Henrywood and Agarwal, Equation (9a)

Average Error: 14.5 → 9.9

Time: 20.7s

Precision: binary64

Cost: 28296

\[ \begin{array}{c}[M, D] = \mathsf{sort}([M, D])\\ \end{array} \]

Math

\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

↓

\[\begin{array}{l} t_0 := {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{h \cdot \frac{M}{d}}{\frac{\ell}{D}}}{\frac{4}{D} \cdot \frac{d}{M}}}\\ \mathbf{elif}\;t_0 \leq 10^{-29}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot \left(M \cdot h\right)}{\ell} \cdot -0.25\right)}\\ \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))

↓

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))
   (if (<= t_0 (- INFINITY))
     (* w0 (sqrt (- 1.0 (/ (/ (* h (/ M d)) (/ l D)) (* (/ 4.0 D) (/ d M))))))
     (if (<= t_0 1e-29)
       (* w0 (sqrt (- 1.0 t_0)))
       (*
        w0
        (sqrt
         (+ 1.0 (* (* (/ D d) (/ D d)) (* (/ (* M (* M h)) l) -0.25)))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}

↓

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = pow(((M * D) / (2.0 * d)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = w0 * sqrt((1.0 - (((h * (M / d)) / (l / D)) / ((4.0 / D) * (d / M)))));
	} else if (t_0 <= 1e-29) {
		tmp = w0 * sqrt((1.0 - t_0));
	} else {
		tmp = w0 * sqrt((1.0 + (((D / d) * (D / d)) * (((M * (M * h)) / l) * -0.25))));
	}
	return tmp;
}

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}

↓

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = w0 * Math.sqrt((1.0 - (((h * (M / d)) / (l / D)) / ((4.0 / D) * (d / M)))));
	} else if (t_0 <= 1e-29) {
		tmp = w0 * Math.sqrt((1.0 - t_0));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (((D / d) * (D / d)) * (((M * (M * h)) / l) * -0.25))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))

↓

def code(w0, M, D, h, l, d):
	t_0 = math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = w0 * math.sqrt((1.0 - (((h * (M / d)) / (l / D)) / ((4.0 / D) * (d / M)))))
	elif t_0 <= 1e-29:
		tmp = w0 * math.sqrt((1.0 - t_0))
	else:
		tmp = w0 * math.sqrt((1.0 + (((D / d) * (D / d)) * (((M * (M * h)) / l) * -0.25))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))))
end

↓

function code(w0, M, D, h, l, d)
	t_0 = Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(h * Float64(M / d)) / Float64(l / D)) / Float64(Float64(4.0 / D) * Float64(d / M))))));
	elseif (t_0 <= 1e-29)
		tmp = Float64(w0 * sqrt(Float64(1.0 - t_0)));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(Float64(M * Float64(M * h)) / l) * -0.25)))));
	end
	return tmp
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))));
end

↓

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = (((M * D) / (2.0 * d)) ^ 2.0) * (h / l);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = w0 * sqrt((1.0 - (((h * (M / d)) / (l / D)) / ((4.0 / D) * (d / M)))));
	elseif (t_0 <= 1e-29)
		tmp = w0 * sqrt((1.0 - t_0));
	else
		tmp = w0 * sqrt((1.0 + (((D / d) * (D / d)) * (((M * (M * h)) / l) * -0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(h * N[(M / d), $MachinePrecision]), $MachinePrecision] / N[(l / D), $MachinePrecision]), $MachinePrecision] / N[(N[(4.0 / D), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-29], N[(w0 * N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < -inf.0

   1. Initial program 64.0

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   2. Simplified 62.1

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}} \]
      Proof

      [Start] 64.0	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      times-frac [=>] 62.1	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}} \]

   3. Applied egg-rr 60.4

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(M \cdot \left(0.5 \cdot D\right)\right)}{\frac{\ell}{h} \cdot \left(2 \cdot d\right)}}} \]

