Henrywood and Agarwal, Equation (9a)

Average Accuracy: 79.0% → 86.2%

Time: 22.9s

Precision: binary64

Cost: 14856

Math

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

↓

\[\begin{array}{l} t_0 := \frac{M \cdot D}{2 \cdot d}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+98}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D \cdot \left(\frac{M}{d} \cdot -0.25\right)}{d \cdot \frac{\frac{\frac{\ell}{h}}{M}}{D}}}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+149}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell \cdot {\left(2 \cdot \frac{d}{M \cdot D}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + M \cdot \frac{M \cdot \left(\frac{D}{d} \cdot -0.5\right)}{\frac{\frac{2}{\frac{h}{\ell}}}{\frac{D}{d}}}}\\ \end{array} \]

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 (/ (* M D) (* 2.0 d))))
   (if (<= t_0 -5e+98)
     (*
      w0
      (sqrt (+ 1.0 (/ (* D (* (/ M d) -0.25)) (* d (/ (/ (/ l h) M) D))))))
     (if (<= t_0 2e+149)
       (* w0 (sqrt (- 1.0 (/ h (* l (pow (* 2.0 (/ d (* M D))) 2.0))))))
       (*
        w0
        (sqrt
         (+
          1.0
          (* M (/ (* M (* (/ D d) -0.5)) (/ (/ 2.0 (/ h l)) (/ D d)))))))))))

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 = (M * D) / (2.0 * d);
	double tmp;
	if (t_0 <= -5e+98) {
		tmp = w0 * sqrt((1.0 + ((D * ((M / d) * -0.25)) / (d * (((l / h) / M) / D)))));
	} else if (t_0 <= 2e+149) {
		tmp = w0 * sqrt((1.0 - (h / (l * pow((2.0 * (d / (M * D))), 2.0)))));
	} else {
		tmp = w0 * sqrt((1.0 + (M * ((M * ((D / d) * -0.5)) / ((2.0 / (h / l)) / (D / d))))));
	}
	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) :: tmp
    t_0 = (m * d) / (2.0d0 * d_1)
    if (t_0 <= (-5d+98)) then
        tmp = w0 * sqrt((1.0d0 + ((d * ((m / d_1) * (-0.25d0))) / (d_1 * (((l / h) / m) / d)))))
    else if (t_0 <= 2d+149) then
        tmp = w0 * sqrt((1.0d0 - (h / (l * ((2.0d0 * (d_1 / (m * d))) ** 2.0d0)))))
    else
        tmp = w0 * sqrt((1.0d0 + (m * ((m * ((d / d_1) * (-0.5d0))) / ((2.0d0 / (h / l)) / (d / d_1))))))
    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 = (M * D) / (2.0 * d);
	double tmp;
	if (t_0 <= -5e+98) {
		tmp = w0 * Math.sqrt((1.0 + ((D * ((M / d) * -0.25)) / (d * (((l / h) / M) / D)))));
	} else if (t_0 <= 2e+149) {
		tmp = w0 * Math.sqrt((1.0 - (h / (l * Math.pow((2.0 * (d / (M * D))), 2.0)))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (M * ((M * ((D / d) * -0.5)) / ((2.0 / (h / l)) / (D / d))))));
	}
	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 = (M * D) / (2.0 * d)
	tmp = 0
	if t_0 <= -5e+98:
		tmp = w0 * math.sqrt((1.0 + ((D * ((M / d) * -0.25)) / (d * (((l / h) / M) / D)))))
	elif t_0 <= 2e+149:
		tmp = w0 * math.sqrt((1.0 - (h / (l * math.pow((2.0 * (d / (M * D))), 2.0)))))
	else:
		tmp = w0 * math.sqrt((1.0 + (M * ((M * ((D / d) * -0.5)) / ((2.0 / (h / l)) / (D / d))))))
	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))
	tmp = 0.0
	if (t_0 <= -5e+98)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(D * Float64(Float64(M / d) * -0.25)) / Float64(d * Float64(Float64(Float64(l / h) / M) / D))))));
	elseif (t_0 <= 2e+149)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h / Float64(l * (Float64(2.0 * Float64(d / Float64(M * D))) ^ 2.0))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(M * Float64(Float64(M * Float64(Float64(D / d) * -0.5)) / Float64(Float64(2.0 / Float64(h / l)) / Float64(D / d)))))));
	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);
	tmp = 0.0;
	if (t_0 <= -5e+98)
		tmp = w0 * sqrt((1.0 + ((D * ((M / d) * -0.25)) / (d * (((l / h) / M) / D)))));
	elseif (t_0 <= 2e+149)
		tmp = w0 * sqrt((1.0 - (h / (l * ((2.0 * (d / (M * D))) ^ 2.0)))));
	else
		tmp = w0 * sqrt((1.0 + (M * ((M * ((D / d) * -0.5)) / ((2.0 / (h / l)) / (D / d))))));
	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[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+98], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(D * N[(N[(M / d), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] / N[(d * N[(N[(N[(l / h), $MachinePrecision] / M), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+149], N[(w0 * N[Sqrt[N[(1.0 - N[(h / N[(l * N[Power[N[(2.0 * N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(M * N[(N[(M * N[(N[(D / d), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 / N[(h / l), $MachinePrecision]), $MachinePrecision] / N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 M D) (*.f64 2 d)) < -4.9999999999999998e98

