Henrywood and Agarwal, Equation (13)

Average Error: 59.5 → 15.9

Time: 29.0s

Precision: binary64

Cost: 30284

Math

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

↓

\[\begin{array}{l} t_0 := \left(w \cdot h\right) \cdot \frac{D}{c0}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := \frac{c0}{2 \cdot w} \cdot \left(t_1 + \sqrt{t_1 \cdot t_1 - M \cdot M}\right)\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{-50}:\\ \;\;\;\;d \cdot \frac{d}{\left(D \cdot t_0\right) \cdot \frac{w}{c0}}\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{M}{\frac{d}{D}}\right)\right) \cdot \left(h \cdot M\right)\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\frac{d}{\frac{D}{\frac{c0}{w} \cdot \frac{d}{t_0}}}\\ \mathbf{else}:\\ \;\;\;\;M \cdot \frac{0.25 \cdot \left(h \cdot \left(M \cdot \frac{D}{d}\right)\right)}{\frac{d}{D}}\\ \end{array} \]

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 (* (* w h) (/ D c0)))
        (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_2 (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))))
   (if (<= t_2 -1e-50)
     (* d (/ d (* (* D t_0) (/ w c0))))
     (if (<= t_2 0.0)
       (* (* 0.25 (* (/ D d) (/ M (/ d D)))) (* h M))
       (if (<= t_2 INFINITY)
         (/ d (/ D (* (/ c0 w) (/ d t_0))))
         (* M (/ (* 0.25 (* h (* M (/ D d)))) (/ d D))))))))

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 = (w * h) * (D / c0);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = (c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -1e-50) {
		tmp = d * (d / ((D * t_0) * (w / c0)));
	} else if (t_2 <= 0.0) {
		tmp = (0.25 * ((D / d) * (M / (d / D)))) * (h * M);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = d / (D / ((c0 / w) * (d / t_0)));
	} else {
		tmp = M * ((0.25 * (h * (M * (D / d)))) / (d / D));
	}
	return tmp;
}

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 = (w * h) * (D / c0);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = (c0 / (2.0 * w)) * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -1e-50) {
		tmp = d * (d / ((D * t_0) * (w / c0)));
	} else if (t_2 <= 0.0) {
		tmp = (0.25 * ((D / d) * (M / (d / D)))) * (h * M);
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = d / (D / ((c0 / w) * (d / t_0)));
	} else {
		tmp = M * ((0.25 * (h * (M * (D / d)))) / (d / D));
	}
	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 = (w * h) * (D / c0)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_2 = (c0 / (2.0 * w)) * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))
	tmp = 0
	if t_2 <= -1e-50:
		tmp = d * (d / ((D * t_0) * (w / c0)))
	elif t_2 <= 0.0:
		tmp = (0.25 * ((D / d) * (M / (d / D)))) * (h * M)
	elif t_2 <= math.inf:
		tmp = d / (D / ((c0 / w) * (d / t_0)))
	else:
		tmp = M * ((0.25 * (h * (M * (D / d)))) / (d / D))
	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(Float64(w * h) * Float64(D / c0))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_2 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))))
	tmp = 0.0
	if (t_2 <= -1e-50)
		tmp = Float64(d * Float64(d / Float64(Float64(D * t_0) * Float64(w / c0))));
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(0.25 * Float64(Float64(D / d) * Float64(M / Float64(d / D)))) * Float64(h * M));
	elseif (t_2 <= Inf)
		tmp = Float64(d / Float64(D / Float64(Float64(c0 / w) * Float64(d / t_0))));
	else
		tmp = Float64(M * Float64(Float64(0.25 * Float64(h * Float64(M * Float64(D / d)))) / Float64(d / D)));
	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 = (w * h) * (D / c0);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_2 = (c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	tmp = 0.0;
	if (t_2 <= -1e-50)
		tmp = d * (d / ((D * t_0) * (w / c0)));
	elseif (t_2 <= 0.0)
		tmp = (0.25 * ((D / d) * (M / (d / D)))) * (h * M);
	elseif (t_2 <= Inf)
		tmp = d / (D / ((c0 / w) * (d / t_0)));
	else
		tmp = M * ((0.25 * (h * (M * (D / d)))) / (d / D));
	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[(N[(w * h), $MachinePrecision] * N[(D / c0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-50], N[(d * N[(d / N[(N[(D * t$95$0), $MachinePrecision] * N[(w / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(0.25 * N[(N[(D / d), $MachinePrecision] * N[(M / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * M), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(d / N[(D / N[(N[(c0 / w), $MachinePrecision] * N[(d / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(M * N[(N[(0.25 * N[(h * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < -1.00000000000000001e-50

   1. Initial program 53.9

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

   2. Simplified 58.1

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

      [Start] 53.9	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*l/ [<=] 54.4	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \left(d \cdot d\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      *-commutative [=>] 54.4	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(d \cdot d\right) \cdot \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-def [=>] 54.4	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]
      associate-*l* [=>] 54.8	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\color{blue}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-/r* [=>] 54.8	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \color{blue}{\frac{\frac{c0}{w}}{h \cdot \left(D \cdot D\right)}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*r* [=>] 54.8	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{\color{blue}{\left(h \cdot D\right) \cdot D}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      *-commutative [=>] 54.8	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{\color{blue}{D \cdot \left(h \cdot D\right)}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 55.6	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\color{blue}{\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*l* [=>] 56.9	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\color{blue}{\frac{c0}{w \cdot h} \cdot \left(\frac{d \cdot d}{D \cdot D} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)} - M \cdot M}\right) \]
      fma-neg [=>] 56.9	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\color{blue}{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{D \cdot D} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, -M \cdot M\right)}}\right) \]
      times-frac [=>] 56.7	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{D \cdot D} \cdot \color{blue}{\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right)}, -M \cdot M\right)}\right) \]
      *-commutative [=>] 56.7	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{D \cdot D} \cdot \color{blue}{\left(\frac{d \cdot d}{D \cdot D} \cdot \frac{c0}{w \cdot h}\right)}, -M \cdot M\right)}\right) \]
      associate-*r* [=>] 58.4	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \color{blue}{\left(\frac{d \cdot d}{D \cdot D} \cdot \frac{d \cdot d}{D \cdot D}\right) \cdot \frac{c0}{w \cdot h}}, -M \cdot M\right)}\right) \]
      times-frac [=>] 58.4	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \left(\color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)} \cdot \frac{d \cdot d}{D \cdot D}\right) \cdot \frac{c0}{w \cdot h}, -M \cdot M\right)}\right) \]
      associate-*l* [=>] 58.4	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \color{blue}{\left(\frac{d}{D} \cdot \left(\frac{d}{D} \cdot \frac{d \cdot d}{D \cdot D}\right)\right)} \cdot \frac{c0}{w \cdot h}, -M \cdot M\right)}\right) \]
      times-frac [=>] 58.1	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \left(\frac{d}{D} \cdot \left(\frac{d}{D} \cdot \color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)}\right)\right) \cdot \frac{c0}{w \cdot h}, -M \cdot M\right)}\right) \]
      cube-unmult [=>] 58.1	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \left(\frac{d}{D} \cdot \color{blue}{{\left(\frac{d}{D}\right)}^{3}}\right) \cdot \frac{c0}{w \cdot h}, -M \cdot M\right)}\right) \]

