Henrywood and Agarwal, Equation (13)

Average Error: 59.7 → 14.8

Time: 27.2s

Precision: binary64

Cost: 30540

Math

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

↓

\[\begin{array}{l} t_0 := \frac{c0}{D \cdot \left(w \cdot \frac{h}{d}\right)}\\ t_1 := c0 \cdot \frac{0.5}{w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_3 := \frac{c0}{2 \cdot w} \cdot \left(t_2 + \sqrt{t_2 \cdot t_2 - M \cdot M}\right)\\ \mathbf{if}\;t_3 \leq -5 \cdot 10^{+52}:\\ \;\;\;\;\left(\frac{d}{D} \cdot \left(t_0 + t_0\right)\right) \cdot t_1\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;0.25 \cdot \left(\frac{h \cdot M}{\frac{d}{D}} \cdot \left(D \cdot \frac{M}{d}\right)\right)\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;2 \cdot \frac{\frac{d}{D} \cdot t_1}{\frac{h}{d} \cdot \left(w \cdot \frac{D}{c0}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(h \cdot \left(M \cdot \frac{D}{d}\right)\right) \cdot \frac{M}{\frac{d}{D}}\right)\\ \end{array} \]

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 (* D (* w (/ h d)))))
        (t_1 (* c0 (/ 0.5 w)))
        (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_3 (* (/ c0 (* 2.0 w)) (+ t_2 (sqrt (- (* t_2 t_2) (* M M)))))))
   (if (<= t_3 -5e+52)
     (* (* (/ d D) (+ t_0 t_0)) t_1)
     (if (<= t_3 0.0)
       (* 0.25 (* (/ (* h M) (/ d D)) (* D (/ M d))))
       (if (<= t_3 INFINITY)
         (* 2.0 (/ (* (/ d D) t_1) (* (/ h d) (* w (/ D c0)))))
         (* 0.25 (* (* h (* M (/ D d))) (/ M (/ 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 = c0 / (D * (w * (h / d)));
	double t_1 = c0 * (0.5 / w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_3 = (c0 / (2.0 * w)) * (t_2 + sqrt(((t_2 * t_2) - (M * M))));
	double tmp;
	if (t_3 <= -5e+52) {
		tmp = ((d / D) * (t_0 + t_0)) * t_1;
	} else if (t_3 <= 0.0) {
		tmp = 0.25 * (((h * M) / (d / D)) * (D * (M / d)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = 2.0 * (((d / D) * t_1) / ((h / d) * (w * (D / c0))));
	} else {
		tmp = 0.25 * ((h * (M * (D / d))) * (M / (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 = c0 / (D * (w * (h / d)));
	double t_1 = c0 * (0.5 / w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_3 = (c0 / (2.0 * w)) * (t_2 + Math.sqrt(((t_2 * t_2) - (M * M))));
	double tmp;
	if (t_3 <= -5e+52) {
		tmp = ((d / D) * (t_0 + t_0)) * t_1;
	} else if (t_3 <= 0.0) {
		tmp = 0.25 * (((h * M) / (d / D)) * (D * (M / d)));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = 2.0 * (((d / D) * t_1) / ((h / d) * (w * (D / c0))));
	} else {
		tmp = 0.25 * ((h * (M * (D / d))) * (M / (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 = c0 / (D * (w * (h / d)))
	t_1 = c0 * (0.5 / w)
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_3 = (c0 / (2.0 * w)) * (t_2 + math.sqrt(((t_2 * t_2) - (M * M))))
	tmp = 0
	if t_3 <= -5e+52:
		tmp = ((d / D) * (t_0 + t_0)) * t_1
	elif t_3 <= 0.0:
		tmp = 0.25 * (((h * M) / (d / D)) * (D * (M / d)))
	elif t_3 <= math.inf:
		tmp = 2.0 * (((d / D) * t_1) / ((h / d) * (w * (D / c0))))
	else:
		tmp = 0.25 * ((h * (M * (D / d))) * (M / (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(c0 / Float64(D * Float64(w * Float64(h / d))))
	t_1 = Float64(c0 * Float64(0.5 / w))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_3 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M)))))
	tmp = 0.0
	if (t_3 <= -5e+52)
		tmp = Float64(Float64(Float64(d / D) * Float64(t_0 + t_0)) * t_1);
	elseif (t_3 <= 0.0)
		tmp = Float64(0.25 * Float64(Float64(Float64(h * M) / Float64(d / D)) * Float64(D * Float64(M / d))));
	elseif (t_3 <= Inf)
		tmp = Float64(2.0 * Float64(Float64(Float64(d / D) * t_1) / Float64(Float64(h / d) * Float64(w * Float64(D / c0)))));
	else
		tmp = Float64(0.25 * Float64(Float64(h * Float64(M * Float64(D / d))) * Float64(M / 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 = c0 / (D * (w * (h / d)));
	t_1 = c0 * (0.5 / w);
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_3 = (c0 / (2.0 * w)) * (t_2 + sqrt(((t_2 * t_2) - (M * M))));
	tmp = 0.0;
	if (t_3 <= -5e+52)
		tmp = ((d / D) * (t_0 + t_0)) * t_1;
	elseif (t_3 <= 0.0)
		tmp = 0.25 * (((h * M) / (d / D)) * (D * (M / d)));
	elseif (t_3 <= Inf)
		tmp = 2.0 * (((d / D) * t_1) / ((h / d) * (w * (D / c0))));
	else
		tmp = 0.25 * ((h * (M * (D / d))) * (M / (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[(c0 / N[(D * N[(w * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 * N[(0.5 / w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+52], N[(N[(N[(d / D), $MachinePrecision] * N[(t$95$0 + t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(0.25 * N[(N[(N[(h * M), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(2.0 * N[(N[(N[(d / D), $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[(h / d), $MachinePrecision] * N[(w * N[(D / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(h * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M / N[(d / D), $MachinePrecision]), $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))))) < -5e52