   1. Initial program 19.3%

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

   2. Simplified 20.4%

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

      [Start] 19.3	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      associate-*l/ [<=] 20.4	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2 \cdot d} \cdot D\right)}}^{2} \cdot \frac{h}{\ell}} \]
      *-commutative [=>] 20.4	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(D \cdot \frac{M}{2 \cdot d}\right)}}^{2} \cdot \frac{h}{\ell}} \]

   3. Applied egg-rr 11.8%

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

      [Start] 20.4	\[ w0 \cdot \sqrt{1 - {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      associate-*r/ [=>] 18.7	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot h}{\ell}}} \]
      associate-/l* [=>] 20.4	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}} \]
      unpow2 [=>] 20.4	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(D \cdot \frac{M}{2 \cdot d}\right) \cdot \left(D \cdot \frac{M}{2 \cdot d}\right)}}{\frac{\ell}{h}}} \]
      associate-*r/ [=>] 15.8	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{D \cdot M}{2 \cdot d}} \cdot \left(D \cdot \frac{M}{2 \cdot d}\right)}{\frac{\ell}{h}}} \]
      associate-*r/ [=>] 19.2	\[ w0 \cdot \sqrt{1 - \frac{\frac{D \cdot M}{2 \cdot d} \cdot \color{blue}{\frac{D \cdot M}{2 \cdot d}}}{\frac{\ell}{h}}} \]
      frac-times [=>] 8.1	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}}}{\frac{\ell}{h}}} \]
      associate-/l/ [=>] 11.8	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\frac{\ell}{h} \cdot \left(\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)\right)}}} \]
      *-commutative [=>] 11.8	\[ w0 \cdot \sqrt{1 - \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\frac{\ell}{h} \cdot \left(\color{blue}{\left(d \cdot 2\right)} \cdot \left(2 \cdot d\right)\right)}} \]
      *-commutative [=>] 11.8	\[ w0 \cdot \sqrt{1 - \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\frac{\ell}{h} \cdot \left(\left(d \cdot 2\right) \cdot \color{blue}{\left(d \cdot 2\right)}\right)}} \]
      swap-sqr [=>] 11.8	\[ w0 \cdot \sqrt{1 - \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\frac{\ell}{h} \cdot \color{blue}{\left(\left(d \cdot d\right) \cdot \left(2 \cdot 2\right)\right)}}} \]
      metadata-eval [=>] 11.8	\[ w0 \cdot \sqrt{1 - \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\frac{\ell}{h} \cdot \left(\left(d \cdot d\right) \cdot \color{blue}{4}\right)}} \]

   4. Simplified 5.2%

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

      [Start] 11.8	\[ w0 \cdot \sqrt{1 - \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\frac{\ell}{h} \cdot \left(\left(d \cdot d\right) \cdot 4\right)}} \]
      associate-/r* [=>] 11.8	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\frac{\ell}{h}}}{\left(d \cdot d\right) \cdot 4}}} \]
      swap-sqr [=>] 5.2	\[ w0 \cdot \sqrt{1 - \frac{\frac{\color{blue}{\left(D \cdot D\right) \cdot \left(M \cdot M\right)}}{\frac{\ell}{h}}}{\left(d \cdot d\right) \cdot 4}} \]
      associate-*l* [=>] 5.2	\[ w0 \cdot \sqrt{1 - \frac{\frac{\left(D \cdot D\right) \cdot \left(M \cdot M\right)}{\frac{\ell}{h}}}{\color{blue}{d \cdot \left(d \cdot 4\right)}}} \]

