	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{{d}^{2}}{{D}^{2}} \cdot \frac{c0}{w \cdot h}\right)}\right) \]
      unpow2 [=>] 58.5	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\color{blue}{d \cdot d}}{{D}^{2}} \cdot \frac{c0}{w \cdot h}\right)\right) \]
      unpow2 [=>] 58.5	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{d \cdot d}{\color{blue}{D \cdot D}} \cdot \frac{c0}{w \cdot h}\right)\right) \]
      associate-/r* [=>] 57.6	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{d \cdot d}{D \cdot D} \cdot \color{blue}{\frac{\frac{c0}{w}}{h}}\right)\right) \]
      times-frac [<=] 56.9	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{\left(d \cdot d\right) \cdot \frac{c0}{w}}{\left(D \cdot D\right) \cdot h}}\right) \]
      *-commutative [=>] 56.9	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\color{blue}{\frac{c0}{w} \cdot \left(d \cdot d\right)}}{\left(D \cdot D\right) \cdot h}\right) \]
      associate-/l* [=>] 57.9	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{\frac{c0}{w}}{\frac{\left(D \cdot D\right) \cdot h}{d \cdot d}}}\right) \]
      associate-*l* [=>] 57.4	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{w}}{\frac{\color{blue}{D \cdot \left(D \cdot h\right)}}{d \cdot d}}\right) \]

   5. Applied egg-rr 52.7

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

   if 3.0999999999999998e-146 < c0

   1. Initial program 59.8

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

   2. Simplified 62.1

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot {\left(\frac{d}{D}\right)}^{3}\right), -M \cdot M\right)}\right)} \]
      Proof

      [Start] 59.8	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*l/ [<=] 60.4	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \left(d \cdot d\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      *-commutative [=>] 60.4	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(d \cdot d\right) \cdot \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-def [=>] 61.0	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]
      times-frac [=>] 61.4	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\color{blue}{\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 61.1	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right) \cdot \color{blue}{\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right)} - M \cdot M}\right) \]
      swap-sqr [=>] 62.4	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\color{blue}{\left(\frac{c0}{w \cdot h} \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\frac{d \cdot d}{D \cdot D} \cdot \frac{d \cdot d}{D \cdot D}\right)} - M \cdot M}\right) \]
      associate-*l* [=>] 62.1	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\color{blue}{\frac{c0}{w \cdot h} \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d \cdot d}{D \cdot D} \cdot \frac{d \cdot d}{D \cdot D}\right)\right)} - M \cdot M}\right) \]
      fma-neg [=>] 62.1	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\color{blue}{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{c0}{w \cdot h} \cdot \left(\frac{d \cdot d}{D \cdot D} \cdot \frac{d \cdot d}{D \cdot D}\right), -M \cdot M\right)}}\right) \]
      times-frac [=>] 62.0	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{c0}{w \cdot h} \cdot \left(\color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)} \cdot \frac{d \cdot d}{D \cdot D}\right), -M \cdot M\right)}\right) \]
      associate-*l* [=>] 62.0	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{c0}{w \cdot h} \cdot \color{blue}{\left(\frac{d}{D} \cdot \left(\frac{d}{D} \cdot \frac{d \cdot d}{D \cdot D}\right)\right)}, -M \cdot M\right)}\right) \]
      times-frac [=>] 62.1	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \left(\frac{d}{D} \cdot \color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)}\right)\right), -M \cdot M\right)}\right) \]
      cube-unmult [=>] 62.1	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \color{blue}{{\left(\frac{d}{D}\right)}^{3}}\right), -M \cdot M\right)}\right) \]

   3. Taylor expanded in c0 around -inf 60.3

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w} + 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]

   4. Simplified 40.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{0}{w} \cdot \left(c0 \cdot c0\right)\right)} \]
      Proof

