Henrywood and Agarwal, Equation (9a)

Average Accuracy: 65.8% → 72.2%

Time: 18.6s

Precision: binary64

Cost: 20804

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 10^{+81}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - M \cdot \left(D \cdot \frac{\frac{0.5}{d} \cdot \left(D \cdot \left(M \cdot h\right)\right)}{2 \cdot \left(d \cdot \ell\right)}\right)}\\ \end{array} \]

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
 (if (<= (pow (/ (* M D) (* 2.0 d)) 2.0) 1e+81)
   (* w0 (sqrt (- 1.0 (* h (/ (pow (* D (/ M (/ d 0.5))) 2.0) l)))))
   (*
    w0
    (sqrt
     (- 1.0 (* M (* D (/ (* (/ 0.5 d) (* D (* M h))) (* 2.0 (* d l))))))))))

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 tmp;
	if (pow(((M * D) / (2.0 * d)), 2.0) <= 1e+81) {
		tmp = w0 * sqrt((1.0 - (h * (pow((D * (M / (d / 0.5))), 2.0) / l))));
	} else {
		tmp = w0 * sqrt((1.0 - (M * (D * (((0.5 / d) * (D * (M * h))) / (2.0 * (d * l)))))));
	}
	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) :: tmp
    if ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) <= 1d+81) then
        tmp = w0 * sqrt((1.0d0 - (h * (((d * (m / (d_1 / 0.5d0))) ** 2.0d0) / l))))
    else
        tmp = w0 * sqrt((1.0d0 - (m * (d * (((0.5d0 / d_1) * (d * (m * h))) / (2.0d0 * (d_1 * l)))))))
    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 tmp;
	if (Math.pow(((M * D) / (2.0 * d)), 2.0) <= 1e+81) {
		tmp = w0 * Math.sqrt((1.0 - (h * (Math.pow((D * (M / (d / 0.5))), 2.0) / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (M * (D * (((0.5 / d) * (D * (M * h))) / (2.0 * (d * l)))))));
	}
	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):
	tmp = 0
	if math.pow(((M * D) / (2.0 * d)), 2.0) <= 1e+81:
		tmp = w0 * math.sqrt((1.0 - (h * (math.pow((D * (M / (d / 0.5))), 2.0) / l))))
	else:
		tmp = w0 * math.sqrt((1.0 - (M * (D * (((0.5 / d) * (D * (M * h))) / (2.0 * (d * l)))))))
	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)
	tmp = 0.0
	if ((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) <= 1e+81)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64((Float64(D * Float64(M / Float64(d / 0.5))) ^ 2.0) / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(M * Float64(D * Float64(Float64(Float64(0.5 / d) * Float64(D * Float64(M * h))) / Float64(2.0 * Float64(d * l))))))));
	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)
	tmp = 0.0;
	if ((((M * D) / (2.0 * d)) ^ 2.0) <= 1e+81)
		tmp = w0 * sqrt((1.0 - (h * (((D * (M / (d / 0.5))) ^ 2.0) / l))));
	else
		tmp = w0 * sqrt((1.0 - (M * (D * (((0.5 / d) * (D * (M * h))) / (2.0 * (d * l)))))));
	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_] := If[LessEqual[N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 1e+81], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(D * N[(M / N[(d / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(M * N[(D * N[(N[(N[(0.5 / d), $MachinePrecision] * N[(D * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 9.99999999999999921e80

   1. Initial program 94.2%

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

   2. Applied egg-rr 93.6%

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

      [Start] 94.2	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      clear-num [=>] 94.2	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}} \]
      un-div-inv [=>] 94.8	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}} \]
      div-inv [=>] 94.8	\[ w0 \cdot \sqrt{1 - \frac{{\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}}^{2}}{\frac{\ell}{h}}} \]
      associate-*l* [=>] 93.6	\[ w0 \cdot \sqrt{1 - \frac{{\color{blue}{\left(M \cdot \left(D \cdot \frac{1}{2 \cdot d}\right)\right)}}^{2}}{\frac{\ell}{h}}} \]
      associate-/r* [=>] 93.6	\[ w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \left(D \cdot \color{blue}{\frac{\frac{1}{2}}{d}}\right)\right)}^{2}}{\frac{\ell}{h}}} \]
      metadata-eval [=>] 93.6	\[ w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \left(D \cdot \frac{\color{blue}{0.5}}{d}\right)\right)}^{2}}{\frac{\ell}{h}}} \]