   5. Applied egg-rr 18.9%

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

      [Start] 5.2	\[ w0 \cdot \sqrt{1 - \frac{\frac{\left(D \cdot D\right) \cdot \left(M \cdot M\right)}{\frac{\ell}{h}}}{d \cdot \left(d \cdot 4\right)}} \]
      associate-/l/ [=>] 5.2	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(D \cdot D\right) \cdot \left(M \cdot M\right)}{\left(d \cdot \left(d \cdot 4\right)\right) \cdot \frac{\ell}{h}}}} \]
      unswap-sqr [=>] 11.8	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}{\left(d \cdot \left(d \cdot 4\right)\right) \cdot \frac{\ell}{h}}} \]
      times-frac [=>] 18.9	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{D \cdot M}{d \cdot \left(d \cdot 4\right)} \cdot \frac{D \cdot M}{\frac{\ell}{h}}}} \]

   6. Applied egg-rr 29.8%

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

      [Start] 18.9	\[ w0 \cdot \sqrt{1 - \frac{D \cdot M}{d \cdot \left(d \cdot 4\right)} \cdot \frac{D \cdot M}{\frac{\ell}{h}}} \]
      associate-*r/ [=>] 13.0	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{D \cdot M}{d \cdot \left(d \cdot 4\right)} \cdot \left(D \cdot M\right)}{\frac{\ell}{h}}}} \]
      associate-/l* [=>] 18.9	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{D \cdot M}{d \cdot \left(d \cdot 4\right)}}{\frac{\frac{\ell}{h}}{D \cdot M}}}} \]
      times-frac [=>] 16.2	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{D}{d} \cdot \frac{M}{d \cdot 4}}}{\frac{\frac{\ell}{h}}{D \cdot M}}} \]
      associate-*l/ [=>] 17.8	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{D \cdot \frac{M}{d \cdot 4}}{d}}}{\frac{\frac{\ell}{h}}{D \cdot M}}} \]
      associate-/l/ [=>] 25.8	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{D \cdot \frac{M}{d \cdot 4}}{\frac{\frac{\ell}{h}}{D \cdot M} \cdot d}}} \]
      *-un-lft-identity [=>] 25.8	\[ w0 \cdot \sqrt{1 - \frac{D \cdot \frac{\color{blue}{1 \cdot M}}{d \cdot 4}}{\frac{\frac{\ell}{h}}{D \cdot M} \cdot d}} \]
      *-commutative [=>] 25.8	\[ w0 \cdot \sqrt{1 - \frac{D \cdot \frac{1 \cdot M}{\color{blue}{4 \cdot d}}}{\frac{\frac{\ell}{h}}{D \cdot M} \cdot d}} \]
      times-frac [=>] 25.8	\[ w0 \cdot \sqrt{1 - \frac{D \cdot \color{blue}{\left(\frac{1}{4} \cdot \frac{M}{d}\right)}}{\frac{\frac{\ell}{h}}{D \cdot M} \cdot d}} \]
      metadata-eval [=>] 25.8	\[ w0 \cdot \sqrt{1 - \frac{D \cdot \left(\color{blue}{0.25} \cdot \frac{M}{d}\right)}{\frac{\frac{\ell}{h}}{D \cdot M} \cdot d}} \]
      associate-/l/ [<=] 29.8	\[ w0 \cdot \sqrt{1 - \frac{D \cdot \left(0.25 \cdot \frac{M}{d}\right)}{\color{blue}{\frac{\frac{\frac{\ell}{h}}{M}}{D}} \cdot d}} \]

   if -4.9999999999999998e98 < (/.f64 (*.f64 M D) (*.f64 2 d)) < 2.0000000000000001e149