   3. Taylor expanded in d around inf 48.7

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

   4. Simplified 36.0

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

      [Start] 48.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]
      times-frac [=>] 47.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{{d}^{2}}{{D}^{2}} \cdot \frac{c0}{w \cdot h}\right)}\right) \]
      associate-/r* [=>] 46.1	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{{d}^{2}}{{D}^{2}} \cdot \color{blue}{\frac{\frac{c0}{w}}{h}}\right)\right) \]
      times-frac [<=] 46.6	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{{d}^{2} \cdot \frac{c0}{w}}{{D}^{2} \cdot h}}\right) \]
      unpow2 [=>] 46.6	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot \frac{c0}{w}}{\color{blue}{\left(D \cdot D\right)} \cdot h}\right) \]
      associate-*r* [<=] 43.1	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot \frac{c0}{w}}{\color{blue}{D \cdot \left(D \cdot h\right)}}\right) \]
      times-frac [=>] 38.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{{d}^{2}}{D} \cdot \frac{\frac{c0}{w}}{D \cdot h}\right)}\right) \]
      unpow2 [=>] 38.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\color{blue}{d \cdot d}}{D} \cdot \frac{\frac{c0}{w}}{D \cdot h}\right)\right) \]
      *-commutative [=>] 38.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \frac{d \cdot d}{D}\right)}\right) \]
      associate-/l* [=>] 36.0	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \color{blue}{\frac{d}{\frac{D}{d}}}\right)\right) \]

   5. Applied egg-rr 62.9

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right) \cdot \left(2 \cdot \left(c0 \cdot \frac{0.5}{w}\right)\right)\right)} - 1} \]

