Henrywood and Agarwal, Equation (9a)

Average Accuracy: 78.1% → 87.2%

Time: 21.9s

Precision: binary64

Cost: 27784

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

Math

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

↓

\[\begin{array}{l} t_0 := {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\\ t_1 := M \cdot \left(D \cdot \frac{0.5}{d}\right)\\ \mathbf{if}\;t_0 \leq 10^{-318}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{t_1 \cdot \left(t_1 \cdot h\right)}{\ell}}\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{+306}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0 \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.5 \cdot \frac{D}{\frac{d}{M}}}{\frac{\frac{2 \cdot d}{\frac{D}{\ell}}}{M \cdot h}}}\\ \end{array} \]

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)) (t_1 (* M (* D (/ 0.5 d)))))
   (if (<= t_0 1e-318)
     (* w0 (sqrt (- 1.0 (/ (* t_1 (* t_1 h)) l))))
     (if (<= t_0 4e+306)
       (* w0 (sqrt (- 1.0 (* t_0 (/ h l)))))
       (*
        w0
        (sqrt
         (-
          1.0
          (/ (* 0.5 (/ D (/ d M))) (/ (/ (* 2.0 d) (/ D l)) (* M h))))))))))

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);
	double t_1 = M * (D * (0.5 / d));
	double tmp;
	if (t_0 <= 1e-318) {
		tmp = w0 * sqrt((1.0 - ((t_1 * (t_1 * h)) / l)));
	} else if (t_0 <= 4e+306) {
		tmp = w0 * sqrt((1.0 - (t_0 * (h / l))));
	} else {
		tmp = w0 * sqrt((1.0 - ((0.5 * (D / (d / M))) / (((2.0 * d) / (D / l)) / (M * h)))));
	}
	return tmp;
}

Fortran

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))
end function

↓

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((m * d) / (2.0d0 * d_1)) ** 2.0d0
    t_1 = m * (d * (0.5d0 / d_1))
    if (t_0 <= 1d-318) then
        tmp = w0 * sqrt((1.0d0 - ((t_1 * (t_1 * h)) / l)))
    else if (t_0 <= 4d+306) then
        tmp = w0 * sqrt((1.0d0 - (t_0 * (h / l))))
    else
        tmp = w0 * sqrt((1.0d0 - ((0.5d0 * (d / (d_1 / m))) / (((2.0d0 * d_1) / (d / l)) / (m * h)))))
    end if
    code = tmp
end function

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);
	double t_1 = M * (D * (0.5 / d));
	double tmp;
	if (t_0 <= 1e-318) {
		tmp = w0 * Math.sqrt((1.0 - ((t_1 * (t_1 * h)) / l)));
	} else if (t_0 <= 4e+306) {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * (h / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((0.5 * (D / (d / M))) / (((2.0 * d) / (D / l)) / (M * h)))));
	}
	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)
	t_1 = M * (D * (0.5 / d))
	tmp = 0
	if t_0 <= 1e-318:
		tmp = w0 * math.sqrt((1.0 - ((t_1 * (t_1 * h)) / l)))
	elif t_0 <= 4e+306:
		tmp = w0 * math.sqrt((1.0 - (t_0 * (h / l))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((0.5 * (D / (d / M))) / (((2.0 * d) / (D / l)) / (M * h)))))
	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(M * D) / Float64(2.0 * d)) ^ 2.0
	t_1 = Float64(M * Float64(D * Float64(0.5 / d)))
	tmp = 0.0
	if (t_0 <= 1e-318)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(t_1 * Float64(t_1 * h)) / l))));
	elseif (t_0 <= 4e+306)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(h / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.5 * Float64(D / Float64(d / M))) / Float64(Float64(Float64(2.0 * d) / Float64(D / l)) / Float64(M * h))))));
	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;
	t_1 = M * (D * (0.5 / d));
	tmp = 0.0;
	if (t_0 <= 1e-318)
		tmp = w0 * sqrt((1.0 - ((t_1 * (t_1 * h)) / l)));
	elseif (t_0 <= 4e+306)
		tmp = w0 * sqrt((1.0 - (t_0 * (h / l))));
	else
		tmp = w0 * sqrt((1.0 - ((0.5 * (D / (d / M))) / (((2.0 * d) / (D / l)) / (M * h)))));
	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[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-318], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(t$95$1 * N[(t$95$1 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+306], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.5 * N[(D / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(2.0 * d), $MachinePrecision] / N[(D / l), $MachinePrecision]), $MachinePrecision] / N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 9.9999875e-319