   1. Initial program 90.4%

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

   2. Applied egg-rr 90.9%

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

      [Start] 90.4	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      associate-*r/ [=>] 96.1	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}}} \]
      associate-/l* [=>] 91.2	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}} \]
      unpow2 [=>] 91.2	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}}{\frac{\ell}{h}}} \]
      clear-num [=>] 91.2	\[ w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{1}{\frac{2 \cdot d}{M \cdot D}}}}{\frac{\ell}{h}}} \]
      clear-num [=>] 91.2	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{1}{\frac{2 \cdot d}{M \cdot D}}} \cdot \frac{1}{\frac{2 \cdot d}{M \cdot D}}}{\frac{\ell}{h}}} \]
      frac-times [=>] 91.2	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{1 \cdot 1}{\frac{2 \cdot d}{M \cdot D} \cdot \frac{2 \cdot d}{M \cdot D}}}}{\frac{\ell}{h}}} \]
      metadata-eval [=>] 91.2	\[ w0 \cdot \sqrt{1 - \frac{\frac{\color{blue}{1}}{\frac{2 \cdot d}{M \cdot D} \cdot \frac{2 \cdot d}{M \cdot D}}}{\frac{\ell}{h}}} \]
      associate-/l/ [=>] 91.2	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{1}{\frac{\ell}{h} \cdot \left(\frac{2 \cdot d}{M \cdot D} \cdot \frac{2 \cdot d}{M \cdot D}\right)}}} \]
      *-commutative [=>] 91.2	\[ w0 \cdot \sqrt{1 - \frac{1}{\frac{\ell}{h} \cdot \left(\frac{2 \cdot d}{\color{blue}{D \cdot M}} \cdot \frac{2 \cdot d}{M \cdot D}\right)}} \]
      times-frac [=>] 90.8	\[ w0 \cdot \sqrt{1 - \frac{1}{\frac{\ell}{h} \cdot \left(\color{blue}{\left(\frac{2}{D} \cdot \frac{d}{M}\right)} \cdot \frac{2 \cdot d}{M \cdot D}\right)}} \]
      *-commutative [=>] 90.8	\[ w0 \cdot \sqrt{1 - \frac{1}{\frac{\ell}{h} \cdot \left(\left(\frac{2}{D} \cdot \frac{d}{M}\right) \cdot \frac{2 \cdot d}{\color{blue}{D \cdot M}}\right)}} \]
      times-frac [=>] 90.9	\[ w0 \cdot \sqrt{1 - \frac{1}{\frac{\ell}{h} \cdot \left(\left(\frac{2}{D} \cdot \frac{d}{M}\right) \cdot \color{blue}{\left(\frac{2}{D} \cdot \frac{d}{M}\right)}\right)}} \]

   3. Applied egg-rr 89.7%

      \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{\frac{h}{\ell}}{{\left(\frac{2}{D} \cdot \frac{d}{M}\right)}^{2}}}\right)} - 1\right)} \]
      Proof

      [Start] 90.9	\[ w0 \cdot \sqrt{1 - \frac{1}{\frac{\ell}{h} \cdot \left(\left(\frac{2}{D} \cdot \frac{d}{M}\right) \cdot \left(\frac{2}{D} \cdot \frac{d}{M}\right)\right)}} \]
      expm1-log1p-u [=>] 90.5	\[ w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{1}{\frac{\ell}{h} \cdot \left(\left(\frac{2}{D} \cdot \frac{d}{M}\right) \cdot \left(\frac{2}{D} \cdot \frac{d}{M}\right)\right)}}\right)\right)} \]
      expm1-udef [=>] 90.5	\[ w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{1}{\frac{\ell}{h} \cdot \left(\left(\frac{2}{D} \cdot \frac{d}{M}\right) \cdot \left(\frac{2}{D} \cdot \frac{d}{M}\right)\right)}}\right)} - 1\right)} \]
      associate-/r* [=>] 89.7	\[ w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \color{blue}{\frac{\frac{1}{\frac{\ell}{h}}}{\left(\frac{2}{D} \cdot \frac{d}{M}\right) \cdot \left(\frac{2}{D} \cdot \frac{d}{M}\right)}}}\right)} - 1\right) \]
      clear-num [<=] 89.7	\[ w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{\color{blue}{\frac{h}{\ell}}}{\left(\frac{2}{D} \cdot \frac{d}{M}\right) \cdot \left(\frac{2}{D} \cdot \frac{d}{M}\right)}}\right)} - 1\right) \]
      pow2 [=>] 89.7	\[ w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{\frac{h}{\ell}}{\color{blue}{{\left(\frac{2}{D} \cdot \frac{d}{M}\right)}^{2}}}}\right)} - 1\right) \]

