Henrywood and Agarwal, Equation (13)

Average Error: 59.7 → 19.0

Time: 41.8s

Precision: binary64

Cost: 46088

Math

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

↓

\[\begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := t_0 \cdot \left(t_1 + \sqrt{t_1 \cdot t_1 - M \cdot M}\right)\\ \mathbf{if}\;t_2 \leq 10^{+301}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \left(d \cdot \frac{\frac{\frac{c0}{w}}{h \cdot \frac{D}{d}}}{D}\right)\right)\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;{\left(\frac{\sqrt{0.5} \cdot \left(c0 \cdot d\right)}{\frac{D}{\sqrt{2}}} \cdot \frac{\sqrt{\frac{1}{h}}}{w}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{\frac{h \cdot D}{\frac{d}{M}}}{\frac{d}{M}}\right)\\ \end{array} \]

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
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_2 (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))))
   (if (<= t_2 1e+301)
     (* t_0 (* 2.0 (* d (/ (/ (/ c0 w) (* h (/ D d))) D))))
     (if (<= t_2 INFINITY)
       (pow
        (* (/ (* (sqrt 0.5) (* c0 d)) (/ D (sqrt 2.0))) (/ (sqrt (/ 1.0 h)) w))
        2.0)
       (* 0.25 (* D (/ (/ (* h D) (/ d M)) (/ d M))))))))

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 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= 1e+301) {
		tmp = t_0 * (2.0 * (d * (((c0 / w) / (h * (D / d))) / D)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = pow((((sqrt(0.5) * (c0 * d)) / (D / sqrt(2.0))) * (sqrt((1.0 / h)) / w)), 2.0);
	} else {
		tmp = 0.25 * (D * (((h * D) / (d / M)) / (d / M)));
	}
	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 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= 1e+301) {
		tmp = t_0 * (2.0 * (d * (((c0 / w) / (h * (D / d))) / D)));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = Math.pow((((Math.sqrt(0.5) * (c0 * d)) / (D / Math.sqrt(2.0))) * (Math.sqrt((1.0 / h)) / w)), 2.0);
	} else {
		tmp = 0.25 * (D * (((h * D) / (d / M)) / (d / M)));
	}
	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 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_2 = t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))
	tmp = 0
	if t_2 <= 1e+301:
		tmp = t_0 * (2.0 * (d * (((c0 / w) / (h * (D / d))) / D)))
	elif t_2 <= math.inf:
		tmp = math.pow((((math.sqrt(0.5) * (c0 * d)) / (D / math.sqrt(2.0))) * (math.sqrt((1.0 / h)) / w)), 2.0)
	else:
		tmp = 0.25 * (D * (((h * D) / (d / M)) / (d / M)))
	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(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_2 = Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))))
	tmp = 0.0
	if (t_2 <= 1e+301)
		tmp = Float64(t_0 * Float64(2.0 * Float64(d * Float64(Float64(Float64(c0 / w) / Float64(h * Float64(D / d))) / D))));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(Float64(sqrt(0.5) * Float64(c0 * d)) / Float64(D / sqrt(2.0))) * Float64(sqrt(Float64(1.0 / h)) / w)) ^ 2.0;
	else
		tmp = Float64(0.25 * Float64(D * Float64(Float64(Float64(h * D) / Float64(d / M)) / Float64(d / M))));
	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 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	tmp = 0.0;
	if (t_2 <= 1e+301)
		tmp = t_0 * (2.0 * (d * (((c0 / w) / (h * (D / d))) / D)));
	elseif (t_2 <= Inf)
		tmp = (((sqrt(0.5) * (c0 * d)) / (D / sqrt(2.0))) * (sqrt((1.0 / h)) / w)) ^ 2.0;
	else
		tmp = 0.25 * (D * (((h * D) / (d / M)) / (d / M)));
	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[(c0 / N[(2.0 * w), $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[(t$95$0 * 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+301], N[(t$95$0 * N[(2.0 * N[(d * N[(N[(N[(c0 / w), $MachinePrecision] / N[(h * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Power[N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[(c0 * d), $MachinePrecision]), $MachinePrecision] / N[(D / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(0.25 * N[(D * N[(N[(N[(h * D), $MachinePrecision] / N[(d / M), $MachinePrecision]), $MachinePrecision] / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 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))))) < 1.00000000000000005e301

