Henrywood and Agarwal, Equation (9a)

Average Accuracy: 78.0% → 86.7%

Time: 31.0s

Precision: binary64

Cost: 27784

Math

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

↓

\[\begin{array}{l} t_0 := {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-24}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot h}{\ell}}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0 \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{M \cdot \left(\frac{D}{d} \cdot -0.5\right)}{\frac{\ell}{h} \cdot \left(\frac{2}{D} \cdot \frac{d}{M}\right)}}\\ \end{array} \]

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)))
   (if (<= t_0 2e-24)
     (* w0 (sqrt (- 1.0 (/ (* (pow (* (/ D d) (* M 0.5)) 2.0) h) l))))
     (if (<= t_0 2e+304)
       (* w0 (sqrt (- 1.0 (* t_0 (/ h l)))))
       (*
        w0
        (sqrt
         (+
          1.0
          (/ (* M (* (/ D d) -0.5)) (* (/ l h) (* (/ 2.0 D) (/ d M)))))))))))

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 tmp;
	if (t_0 <= 2e-24) {
		tmp = w0 * sqrt((1.0 - ((pow(((D / d) * (M * 0.5)), 2.0) * h) / l)));
	} else if (t_0 <= 2e+304) {
		tmp = w0 * sqrt((1.0 - (t_0 * (h / l))));
	} else {
		tmp = w0 * sqrt((1.0 + ((M * ((D / d) * -0.5)) / ((l / h) * ((2.0 / D) * (d / M))))));
	}
	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)) ** 2.0d0
    if (t_0 <= 2d-24) then
        tmp = w0 * sqrt((1.0d0 - (((((d / d_1) * (m * 0.5d0)) ** 2.0d0) * h) / l)))
    else if (t_0 <= 2d+304) then
        tmp = w0 * sqrt((1.0d0 - (t_0 * (h / l))))
    else
        tmp = w0 * sqrt((1.0d0 + ((m * ((d / d_1) * (-0.5d0))) / ((l / h) * ((2.0d0 / d) * (d_1 / m))))))
    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 tmp;
	if (t_0 <= 2e-24) {
		tmp = w0 * Math.sqrt((1.0 - ((Math.pow(((D / d) * (M * 0.5)), 2.0) * h) / l)));
	} else if (t_0 <= 2e+304) {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * (h / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + ((M * ((D / d) * -0.5)) / ((l / h) * ((2.0 / D) * (d / M))))));
	}
	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)
	tmp = 0
	if t_0 <= 2e-24:
		tmp = w0 * math.sqrt((1.0 - ((math.pow(((D / d) * (M * 0.5)), 2.0) * h) / l)))
	elif t_0 <= 2e+304:
		tmp = w0 * math.sqrt((1.0 - (t_0 * (h / l))))
	else:
		tmp = w0 * math.sqrt((1.0 + ((M * ((D / d) * -0.5)) / ((l / h) * ((2.0 / D) * (d / M))))))
	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
	tmp = 0.0
	if (t_0 <= 2e-24)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64((Float64(Float64(D / d) * Float64(M * 0.5)) ^ 2.0) * h) / l))));
	elseif (t_0 <= 2e+304)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(h / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(M * Float64(Float64(D / d) * -0.5)) / Float64(Float64(l / h) * Float64(Float64(2.0 / D) * Float64(d / M)))))));
	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;
	tmp = 0.0;
	if (t_0 <= 2e-24)
		tmp = w0 * sqrt((1.0 - (((((D / d) * (M * 0.5)) ^ 2.0) * h) / l)));
	elseif (t_0 <= 2e+304)
		tmp = w0 * sqrt((1.0 - (t_0 * (h / l))));
	else
		tmp = w0 * sqrt((1.0 + ((M * ((D / d) * -0.5)) / ((l / h) * ((2.0 / D) * (d / M))))));
	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]}, If[LessEqual[t$95$0, 2e-24], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[Power[N[(N[(D / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+304], 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[(M * N[(N[(D / d), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / N[(N[(l / h), $MachinePrecision] * N[(N[(2.0 / D), $MachinePrecision] * N[(d / M), $MachinePrecision]), $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) < 1.99999999999999985e-24

   1. Initial program 90.3%

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

   2. Simplified 90.3%

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

      [Start] 90.3	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      associate-*l/ [<=] 90.3	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2 \cdot d} \cdot D\right)}}^{2} \cdot \frac{h}{\ell}} \]
      *-commutative [=>] 90.3	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(D \cdot \frac{M}{2 \cdot d}\right)}}^{2} \cdot \frac{h}{\ell}} \]

   3. Applied egg-rr 98.9%

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

      [Start] 90.3	\[ w0 \cdot \sqrt{1 - {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      associate-*r/ [=>] 99.0	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot h}{\ell}}} \]
      associate-*r/ [=>] 98.9	\[ w0 \cdot \sqrt{1 - \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}} \]
      *-commutative [=>] 98.9	\[ w0 \cdot \sqrt{1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot h}{\ell}} \]
      times-frac [=>] 98.9	\[ w0 \cdot \sqrt{1 - \frac{{\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{2} \cdot h}{\ell}} \]
      div-inv [=>] 98.9	\[ w0 \cdot \sqrt{1 - \frac{{\left(\frac{D}{d} \cdot \color{blue}{\left(M \cdot \frac{1}{2}\right)}\right)}^{2} \cdot h}{\ell}} \]
      metadata-eval [=>] 98.9	\[ w0 \cdot \sqrt{1 - \frac{{\left(\frac{D}{d} \cdot \left(M \cdot \color{blue}{0.5}\right)\right)}^{2} \cdot h}{\ell}} \]

   if 1.99999999999999985e-24 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 1.9999999999999999e304