   6. Simplified 32.4

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

      [Start] 62.9	\[ e^{\mathsf{log1p}\left(\left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right) \cdot \left(2 \cdot \left(c0 \cdot \frac{0.5}{w}\right)\right)\right)} - 1 \]
      expm1-def [=>] 58.4	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right) \cdot \left(2 \cdot \left(c0 \cdot \frac{0.5}{w}\right)\right)\right)\right)} \]
      expm1-log1p [=>] 35.9	\[ \color{blue}{\left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right) \cdot \left(2 \cdot \left(c0 \cdot \frac{0.5}{w}\right)\right)} \]
      *-commutative [=>] 35.9	\[ \color{blue}{\left(2 \cdot \left(c0 \cdot \frac{0.5}{w}\right)\right) \cdot \left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right)} \]
      associate-*r* [=>] 35.9	\[ \color{blue}{\left(\left(2 \cdot c0\right) \cdot \frac{0.5}{w}\right)} \cdot \left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right) \]
      *-commutative [<=] 35.9	\[ \left(\color{blue}{\left(c0 \cdot 2\right)} \cdot \frac{0.5}{w}\right) \cdot \left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right) \]
      *-commutative [=>] 35.9	\[ \color{blue}{\left(\frac{0.5}{w} \cdot \left(c0 \cdot 2\right)\right)} \cdot \left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right) \]
      metadata-eval [<=] 35.9	\[ \left(\frac{\color{blue}{\frac{1}{2}}}{w} \cdot \left(c0 \cdot 2\right)\right) \cdot \left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right) \]
      associate-/r* [<=] 35.9	\[ \left(\color{blue}{\frac{1}{2 \cdot w}} \cdot \left(c0 \cdot 2\right)\right) \cdot \left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right) \]
      *-commutative [<=] 35.9	\[ \left(\frac{1}{\color{blue}{w \cdot 2}} \cdot \left(c0 \cdot 2\right)\right) \cdot \left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right) \]
      associate-*r* [<=] 38.9	\[ \color{blue}{\frac{1}{w \cdot 2} \cdot \left(\left(c0 \cdot 2\right) \cdot \left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right)\right)} \]
      associate-/r/ [<=] 38.9	\[ \color{blue}{\frac{1}{\frac{w \cdot 2}{\left(c0 \cdot 2\right) \cdot \left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right)}}} \]
      associate-/r* [=>] 36.0	\[ \frac{1}{\color{blue}{\frac{\frac{w \cdot 2}{c0 \cdot 2}}{\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)}}} \]
      *-commutative [=>] 36.0	\[ \frac{1}{\frac{\frac{w \cdot 2}{c0 \cdot 2}}{\color{blue}{\left(d \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{D \cdot h}}}} \]
      associate-/r* [=>] 35.2	\[ \frac{1}{\color{blue}{\frac{\frac{\frac{w \cdot 2}{c0 \cdot 2}}{d \cdot \frac{d}{D}}}{\frac{\frac{c0}{w}}{D \cdot h}}}} \]
      associate-/r* [=>] 35.2	\[ \frac{1}{\frac{\frac{\color{blue}{\frac{\frac{w \cdot 2}{c0}}{2}}}{d \cdot \frac{d}{D}}}{\frac{\frac{c0}{w}}{D \cdot h}}} \]
      *-commutative [=>] 35.2	\[ \frac{1}{\frac{\frac{\frac{\frac{\color{blue}{2 \cdot w}}{c0}}{2}}{d \cdot \frac{d}{D}}}{\frac{\frac{c0}{w}}{D \cdot h}}} \]
      associate-*r/ [<=] 35.2	\[ \frac{1}{\frac{\frac{\frac{\color{blue}{2 \cdot \frac{w}{c0}}}{2}}{d \cdot \frac{d}{D}}}{\frac{\frac{c0}{w}}{D \cdot h}}} \]
      associate-/r* [<=] 35.2	\[ \frac{1}{\frac{\color{blue}{\frac{2 \cdot \frac{w}{c0}}{2 \cdot \left(d \cdot \frac{d}{D}\right)}}}{\frac{\frac{c0}{w}}{D \cdot h}}} \]
      *-commutative [<=] 35.2	\[ \frac{1}{\frac{\frac{2 \cdot \frac{w}{c0}}{\color{blue}{\left(d \cdot \frac{d}{D}\right) \cdot 2}}}{\frac{\frac{c0}{w}}{D \cdot h}}} \]
      associate-/l* [<=] 35.2	\[ \color{blue}{\frac{1 \cdot \frac{\frac{c0}{w}}{D \cdot h}}{\frac{2 \cdot \frac{w}{c0}}{\left(d \cdot \frac{d}{D}\right) \cdot 2}}} \]
      *-lft-identity [=>] 35.2	\[ \frac{\color{blue}{\frac{\frac{c0}{w}}{D \cdot h}}}{\frac{2 \cdot \frac{w}{c0}}{\left(d \cdot \frac{d}{D}\right) \cdot 2}} \]
      associate-/l* [<=] 36.0	\[ \color{blue}{\frac{\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(\left(d \cdot \frac{d}{D}\right) \cdot 2\right)}{2 \cdot \frac{w}{c0}}} \]
      associate-*r* [=>] 36.0	\[ \frac{\color{blue}{\left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right) \cdot 2}}{2 \cdot \frac{w}{c0}} \]
      associate-/l* [=>] 36.0	\[ \color{blue}{\frac{\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)}{\frac{2 \cdot \frac{w}{c0}}{2}}} \]
      associate-*r/ [=>] 36.0	\[ \frac{\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)}{\frac{\color{blue}{\frac{2 \cdot w}{c0}}}{2}} \]
      *-commutative [<=] 36.0	\[ \frac{\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)}{\frac{\frac{\color{blue}{w \cdot 2}}{c0}}{2}} \]
      associate-/r* [<=] 36.0	\[ \frac{\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)}{\color{blue}{\frac{w \cdot 2}{c0 \cdot 2}}} \]
      associate-/l* [<=] 38.9	\[ \color{blue}{\frac{\left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right) \cdot \left(c0 \cdot 2\right)}{w \cdot 2}} \]
      *-commutative [<=] 38.9	\[ \frac{\color{blue}{\left(c0 \cdot 2\right) \cdot \left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right)}}{w \cdot 2} \]
      associate-/l* [=>] 40.3	\[ \color{blue}{\frac{c0 \cdot 2}{\frac{w \cdot 2}{\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)}}} \]
      associate-/l* [=>] 40.3	\[ \color{blue}{\frac{c0}{\frac{\frac{w \cdot 2}{\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)}}{2}}} \]
      associate-/r* [<=] 40.3	\[ \frac{c0}{\color{blue}{\frac{w \cdot 2}{\left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right) \cdot 2}}} \]
      associate-*r* [<=] 40.3	\[ \frac{c0}{\frac{w \cdot 2}{\color{blue}{\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(\left(d \cdot \frac{d}{D}\right) \cdot 2\right)}}} \]
      associate-/l* [=>] 40.3	\[ \frac{c0}{\color{blue}{\frac{w}{\frac{\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(\left(d \cdot \frac{d}{D}\right) \cdot 2\right)}{2}}}} \]
      associate-*r* [=>] 40.3	\[ \frac{c0}{\frac{w}{\frac{\color{blue}{\left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right) \cdot 2}}{2}}} \]
      associate-/l* [=>] 40.3	\[ \frac{c0}{\frac{w}{\color{blue}{\frac{\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)}{\frac{2}{2}}}}} \]
      associate-*r* [=>] 36.2	\[ \frac{c0}{\frac{w}{\frac{\color{blue}{\left(\frac{\frac{c0}{w}}{D \cdot h} \cdot d\right) \cdot \frac{d}{D}}}{\frac{2}{2}}}} \]
      metadata-eval [=>] 36.2	\[ \frac{c0}{\frac{w}{\frac{\left(\frac{\frac{c0}{w}}{D \cdot h} \cdot d\right) \cdot \frac{d}{D}}{\color{blue}{1}}}} \]
      associate-*l/ [<=] 36.2	\[ \frac{c0}{\frac{w}{\color{blue}{\frac{\frac{\frac{c0}{w}}{D \cdot h} \cdot d}{1} \cdot \frac{d}{D}}}} \]
      /-rgt-identity [=>] 36.2	\[ \frac{c0}{\frac{w}{\color{blue}{\left(\frac{\frac{c0}{w}}{D \cdot h} \cdot d\right)} \cdot \frac{d}{D}}} \]
      associate-*r* [<=] 40.3	\[ \frac{c0}{\frac{w}{\color{blue}{\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)}}} \]
      associate-/r/ [=>] 36.0	\[ \color{blue}{\frac{c0}{w} \cdot \left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right)} \]
      associate-*r* [=>] 31.0	\[ \frac{c0}{w} \cdot \color{blue}{\left(\left(\frac{\frac{c0}{w}}{D \cdot h} \cdot d\right) \cdot \frac{d}{D}\right)} \]
      *-commutative [=>] 31.0	\[ \frac{c0}{w} \cdot \color{blue}{\left(\frac{d}{D} \cdot \left(\frac{\frac{c0}{w}}{D \cdot h} \cdot d\right)\right)} \]
      *-commutative [=>] 31.0	\[ \frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \color{blue}{\left(d \cdot \frac{\frac{c0}{w}}{D \cdot h}\right)}\right) \]
      associate-/l/ [=>] 32.7	\[ \frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \left(d \cdot \color{blue}{\frac{c0}{\left(D \cdot h\right) \cdot w}}\right)\right) \]
      associate-*r* [<=] 32.4	\[ \frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \left(d \cdot \frac{c0}{\color{blue}{D \cdot \left(h \cdot w\right)}}\right)\right) \]
      *-commutative [<=] 32.4	\[ \frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \left(d \cdot \frac{c0}{D \cdot \color{blue}{\left(w \cdot h\right)}}\right)\right) \]

   7. Applied egg-rr 30.3

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

   8. Applied egg-rr 34.2

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

   if -1.00000000000000001e-50 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < -0.0