   1. Initial program 34.7

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

   2. Applied egg-rr 34.9

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

   3. Taylor expanded in c0 around inf 37.4

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

   4. Simplified 30.2

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

      [Start] 37.4	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]
      times-frac [=>] 41.5	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{{d}^{2}}{{D}^{2}} \cdot \frac{c0}{w \cdot h}\right)}\right) \]
      *-commutative [=>] 41.5	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{c0}{w \cdot h} \cdot \frac{{d}^{2}}{{D}^{2}}\right)}\right) \]
      associate-/r* [=>] 41.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\color{blue}{\frac{\frac{c0}{w}}{h}} \cdot \frac{{d}^{2}}{{D}^{2}}\right)\right) \]
      unpow2 [=>] 41.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{c0}{w}}{h} \cdot \frac{\color{blue}{d \cdot d}}{{D}^{2}}\right)\right) \]
      unpow2 [=>] 41.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{c0}{w}}{h} \cdot \frac{d \cdot d}{\color{blue}{D \cdot D}}\right)\right) \]
      associate-/r* [=>] 39.6	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{c0}{w}}{h} \cdot \color{blue}{\frac{\frac{d \cdot d}{D}}{D}}\right)\right) \]
      associate-/l* [=>] 38.6	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{c0}{w}}{h} \cdot \frac{\color{blue}{\frac{d}{\frac{D}{d}}}}{D}\right)\right) \]
      times-frac [<=] 36.0	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{\frac{c0}{w} \cdot \frac{d}{\frac{D}{d}}}{h \cdot D}}\right) \]
      associate-*l/ [<=] 36.0	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{\frac{c0}{w}}{h \cdot D} \cdot \frac{d}{\frac{D}{d}}\right)}\right) \]
      associate-/r* [<=] 36.5	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\color{blue}{\frac{c0}{w \cdot \left(h \cdot D\right)}} \cdot \frac{d}{\frac{D}{d}}\right)\right) \]
      associate-*r/ [=>] 32.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\frac{\frac{c0}{w \cdot \left(h \cdot D\right)} \cdot d}{\frac{D}{d}}}\right) \]
      associate-/r/ [=>] 32.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(\frac{\frac{c0}{w \cdot \left(h \cdot D\right)} \cdot d}{D} \cdot d\right)}\right) \]
      *-commutative [=>] 32.3	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \color{blue}{\left(d \cdot \frac{\frac{c0}{w \cdot \left(h \cdot D\right)} \cdot d}{D}\right)}\right) \]
      associate-/r* [=>] 31.9	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(d \cdot \frac{\color{blue}{\frac{\frac{c0}{w}}{h \cdot D}} \cdot d}{D}\right)\right) \]
      associate-*l/ [=>] 30.4	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(d \cdot \frac{\color{blue}{\frac{\frac{c0}{w} \cdot d}{h \cdot D}}}{D}\right)\right) \]
      associate-/l* [=>] 31.1	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(d \cdot \frac{\color{blue}{\frac{\frac{c0}{w}}{\frac{h \cdot D}{d}}}}{D}\right)\right) \]
      *-commutative [=>] 31.1	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(d \cdot \frac{\frac{\frac{c0}{w}}{\frac{\color{blue}{D \cdot h}}{d}}}{D}\right)\right) \]
      associate-*l/ [<=] 30.2	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(d \cdot \frac{\frac{\frac{c0}{w}}{\color{blue}{\frac{D}{d} \cdot h}}}{D}\right)\right) \]
      *-commutative [=>] 30.2	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(d \cdot \frac{\frac{\frac{c0}{w}}{\color{blue}{h \cdot \frac{D}{d}}}}{D}\right)\right) \]

   if 1.00000000000000005e301 < (*.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 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 62.7