   1. Initial program 55.6

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

   2. Simplified 55.0

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

      [Start] 55.6	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 56.3	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-neg [=>] 56.3	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\color{blue}{\mathsf{fma}\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, -M \cdot M\right)}}\right) \]
      times-frac [=>] 56.3	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}}, \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, -M \cdot M\right)}\right) \]
      times-frac [=>] 55.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, \color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}}, -M \cdot M\right)}\right) \]

   3. Taylor expanded in c0 around inf 52.1

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

   4. Simplified 50.6

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

      [Start] 52.1	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]
      *-commutative [=>] 52.1	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \frac{\color{blue}{c0 \cdot {d}^{2}}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]
      unpow2 [=>] 52.1	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]
      *-commutative [=>] 52.1	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(w \cdot h\right) \cdot {D}^{2}}}\right) \]
      unpow2 [=>] 52.1	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}}\right) \]
      associate-/r* [=>] 50.6	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \color{blue}{\frac{\frac{c0 \cdot \left(d \cdot d\right)}{w \cdot h}}{D \cdot D}}\right) \]

   5. Applied egg-rr 51.8

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

   6. Applied egg-rr 49.8

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

   7. Simplified 35.9

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

      [Start] 49.8	\[ c0 \cdot \left(\frac{0.5}{w} \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right) + c0 \cdot \left(\frac{0.5}{w} \cdot \frac{\frac{c0}{w}}{D \cdot \frac{\frac{h}{d}}{\frac{d}{D}}}\right) \]
      associate-*r* [=>] 49.8	\[ \color{blue}{\left(c0 \cdot \frac{0.5}{w}\right) \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)} + c0 \cdot \left(\frac{0.5}{w} \cdot \frac{\frac{c0}{w}}{D \cdot \frac{\frac{h}{d}}{\frac{d}{D}}}\right) \]
      associate-*r* [=>] 48.0	\[ \left(c0 \cdot \frac{0.5}{w}\right) \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) + \color{blue}{\left(c0 \cdot \frac{0.5}{w}\right) \cdot \frac{\frac{c0}{w}}{D \cdot \frac{\frac{h}{d}}{\frac{d}{D}}}} \]
      distribute-lft-in [<=] 48.0	\[ \color{blue}{\left(c0 \cdot \frac{0.5}{w}\right) \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + \frac{\frac{c0}{w}}{D \cdot \frac{\frac{h}{d}}{\frac{d}{D}}}\right)} \]
      fma-udef [<=] 48.0	\[ \left(c0 \cdot \frac{0.5}{w}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d}{D} \cdot \frac{d}{D}, \frac{\frac{c0}{w}}{D \cdot \frac{\frac{h}{d}}{\frac{d}{D}}}\right)} \]
      *-commutative [=>] 48.0	\[ \color{blue}{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d}{D} \cdot \frac{d}{D}, \frac{\frac{c0}{w}}{D \cdot \frac{\frac{h}{d}}{\frac{d}{D}}}\right) \cdot \left(c0 \cdot \frac{0.5}{w}\right)} \]