   4. Applied egg-rr 64.0

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{\left(D \cdot 0.5\right) \cdot \frac{\left(M \cdot D\right) \cdot M}{d}}{d \cdot 2} + 0\right)}} \]

   5. Simplified 55.1

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{D}{4} \cdot \left(\frac{M}{d} \cdot \left(\frac{h}{\ell} \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)}} \]
      Proof

      [Start] 64.0	\[ w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{\left(D \cdot 0.5\right) \cdot \frac{\left(M \cdot D\right) \cdot M}{d}}{d \cdot 2} + 0\right)} \]
      +-rgt-identity [=>] 64.0	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{h}{\ell} \cdot \frac{\left(D \cdot 0.5\right) \cdot \frac{\left(M \cdot D\right) \cdot M}{d}}{d \cdot 2}}} \]
      *-commutative [=>] 64.0	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(D \cdot 0.5\right) \cdot \frac{\left(M \cdot D\right) \cdot M}{d}}{d \cdot 2} \cdot \frac{h}{\ell}}} \]
      associate-/r* [=>] 64.0	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{\left(D \cdot 0.5\right) \cdot \frac{\left(M \cdot D\right) \cdot M}{d}}{d}}{2}} \cdot \frac{h}{\ell}} \]
      associate-*r/ [=>] 63.9	\[ w0 \cdot \sqrt{1 - \frac{\frac{\color{blue}{\frac{\left(D \cdot 0.5\right) \cdot \left(\left(M \cdot D\right) \cdot M\right)}{d}}}{d}}{2} \cdot \frac{h}{\ell}} \]
      *-commutative [=>] 63.9	\[ w0 \cdot \sqrt{1 - \frac{\frac{\frac{\left(D \cdot 0.5\right) \cdot \color{blue}{\left(M \cdot \left(M \cdot D\right)\right)}}{d}}{d}}{2} \cdot \frac{h}{\ell}} \]
      associate-*r* [=>] 64.0	\[ w0 \cdot \sqrt{1 - \frac{\frac{\frac{\color{blue}{\left(\left(D \cdot 0.5\right) \cdot M\right) \cdot \left(M \cdot D\right)}}{d}}{d}}{2} \cdot \frac{h}{\ell}} \]
      associate-*r* [<=] 64.0	\[ w0 \cdot \sqrt{1 - \frac{\frac{\frac{\color{blue}{\left(D \cdot \left(0.5 \cdot M\right)\right)} \cdot \left(M \cdot D\right)}{d}}{d}}{2} \cdot \frac{h}{\ell}} \]
      *-commutative [<=] 64.0	\[ w0 \cdot \sqrt{1 - \frac{\frac{\frac{\left(D \cdot \color{blue}{\left(M \cdot 0.5\right)}\right) \cdot \left(M \cdot D\right)}{d}}{d}}{2} \cdot \frac{h}{\ell}} \]
      associate-*r/ [<=] 64.0	\[ w0 \cdot \sqrt{1 - \frac{\frac{\color{blue}{\left(D \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{M \cdot D}{d}}}{d}}{2} \cdot \frac{h}{\ell}} \]
      associate-*l/ [<=] 64.0	\[ w0 \cdot \sqrt{1 - \frac{\frac{\left(D \cdot \left(M \cdot 0.5\right)\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{d}}{2} \cdot \frac{h}{\ell}} \]
      *-commutative [<=] 64.0	\[ w0 \cdot \sqrt{1 - \frac{\frac{\color{blue}{\left(\frac{M}{d} \cdot D\right) \cdot \left(D \cdot \left(M \cdot 0.5\right)\right)}}{d}}{2} \cdot \frac{h}{\ell}} \]