   1. Initial program 88.4%

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

   2. Applied egg-rr 89.7%

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

      [Start] 88.4	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      clear-num [=>] 88.4	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}} \]
      un-div-inv [=>] 89.6	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}} \]
      div-inv [=>] 89.6	\[ w0 \cdot \sqrt{1 - \frac{{\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}}^{2}}{\frac{\ell}{h}}} \]
      associate-*l* [=>] 89.7	\[ w0 \cdot \sqrt{1 - \frac{{\color{blue}{\left(M \cdot \left(D \cdot \frac{1}{2 \cdot d}\right)\right)}}^{2}}{\frac{\ell}{h}}} \]
      associate-/r* [=>] 89.7	\[ w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \left(D \cdot \color{blue}{\frac{\frac{1}{2}}{d}}\right)\right)}^{2}}{\frac{\ell}{h}}} \]
      metadata-eval [=>] 89.7	\[ w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \left(D \cdot \frac{\color{blue}{0.5}}{d}\right)\right)}^{2}}{\frac{\ell}{h}}} \]

   3. Simplified 99.2%

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

      [Start] 89.7	\[ w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{h}}} \]
      associate-/r/ [=>] 99.2	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell} \cdot h}} \]
      *-commutative [=>] 99.2	\[ w0 \cdot \sqrt{1 - \color{blue}{h \cdot \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}} \]
      *-commutative [=>] 99.2	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(M \cdot \color{blue}{\left(\frac{0.5}{d} \cdot D\right)}\right)}^{2}}{\ell}} \]
      associate-/r/ [<=] 99.2	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(M \cdot \color{blue}{\frac{0.5}{\frac{d}{D}}}\right)}^{2}}{\ell}} \]
      metadata-eval [<=] 99.2	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(M \cdot \frac{\color{blue}{\frac{1}{2}}}{\frac{d}{D}}\right)}^{2}}{\ell}} \]
      associate-/r* [<=] 99.2	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(M \cdot \color{blue}{\frac{1}{2 \cdot \frac{d}{D}}}\right)}^{2}}{\ell}} \]
      associate-*r/ [=>] 99.2	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\color{blue}{\left(\frac{M \cdot 1}{2 \cdot \frac{d}{D}}\right)}}^{2}}{\ell}} \]
      *-rgt-identity [=>] 99.2	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(\frac{\color{blue}{M}}{2 \cdot \frac{d}{D}}\right)}^{2}}{\ell}} \]
      associate-*r/ [=>] 99.2	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(\frac{M}{\color{blue}{\frac{2 \cdot d}{D}}}\right)}^{2}}{\ell}} \]
      associate-/r/ [=>] 99.2	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\color{blue}{\left(\frac{M}{2 \cdot d} \cdot D\right)}}^{2}}{\ell}} \]
      *-commutative [<=] 99.2	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\color{blue}{\left(D \cdot \frac{M}{2 \cdot d}\right)}}^{2}}{\ell}} \]