   if -5e52 < (*.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 28.5

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

   2. Simplified 33.5

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

      [Start] 28.5	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 39.4	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-neg [=>] 39.4	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\color{blue}{\mathsf{fma}\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, -M \cdot M\right)}}\right) \]
      times-frac [=>] 37.2	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}}, \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, -M \cdot M\right)}\right) \]
      times-frac [=>] 33.5	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, \color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}}, -M \cdot M\right)}\right) \]

   3. Taylor expanded in c0 around -inf 32.7

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

   4. Simplified 34.0

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

      [Start] 32.7	\[ \frac{c0}{2 \cdot w} \cdot \left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right) \]
      fma-def [=>] 32.7	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0}, -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right)} \]

   5. Taylor expanded in c0 around 0 28.6

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

   6. Simplified 23.6

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

      [Start] 28.6	\[ 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}} \]
      *-commutative [=>] 28.6	\[ 0.25 \cdot \frac{\color{blue}{\left(h \cdot {M}^{2}\right) \cdot {D}^{2}}}{{d}^{2}} \]
      unpow2 [=>] 28.6	\[ 0.25 \cdot \frac{\left(h \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot {D}^{2}}{{d}^{2}} \]
      associate-/l* [=>] 28.5	\[ 0.25 \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{\frac{{d}^{2}}{{D}^{2}}}} \]
      associate-*r* [=>] 26.7	\[ 0.25 \cdot \frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{\frac{{d}^{2}}{{D}^{2}}} \]
      *-commutative [=>] 26.7	\[ 0.25 \cdot \frac{\color{blue}{M \cdot \left(h \cdot M\right)}}{\frac{{d}^{2}}{{D}^{2}}} \]
      unpow2 [=>] 26.7	\[ 0.25 \cdot \frac{M \cdot \left(h \cdot M\right)}{\frac{\color{blue}{d \cdot d}}{{D}^{2}}} \]
      unpow2 [=>] 26.7	\[ 0.25 \cdot \frac{M \cdot \left(h \cdot M\right)}{\frac{d \cdot d}{\color{blue}{D \cdot D}}} \]
      times-frac [=>] 23.6	\[ 0.25 \cdot \frac{M \cdot \left(h \cdot M\right)}{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}} \]
      unpow2 [<=] 23.6	\[ 0.25 \cdot \frac{M \cdot \left(h \cdot M\right)}{\color{blue}{{\left(\frac{d}{D}\right)}^{2}}} \]

   7. Applied egg-rr 17.2

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

   8. Applied egg-rr 17.7

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

   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 47.0

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

   2. Simplified 46.9

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

      [Start] 47.0	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      times-frac [=>] 48.4	\[ \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
      fma-neg [=>] 48.4	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\color{blue}{\mathsf{fma}\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, -M \cdot M\right)}}\right) \]
      times-frac [=>] 48.4	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}}, \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, -M \cdot M\right)}\right) \]
      times-frac [=>] 46.9	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, \color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}}, -M \cdot M\right)}\right) \]

   3. Taylor expanded in c0 around inf 44.6

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

   4. Simplified 44.3

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

      [Start] 44.6	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]
      *-commutative [=>] 44.6	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \frac{\color{blue}{c0 \cdot {d}^{2}}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]
      unpow2 [=>] 44.6	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \]
      *-commutative [=>] 44.6	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(w \cdot h\right) \cdot {D}^{2}}}\right) \]
      unpow2 [=>] 44.6	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}}\right) \]
      associate-/r* [=>] 44.3	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \color{blue}{\frac{\frac{c0 \cdot \left(d \cdot d\right)}{w \cdot h}}{D \cdot D}}\right) \]