   3. Simplified 98.0%

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

      [Start] 93.6	\[ w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{h}}} \]
      associate-/r/ [=>] 97.1	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell} \cdot h}} \]
      *-commutative [=>] 97.1	\[ w0 \cdot \sqrt{1 - \color{blue}{h \cdot \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}} \]
      *-commutative [=>] 97.1	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\color{blue}{\left(\left(D \cdot \frac{0.5}{d}\right) \cdot M\right)}}^{2}}{\ell}} \]
      associate-*r* [<=] 98.0	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\color{blue}{\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}}^{2}}{\ell}} \]
      *-commutative [<=] 98.0	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(D \cdot \color{blue}{\left(M \cdot \frac{0.5}{d}\right)}\right)}^{2}}{\ell}} \]
      associate-*r/ [=>] 98.0	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(D \cdot \color{blue}{\frac{M \cdot 0.5}{d}}\right)}^{2}}{\ell}} \]
      associate-/l* [=>] 98.0	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(D \cdot \color{blue}{\frac{M}{\frac{d}{0.5}}}\right)}^{2}}{\ell}} \]

   if 9.99999999999999921e80 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)

   1. Initial program 16.8%

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

   2. Applied egg-rr 19.2%

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

      [Start] 16.8	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      clear-num [=>] 16.8	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}} \]
      un-div-inv [=>] 16.8	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}} \]
      div-inv [=>] 16.8	\[ w0 \cdot \sqrt{1 - \frac{{\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}}^{2}}{\frac{\ell}{h}}} \]
      associate-*l* [=>] 19.2	\[ w0 \cdot \sqrt{1 - \frac{{\color{blue}{\left(M \cdot \left(D \cdot \frac{1}{2 \cdot d}\right)\right)}}^{2}}{\frac{\ell}{h}}} \]
      associate-/r* [=>] 19.2	\[ w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \left(D \cdot \color{blue}{\frac{\frac{1}{2}}{d}}\right)\right)}^{2}}{\frac{\ell}{h}}} \]
      metadata-eval [=>] 19.2	\[ w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \left(D \cdot \frac{\color{blue}{0.5}}{d}\right)\right)}^{2}}{\frac{\ell}{h}}} \]

   3. Simplified 16.8%

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

      [Start] 19.2	\[ w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{h}}} \]
      associate-/r/ [=>] 16.8	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell} \cdot h}} \]
      *-commutative [=>] 16.8	\[ w0 \cdot \sqrt{1 - \color{blue}{h \cdot \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}} \]
      *-commutative [=>] 16.8	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\color{blue}{\left(\left(D \cdot \frac{0.5}{d}\right) \cdot M\right)}}^{2}}{\ell}} \]
      associate-*r* [<=] 16.8	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\color{blue}{\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}}^{2}}{\ell}} \]
      *-commutative [<=] 16.8	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(D \cdot \color{blue}{\left(M \cdot \frac{0.5}{d}\right)}\right)}^{2}}{\ell}} \]
      associate-*r/ [=>] 16.8	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(D \cdot \color{blue}{\frac{M \cdot 0.5}{d}}\right)}^{2}}{\ell}} \]
      associate-/l* [=>] 16.8	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(D \cdot \color{blue}{\frac{M}{\frac{d}{0.5}}}\right)}^{2}}{\ell}} \]