   4. Applied egg-rr 99.8%

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

      [Start] 99.2	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}{\ell}} \]
      unpow2 [=>] 99.2	\[ w0 \cdot \sqrt{1 - h \cdot \frac{\color{blue}{\left(D \cdot \frac{M}{2 \cdot d}\right) \cdot \left(D \cdot \frac{M}{2 \cdot d}\right)}}{\ell}} \]
      *-un-lft-identity [=>] 99.2	\[ w0 \cdot \sqrt{1 - h \cdot \frac{\left(D \cdot \frac{M}{2 \cdot d}\right) \cdot \left(D \cdot \frac{M}{2 \cdot d}\right)}{\color{blue}{1 \cdot \ell}}} \]
      times-frac [=>] 99.9	\[ w0 \cdot \sqrt{1 - h \cdot \color{blue}{\left(\frac{D \cdot \frac{M}{2 \cdot d}}{1} \cdot \frac{D \cdot \frac{M}{2 \cdot d}}{\ell}\right)}} \]
      *-commutative [=>] 99.9	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{\color{blue}{\frac{M}{2 \cdot d} \cdot D}}{1} \cdot \frac{D \cdot \frac{M}{2 \cdot d}}{\ell}\right)} \]
      div-inv [=>] 99.9	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{\color{blue}{\left(M \cdot \frac{1}{2 \cdot d}\right)} \cdot D}{1} \cdot \frac{D \cdot \frac{M}{2 \cdot d}}{\ell}\right)} \]
      associate-*l* [=>] 99.7	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{\color{blue}{M \cdot \left(\frac{1}{2 \cdot d} \cdot D\right)}}{1} \cdot \frac{D \cdot \frac{M}{2 \cdot d}}{\ell}\right)} \]
      associate-/r* [=>] 99.7	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{M \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot D\right)}{1} \cdot \frac{D \cdot \frac{M}{2 \cdot d}}{\ell}\right)} \]
      metadata-eval [=>] 99.7	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{M \cdot \left(\frac{\color{blue}{0.5}}{d} \cdot D\right)}{1} \cdot \frac{D \cdot \frac{M}{2 \cdot d}}{\ell}\right)} \]
      *-commutative [=>] 99.7	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{1} \cdot \frac{\color{blue}{\frac{M}{2 \cdot d} \cdot D}}{\ell}\right)} \]
      div-inv [=>] 99.7	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{1} \cdot \frac{\color{blue}{\left(M \cdot \frac{1}{2 \cdot d}\right)} \cdot D}{\ell}\right)} \]
      associate-*l* [=>] 99.8	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{1} \cdot \frac{\color{blue}{M \cdot \left(\frac{1}{2 \cdot d} \cdot D\right)}}{\ell}\right)} \]
      associate-/r* [=>] 99.8	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{1} \cdot \frac{M \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot D\right)}{\ell}\right)} \]
      metadata-eval [=>] 99.8	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{1} \cdot \frac{M \cdot \left(\frac{\color{blue}{0.5}}{d} \cdot D\right)}{\ell}\right)} \]

   5. Applied egg-rr 99.8%

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

      [Start] 99.8	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{1} \cdot \frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{\ell}\right)} \]
      /-rgt-identity [=>] 99.8	\[ w0 \cdot \sqrt{1 - h \cdot \left(\color{blue}{\left(M \cdot \left(\frac{0.5}{d} \cdot D\right)\right)} \cdot \frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{\ell}\right)} \]
      associate-*r* [=>] 99.8	\[ w0 \cdot \sqrt{1 - \color{blue}{\left(h \cdot \left(M \cdot \left(\frac{0.5}{d} \cdot D\right)\right)\right) \cdot \frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{\ell}}} \]
      associate-*r/ [=>] 99.8	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(h \cdot \left(M \cdot \left(\frac{0.5}{d} \cdot D\right)\right)\right) \cdot \left(M \cdot \left(\frac{0.5}{d} \cdot D\right)\right)}{\ell}}} \]
      *-commutative [=>] 99.8	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\left(M \cdot \left(\frac{0.5}{d} \cdot D\right)\right) \cdot h\right)} \cdot \left(M \cdot \left(\frac{0.5}{d} \cdot D\right)\right)}{\ell}} \]

   if 9.9999875e-319 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 4.00000000000000007e306

   1. Initial program 90.7%

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

   if 4.00000000000000007e306 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)

   1. Initial program 0.2%

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

   2. Applied egg-rr 5.7%

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

      [Start] 0.2	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      clear-num [=>] 0.2	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}} \]
      un-div-inv [=>] 0.2	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}} \]
      div-inv [=>] 0.2	\[ w0 \cdot \sqrt{1 - \frac{{\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}}^{2}}{\frac{\ell}{h}}} \]
      associate-*l* [=>] 5.7	\[ w0 \cdot \sqrt{1 - \frac{{\color{blue}{\left(M \cdot \left(D \cdot \frac{1}{2 \cdot d}\right)\right)}}^{2}}{\frac{\ell}{h}}} \]
      associate-/r* [=>] 5.7	\[ w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \left(D \cdot \color{blue}{\frac{\frac{1}{2}}{d}}\right)\right)}^{2}}{\frac{\ell}{h}}} \]
      metadata-eval [=>] 5.7	\[ w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \left(D \cdot \frac{\color{blue}{0.5}}{d}\right)\right)}^{2}}{\frac{\ell}{h}}} \]