   1. Initial program 28.8

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

   2. Simplified 54.3

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

      [Start] 28.8	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*l/ [<=] 35.7	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \left(d \cdot d\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      *-commutative [=>] 35.7	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(d \cdot d\right) \cdot \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-def [=>] 42.6	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]
      associate-*l* [=>] 46.3	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\color{blue}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-/r* [=>] 46.4	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \color{blue}{\frac{\frac{c0}{w}}{h \cdot \left(D \cdot D\right)}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*r* [=>] 47.1	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{\color{blue}{\left(h \cdot D\right) \cdot D}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      *-commutative [=>] 47.1	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{\color{blue}{D \cdot \left(h \cdot D\right)}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 50.6	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\color{blue}{\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*l* [=>] 51.0	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\color{blue}{\frac{c0}{w \cdot h} \cdot \left(\frac{d \cdot d}{D \cdot D} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)} - M \cdot M}\right) \]
      fma-neg [=>] 51.0	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\color{blue}{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{D \cdot D} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, -M \cdot M\right)}}\right) \]
      times-frac [=>] 50.6	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{D \cdot D} \cdot \color{blue}{\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right)}, -M \cdot M\right)}\right) \]
      *-commutative [=>] 50.6	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{D \cdot D} \cdot \color{blue}{\left(\frac{d \cdot d}{D \cdot D} \cdot \frac{c0}{w \cdot h}\right)}, -M \cdot M\right)}\right) \]
      associate-*r* [=>] 54.3	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \color{blue}{\left(\frac{d \cdot d}{D \cdot D} \cdot \frac{d \cdot d}{D \cdot D}\right) \cdot \frac{c0}{w \cdot h}}, -M \cdot M\right)}\right) \]
      times-frac [=>] 54.3	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \left(\color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)} \cdot \frac{d \cdot d}{D \cdot D}\right) \cdot \frac{c0}{w \cdot h}, -M \cdot M\right)}\right) \]
      associate-*l* [=>] 54.3	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \color{blue}{\left(\frac{d}{D} \cdot \left(\frac{d}{D} \cdot \frac{d \cdot d}{D \cdot D}\right)\right)} \cdot \frac{c0}{w \cdot h}, -M \cdot M\right)}\right) \]
      times-frac [=>] 54.3	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \left(\frac{d}{D} \cdot \left(\frac{d}{D} \cdot \color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)}\right)\right) \cdot \frac{c0}{w \cdot h}, -M \cdot M\right)}\right) \]
      cube-unmult [=>] 54.3	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \left(\frac{d}{D} \cdot \color{blue}{{\left(\frac{d}{D}\right)}^{3}}\right) \cdot \frac{c0}{w \cdot h}, -M \cdot M\right)}\right) \]

   3. Taylor expanded in c0 around -inf 33.4

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

   4. Simplified 28.2

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

      [Start] 33.4	\[ -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w} + 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]
      +-commutative [=>] 33.4	\[ \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} + -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}} \]
      fma-def [=>] 33.4	\[ \color{blue}{\mathsf{fma}\left(0.25, \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}, -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right)} \]
      associate-/l* [=>] 33.3	\[ \mathsf{fma}\left(0.25, \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{{M}^{2} \cdot h}}}, -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right) \]
      associate-/r/ [=>] 32.6	\[ \mathsf{fma}\left(0.25, \color{blue}{\frac{{D}^{2}}{{d}^{2}} \cdot \left({M}^{2} \cdot h\right)}, -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right) \]
      unpow2 [=>] 32.6	\[ \mathsf{fma}\left(0.25, \frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \left({M}^{2} \cdot h\right), -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right) \]
      unpow2 [=>] 32.6	\[ \mathsf{fma}\left(0.25, \frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \left({M}^{2} \cdot h\right), -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right) \]
      times-frac [=>] 31.0	\[ \mathsf{fma}\left(0.25, \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left({M}^{2} \cdot h\right), -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right) \]
      unpow2 [=>] 31.0	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right), -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right) \]
      associate-*l* [=>] 29.4	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(M \cdot \left(M \cdot h\right)\right)}, -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right) \]
      associate-/l* [=>] 29.8	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), -0.5 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{\frac{w}{{c0}^{2}}}}\right) \]
      associate-*r/ [=>] 29.8	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \color{blue}{\frac{-0.5 \cdot \left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}{\frac{w}{{c0}^{2}}}}\right) \]
      distribute-rgt1-in [=>] 29.8	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{-0.5 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}}{\frac{w}{{c0}^{2}}}\right) \]
      metadata-eval [=>] 29.8	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{-0.5 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}{\frac{w}{{c0}^{2}}}\right) \]
      mul0-lft [=>] 28.6	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{-0.5 \cdot \color{blue}{0}}{\frac{w}{{c0}^{2}}}\right) \]
      metadata-eval [=>] 28.6	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{\color{blue}{0}}{\frac{w}{{c0}^{2}}}\right) \]
      mul0-lft [<=] 29.8	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{\color{blue}{0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}}{\frac{w}{{c0}^{2}}}\right) \]
      metadata-eval [<=] 29.8	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{\color{blue}{\left(-1 + 1\right)} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{\frac{w}{{c0}^{2}}}\right) \]
      distribute-rgt1-in [<=] 29.8	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{\color{blue}{\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}}{\frac{w}{{c0}^{2}}}\right) \]
      associate-/r/ [=>] 29.4	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{w} \cdot {c0}^{2}}\right) \]
      distribute-rgt1-in [=>] 29.4	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{\color{blue}{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}}{w} \cdot {c0}^{2}\right) \]
      metadata-eval [=>] 29.4	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{w} \cdot {c0}^{2}\right) \]
      mul0-lft [=>] 28.2	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{\color{blue}{0}}{w} \cdot {c0}^{2}\right) \]
      unpow2 [=>] 28.2	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{0}{w} \cdot \color{blue}{\left(c0 \cdot c0\right)}\right) \]