   4. Applied egg-rr 24.1%

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

      [Start] 16.8	\[ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right)}^{2}}{\ell}} \]
      unpow2 [=>] 16.8	\[ w0 \cdot \sqrt{1 - h \cdot \frac{\color{blue}{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right) \cdot \left(D \cdot \frac{M}{\frac{d}{0.5}}\right)}}{\ell}} \]
      *-un-lft-identity [=>] 16.8	\[ w0 \cdot \sqrt{1 - h \cdot \frac{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right) \cdot \left(D \cdot \frac{M}{\frac{d}{0.5}}\right)}{\color{blue}{1 \cdot \ell}}} \]
      times-frac [=>] 20.5	\[ w0 \cdot \sqrt{1 - h \cdot \color{blue}{\left(\frac{D \cdot \frac{M}{\frac{d}{0.5}}}{1} \cdot \frac{D \cdot \frac{M}{\frac{d}{0.5}}}{\ell}\right)}} \]
      *-commutative [=>] 20.5	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{\color{blue}{\frac{M}{\frac{d}{0.5}} \cdot D}}{1} \cdot \frac{D \cdot \frac{M}{\frac{d}{0.5}}}{\ell}\right)} \]
      div-inv [=>] 20.5	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{\color{blue}{\left(M \cdot \frac{1}{\frac{d}{0.5}}\right)} \cdot D}{1} \cdot \frac{D \cdot \frac{M}{\frac{d}{0.5}}}{\ell}\right)} \]
      clear-num [<=] 20.5	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{\left(M \cdot \color{blue}{\frac{0.5}{d}}\right) \cdot D}{1} \cdot \frac{D \cdot \frac{M}{\frac{d}{0.5}}}{\ell}\right)} \]
      associate-*l* [=>] 19.2	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{\color{blue}{M \cdot \left(\frac{0.5}{d} \cdot D\right)}}{1} \cdot \frac{D \cdot \frac{M}{\frac{d}{0.5}}}{\ell}\right)} \]
      *-commutative [=>] 19.2	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{1} \cdot \frac{\color{blue}{\frac{M}{\frac{d}{0.5}} \cdot D}}{\ell}\right)} \]
      div-inv [=>] 19.2	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{1} \cdot \frac{\color{blue}{\left(M \cdot \frac{1}{\frac{d}{0.5}}\right)} \cdot D}{\ell}\right)} \]
      clear-num [<=] 19.2	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{1} \cdot \frac{\left(M \cdot \color{blue}{\frac{0.5}{d}}\right) \cdot D}{\ell}\right)} \]
      associate-*l* [=>] 24.1	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{1} \cdot \frac{\color{blue}{M \cdot \left(\frac{0.5}{d} \cdot D\right)}}{\ell}\right)} \]