      associate-*r/ [<=] 64.0	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right) \cdot \frac{D \cdot \left(M \cdot 0.5\right)}{d}}}{2} \cdot \frac{h}{\ell}} \]
      *-commutative [=>] 64.0	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{D \cdot \left(M \cdot 0.5\right)}{d} \cdot \left(\frac{M}{d} \cdot D\right)}}{2} \cdot \frac{h}{\ell}} \]
      associate-*r* [=>] 64.0	\[ w0 \cdot \sqrt{1 - \frac{\frac{\color{blue}{\left(D \cdot M\right) \cdot 0.5}}{d} \cdot \left(\frac{M}{d} \cdot D\right)}{2} \cdot \frac{h}{\ell}} \]
      *-commutative [<=] 64.0	\[ w0 \cdot \sqrt{1 - \frac{\frac{\color{blue}{\left(M \cdot D\right)} \cdot 0.5}{d} \cdot \left(\frac{M}{d} \cdot D\right)}{2} \cdot \frac{h}{\ell}} \]
      associate-*l* [=>] 64.0	\[ w0 \cdot \sqrt{1 - \frac{\frac{\color{blue}{M \cdot \left(D \cdot 0.5\right)}}{d} \cdot \left(\frac{M}{d} \cdot D\right)}{2} \cdot \frac{h}{\ell}} \]
      associate-*l/ [<=] 62.1	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right)} \cdot \left(\frac{M}{d} \cdot D\right)}{2} \cdot \frac{h}{\ell}} \]
      associate-*l/ [<=] 62.1	\[ w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{\frac{M}{d} \cdot \left(D \cdot 0.5\right)}{2} \cdot \left(\frac{M}{d} \cdot D\right)\right)} \cdot \frac{h}{\ell}} \]
      associate-*r/ [<=] 62.1	\[ w0 \cdot \sqrt{1 - \left(\color{blue}{\left(\frac{M}{d} \cdot \frac{D \cdot 0.5}{2}\right)} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{h}{\ell}} \]
      associate-*r* [<=] 54.9	\[ w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{M}{d} \cdot \frac{D \cdot 0.5}{2}\right) \cdot \left(\left(\frac{M}{d} \cdot D\right) \cdot \frac{h}{\ell}\right)}} \]
      *-commutative [=>] 54.9	\[ w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{D \cdot 0.5}{2} \cdot \frac{M}{d}\right)} \cdot \left(\left(\frac{M}{d} \cdot D\right) \cdot \frac{h}{\ell}\right)} \]
      associate-*l* [=>] 55.1	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{D \cdot 0.5}{2} \cdot \left(\frac{M}{d} \cdot \left(\left(\frac{M}{d} \cdot D\right) \cdot \frac{h}{\ell}\right)\right)}} \]
      associate-/l* [=>] 55.1	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{D}{\frac{2}{0.5}}} \cdot \left(\frac{M}{d} \cdot \left(\left(\frac{M}{d} \cdot D\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 55.1	\[ w0 \cdot \sqrt{1 - \frac{D}{\color{blue}{4}} \cdot \left(\frac{M}{d} \cdot \left(\left(\frac{M}{d} \cdot D\right) \cdot \frac{h}{\ell}\right)\right)} \]
      *-commutative [=>] 55.1	\[ w0 \cdot \sqrt{1 - \frac{D}{4} \cdot \left(\frac{M}{d} \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \left(\frac{M}{d} \cdot D\right)\right)}\right)} \]
      *-commutative [=>] 55.1	\[ w0 \cdot \sqrt{1 - \frac{D}{4} \cdot \left(\frac{M}{d} \cdot \left(\frac{h}{\ell} \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}\right)\right)} \]