   3. Simplified 5.2%

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

      [Start] 5.7	\[ w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{h}}} \]
      associate-/r/ [=>] 5.1	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell} \cdot h}} \]
      *-commutative [=>] 5.1	\[ w0 \cdot \sqrt{1 - \color{blue}{h \cdot \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}} \]
      *-commutative [=>] 5.1	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(M \cdot \color{blue}{\left(\frac{0.5}{d} \cdot D\right)}\right)}^{2}}{\ell}} \]
      associate-/r/ [<=] 5.1	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(M \cdot \color{blue}{\frac{0.5}{\frac{d}{D}}}\right)}^{2}}{\ell}} \]
      metadata-eval [<=] 5.1	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(M \cdot \frac{\color{blue}{\frac{1}{2}}}{\frac{d}{D}}\right)}^{2}}{\ell}} \]
      associate-/r* [<=] 5.1	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(M \cdot \color{blue}{\frac{1}{2 \cdot \frac{d}{D}}}\right)}^{2}}{\ell}} \]
      associate-*r/ [=>] 5.1	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\color{blue}{\left(\frac{M \cdot 1}{2 \cdot \frac{d}{D}}\right)}}^{2}}{\ell}} \]
      *-rgt-identity [=>] 5.1	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(\frac{\color{blue}{M}}{2 \cdot \frac{d}{D}}\right)}^{2}}{\ell}} \]
      associate-*r/ [=>] 5.1	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(\frac{M}{\color{blue}{\frac{2 \cdot d}{D}}}\right)}^{2}}{\ell}} \]
      associate-/r/ [=>] 5.2	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\color{blue}{\left(\frac{M}{2 \cdot d} \cdot D\right)}}^{2}}{\ell}} \]
      *-commutative [<=] 5.2	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\color{blue}{\left(D \cdot \frac{M}{2 \cdot d}\right)}}^{2}}{\ell}} \]

   4. Applied egg-rr 20.9%

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

      [Start] 5.2	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}{\ell}} \]
      unpow2 [=>] 5.2	\[ w0 \cdot \sqrt{1 - h \cdot \frac{\color{blue}{\left(D \cdot \frac{M}{2 \cdot d}\right) \cdot \left(D \cdot \frac{M}{2 \cdot d}\right)}}{\ell}} \]
      *-un-lft-identity [=>] 5.2	\[ w0 \cdot \sqrt{1 - h \cdot \frac{\left(D \cdot \frac{M}{2 \cdot d}\right) \cdot \left(D \cdot \frac{M}{2 \cdot d}\right)}{\color{blue}{1 \cdot \ell}}} \]
      times-frac [=>] 21.6	\[ w0 \cdot \sqrt{1 - h \cdot \color{blue}{\left(\frac{D \cdot \frac{M}{2 \cdot d}}{1} \cdot \frac{D \cdot \frac{M}{2 \cdot d}}{\ell}\right)}} \]
      *-commutative [=>] 21.6	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{\color{blue}{\frac{M}{2 \cdot d} \cdot D}}{1} \cdot \frac{D \cdot \frac{M}{2 \cdot d}}{\ell}\right)} \]
      div-inv [=>] 21.6	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{\color{blue}{\left(M \cdot \frac{1}{2 \cdot d}\right)} \cdot D}{1} \cdot \frac{D \cdot \frac{M}{2 \cdot d}}{\ell}\right)} \]
      associate-*l* [=>] 18.0	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{\color{blue}{M \cdot \left(\frac{1}{2 \cdot d} \cdot D\right)}}{1} \cdot \frac{D \cdot \frac{M}{2 \cdot d}}{\ell}\right)} \]
      associate-/r* [=>] 18.0	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{M \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot D\right)}{1} \cdot \frac{D \cdot \frac{M}{2 \cdot d}}{\ell}\right)} \]
      metadata-eval [=>] 18.0	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{M \cdot \left(\frac{\color{blue}{0.5}}{d} \cdot D\right)}{1} \cdot \frac{D \cdot \frac{M}{2 \cdot d}}{\ell}\right)} \]
      *-commutative [=>] 18.0	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{1} \cdot \frac{\color{blue}{\frac{M}{2 \cdot d} \cdot D}}{\ell}\right)} \]
      div-inv [=>] 18.0	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{1} \cdot \frac{\color{blue}{\left(M \cdot \frac{1}{2 \cdot d}\right)} \cdot D}{\ell}\right)} \]
      associate-*l* [=>] 20.9	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{1} \cdot \frac{\color{blue}{M \cdot \left(\frac{1}{2 \cdot d} \cdot D\right)}}{\ell}\right)} \]
      associate-/r* [=>] 20.9	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{1} \cdot \frac{M \cdot \left(\color{blue}{\frac{\frac{1}{2}}{d}} \cdot D\right)}{\ell}\right)} \]
      metadata-eval [=>] 20.9	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{1} \cdot \frac{M \cdot \left(\frac{\color{blue}{0.5}}{d} \cdot D\right)}{\ell}\right)} \]