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

      [Start] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-neg [=>] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\color{blue}{\mathsf{fma}\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, -M \cdot M\right)}}\right) \]
      times-frac [=>] 64.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}}, \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, -M \cdot M\right)}\right) \]
      times-frac [=>] 62.7	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, \color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}}, -M \cdot M\right)}\right) \]

   3. Taylor expanded in c0 around inf 57.9

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

   4. Simplified 49.6

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

      [Start] 57.9	\[ \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]
      *-commutative [=>] 57.9	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)} \cdot 2\right)} \]
      times-frac [=>] 56.3	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(\frac{{d}^{2}}{{D}^{2}} \cdot \frac{c0}{w \cdot h}\right)} \cdot 2\right) \]
      unpow2 [=>] 56.3	\[ \frac{c0}{2 \cdot w} \cdot \left(\left(\frac{\color{blue}{d \cdot d}}{{D}^{2}} \cdot \frac{c0}{w \cdot h}\right) \cdot 2\right) \]
      unpow2 [=>] 56.3	\[ \frac{c0}{2 \cdot w} \cdot \left(\left(\frac{d \cdot d}{\color{blue}{D \cdot D}} \cdot \frac{c0}{w \cdot h}\right) \cdot 2\right) \]
      associate-*l/ [<=] 53.4	\[ \frac{c0}{2 \cdot w} \cdot \left(\left(\color{blue}{\left(\frac{d}{D \cdot D} \cdot d\right)} \cdot \frac{c0}{w \cdot h}\right) \cdot 2\right) \]
      *-commutative [<=] 53.4	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D \cdot D} \cdot d\right)\right)} \cdot 2\right) \]
      associate-*l* [=>] 53.4	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\frac{c0}{w \cdot h} \cdot \left(\left(\frac{d}{D \cdot D} \cdot d\right) \cdot 2\right)\right)} \]
      *-commutative [=>] 53.4	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{\color{blue}{h \cdot w}} \cdot \left(\left(\frac{d}{D \cdot D} \cdot d\right) \cdot 2\right)\right) \]
      associate-/r* [=>] 53.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\frac{c0}{h}}{w}} \cdot \left(\left(\frac{d}{D \cdot D} \cdot d\right) \cdot 2\right)\right) \]
      associate-/r* [=>] 51.3	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\left(\color{blue}{\frac{\frac{d}{D}}{D}} \cdot d\right) \cdot 2\right)\right) \]
      associate-*l/ [=>] 49.9	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\color{blue}{\frac{\frac{d}{D} \cdot d}{D}} \cdot 2\right)\right) \]
      associate-*r/ [<=] 49.6	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)} \cdot 2\right)\right) \]
      unpow2 [<=] 49.6	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\color{blue}{{\left(\frac{d}{D}\right)}^{2}} \cdot 2\right)\right) \]

   5. Applied egg-rr 42.5

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

   6. Simplified 40.0

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

      [Start] 42.5	\[ {\left(\sqrt{\frac{c0}{\left(\left(2 \cdot \frac{w}{c0}\right) \cdot w\right) \cdot h}} \cdot \left(\frac{d}{D} \cdot \sqrt{2}\right)\right)}^{2} \]
      *-commutative [=>] 42.5	\[ {\color{blue}{\left(\left(\frac{d}{D} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{c0}{\left(\left(2 \cdot \frac{w}{c0}\right) \cdot w\right) \cdot h}}\right)}}^{2} \]
      associate-*l* [=>] 42.5	\[ {\left(\left(\frac{d}{D} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{c0}{\color{blue}{\left(2 \cdot \left(\frac{w}{c0} \cdot w\right)\right)} \cdot h}}\right)}^{2} \]
      associate-*l* [=>] 42.5	\[ {\left(\left(\frac{d}{D} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{c0}{\color{blue}{2 \cdot \left(\left(\frac{w}{c0} \cdot w\right) \cdot h\right)}}}\right)}^{2} \]
      associate-*r* [<=] 40.0	\[ {\left(\left(\frac{d}{D} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{c0}{2 \cdot \color{blue}{\left(\frac{w}{c0} \cdot \left(w \cdot h\right)\right)}}}\right)}^{2} \]