   6. Applied egg-rr 54.0

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{M}{d} \cdot \frac{h \cdot D}{\ell}}{\frac{4}{D} \cdot \frac{d}{M}}}} \]

   7. Applied egg-rr 53.4

      \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{\frac{M}{d} \cdot h}{\frac{\ell}{D}}}}{\frac{4}{D} \cdot \frac{d}{M}}} \]

   if -inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < 9.99999999999999943e-30

   1. Initial program 0.1

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   if 9.99999999999999943e-30 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 56.8

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   2. Applied egg-rr 19.4

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{1}{\frac{\ell}{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}}}} \]

   3. Simplified 19.4

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(h \cdot {\left(\frac{M}{\frac{d}{D \cdot 0.5}}\right)}^{2}\right) \cdot \frac{1}{\ell}}} \]
      Proof

      [Start] 19.4	\[ w0 \cdot \sqrt{1 - \frac{1}{\frac{\ell}{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}}} \]
      associate-/r/ [=>] 19.4	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{1}{\ell} \cdot \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h\right)}} \]
      *-commutative [=>] 19.4	\[ w0 \cdot \sqrt{1 - \color{blue}{\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}}} \]
      *-commutative [=>] 19.4	\[ w0 \cdot \sqrt{1 - \color{blue}{\left(h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)} \cdot \frac{1}{\ell}} \]
      *-commutative [=>] 19.4	\[ w0 \cdot \sqrt{1 - \left(h \cdot {\left(M \cdot \color{blue}{\left(\frac{D}{d} \cdot 0.5\right)}\right)}^{2}\right) \cdot \frac{1}{\ell}} \]
      associate-*l/ [=>] 19.4	\[ w0 \cdot \sqrt{1 - \left(h \cdot {\left(M \cdot \color{blue}{\frac{D \cdot 0.5}{d}}\right)}^{2}\right) \cdot \frac{1}{\ell}} \]
      associate-*r/ [=>] 21.4	\[ w0 \cdot \sqrt{1 - \left(h \cdot {\color{blue}{\left(\frac{M \cdot \left(D \cdot 0.5\right)}{d}\right)}}^{2}\right) \cdot \frac{1}{\ell}} \]
      associate-/l* [=>] 19.4	\[ w0 \cdot \sqrt{1 - \left(h \cdot {\color{blue}{\left(\frac{M}{\frac{d}{D \cdot 0.5}}\right)}}^{2}\right) \cdot \frac{1}{\ell}} \]

   4. Taylor expanded in h around 0 34.4

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}}} \]

   5. Simplified 31.3

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\left(\frac{M \cdot M}{\ell} \cdot h\right) \cdot 0.25\right)}} \]
      Proof

      [Start] 34.4	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}} \]
      *-commutative [=>] 34.4	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}} \cdot 0.25}} \]
      *-commutative [=>] 34.4	\[ w0 \cdot \sqrt{1 - \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{\ell \cdot {d}^{2}} \cdot 0.25} \]
      *-commutative [=>] 34.4	\[ w0 \cdot \sqrt{1 - \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{{d}^{2} \cdot \ell}} \cdot 0.25} \]
      *-commutative [<=] 34.4	\[ w0 \cdot \sqrt{1 - \frac{{D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2} \cdot \ell} \cdot 0.25} \]
      times-frac [=>] 37.5	\[ w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)} \cdot 0.25} \]
      associate-*l* [=>] 37.5	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{D}^{2}}{{d}^{2}} \cdot \left(\frac{h \cdot {M}^{2}}{\ell} \cdot 0.25\right)}} \]
      unpow2 [=>] 37.5	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \left(\frac{h \cdot {M}^{2}}{\ell} \cdot 0.25\right)} \]
      unpow2 [=>] 37.5	\[ w0 \cdot \sqrt{1 - \frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \left(\frac{h \cdot {M}^{2}}{\ell} \cdot 0.25\right)} \]
      times-frac [=>] 31.3	\[ w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(\frac{h \cdot {M}^{2}}{\ell} \cdot 0.25\right)} \]
      *-commutative [=>] 31.3	\[ w0 \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{\color{blue}{{M}^{2} \cdot h}}{\ell} \cdot 0.25\right)} \]
      unpow2 [=>] 31.3	\[ w0 \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell} \cdot 0.25\right)} \]
      associate-/l* [=>] 56.0	\[ w0 \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\color{blue}{\frac{M \cdot M}{\frac{\ell}{h}}} \cdot 0.25\right)} \]
      associate-/r/ [=>] 31.3	\[ w0 \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\color{blue}{\left(\frac{M \cdot M}{\ell} \cdot h\right)} \cdot 0.25\right)} \]