   5. Applied egg-rr 24.5%

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

      [Start] 20.9	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{1} \cdot \frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{\ell}\right)} \]
      /-rgt-identity [=>] 20.9	\[ w0 \cdot \sqrt{1 - h \cdot \left(\color{blue}{\left(M \cdot \left(\frac{0.5}{d} \cdot D\right)\right)} \cdot \frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{\ell}\right)} \]
      associate-*r* [=>] 27.9	\[ w0 \cdot \sqrt{1 - \color{blue}{\left(h \cdot \left(M \cdot \left(\frac{0.5}{d} \cdot D\right)\right)\right) \cdot \frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{\ell}}} \]
      clear-num [=>] 27.9	\[ w0 \cdot \sqrt{1 - \left(h \cdot \left(M \cdot \left(\frac{0.5}{d} \cdot D\right)\right)\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{M \cdot \left(\frac{0.5}{d} \cdot D\right)}}}} \]
      un-div-inv [=>] 28.0	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{h \cdot \left(M \cdot \left(\frac{0.5}{d} \cdot D\right)\right)}{\frac{\ell}{M \cdot \left(\frac{0.5}{d} \cdot D\right)}}}} \]
      *-commutative [=>] 28.0	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(M \cdot \left(\frac{0.5}{d} \cdot D\right)\right) \cdot h}}{\frac{\ell}{M \cdot \left(\frac{0.5}{d} \cdot D\right)}}} \]
      *-commutative [=>] 28.0	\[ w0 \cdot \sqrt{1 - \frac{\left(M \cdot \left(\frac{0.5}{d} \cdot D\right)\right) \cdot h}{\frac{\ell}{\color{blue}{\left(\frac{0.5}{d} \cdot D\right) \cdot M}}}} \]
      associate-/r* [=>] 27.5	\[ w0 \cdot \sqrt{1 - \frac{\left(M \cdot \left(\frac{0.5}{d} \cdot D\right)\right) \cdot h}{\color{blue}{\frac{\frac{\ell}{\frac{0.5}{d} \cdot D}}{M}}}} \]
      *-un-lft-identity [=>] 27.5	\[ w0 \cdot \sqrt{1 - \frac{\left(M \cdot \left(\frac{0.5}{d} \cdot D\right)\right) \cdot h}{\frac{\frac{\color{blue}{1 \cdot \ell}}{\frac{0.5}{d} \cdot D}}{M}}} \]
      times-frac [=>] 24.5	\[ w0 \cdot \sqrt{1 - \frac{\left(M \cdot \left(\frac{0.5}{d} \cdot D\right)\right) \cdot h}{\frac{\color{blue}{\frac{1}{\frac{0.5}{d}} \cdot \frac{\ell}{D}}}{M}}} \]
      clear-num [<=] 24.5	\[ w0 \cdot \sqrt{1 - \frac{\left(M \cdot \left(\frac{0.5}{d} \cdot D\right)\right) \cdot h}{\frac{\color{blue}{\frac{d}{0.5}} \cdot \frac{\ell}{D}}{M}}} \]
      div-inv [=>] 24.5	\[ w0 \cdot \sqrt{1 - \frac{\left(M \cdot \left(\frac{0.5}{d} \cdot D\right)\right) \cdot h}{\frac{\color{blue}{\left(d \cdot \frac{1}{0.5}\right)} \cdot \frac{\ell}{D}}{M}}} \]
      metadata-eval [=>] 24.5	\[ w0 \cdot \sqrt{1 - \frac{\left(M \cdot \left(\frac{0.5}{d} \cdot D\right)\right) \cdot h}{\frac{\left(d \cdot \color{blue}{2}\right) \cdot \frac{\ell}{D}}{M}}} \]