   5. Applied egg-rr 44.5

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

   6. Applied egg-rr 43.9

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

   7. Simplified 25.9

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

      [Start] 43.9	\[ c0 \cdot \left(\frac{0.5}{w} \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right) + c0 \cdot \left(\frac{0.5}{w} \cdot \frac{\frac{c0}{w}}{D \cdot \frac{\frac{h}{d}}{\frac{d}{D}}}\right) \]
      associate-*r* [=>] 43.9	\[ \color{blue}{\left(c0 \cdot \frac{0.5}{w}\right) \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)} + c0 \cdot \left(\frac{0.5}{w} \cdot \frac{\frac{c0}{w}}{D \cdot \frac{\frac{h}{d}}{\frac{d}{D}}}\right) \]
      associate-*r* [=>] 41.7	\[ \left(c0 \cdot \frac{0.5}{w}\right) \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) + \color{blue}{\left(c0 \cdot \frac{0.5}{w}\right) \cdot \frac{\frac{c0}{w}}{D \cdot \frac{\frac{h}{d}}{\frac{d}{D}}}} \]
      distribute-lft-in [<=] 41.7	\[ \color{blue}{\left(c0 \cdot \frac{0.5}{w}\right) \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + \frac{\frac{c0}{w}}{D \cdot \frac{\frac{h}{d}}{\frac{d}{D}}}\right)} \]
      +-commutative [=>] 41.7	\[ \left(c0 \cdot \frac{0.5}{w}\right) \cdot \color{blue}{\left(\frac{\frac{c0}{w}}{D \cdot \frac{\frac{h}{d}}{\frac{d}{D}}} + \frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)} \]
      associate-*r/ [=>] 42.0	\[ \left(c0 \cdot \frac{0.5}{w}\right) \cdot \left(\frac{\frac{c0}{w}}{D \cdot \frac{\frac{h}{d}}{\frac{d}{D}}} + \frac{c0}{w \cdot h} \cdot \color{blue}{\frac{\frac{d}{D} \cdot d}{D}}\right) \]
      associate-*r/ [=>] 40.9	\[ \left(c0 \cdot \frac{0.5}{w}\right) \cdot \left(\frac{\frac{c0}{w}}{D \cdot \frac{\frac{h}{d}}{\frac{d}{D}}} + \color{blue}{\frac{\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot d\right)}{D}}\right) \]
      associate-/r* [=>] 39.9	\[ \left(c0 \cdot \frac{0.5}{w}\right) \cdot \left(\frac{\frac{c0}{w}}{D \cdot \frac{\frac{h}{d}}{\frac{d}{D}}} + \frac{\color{blue}{\frac{\frac{c0}{w}}{h}} \cdot \left(\frac{d}{D} \cdot d\right)}{D}\right) \]
      associate-/r/ [<=] 38.8	\[ \left(c0 \cdot \frac{0.5}{w}\right) \cdot \left(\frac{\frac{c0}{w}}{D \cdot \frac{\frac{h}{d}}{\frac{d}{D}}} + \frac{\color{blue}{\frac{\frac{c0}{w}}{\frac{h}{\frac{d}{D} \cdot d}}}}{D}\right) \]
      associate-/l/ [<=] 35.9	\[ \left(c0 \cdot \frac{0.5}{w}\right) \cdot \left(\frac{\frac{c0}{w}}{D \cdot \frac{\frac{h}{d}}{\frac{d}{D}}} + \frac{\frac{\frac{c0}{w}}{\color{blue}{\frac{\frac{h}{d}}{\frac{d}{D}}}}}{D}\right) \]

   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 63.3

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, \frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, -M \cdot M\right)}\right)} \]
      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 [=>] 63.3	\[ \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}, \color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}}, -M \cdot M\right)}\right) \]

   3. Taylor expanded in c0 around -inf 63.1

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

   4. Simplified 57.4

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

      [Start] 63.1	\[ \frac{c0}{2 \cdot w} \cdot \left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right) \]
      fma-def [=>] 63.1	\[ \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0}, -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right)} \]