   5. Taylor expanded in w around 0 23.0

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

   6. Applied egg-rr 21.1

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

   7. Taylor expanded in D around 0 24.8

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

   8. Simplified 18.6

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

      [Start] 24.8	\[ \left(0.25 \cdot \frac{{D}^{2} \cdot M}{{d}^{2}}\right) \cdot \left(M \cdot h\right) \]
      *-commutative [=>] 24.8	\[ \left(0.25 \cdot \frac{\color{blue}{M \cdot {D}^{2}}}{{d}^{2}}\right) \cdot \left(M \cdot h\right) \]
      unpow2 [=>] 24.8	\[ \left(0.25 \cdot \frac{M \cdot \color{blue}{\left(D \cdot D\right)}}{{d}^{2}}\right) \cdot \left(M \cdot h\right) \]
      unpow2 [=>] 24.8	\[ \left(0.25 \cdot \frac{M \cdot \left(D \cdot D\right)}{\color{blue}{d \cdot d}}\right) \cdot \left(M \cdot h\right) \]
      times-frac [=>] 21.7	\[ \left(0.25 \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{D \cdot D}{d}\right)}\right) \cdot \left(M \cdot h\right) \]
      associate-/l* [=>] 20.8	\[ \left(0.25 \cdot \left(\frac{M}{d} \cdot \color{blue}{\frac{D}{\frac{d}{D}}}\right)\right) \cdot \left(M \cdot h\right) \]
      times-frac [<=] 21.0	\[ \left(0.25 \cdot \color{blue}{\frac{M \cdot D}{d \cdot \frac{d}{D}}}\right) \cdot \left(M \cdot h\right) \]
      *-commutative [<=] 21.0	\[ \left(0.25 \cdot \frac{\color{blue}{D \cdot M}}{d \cdot \frac{d}{D}}\right) \cdot \left(M \cdot h\right) \]
      times-frac [=>] 18.6	\[ \left(0.25 \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{M}{\frac{d}{D}}\right)}\right) \cdot \left(M \cdot h\right) \]

   if -0.0 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 48.9

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

   2. Simplified 56.5

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

      [Start] 48.9	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*l/ [<=] 49.3	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \left(d \cdot d\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      *-commutative [=>] 49.3	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(d \cdot d\right) \cdot \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-def [=>] 49.3	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]
      associate-*l* [=>] 50.5	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\color{blue}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-/r* [=>] 50.5	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \color{blue}{\frac{\frac{c0}{w}}{h \cdot \left(D \cdot D\right)}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*r* [=>] 50.5	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{\color{blue}{\left(h \cdot D\right) \cdot D}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      *-commutative [=>] 50.5	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{\color{blue}{D \cdot \left(h \cdot D\right)}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 52.8	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\color{blue}{\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*l* [=>] 53.8	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\color{blue}{\frac{c0}{w \cdot h} \cdot \left(\frac{d \cdot d}{D \cdot D} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)} - M \cdot M}\right) \]
      fma-neg [=>] 53.8	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\color{blue}{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{D \cdot D} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, -M \cdot M\right)}}\right) \]
      times-frac [=>] 53.5	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{D \cdot D} \cdot \color{blue}{\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right)}, -M \cdot M\right)}\right) \]
      *-commutative [=>] 53.5	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{D \cdot D} \cdot \color{blue}{\left(\frac{d \cdot d}{D \cdot D} \cdot \frac{c0}{w \cdot h}\right)}, -M \cdot M\right)}\right) \]
      associate-*r* [=>] 56.5	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \color{blue}{\left(\frac{d \cdot d}{D \cdot D} \cdot \frac{d \cdot d}{D \cdot D}\right) \cdot \frac{c0}{w \cdot h}}, -M \cdot M\right)}\right) \]
      times-frac [=>] 56.5	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \left(\color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)} \cdot \frac{d \cdot d}{D \cdot D}\right) \cdot \frac{c0}{w \cdot h}, -M \cdot M\right)}\right) \]
      associate-*l* [=>] 56.5	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \color{blue}{\left(\frac{d}{D} \cdot \left(\frac{d}{D} \cdot \frac{d \cdot d}{D \cdot D}\right)\right)} \cdot \frac{c0}{w \cdot h}, -M \cdot M\right)}\right) \]
      times-frac [=>] 56.5	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \left(\frac{d}{D} \cdot \left(\frac{d}{D} \cdot \color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)}\right)\right) \cdot \frac{c0}{w \cdot h}, -M \cdot M\right)}\right) \]
      cube-unmult [=>] 56.5	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \left(\frac{d}{D} \cdot \color{blue}{{\left(\frac{d}{D}\right)}^{3}}\right) \cdot \frac{c0}{w \cdot h}, -M \cdot M\right)}\right) \]

   3. Taylor expanded in d around inf 42.2

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

   4. Simplified 35.0

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

      [Start] 42.2	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]
      times-frac [=>] 43.7	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{{d}^{2}}{{D}^{2}} \cdot \frac{c0}{w \cdot h}\right)}\right) \]
      associate-/r* [=>] 43.0	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{{d}^{2}}{{D}^{2}} \cdot \color{blue}{\frac{\frac{c0}{w}}{h}}\right)\right) \]
      times-frac [<=] 42.1	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{{d}^{2} \cdot \frac{c0}{w}}{{D}^{2} \cdot h}}\right) \]
      unpow2 [=>] 42.1	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot \frac{c0}{w}}{\color{blue}{\left(D \cdot D\right)} \cdot h}\right) \]
      associate-*r* [<=] 39.8	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot \frac{c0}{w}}{\color{blue}{D \cdot \left(D \cdot h\right)}}\right) \]
      times-frac [=>] 36.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{{d}^{2}}{D} \cdot \frac{\frac{c0}{w}}{D \cdot h}\right)}\right) \]
      unpow2 [=>] 36.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\color{blue}{d \cdot d}}{D} \cdot \frac{\frac{c0}{w}}{D \cdot h}\right)\right) \]
      *-commutative [=>] 36.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \frac{d \cdot d}{D}\right)}\right) \]
      associate-/l* [=>] 35.0	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \color{blue}{\frac{d}{\frac{D}{d}}}\right)\right) \]

   5. Applied egg-rr 44.1

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right) \cdot \left(2 \cdot \left(c0 \cdot \frac{0.5}{w}\right)\right)\right)} - 1} \]