   6. Simplified 26.1%

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

      [Start] 24.5	\[ w0 \cdot \sqrt{1 - \frac{\left(M \cdot \left(\frac{0.5}{d} \cdot D\right)\right) \cdot h}{\frac{\left(d \cdot 2\right) \cdot \frac{\ell}{D}}{M}}} \]
      associate-/l* [=>] 25.3	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{\frac{\frac{\left(d \cdot 2\right) \cdot \frac{\ell}{D}}{M}}{h}}}} \]
      *-commutative [=>] 25.3	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\frac{0.5}{d} \cdot D\right) \cdot M}}{\frac{\frac{\left(d \cdot 2\right) \cdot \frac{\ell}{D}}{M}}{h}}} \]
      associate-*l/ [=>] 25.3	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{0.5 \cdot D}{d}} \cdot M}{\frac{\frac{\left(d \cdot 2\right) \cdot \frac{\ell}{D}}{M}}{h}}} \]
      associate-/r/ [<=] 27.4	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{0.5 \cdot D}{\frac{d}{M}}}}{\frac{\frac{\left(d \cdot 2\right) \cdot \frac{\ell}{D}}{M}}{h}}} \]
      associate-*r/ [<=] 27.4	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{0.5 \cdot \frac{D}{\frac{d}{M}}}}{\frac{\frac{\left(d \cdot 2\right) \cdot \frac{\ell}{D}}{M}}{h}}} \]
      associate-/l/ [=>] 26.1	\[ w0 \cdot \sqrt{1 - \frac{0.5 \cdot \frac{D}{\frac{d}{M}}}{\color{blue}{\frac{\left(d \cdot 2\right) \cdot \frac{\ell}{D}}{h \cdot M}}}} \]
      associate-*r/ [=>] 25.3	\[ w0 \cdot \sqrt{1 - \frac{0.5 \cdot \frac{D}{\frac{d}{M}}}{\frac{\color{blue}{\frac{\left(d \cdot 2\right) \cdot \ell}{D}}}{h \cdot M}}} \]
      associate-/l* [=>] 26.1	\[ w0 \cdot \sqrt{1 - \frac{0.5 \cdot \frac{D}{\frac{d}{M}}}{\frac{\color{blue}{\frac{d \cdot 2}{\frac{D}{\ell}}}}{h \cdot M}}} \]
      *-commutative [=>] 26.1	\[ w0 \cdot \sqrt{1 - \frac{0.5 \cdot \frac{D}{\frac{d}{M}}}{\frac{\frac{d \cdot 2}{\frac{D}{\ell}}}{\color{blue}{M \cdot h}}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 10^{-318}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right) \cdot \left(\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right) \cdot h\right)}{\ell}}\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 4 \cdot 10^{+306}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.5 \cdot \frac{D}{\frac{d}{M}}}{\frac{\frac{2 \cdot d}{\frac{D}{\ell}}}{M \cdot h}}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	86.2%
Cost	8393

\[\begin{array}{l} t_0 := M \cdot \left(D \cdot \frac{0.5}{d}\right)\\ \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{+87} \lor \neg \left(\frac{h}{\ell} \leq -1 \cdot 10^{-105}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{t_0 \cdot \left(t_0 \cdot h\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \frac{h}{\ell}}{\frac{d}{D}}\right)}\\ \end{array} \]

Alternative 2
Accuracy	84.3%
Cost	8264

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{+265}:\\ \;\;\;\;w0\\ \mathbf{elif}\;\frac{h}{\ell} \leq -5 \cdot 10^{-279}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \left(\frac{D}{d} \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 3
Accuracy	84.7%
Cost	8264

\[\begin{array}{l} t_0 := M \cdot \frac{D}{d}\\ \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{+294}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(M \cdot h\right) \cdot \frac{D \cdot t_0}{d \cdot \ell}\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -5 \cdot 10^{-279}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(t_0 \cdot \left(\frac{D}{d} \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 4
Accuracy	85.0%
Cost	8264

\[\begin{array}{l} t_0 := M \cdot \frac{D}{d}\\ \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\frac{\left(M \cdot h\right) \cdot t_0}{\ell}}{\frac{d}{D}}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -5 \cdot 10^{-279}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(t_0 \cdot \left(\frac{D}{d} \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 5
Accuracy	85.0%
Cost	8264

\[\begin{array}{l} t_0 := M \cdot \frac{D}{d}\\ \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\frac{\left(M \cdot h\right) \cdot t_0}{\ell}}{\frac{d}{D}}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -5 \cdot 10^{-279}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(t_0 \cdot \frac{M \cdot \frac{h}{\ell}}{\frac{d}{D}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 6
Accuracy	87.0%
Cost	8137

\[\begin{array}{l} t_0 := M \cdot \left(D \cdot \frac{0.5}{d}\right)\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{-22} \lor \neg \left(\ell \leq 10^{+78}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(t_0 \cdot \frac{t_0}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{t_0 \cdot \left(t_0 \cdot h\right)}{\ell}}\\ \end{array} \]

Alternative 7
Accuracy	79.0%
Cost	64

\[w0 \]

Reproduce

herbie shell --seed 2023146 
(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))))))