      [Start] 60.3	\[ -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w} + 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]
      +-commutative [=>] 60.3	\[ \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} + -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}} \]
      fma-def [=>] 60.3	\[ \color{blue}{\mathsf{fma}\left(0.25, \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}, -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right)} \]
      *-commutative [<=] 60.3	\[ \mathsf{fma}\left(0.25, \frac{{D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2}}, -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right) \]
      associate-/l* [=>] 60.3	\[ \mathsf{fma}\left(0.25, \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}}, -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right) \]
      associate-/r/ [=>] 60.2	\[ \mathsf{fma}\left(0.25, \color{blue}{\frac{{D}^{2}}{{d}^{2}} \cdot \left(h \cdot {M}^{2}\right)}, -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right) \]
      unpow2 [=>] 60.2	\[ \mathsf{fma}\left(0.25, \frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \left(h \cdot {M}^{2}\right), -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right) \]
      unpow2 [=>] 60.2	\[ \mathsf{fma}\left(0.25, \frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \left(h \cdot {M}^{2}\right), -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right) \]
      times-frac [=>] 59.9	\[ \mathsf{fma}\left(0.25, \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(h \cdot {M}^{2}\right), -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right) \]
      *-commutative [=>] 59.9	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\left({M}^{2} \cdot h\right)}, -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right) \]
      unpow2 [=>] 59.9	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right), -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right) \]
      associate-*l* [=>] 59.6	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(M \cdot \left(M \cdot h\right)\right)}, -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right) \]
      associate-/l* [=>] 60.0	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), -0.5 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{\frac{w}{{c0}^{2}}}}\right) \]
      associate-*r/ [=>] 60.0	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \color{blue}{\frac{-0.5 \cdot \left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}{\frac{w}{{c0}^{2}}}}\right) \]
      distribute-rgt1-in [=>] 60.0	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{-0.5 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}}{\frac{w}{{c0}^{2}}}\right) \]
      metadata-eval [=>] 60.0	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{-0.5 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}{\frac{w}{{c0}^{2}}}\right) \]
      mul0-lft [=>] 43.4	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{-0.5 \cdot \color{blue}{0}}{\frac{w}{{c0}^{2}}}\right) \]
      metadata-eval [=>] 43.4	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{\color{blue}{0}}{\frac{w}{{c0}^{2}}}\right) \]
      mul0-lft [<=] 60.0	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{\color{blue}{0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}}{\frac{w}{{c0}^{2}}}\right) \]
      metadata-eval [<=] 60.0	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{\color{blue}{\left(-1 + 1\right)} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{\frac{w}{{c0}^{2}}}\right) \]
      distribute-rgt1-in [<=] 60.0	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{\color{blue}{\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}}{\frac{w}{{c0}^{2}}}\right) \]
      associate-/r/ [=>] 59.6	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{w} \cdot {c0}^{2}}\right) \]
      distribute-rgt1-in [=>] 59.6	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{\color{blue}{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}}{w} \cdot {c0}^{2}\right) \]
      metadata-eval [=>] 59.6	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{w} \cdot {c0}^{2}\right) \]
      mul0-lft [=>] 40.3	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{\color{blue}{0}}{w} \cdot {c0}^{2}\right) \]
      unpow2 [=>] 40.3	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{0}{w} \cdot \color{blue}{\left(c0 \cdot c0\right)}\right) \]

   5. Taylor expanded in w around 0 26.0

      \[\leadsto \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \color{blue}{0}\right) \]

   6. Applied egg-rr 18.2

      \[\leadsto \mathsf{fma}\left(0.25, \color{blue}{\frac{\frac{D}{d} \cdot M}{\frac{\frac{d}{D}}{M \cdot h}}}, 0\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 20.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq 2.7 \cdot 10^{-266}:\\ \;\;\;\;0.25 \cdot \frac{\frac{D}{d} \cdot \left(h \cdot M\right)}{\frac{\frac{d}{D}}{M}}\\ \mathbf{elif}\;c0 \leq 9 \cdot 10^{-216}:\\ \;\;\;\;\frac{{\left(c0 \cdot d\right)}^{2}}{h \cdot \left(\left(D \cdot w\right) \cdot \left(D \cdot w\right)\right)}\\ \mathbf{elif}\;c0 \leq 3.7 \cdot 10^{-175}:\\ \;\;\;\;\mathsf{fma}\left(0.25, h \cdot \frac{\frac{D}{\frac{d}{M}}}{\frac{d}{D \cdot M}}, 0\right)\\ \mathbf{elif}\;c0 \leq 3.1 \cdot 10^{-146}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D \cdot h}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.25, \frac{\frac{D}{d} \cdot M}{\frac{\frac{d}{D}}{h \cdot M}}, 0\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	20.8
Cost	7756