   6. Simplified 28.1

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

      [Start] 44.1	\[ e^{\mathsf{log1p}\left(\left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right) \cdot \left(2 \cdot \left(c0 \cdot \frac{0.5}{w}\right)\right)\right)} - 1 \]
      expm1-def [=>] 36.6	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right) \cdot \left(2 \cdot \left(c0 \cdot \frac{0.5}{w}\right)\right)\right)\right)} \]
      expm1-log1p [=>] 35.0	\[ \color{blue}{\left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right) \cdot \left(2 \cdot \left(c0 \cdot \frac{0.5}{w}\right)\right)} \]
      *-commutative [=>] 35.0	\[ \color{blue}{\left(2 \cdot \left(c0 \cdot \frac{0.5}{w}\right)\right) \cdot \left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right)} \]
      associate-*r* [=>] 35.0	\[ \color{blue}{\left(\left(2 \cdot c0\right) \cdot \frac{0.5}{w}\right)} \cdot \left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right) \]
      *-commutative [<=] 35.0	\[ \left(\color{blue}{\left(c0 \cdot 2\right)} \cdot \frac{0.5}{w}\right) \cdot \left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right) \]
      *-commutative [=>] 35.0	\[ \color{blue}{\left(\frac{0.5}{w} \cdot \left(c0 \cdot 2\right)\right)} \cdot \left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right) \]
      metadata-eval [<=] 35.0	\[ \left(\frac{\color{blue}{\frac{1}{2}}}{w} \cdot \left(c0 \cdot 2\right)\right) \cdot \left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right) \]
      associate-/r* [<=] 35.0	\[ \left(\color{blue}{\frac{1}{2 \cdot w}} \cdot \left(c0 \cdot 2\right)\right) \cdot \left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right) \]
      *-commutative [<=] 35.0	\[ \left(\frac{1}{\color{blue}{w \cdot 2}} \cdot \left(c0 \cdot 2\right)\right) \cdot \left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right) \]
      associate-*r* [<=] 37.9	\[ \color{blue}{\frac{1}{w \cdot 2} \cdot \left(\left(c0 \cdot 2\right) \cdot \left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right)\right)} \]
      associate-/r/ [<=] 37.9	\[ \color{blue}{\frac{1}{\frac{w \cdot 2}{\left(c0 \cdot 2\right) \cdot \left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right)}}} \]
      associate-/r* [=>] 35.1	\[ \frac{1}{\color{blue}{\frac{\frac{w \cdot 2}{c0 \cdot 2}}{\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)}}} \]
      *-commutative [=>] 35.1	\[ \frac{1}{\frac{\frac{w \cdot 2}{c0 \cdot 2}}{\color{blue}{\left(d \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{D \cdot h}}}} \]
      associate-/r* [=>] 34.3	\[ \frac{1}{\color{blue}{\frac{\frac{\frac{w \cdot 2}{c0 \cdot 2}}{d \cdot \frac{d}{D}}}{\frac{\frac{c0}{w}}{D \cdot h}}}} \]
      associate-/r* [=>] 34.3	\[ \frac{1}{\frac{\frac{\color{blue}{\frac{\frac{w \cdot 2}{c0}}{2}}}{d \cdot \frac{d}{D}}}{\frac{\frac{c0}{w}}{D \cdot h}}} \]
      *-commutative [=>] 34.3	\[ \frac{1}{\frac{\frac{\frac{\frac{\color{blue}{2 \cdot w}}{c0}}{2}}{d \cdot \frac{d}{D}}}{\frac{\frac{c0}{w}}{D \cdot h}}} \]
      associate-*r/ [<=] 34.3	\[ \frac{1}{\frac{\frac{\frac{\color{blue}{2 \cdot \frac{w}{c0}}}{2}}{d \cdot \frac{d}{D}}}{\frac{\frac{c0}{w}}{D \cdot h}}} \]
      associate-/r* [<=] 34.3	\[ \frac{1}{\frac{\color{blue}{\frac{2 \cdot \frac{w}{c0}}{2 \cdot \left(d \cdot \frac{d}{D}\right)}}}{\frac{\frac{c0}{w}}{D \cdot h}}} \]
      *-commutative [<=] 34.3	\[ \frac{1}{\frac{\frac{2 \cdot \frac{w}{c0}}{\color{blue}{\left(d \cdot \frac{d}{D}\right) \cdot 2}}}{\frac{\frac{c0}{w}}{D \cdot h}}} \]
      associate-/l* [<=] 34.2	\[ \color{blue}{\frac{1 \cdot \frac{\frac{c0}{w}}{D \cdot h}}{\frac{2 \cdot \frac{w}{c0}}{\left(d \cdot \frac{d}{D}\right) \cdot 2}}} \]
      *-lft-identity [=>] 34.2	\[ \frac{\color{blue}{\frac{\frac{c0}{w}}{D \cdot h}}}{\frac{2 \cdot \frac{w}{c0}}{\left(d \cdot \frac{d}{D}\right) \cdot 2}} \]
      associate-/l* [<=] 35.1	\[ \color{blue}{\frac{\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(\left(d \cdot \frac{d}{D}\right) \cdot 2\right)}{2 \cdot \frac{w}{c0}}} \]
      associate-*r* [=>] 35.1	\[ \frac{\color{blue}{\left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right) \cdot 2}}{2 \cdot \frac{w}{c0}} \]
      associate-/l* [=>] 35.1	\[ \color{blue}{\frac{\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)}{\frac{2 \cdot \frac{w}{c0}}{2}}} \]
      associate-*r/ [=>] 35.1	\[ \frac{\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)}{\frac{\color{blue}{\frac{2 \cdot w}{c0}}}{2}} \]
      *-commutative [<=] 35.1	\[ \frac{\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)}{\frac{\frac{\color{blue}{w \cdot 2}}{c0}}{2}} \]
      associate-/r* [<=] 35.1	\[ \frac{\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)}{\color{blue}{\frac{w \cdot 2}{c0 \cdot 2}}} \]
      associate-/l* [<=] 37.9	\[ \color{blue}{\frac{\left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right) \cdot \left(c0 \cdot 2\right)}{w \cdot 2}} \]
      *-commutative [<=] 37.9	\[ \frac{\color{blue}{\left(c0 \cdot 2\right) \cdot \left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right)}}{w \cdot 2} \]
      associate-/l* [=>] 38.5	\[ \color{blue}{\frac{c0 \cdot 2}{\frac{w \cdot 2}{\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)}}} \]
      associate-/l* [=>] 38.5	\[ \color{blue}{\frac{c0}{\frac{\frac{w \cdot 2}{\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)}}{2}}} \]
      associate-/r* [<=] 38.5	\[ \frac{c0}{\color{blue}{\frac{w \cdot 2}{\left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right) \cdot 2}}} \]
      associate-*r* [<=] 38.5	\[ \frac{c0}{\frac{w \cdot 2}{\color{blue}{\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(\left(d \cdot \frac{d}{D}\right) \cdot 2\right)}}} \]
      associate-/l* [=>] 38.5	\[ \frac{c0}{\color{blue}{\frac{w}{\frac{\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(\left(d \cdot \frac{d}{D}\right) \cdot 2\right)}{2}}}} \]
      associate-*r* [=>] 38.5	\[ \frac{c0}{\frac{w}{\frac{\color{blue}{\left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right) \cdot 2}}{2}}} \]
      associate-/l* [=>] 38.5	\[ \frac{c0}{\frac{w}{\color{blue}{\frac{\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)}{\frac{2}{2}}}}} \]
      associate-*r* [=>] 32.8	\[ \frac{c0}{\frac{w}{\frac{\color{blue}{\left(\frac{\frac{c0}{w}}{D \cdot h} \cdot d\right) \cdot \frac{d}{D}}}{\frac{2}{2}}}} \]
      metadata-eval [=>] 32.8	\[ \frac{c0}{\frac{w}{\frac{\left(\frac{\frac{c0}{w}}{D \cdot h} \cdot d\right) \cdot \frac{d}{D}}{\color{blue}{1}}}} \]
      associate-*l/ [<=] 32.8	\[ \frac{c0}{\frac{w}{\color{blue}{\frac{\frac{\frac{c0}{w}}{D \cdot h} \cdot d}{1} \cdot \frac{d}{D}}}} \]
      /-rgt-identity [=>] 32.8	\[ \frac{c0}{\frac{w}{\color{blue}{\left(\frac{\frac{c0}{w}}{D \cdot h} \cdot d\right)} \cdot \frac{d}{D}}} \]
      associate-*r* [<=] 38.5	\[ \frac{c0}{\frac{w}{\color{blue}{\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)}}} \]
      associate-/r/ [=>] 35.0	\[ \color{blue}{\frac{c0}{w} \cdot \left(\frac{\frac{c0}{w}}{D \cdot h} \cdot \left(d \cdot \frac{d}{D}\right)\right)} \]
      associate-*r* [=>] 28.0	\[ \frac{c0}{w} \cdot \color{blue}{\left(\left(\frac{\frac{c0}{w}}{D \cdot h} \cdot d\right) \cdot \frac{d}{D}\right)} \]
      *-commutative [=>] 28.0	\[ \frac{c0}{w} \cdot \color{blue}{\left(\frac{d}{D} \cdot \left(\frac{\frac{c0}{w}}{D \cdot h} \cdot d\right)\right)} \]
      *-commutative [=>] 28.0	\[ \frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \color{blue}{\left(d \cdot \frac{\frac{c0}{w}}{D \cdot h}\right)}\right) \]
      associate-/l/ [=>] 29.1	\[ \frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \left(d \cdot \color{blue}{\frac{c0}{\left(D \cdot h\right) \cdot w}}\right)\right) \]
      associate-*r* [<=] 28.1	\[ \frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \left(d \cdot \frac{c0}{\color{blue}{D \cdot \left(h \cdot w\right)}}\right)\right) \]
      *-commutative [<=] 28.1	\[ \frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \left(d \cdot \frac{c0}{D \cdot \color{blue}{\left(w \cdot h\right)}}\right)\right) \]