   5. Taylor expanded in c0 around 0 34.3

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

   6. Simplified 22.6

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

      [Start] 34.3	\[ 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}} \]
      *-commutative [=>] 34.3	\[ 0.25 \cdot \frac{\color{blue}{\left(h \cdot {M}^{2}\right) \cdot {D}^{2}}}{{d}^{2}} \]
      unpow2 [=>] 34.3	\[ 0.25 \cdot \frac{\left(h \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot {D}^{2}}{{d}^{2}} \]
      associate-/l* [=>] 34.3	\[ 0.25 \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{\frac{{d}^{2}}{{D}^{2}}}} \]
      associate-*r* [=>] 32.3	\[ 0.25 \cdot \frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{\frac{{d}^{2}}{{D}^{2}}} \]
      *-commutative [=>] 32.3	\[ 0.25 \cdot \frac{\color{blue}{M \cdot \left(h \cdot M\right)}}{\frac{{d}^{2}}{{D}^{2}}} \]
      unpow2 [=>] 32.3	\[ 0.25 \cdot \frac{M \cdot \left(h \cdot M\right)}{\frac{\color{blue}{d \cdot d}}{{D}^{2}}} \]
      unpow2 [=>] 32.3	\[ 0.25 \cdot \frac{M \cdot \left(h \cdot M\right)}{\frac{d \cdot d}{\color{blue}{D \cdot D}}} \]
      times-frac [=>] 22.6	\[ 0.25 \cdot \frac{M \cdot \left(h \cdot M\right)}{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}} \]
      unpow2 [<=] 22.6	\[ 0.25 \cdot \frac{M \cdot \left(h \cdot M\right)}{\color{blue}{{\left(\frac{d}{D}\right)}^{2}}} \]

   7. Applied egg-rr 15.1

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

   8. Applied egg-rr 12.2

      \[\leadsto 0.25 \cdot \left(\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot h\right)} \cdot \frac{M}{\frac{d}{D}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 14.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -5 \cdot 10^{+52}:\\ \;\;\;\;\left(\frac{d}{D} \cdot \left(\frac{c0}{D \cdot \left(w \cdot \frac{h}{d}\right)} + \frac{c0}{D \cdot \left(w \cdot \frac{h}{d}\right)}\right)\right) \cdot \left(c0 \cdot \frac{0.5}{w}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;0.25 \cdot \left(\frac{h \cdot M}{\frac{d}{D}} \cdot \left(D \cdot \frac{M}{d}\right)\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;2 \cdot \frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{0.5}{w}\right)}{\frac{h}{d} \cdot \left(w \cdot \frac{D}{c0}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(h \cdot \left(M \cdot \frac{D}{d}\right)\right) \cdot \frac{M}{\frac{d}{D}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	17.6
Cost	1865

\[\begin{array}{l} \mathbf{if}\;D \cdot D \leq 10^{+145} \lor \neg \left(D \cdot D \leq 4 \cdot 10^{+170}\right):\\ \;\;\;\;0.25 \cdot \left(\left(h \cdot \left(M \cdot \frac{D}{d}\right)\right) \cdot \frac{M}{\frac{d}{D}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{c0}{2}}{w} \cdot \left(\frac{d}{D} \cdot \frac{c0}{D \cdot \left(w \cdot \frac{h}{d}\right)}\right)\right)\\ \end{array} \]

Alternative 2
Error	18.1
Cost	1865

\[\begin{array}{l} \mathbf{if}\;D \cdot D \leq 2 \cdot 10^{+119} \lor \neg \left(D \cdot D \leq 4 \cdot 10^{+170}\right):\\ \;\;\;\;0.25 \cdot \left(\left(h \cdot \left(M \cdot \frac{D}{d}\right)\right) \cdot \frac{M}{\frac{d}{D}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{d}{D} \cdot \left(c0 \cdot \frac{0.5}{w}\right)}{\frac{h}{d} \cdot \left(w \cdot \frac{D}{c0}\right)}\\ \end{array} \]

Alternative 3
Error	23.4
Cost	960

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

Alternative 4
Error	17.0
Cost	960

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

Alternative 5
Error	32.5
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023066 
(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))))))