\[\begin{array}{l} t_0 := 0.25 \cdot \frac{\frac{D}{d} \cdot \left(h \cdot M\right)}{\frac{\frac{d}{D}}{M}}\\ \mathbf{if}\;c0 \leq 8.8 \cdot 10^{-266}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c0 \leq 9 \cdot 10^{-216}:\\ \;\;\;\;\frac{{\left(c0 \cdot d\right)}^{2}}{h \cdot \left(\left(D \cdot w\right) \cdot \left(D \cdot w\right)\right)}\\ \mathbf{elif}\;c0 \leq 2.2 \cdot 10^{-175}:\\ \;\;\;\;\mathsf{fma}\left(0.25, h \cdot \frac{\frac{D}{\frac{d}{M}}}{\frac{d}{D \cdot M}}, 0\right)\\ \mathbf{elif}\;c0 \leq 3.1 \cdot 10^{-146}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D \cdot h}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	21.6
Cost	7560

\[\begin{array}{l} t_0 := 0.25 \cdot \frac{\frac{D}{d} \cdot \left(h \cdot M\right)}{\frac{\frac{d}{D}}{M}}\\ \mathbf{if}\;c0 \leq 8.8 \cdot 10^{-266}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c0 \leq 9 \cdot 10^{-216}:\\ \;\;\;\;\frac{{\left(c0 \cdot d\right)}^{2}}{h \cdot \left(\left(D \cdot w\right) \cdot \left(D \cdot w\right)\right)}\\ \mathbf{elif}\;c0 \leq 6.6 \cdot 10^{-207}:\\ \;\;\;\;0\\ \mathbf{elif}\;c0 \leq 3.1 \cdot 10^{-146}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D \cdot h}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	20.9
Cost	1609

\[\begin{array}{l} \mathbf{if}\;c0 \leq 6.6 \cdot 10^{-207} \lor \neg \left(c0 \leq 3.1 \cdot 10^{-146}\right):\\ \;\;\;\;0.25 \cdot \frac{\frac{D}{d} \cdot \left(h \cdot M\right)}{\frac{\frac{d}{D}}{M}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D \cdot h}\right)\right)\right)\\ \end{array} \]

Alternative 4
Error	26.6
Cost	1225

\[\begin{array}{l} \mathbf{if}\;D \leq -2.5 \cdot 10^{-266} \lor \neg \left(D \leq 1.25 \cdot 10^{-178}\right):\\ \;\;\;\;0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(h \cdot M\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Error	23.9
Cost	1225

\[\begin{array}{l} \mathbf{if}\;M \leq -2.45 \cdot 10^{+169} \lor \neg \left(M \leq 2.35 \cdot 10^{+132}\right):\\ \;\;\;\;0.25 \cdot \frac{D}{\frac{d}{\frac{D}{d} \cdot \left(M \cdot \left(h \cdot M\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{\frac{D}{d} \cdot h}{\frac{\frac{d}{D}}{M \cdot M}}\\ \end{array} \]

Alternative 6
Error	23.9
Cost	1225

\[\begin{array}{l} \mathbf{if}\;M \leq -2.45 \cdot 10^{+169} \lor \neg \left(M \leq 4.5 \cdot 10^{+134}\right):\\ \;\;\;\;0.25 \cdot \frac{D}{\frac{d}{\frac{D}{d} \cdot \left(M \cdot \left(h \cdot M\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{\frac{D}{d} \cdot \left(M \cdot M\right)}{\frac{\frac{d}{D}}{h}}\\ \end{array} \]

Alternative 7
Error	23.9
Cost	1224

\[\begin{array}{l} t_0 := \frac{D}{d} \cdot \left(M \cdot \left(h \cdot M\right)\right)\\ \mathbf{if}\;M \leq -2.45 \cdot 10^{+169}:\\ \;\;\;\;0.25 \cdot \frac{D}{\frac{d}{t_0}}\\ \mathbf{elif}\;M \leq -5.2 \cdot 10^{-161}:\\ \;\;\;\;0.25 \cdot \frac{\frac{D}{d} \cdot h}{\frac{\frac{d}{D}}{M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{t_0}{\frac{d}{D}}\\ \end{array} \]

Alternative 8
Error	25.3
Cost	1220

\[\begin{array}{l} t_0 := M \cdot \left(h \cdot M\right)\\ \mathbf{if}\;D \cdot D \leq 1.5 \cdot 10^{+176}:\\ \;\;\;\;0.25 \cdot \frac{D}{\frac{d}{\frac{D}{d} \cdot t_0}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot t_0\right)\\ \end{array} \]

Alternative 9
Error	19.5
Cost	960

\[0.25 \cdot \frac{\frac{D}{d} \cdot \left(h \cdot M\right)}{\frac{\frac{d}{D}}{M}} \]

Alternative 10
Error	32.3
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022356 
(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))))))