   4. Simplified 96.2%

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

      [Start] 89.7	\[ w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{\frac{h}{\ell}}{{\left(\frac{2}{D} \cdot \frac{d}{M}\right)}^{2}}}\right)} - 1\right) \]
      expm1-def [=>] 89.7	\[ w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{\frac{h}{\ell}}{{\left(\frac{2}{D} \cdot \frac{d}{M}\right)}^{2}}}\right)\right)} \]
      expm1-log1p [=>] 90.1	\[ w0 \cdot \color{blue}{\sqrt{1 - \frac{\frac{h}{\ell}}{{\left(\frac{2}{D} \cdot \frac{d}{M}\right)}^{2}}}} \]
      associate-/r* [<=] 95.9	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{h}{\ell \cdot {\left(\frac{2}{D} \cdot \frac{d}{M}\right)}^{2}}}} \]
      times-frac [<=] 96.2	\[ w0 \cdot \sqrt{1 - \frac{h}{\ell \cdot {\color{blue}{\left(\frac{2 \cdot d}{D \cdot M}\right)}}^{2}}} \]
      associate-*r/ [<=] 96.2	\[ w0 \cdot \sqrt{1 - \frac{h}{\ell \cdot {\color{blue}{\left(2 \cdot \frac{d}{D \cdot M}\right)}}^{2}}} \]

   if 2.0000000000000001e149 < (/.f64 (*.f64 M D) (*.f64 2 d))

   1. Initial program 0.6%

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

   2. Applied egg-rr 13.5%

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

      [Start] 0.6	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      associate-*r/ [=>] 0.0	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}}} \]
      associate-/l* [=>] 0.6	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}} \]
      unpow2 [=>] 0.6	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}}{\frac{\ell}{h}}} \]
      associate-/l* [=>] 0.2	\[ w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{M}{\frac{2 \cdot d}{D}}}}{\frac{\ell}{h}}} \]
      associate-*r/ [=>] 0.2	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{\frac{M \cdot D}{2 \cdot d} \cdot M}{\frac{2 \cdot d}{D}}}}{\frac{\ell}{h}}} \]
      associate-/l/ [=>] 7.2	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d} \cdot M}{\frac{\ell}{h} \cdot \frac{2 \cdot d}{D}}}} \]
      associate-/l* [=>] 13.7	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{M}{\frac{2 \cdot d}{D}}} \cdot M}{\frac{\ell}{h} \cdot \frac{2 \cdot d}{D}}} \]
      div-inv [=>] 13.3	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(M \cdot \frac{1}{\frac{2 \cdot d}{D}}\right)} \cdot M}{\frac{\ell}{h} \cdot \frac{2 \cdot d}{D}}} \]
      associate-/l* [=>] 13.3	\[ w0 \cdot \sqrt{1 - \frac{\left(M \cdot \frac{1}{\color{blue}{\frac{2}{\frac{D}{d}}}}\right) \cdot M}{\frac{\ell}{h} \cdot \frac{2 \cdot d}{D}}} \]
      associate-/r/ [=>] 13.3	\[ w0 \cdot \sqrt{1 - \frac{\left(M \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{D}{d}\right)}\right) \cdot M}{\frac{\ell}{h} \cdot \frac{2 \cdot d}{D}}} \]
      metadata-eval [=>] 13.3	\[ w0 \cdot \sqrt{1 - \frac{\left(M \cdot \left(\color{blue}{0.5} \cdot \frac{D}{d}\right)\right) \cdot M}{\frac{\ell}{h} \cdot \frac{2 \cdot d}{D}}} \]
      *-un-lft-identity [=>] 13.3	\[ w0 \cdot \sqrt{1 - \frac{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot M}{\frac{\ell}{h} \cdot \frac{2 \cdot d}{\color{blue}{1 \cdot D}}}} \]
      times-frac [=>] 13.5	\[ w0 \cdot \sqrt{1 - \frac{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot M}{\frac{\ell}{h} \cdot \color{blue}{\left(\frac{2}{1} \cdot \frac{d}{D}\right)}}} \]
      metadata-eval [=>] 13.5	\[ w0 \cdot \sqrt{1 - \frac{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot M}{\frac{\ell}{h} \cdot \left(\color{blue}{2} \cdot \frac{d}{D}\right)}} \]