   7. Taylor expanded in d around 0 20.8

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

   8. Simplified 21.5

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

      [Start] 20.8	\[ {\left(\frac{\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \left(d \cdot c0\right)\right)}{D \cdot w} \cdot \sqrt{\frac{1}{h}}\right)}^{2} \]
      associate-*l/ [=>] 21.9	\[ {\color{blue}{\left(\frac{\left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \left(d \cdot c0\right)\right)\right) \cdot \sqrt{\frac{1}{h}}}{D \cdot w}\right)}}^{2} \]
      times-frac [=>] 21.6	\[ {\color{blue}{\left(\frac{\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \left(d \cdot c0\right)\right)}{D} \cdot \frac{\sqrt{\frac{1}{h}}}{w}\right)}}^{2} \]
      *-commutative [=>] 21.6	\[ {\left(\frac{\color{blue}{\left(\sqrt{0.5} \cdot \left(d \cdot c0\right)\right) \cdot \sqrt{2}}}{D} \cdot \frac{\sqrt{\frac{1}{h}}}{w}\right)}^{2} \]
      associate-/l* [=>] 21.5	\[ {\left(\color{blue}{\frac{\sqrt{0.5} \cdot \left(d \cdot c0\right)}{\frac{D}{\sqrt{2}}}} \cdot \frac{\sqrt{\frac{1}{h}}}{w}\right)}^{2} \]
      *-commutative [=>] 21.5	\[ {\left(\frac{\sqrt{0.5} \cdot \color{blue}{\left(c0 \cdot d\right)}}{\frac{D}{\sqrt{2}}} \cdot \frac{\sqrt{\frac{1}{h}}}{w}\right)}^{2} \]

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

   3. Taylor expanded in c0 around -inf 63.0

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

   4. Simplified 32.9

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

      [Start] 63.0	\[ \frac{c0}{2 \cdot w} \cdot \left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right) \]
      fma-def [=>] 63.0	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0}, -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right)} \]

   5. Taylor expanded in c0 around 0 33.9

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

   6. Simplified 33.6

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

      [Start] 33.9	\[ 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]
      *-commutative [<=] 33.9	\[ 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2}} \]
      associate-/l* [=>] 33.9	\[ 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{h \cdot {M}^{2}}}} \]
      unpow2 [=>] 33.9	\[ 0.25 \cdot \frac{\color{blue}{D \cdot D}}{\frac{{d}^{2}}{h \cdot {M}^{2}}} \]
      unpow2 [=>] 33.9	\[ 0.25 \cdot \frac{D \cdot D}{\frac{\color{blue}{d \cdot d}}{h \cdot {M}^{2}}} \]
      unpow2 [=>] 33.9	\[ 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \color{blue}{\left(M \cdot M\right)}}} \]
      associate-/r* [=>] 33.6	\[ 0.25 \cdot \frac{D \cdot D}{\color{blue}{\frac{\frac{d \cdot d}{h}}{M \cdot M}}} \]