   6. Applied egg-rr 25.6

      \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\color{blue}{\frac{M \cdot \left(M \cdot h\right)}{\ell}} \cdot 0.25\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 9.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{h \cdot \frac{M}{d}}{\frac{\ell}{D}}}{\frac{4}{D} \cdot \frac{d}{M}}}\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{-29}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot \left(M \cdot h\right)}{\ell} \cdot -0.25\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.3
Cost	8264

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot \left(M \cdot h\right)}{\ell} \cdot -0.25\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -5 \cdot 10^{-217}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{4} \cdot \left(\frac{M}{d} \cdot \left(\frac{h}{\ell} \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 2
Error	10.6
Cost	8264

\[\begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot \left(M \cdot h\right)}{\ell} \cdot -0.25\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-110}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{t_0 \cdot t_0}{4 \cdot \frac{\ell}{h}}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 3
Error	10.8
Cost	8140

\[\begin{array}{l} t_0 := w0 \cdot \sqrt{1 - \frac{\frac{M}{d} \cdot \frac{D \cdot h}{\ell}}{\frac{4}{D} \cdot \frac{d}{M}}}\\ \mathbf{if}\;D \leq -2 \cdot 10^{-227}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;D \leq 10^{-190}:\\ \;\;\;\;w0\\ \mathbf{elif}\;D \leq 10^{-48}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{4} \cdot \left(\frac{M}{d} \cdot \left(\frac{h}{\ell} \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)}\\ \end{array} \]

Alternative 4
Error	10.0
Cost	8140

\[\begin{array}{l} t_0 := \frac{4}{D} \cdot \frac{d}{M}\\ \mathbf{if}\;D \leq -4.5 \cdot 10^{-228}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{M}{d} \cdot \frac{D \cdot h}{\ell}}{t_0}}\\ \mathbf{elif}\;D \leq 2 \cdot 10^{-215}:\\ \;\;\;\;w0\\ \mathbf{elif}\;D \leq 1.1 \cdot 10^{+165}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{h \cdot \frac{M}{d}}{\frac{\ell}{D}}}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{4} \cdot \left(\frac{M}{d} \cdot \left(\frac{h}{\ell} \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)}\\ \end{array} \]

Alternative 5
Error	10.7
Cost	8140

\[\begin{array}{l} t_0 := \frac{4}{D} \cdot \frac{d}{M}\\ \mathbf{if}\;D \leq -2 \cdot 10^{-227}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{M}{d} \cdot \frac{D \cdot h}{\ell}}{t_0}}\\ \mathbf{elif}\;D \leq 2 \cdot 10^{-216}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{M}{\frac{\frac{\ell}{M}}{D \cdot h}}}{d \cdot \frac{d \cdot -4}{D}}}\\ \mathbf{elif}\;D \leq 1.02 \cdot 10^{+165}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{h \cdot \frac{M}{d}}{\frac{\ell}{D}}}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{4} \cdot \left(\frac{M}{d} \cdot \left(\frac{h}{\ell} \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)}\\ \end{array} \]

Alternative 6
Error	14.4
Cost	8008

\[\begin{array}{l} \mathbf{if}\;M \leq -3.5 \cdot 10^{+101}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq -9.5 \cdot 10^{-171}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot \left(M \cdot h\right)}{\ell} \cdot -0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 7
Error	14.0
Cost	8004

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{+211}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\left(h \cdot \frac{M \cdot M}{\ell}\right) \cdot -0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 8
Error	13.8
Cost	64

\[w0 \]

Reproduce

herbie shell --seed 2022354 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  :precision binary64
  (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))