   5. Applied egg-rr 25.4%

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

      [Start] 24.1	\[ w0 \cdot \sqrt{1 - h \cdot \left(\frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{1} \cdot \frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{\ell}\right)} \]
      /-rgt-identity [=>] 24.1	\[ w0 \cdot \sqrt{1 - h \cdot \left(\color{blue}{\left(M \cdot \left(\frac{0.5}{d} \cdot D\right)\right)} \cdot \frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{\ell}\right)} \]
      associate-*r* [=>] 28.9	\[ w0 \cdot \sqrt{1 - \color{blue}{\left(h \cdot \left(M \cdot \left(\frac{0.5}{d} \cdot D\right)\right)\right) \cdot \frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{\ell}}} \]
      clear-num [=>] 28.8	\[ w0 \cdot \sqrt{1 - \left(h \cdot \left(M \cdot \left(\frac{0.5}{d} \cdot D\right)\right)\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{M \cdot \left(\frac{0.5}{d} \cdot D\right)}}}} \]
      un-div-inv [=>] 29.2	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{h \cdot \left(M \cdot \left(\frac{0.5}{d} \cdot D\right)\right)}{\frac{\ell}{M \cdot \left(\frac{0.5}{d} \cdot D\right)}}}} \]
      associate-*r* [=>] 27.0	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(h \cdot M\right) \cdot \left(\frac{0.5}{d} \cdot D\right)}}{\frac{\ell}{M \cdot \left(\frac{0.5}{d} \cdot D\right)}}} \]
      *-commutative [=>] 27.0	\[ w0 \cdot \sqrt{1 - \frac{\left(h \cdot M\right) \cdot \left(\frac{0.5}{d} \cdot D\right)}{\frac{\ell}{\color{blue}{\left(\frac{0.5}{d} \cdot D\right) \cdot M}}}} \]
      associate-/r* [=>] 26.9	\[ w0 \cdot \sqrt{1 - \frac{\left(h \cdot M\right) \cdot \left(\frac{0.5}{d} \cdot D\right)}{\color{blue}{\frac{\frac{\ell}{\frac{0.5}{d} \cdot D}}{M}}}} \]
      *-un-lft-identity [=>] 26.9	\[ w0 \cdot \sqrt{1 - \frac{\left(h \cdot M\right) \cdot \left(\frac{0.5}{d} \cdot D\right)}{\frac{\frac{\color{blue}{1 \cdot \ell}}{\frac{0.5}{d} \cdot D}}{M}}} \]
      times-frac [=>] 25.4	\[ w0 \cdot \sqrt{1 - \frac{\left(h \cdot M\right) \cdot \left(\frac{0.5}{d} \cdot D\right)}{\frac{\color{blue}{\frac{1}{\frac{0.5}{d}} \cdot \frac{\ell}{D}}}{M}}} \]
      clear-num [<=] 25.4	\[ w0 \cdot \sqrt{1 - \frac{\left(h \cdot M\right) \cdot \left(\frac{0.5}{d} \cdot D\right)}{\frac{\color{blue}{\frac{d}{0.5}} \cdot \frac{\ell}{D}}{M}}} \]
      div-inv [=>] 25.4	\[ w0 \cdot \sqrt{1 - \frac{\left(h \cdot M\right) \cdot \left(\frac{0.5}{d} \cdot D\right)}{\frac{\color{blue}{\left(d \cdot \frac{1}{0.5}\right)} \cdot \frac{\ell}{D}}{M}}} \]
      metadata-eval [=>] 25.4	\[ w0 \cdot \sqrt{1 - \frac{\left(h \cdot M\right) \cdot \left(\frac{0.5}{d} \cdot D\right)}{\frac{\left(d \cdot \color{blue}{2}\right) \cdot \frac{\ell}{D}}{M}}} \]

   6. Simplified 27.8%

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

      [Start] 25.4	\[ w0 \cdot \sqrt{1 - \frac{\left(h \cdot M\right) \cdot \left(\frac{0.5}{d} \cdot D\right)}{\frac{\left(d \cdot 2\right) \cdot \frac{\ell}{D}}{M}}} \]
      associate-/r/ [=>] 25.4	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(h \cdot M\right) \cdot \left(\frac{0.5}{d} \cdot D\right)}{\left(d \cdot 2\right) \cdot \frac{\ell}{D}} \cdot M}} \]
      *-commutative [=>] 25.4	\[ w0 \cdot \sqrt{1 - \color{blue}{M \cdot \frac{\left(h \cdot M\right) \cdot \left(\frac{0.5}{d} \cdot D\right)}{\left(d \cdot 2\right) \cdot \frac{\ell}{D}}}} \]
      associate-*r/ [=>] 26.6	\[ w0 \cdot \sqrt{1 - M \cdot \frac{\left(h \cdot M\right) \cdot \left(\frac{0.5}{d} \cdot D\right)}{\color{blue}{\frac{\left(d \cdot 2\right) \cdot \ell}{D}}}} \]
      associate-/r/ [=>] 26.6	\[ w0 \cdot \sqrt{1 - M \cdot \color{blue}{\left(\frac{\left(h \cdot M\right) \cdot \left(\frac{0.5}{d} \cdot D\right)}{\left(d \cdot 2\right) \cdot \ell} \cdot D\right)}} \]
      *-commutative [=>] 26.6	\[ w0 \cdot \sqrt{1 - M \cdot \left(\frac{\color{blue}{\left(\frac{0.5}{d} \cdot D\right) \cdot \left(h \cdot M\right)}}{\left(d \cdot 2\right) \cdot \ell} \cdot D\right)} \]
      associate-*l* [=>] 27.8	\[ w0 \cdot \sqrt{1 - M \cdot \left(\frac{\color{blue}{\frac{0.5}{d} \cdot \left(D \cdot \left(h \cdot M\right)\right)}}{\left(d \cdot 2\right) \cdot \ell} \cdot D\right)} \]
      *-commutative [=>] 27.8	\[ w0 \cdot \sqrt{1 - M \cdot \left(\frac{\frac{0.5}{d} \cdot \left(D \cdot \left(h \cdot M\right)\right)}{\color{blue}{\left(2 \cdot d\right)} \cdot \ell} \cdot D\right)} \]
      associate-*l* [=>] 27.8	\[ w0 \cdot \sqrt{1 - M \cdot \left(\frac{\frac{0.5}{d} \cdot \left(D \cdot \left(h \cdot M\right)\right)}{\color{blue}{2 \cdot \left(d \cdot \ell\right)}} \cdot D\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 10^{+81}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - M \cdot \left(D \cdot \frac{\frac{0.5}{d} \cdot \left(D \cdot \left(M \cdot h\right)\right)}{2 \cdot \left(d \cdot \ell\right)}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	70.1%
Cost	8136