   7. Taylor expanded in D around 0 33.9

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

   8. Simplified 22.2

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

      [Start] 33.9	\[ 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \]
      unpow2 [=>] 33.9	\[ 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{d \cdot d}} \]
      associate-*r* [=>] 33.2	\[ 0.25 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{d \cdot d} \]
      unpow2 [=>] 33.2	\[ 0.25 \cdot \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h}{d \cdot d} \]
      unpow2 [=>] 33.2	\[ 0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h}{d \cdot d} \]
      associate-*r/ [<=] 33.8	\[ 0.25 \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}\right)} \]
      associate-*r* [<=] 33.7	\[ 0.25 \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d \cdot d}\right)\right)} \]
      remove-double-neg [<=] 33.7	\[ 0.25 \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(-\left(-\left(M \cdot M\right) \cdot \frac{h}{d \cdot d}\right)\right)}\right) \]
      associate-*r/ [=>] 33.9	\[ 0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(-\left(-\color{blue}{\frac{\left(M \cdot M\right) \cdot h}{d \cdot d}}\right)\right)\right) \]
      times-frac [=>] 29.9	\[ 0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(-\left(-\color{blue}{\frac{M \cdot M}{d} \cdot \frac{h}{d}}\right)\right)\right) \]
      distribute-lft-neg-in [=>] 29.9	\[ 0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(-\color{blue}{\left(-\frac{M \cdot M}{d}\right) \cdot \frac{h}{d}}\right)\right) \]
      associate-*l* [=>] 26.2	\[ 0.25 \cdot \color{blue}{\left(D \cdot \left(D \cdot \left(-\left(-\frac{M \cdot M}{d}\right) \cdot \frac{h}{d}\right)\right)\right)} \]
      distribute-lft-neg-in [<=] 26.2	\[ 0.25 \cdot \left(D \cdot \left(D \cdot \left(-\color{blue}{\left(-\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)}\right)\right)\right) \]
      times-frac [<=] 30.8	\[ 0.25 \cdot \left(D \cdot \left(D \cdot \left(-\left(-\color{blue}{\frac{\left(M \cdot M\right) \cdot h}{d \cdot d}}\right)\right)\right)\right) \]
      associate-*r/ [<=] 30.4	\[ 0.25 \cdot \left(D \cdot \left(D \cdot \left(-\left(-\color{blue}{\left(M \cdot M\right) \cdot \frac{h}{d \cdot d}}\right)\right)\right)\right) \]
      remove-double-neg [=>] 30.4	\[ 0.25 \cdot \left(D \cdot \left(D \cdot \color{blue}{\left(\left(M \cdot M\right) \cdot \frac{h}{d \cdot d}\right)}\right)\right) \]
      associate-*r/ [=>] 30.8	\[ 0.25 \cdot \left(D \cdot \left(D \cdot \color{blue}{\frac{\left(M \cdot M\right) \cdot h}{d \cdot d}}\right)\right) \]

   9. Applied egg-rr 16.8

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

4. Final simplification 19.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 10^{+301}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(d \cdot \frac{\frac{\frac{c0}{w}}{h \cdot \frac{D}{d}}}{D}\right)\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;{\left(\frac{\sqrt{0.5} \cdot \left(c0 \cdot d\right)}{\frac{D}{\sqrt{2}}} \cdot \frac{\sqrt{\frac{1}{h}}}{w}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{\frac{h \cdot D}{\frac{d}{M}}}{\frac{d}{M}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	19.7
Cost	11076

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

Alternative 2
Error	19.7
Cost	1545

\[\begin{array}{l} \mathbf{if}\;M \cdot M \leq 2 \cdot 10^{-322} \lor \neg \left(M \cdot M \leq 2 \cdot 10^{+65}\right):\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{\frac{h \cdot D}{\frac{d}{M}}}{\frac{d}{M}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(M \cdot M\right) \cdot \left(\left(h \cdot \frac{D}{d}\right) \cdot \frac{D}{-d}\right)\right) \cdot -0.25\\ \end{array} \]

Alternative 3
Error	20.9
Cost	1225

\[\begin{array}{l} \mathbf{if}\;M \leq 2.5 \cdot 10^{-178} \lor \neg \left(M \leq 3.4 \cdot 10^{+99}\right):\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{\frac{h \cdot D}{\frac{d}{M}}}{\frac{d}{M}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(M \cdot \left(M \cdot \left(h \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\right)\right)\\ \end{array} \]

Alternative 4
Error	22.3
Cost	960

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

Alternative 5
Error	20.9
Cost	960

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

Alternative 6
Error	31.8
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023038 
(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))))))