   7. Applied egg-rr 26.9

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

   8. Applied egg-rr 24.4

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 64.0

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

   2. Simplified 63.7

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

      [Start] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*l/ [<=] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \left(d \cdot d\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      *-commutative [=>] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(d \cdot d\right) \cdot \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-def [=>] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]
      associate-*l* [=>] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\color{blue}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-/r* [=>] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \color{blue}{\frac{\frac{c0}{w}}{h \cdot \left(D \cdot D\right)}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*r* [=>] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{\color{blue}{\left(h \cdot D\right) \cdot D}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      *-commutative [=>] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{\color{blue}{D \cdot \left(h \cdot D\right)}}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\color{blue}{\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      associate-*l* [=>] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\color{blue}{\frac{c0}{w \cdot h} \cdot \left(\frac{d \cdot d}{D \cdot D} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)} - M \cdot M}\right) \]
      fma-neg [=>] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\color{blue}{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{D \cdot D} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, -M \cdot M\right)}}\right) \]
      times-frac [=>] 63.8	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{D \cdot D} \cdot \color{blue}{\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right)}, -M \cdot M\right)}\right) \]
      *-commutative [=>] 63.8	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{D \cdot D} \cdot \color{blue}{\left(\frac{d \cdot d}{D \cdot D} \cdot \frac{c0}{w \cdot h}\right)}, -M \cdot M\right)}\right) \]
      associate-*r* [=>] 63.8	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \color{blue}{\left(\frac{d \cdot d}{D \cdot D} \cdot \frac{d \cdot d}{D \cdot D}\right) \cdot \frac{c0}{w \cdot h}}, -M \cdot M\right)}\right) \]
      times-frac [=>] 63.8	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \left(\color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)} \cdot \frac{d \cdot d}{D \cdot D}\right) \cdot \frac{c0}{w \cdot h}, -M \cdot M\right)}\right) \]
      associate-*l* [=>] 63.8	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \color{blue}{\left(\frac{d}{D} \cdot \left(\frac{d}{D} \cdot \frac{d \cdot d}{D \cdot D}\right)\right)} \cdot \frac{c0}{w \cdot h}, -M \cdot M\right)}\right) \]
      times-frac [=>] 63.7	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \left(\frac{d}{D} \cdot \left(\frac{d}{D} \cdot \color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)}\right)\right) \cdot \frac{c0}{w \cdot h}, -M \cdot M\right)}\right) \]
      cube-unmult [=>] 63.7	\[ \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{\frac{c0}{w}}{D \cdot \left(h \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \left(\frac{d}{D} \cdot \color{blue}{{\left(\frac{d}{D}\right)}^{3}}\right) \cdot \frac{c0}{w \cdot h}, -M \cdot M\right)}\right) \]