   3. Applied egg-rr 24.1%

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

      [Start] 13.5	\[ w0 \cdot \sqrt{1 - \frac{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot M}{\frac{\ell}{h} \cdot \left(2 \cdot \frac{d}{D}\right)}} \]
      associate-/l* [=>] 24.1	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}{\frac{\frac{\ell}{h} \cdot \left(2 \cdot \frac{d}{D}\right)}{M}}}} \]
      associate-/r/ [=>] 24.1	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}{\frac{\ell}{h} \cdot \left(2 \cdot \frac{d}{D}\right)} \cdot M}} \]
      associate-*r* [=>] 24.1	\[ w0 \cdot \sqrt{1 - \frac{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}{\color{blue}{\left(\frac{\ell}{h} \cdot 2\right) \cdot \frac{d}{D}}} \cdot M} \]
      clear-num [=>] 24.1	\[ w0 \cdot \sqrt{1 - \frac{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}{\left(\frac{\ell}{h} \cdot 2\right) \cdot \color{blue}{\frac{1}{\frac{D}{d}}}} \cdot M} \]
      un-div-inv [=>] 24.1	\[ w0 \cdot \sqrt{1 - \frac{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}{\color{blue}{\frac{\frac{\ell}{h} \cdot 2}{\frac{D}{d}}}} \cdot M} \]
      clear-num [=>] 24.1	\[ w0 \cdot \sqrt{1 - \frac{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}{\frac{\color{blue}{\frac{1}{\frac{h}{\ell}}} \cdot 2}{\frac{D}{d}}} \cdot M} \]
      associate-*l/ [=>] 24.1	\[ w0 \cdot \sqrt{1 - \frac{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}{\frac{\color{blue}{\frac{1 \cdot 2}{\frac{h}{\ell}}}}{\frac{D}{d}}} \cdot M} \]
      metadata-eval [=>] 24.1	\[ w0 \cdot \sqrt{1 - \frac{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}{\frac{\frac{\color{blue}{2}}{\frac{h}{\ell}}}{\frac{D}{d}}} \cdot M} \]

3. Recombined 3 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq -5 \cdot 10^{+98}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D \cdot \left(\frac{M}{d} \cdot -0.25\right)}{d \cdot \frac{\frac{\frac{\ell}{h}}{M}}{D}}}\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \leq 2 \cdot 10^{+149}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell \cdot {\left(2 \cdot \frac{d}{M \cdot D}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + M \cdot \frac{M \cdot \left(\frac{D}{d} \cdot -0.5\right)}{\frac{\frac{2}{\frac{h}{\ell}}}{\frac{D}{d}}}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	81.1%
Cost	8272

\[\begin{array}{l} t_0 := w0 \cdot \sqrt{1 - \frac{h \cdot \left(M \cdot \frac{M}{\frac{d}{D}}\right)}{\ell \cdot \left(\frac{d}{D} \cdot 4\right)}}\\ \mathbf{if}\;h \leq -5 \cdot 10^{-89}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-268}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D \cdot \left(\frac{M}{d} \cdot -0.25\right)}{d \cdot \frac{\frac{\frac{\ell}{h}}{M}}{D}}}\\ \mathbf{elif}\;h \leq 5.8 \cdot 10^{+195}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq 2 \cdot 10^{+255}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot \frac{D}{\ell}\right) \cdot \frac{M \cdot D}{\frac{d}{\frac{\frac{M}{d}}{4}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{h}{\ell \cdot 4}}{\frac{d}{D}} \cdot \left(M \cdot \left(M \cdot \frac{D}{d}\right)\right)}\\ \end{array} \]

Alternative 2
Accuracy	84.0%
Cost	8264

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0\\ \mathbf{elif}\;\frac{h}{\ell} \leq -2 \cdot 10^{-277}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{h}{\ell \cdot 4}}{\frac{d}{D}} \cdot \left(M \cdot \left(M \cdot \frac{D}{d}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 3
Accuracy	77.9%
Cost	8141

\[\begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{-129} \lor \neg \left(d \leq 1.35 \cdot 10^{-137}\right) \land d \leq 10^{+100}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot \frac{D}{\ell}\right) \cdot \frac{M \cdot D}{\frac{d}{\frac{\frac{M}{d}}{4}}}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 4
Accuracy	79.3%
Cost	64

\[w0 \]

Reproduce

herbie shell --seed 2023131 
(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))))))