\[\begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{+239}:\\ \;\;\;\;w0 \cdot \sqrt{1 - M \cdot \left(D \cdot \frac{\frac{0.5}{d} \cdot \left(D \cdot \left(M \cdot h\right)\right)}{2 \cdot \left(d \cdot \ell\right)}\right)}\\ \mathbf{elif}\;d \leq 10^{+14}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \left(M \cdot \frac{D \cdot 0.5}{d}\right)}{\frac{2 \cdot d}{\frac{M}{\frac{\ell}{D}}}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{0.25}{\ell} \cdot \left(M \cdot \left(h \cdot \left(M \cdot \frac{D}{d}\right)\right)\right)}{\frac{d}{D}}}\\ \end{array} \]

Alternative 2
Accuracy	67.4%
Cost	8008

\[\begin{array}{l} \mathbf{if}\;D \leq 10^{-145}:\\ \;\;\;\;w0\\ \mathbf{elif}\;D \leq 1.2 \cdot 10^{+29}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \left(h \cdot \frac{M}{d}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 3
Accuracy	68.6%
Cost	8008

\[\begin{array}{l} \mathbf{if}\;M \leq -1.95 \cdot 10^{+152}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq 10^{-242}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \left(\frac{M \cdot M}{\frac{d}{D}} \cdot \frac{h}{\frac{d}{D}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \left(\frac{M \cdot h}{\frac{d}{D}} \cdot \frac{M}{\frac{d}{D}}\right)}\\ \end{array} \]

Alternative 4
Accuracy	69.8%
Cost	8008

\[\begin{array}{l} \mathbf{if}\;D \leq 5 \cdot 10^{-5}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\frac{0.25}{\ell}}{\frac{\frac{d}{D \cdot h}}{M}}}{\frac{d}{M \cdot D}}}\\ \mathbf{elif}\;D \leq 3.4 \cdot 10^{+190}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \left(\frac{M \cdot h}{\frac{d}{D}} \cdot \frac{M}{\frac{d}{D}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 5
Accuracy	64.6%
Cost	7876

\[\begin{array}{l} \mathbf{if}\;M \leq -2.6 \cdot 10^{+153}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \left(\frac{M \cdot M}{\frac{d}{D}} \cdot \frac{h}{\frac{d}{D}}\right)}\\ \end{array} \]

Alternative 6
Accuracy	70.4%
Cost	7876

\[\begin{array}{l} \mathbf{if}\;M \leq -5 \cdot 10^{+36}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \frac{M \cdot \frac{\frac{0.25}{\ell}}{\frac{\frac{d}{h}}{D}}}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \left(\frac{M \cdot h}{\frac{d}{D}} \cdot \frac{M}{\frac{d}{D}}\right)}\\ \end{array} \]

Alternative 7
Accuracy	72.5%
Cost	7872

\[\begin{array}{l} t_0 := M \cdot \left(D \cdot \frac{0.5}{d}\right)\\ w0 \cdot \sqrt{1 - h \cdot \left(t_0 \cdot \frac{t_0}{\ell}\right)} \end{array} \]

Alternative 8
Accuracy	71.3%
Cost	7744

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

Alternative 9
Accuracy	66.8%
Cost	64

\[w0 \]

Reproduce

herbie shell --seed 2023157 
(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))))))