   3. Taylor expanded in c0 around -inf 63.5

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

   4. Simplified 34.0

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

      [Start] 63.5	\[ -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w} + 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]
      +-commutative [=>] 63.5	\[ \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} + -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}} \]
      fma-def [=>] 63.5	\[ \color{blue}{\mathsf{fma}\left(0.25, \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}, -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right)} \]
      associate-/l* [=>] 63.5	\[ \mathsf{fma}\left(0.25, \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{{M}^{2} \cdot h}}}, -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right) \]
      associate-/r/ [=>] 63.5	\[ \mathsf{fma}\left(0.25, \color{blue}{\frac{{D}^{2}}{{d}^{2}} \cdot \left({M}^{2} \cdot h\right)}, -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right) \]
      unpow2 [=>] 63.5	\[ \mathsf{fma}\left(0.25, \frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \left({M}^{2} \cdot h\right), -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right) \]
      unpow2 [=>] 63.5	\[ \mathsf{fma}\left(0.25, \frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \left({M}^{2} \cdot h\right), -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right) \]
      times-frac [=>] 63.3	\[ \mathsf{fma}\left(0.25, \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left({M}^{2} \cdot h\right), -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right) \]
      unpow2 [=>] 63.3	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right), -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right) \]
      associate-*l* [=>] 63.2	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(M \cdot \left(M \cdot h\right)\right)}, -0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}\right) \]
      associate-/l* [=>] 63.4	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), -0.5 \cdot \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{\frac{w}{{c0}^{2}}}}\right) \]
      associate-*r/ [=>] 63.4	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \color{blue}{\frac{-0.5 \cdot \left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}{\frac{w}{{c0}^{2}}}}\right) \]
      distribute-rgt1-in [=>] 63.4	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{-0.5 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}}{\frac{w}{{c0}^{2}}}\right) \]
      metadata-eval [=>] 63.4	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{-0.5 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}{\frac{w}{{c0}^{2}}}\right) \]
      mul0-lft [=>] 37.0	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{-0.5 \cdot \color{blue}{0}}{\frac{w}{{c0}^{2}}}\right) \]
      metadata-eval [=>] 37.0	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{\color{blue}{0}}{\frac{w}{{c0}^{2}}}\right) \]
      mul0-lft [<=] 63.4	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{\color{blue}{0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}}{\frac{w}{{c0}^{2}}}\right) \]
      metadata-eval [<=] 63.4	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{\color{blue}{\left(-1 + 1\right)} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{\frac{w}{{c0}^{2}}}\right) \]
      distribute-rgt1-in [<=] 63.4	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{\color{blue}{\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}}{\frac{w}{{c0}^{2}}}\right) \]
      associate-/r/ [=>] 63.2	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \color{blue}{\frac{\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{w} \cdot {c0}^{2}}\right) \]
      distribute-rgt1-in [=>] 63.2	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{\color{blue}{\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}}{w} \cdot {c0}^{2}\right) \]
      metadata-eval [=>] 63.2	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{w} \cdot {c0}^{2}\right) \]
      mul0-lft [=>] 34.0	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{\color{blue}{0}}{w} \cdot {c0}^{2}\right) \]
      unpow2 [=>] 34.0	\[ \mathsf{fma}\left(0.25, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right), \frac{0}{w} \cdot \color{blue}{\left(c0 \cdot c0\right)}\right) \]

   5. Taylor expanded in w around 0 21.3

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

   6. Applied egg-rr 14.2

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

   7. Applied egg-rr 14.2

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

   8. Simplified 13.7

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

      [Start] 14.2	\[ \frac{0.25 \cdot \left(\frac{D}{d} \cdot \left(M \cdot \left(-h\right)\right)\right)}{\frac{\frac{d}{D}}{-M}} \]
      associate-/r/ [=>] 16.3	\[ \color{blue}{\frac{0.25 \cdot \left(\frac{D}{d} \cdot \left(M \cdot \left(-h\right)\right)\right)}{\frac{d}{D}} \cdot \left(-M\right)} \]
      associate-*r* [=>] 13.7	\[ \frac{0.25 \cdot \color{blue}{\left(\left(\frac{D}{d} \cdot M\right) \cdot \left(-h\right)\right)}}{\frac{d}{D}} \cdot \left(-M\right) \]
      *-commutative [=>] 13.7	\[ \frac{0.25 \cdot \left(\color{blue}{\left(M \cdot \frac{D}{d}\right)} \cdot \left(-h\right)\right)}{\frac{d}{D}} \cdot \left(-M\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 15.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -1 \cdot 10^{-50}:\\ \;\;\;\;d \cdot \frac{d}{\left(D \cdot \left(\left(w \cdot h\right) \cdot \frac{D}{c0}\right)\right) \cdot \frac{w}{c0}}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{M}{\frac{d}{D}}\right)\right) \cdot \left(h \cdot M\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{d}{\frac{D}{\frac{c0}{w} \cdot \frac{d}{\left(w \cdot h\right) \cdot \frac{D}{c0}}}}\\ \mathbf{else}:\\ \;\;\;\;M \cdot \frac{0.25 \cdot \left(h \cdot \left(M \cdot \frac{D}{d}\right)\right)}{\frac{d}{D}}\\ \end{array} \]

Alternatives

Alternative 1
Error	29.1
Cost	1489

\[\begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{+122}:\\ \;\;\;\;0\\ \mathbf{elif}\;d \leq -8.2 \cdot 10^{-73} \lor \neg \left(d \leq 2 \cdot 10^{-117}\right) \land d \leq 4.7 \cdot 10^{+88}:\\ \;\;\;\;0.25 \cdot \left(\left(M \cdot M\right) \cdot \frac{D \cdot \left(h \cdot D\right)}{d \cdot d}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 2
Error	22.9
Cost	1481

\[\begin{array}{l} \mathbf{if}\;M \cdot M \leq 4 \cdot 10^{+107} \lor \neg \left(M \cdot M \leq 2 \cdot 10^{+229}\right):\\ \;\;\;\;0.25 \cdot \frac{M}{\frac{\frac{d}{D} \cdot \frac{d}{D}}{h \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{D}{\frac{\frac{d}{\frac{h}{\frac{d}{M \cdot M}}}}{D}}\\ \end{array} \]

Alternative 3
Error	29.7
Cost	1480

\[\begin{array}{l} \mathbf{if}\;d \cdot d \leq 1.08 \cdot 10^{-232}:\\ \;\;\;\;0\\ \mathbf{elif}\;d \cdot d \leq 1.65 \cdot 10^{+149}:\\ \;\;\;\;0.25 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4
Error	26.5
Cost	1220

\[\begin{array}{l} \mathbf{if}\;M \cdot M \leq 1.05 \cdot 10^{+270}:\\ \;\;\;\;0.25 \cdot \frac{D}{\frac{\frac{d}{\frac{h}{\frac{d}{M \cdot M}}}}{D}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Error	18.6
Cost	960

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

Alternative 6
Error	18.0
Cost	960

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

Alternative 7
Error	31.3
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022354 
(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))))))