   1. Initial program 85.1%

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

   if 1.9999999999999999e304 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)

   1. Initial program 0.9%

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

   2. Applied egg-rr 21.4%

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

      [Start] 0.9	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      associate-*r/ [=>] 0.7	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}}} \]
      associate-/l* [=>] 0.8	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}} \]
      unpow2 [=>] 0.8	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}}{\frac{\ell}{h}}} \]
      clear-num [=>] 0.8	\[ w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{1}{\frac{2 \cdot d}{M \cdot D}}}}{\frac{\ell}{h}}} \]
      un-div-inv [=>] 0.8	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{2 \cdot d}{M \cdot D}}}}{\frac{\ell}{h}}} \]
      associate-/l/ [=>] 20.7	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{h} \cdot \frac{2 \cdot d}{M \cdot D}}}} \]
      associate-/l* [=>] 16.8	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{M}{\frac{2 \cdot d}{D}}}}{\frac{\ell}{h} \cdot \frac{2 \cdot d}{M \cdot D}}} \]
      div-inv [=>] 16.3	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{M \cdot \frac{1}{\frac{2 \cdot d}{D}}}}{\frac{\ell}{h} \cdot \frac{2 \cdot d}{M \cdot D}}} \]
      associate-/l* [=>] 16.3	\[ w0 \cdot \sqrt{1 - \frac{M \cdot \frac{1}{\color{blue}{\frac{2}{\frac{D}{d}}}}}{\frac{\ell}{h} \cdot \frac{2 \cdot d}{M \cdot D}}} \]
      associate-/r/ [=>] 16.3	\[ w0 \cdot \sqrt{1 - \frac{M \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{D}{d}\right)}}{\frac{\ell}{h} \cdot \frac{2 \cdot d}{M \cdot D}}} \]
      metadata-eval [=>] 16.3	\[ w0 \cdot \sqrt{1 - \frac{M \cdot \left(\color{blue}{0.5} \cdot \frac{D}{d}\right)}{\frac{\ell}{h} \cdot \frac{2 \cdot d}{M \cdot D}}} \]
      *-commutative [=>] 16.3	\[ w0 \cdot \sqrt{1 - \frac{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}{\frac{\ell}{h} \cdot \frac{2 \cdot d}{\color{blue}{D \cdot M}}}} \]
      times-frac [=>] 21.4	\[ w0 \cdot \sqrt{1 - \frac{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}{\frac{\ell}{h} \cdot \color{blue}{\left(\frac{2}{D} \cdot \frac{d}{M}\right)}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 2 \cdot 10^{-24}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot h}{\ell}}\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 2 \cdot 10^{+304}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{M \cdot \left(\frac{D}{d} \cdot -0.5\right)}{\frac{\ell}{h} \cdot \left(\frac{2}{D} \cdot \frac{d}{M}\right)}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	85.9%
Cost	27784

\[\begin{array}{l} t_0 := {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-205}:\\ \;\;\;\;w0 \cdot \sqrt{1 - D \cdot \left(h \cdot \frac{M \cdot \frac{M \cdot \left(D \cdot \frac{-0.25}{d}\right)}{d}}{-\ell}\right)}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0 \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{M \cdot \left(\frac{D}{d} \cdot -0.5\right)}{\frac{\ell}{h} \cdot \left(\frac{2}{D} \cdot \frac{d}{M}\right)}}\\ \end{array} \]

Alternative 2
Accuracy	79.4%
Cost	8536

\[\begin{array}{l} t_0 := w0 \cdot \sqrt{1 + M \cdot \frac{\frac{D}{-4} \cdot \frac{M}{\frac{d}{h}}}{\frac{d}{\frac{D}{\ell}}}}\\ \mathbf{if}\;D \leq -1.1 \cdot 10^{-188}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;D \leq 1.9 \cdot 10^{-43}:\\ \;\;\;\;w0\\ \mathbf{elif}\;D \leq 1.1 \cdot 10^{+61}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{M}{\ell} \cdot \left(\left(\frac{M}{d} \cdot \frac{D \cdot -0.25}{d}\right) \cdot \left(D \cdot h\right)\right)}\\ \mathbf{elif}\;D \leq 3.1 \cdot 10^{+109}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot D}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)} \cdot \left(h \cdot \left(M \cdot D\right)\right)}\\ \mathbf{elif}\;D \leq 4.5 \cdot 10^{+137}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;D \leq 3.3 \cdot 10^{+274}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(\frac{h}{\ell} \cdot \left(M \cdot D\right)\right) \cdot \left(-0.25 \cdot \frac{M \cdot D}{d \cdot d}\right)}\\ \end{array} \]

Alternative 3
Accuracy	76.7%
Cost	8536

\[\begin{array}{l} \mathbf{if}\;D \leq -4.6 \cdot 10^{-188}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M}{\frac{\frac{\ell}{h}}{-D}} \cdot \frac{M}{\left(d \cdot -4\right) \cdot \frac{d}{D}}}\\ \mathbf{elif}\;D \leq 1.7 \cdot 10^{-43}:\\ \;\;\;\;w0\\ \mathbf{elif}\;D \leq 5 \cdot 10^{+60}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{M}{\ell} \cdot \left(\left(\frac{M}{d} \cdot \frac{D \cdot -0.25}{d}\right) \cdot \left(D \cdot h\right)\right)}\\ \mathbf{elif}\;D \leq 4 \cdot 10^{+108}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot D}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)} \cdot \left(h \cdot \left(M \cdot D\right)\right)}\\ \mathbf{elif}\;D \leq 1.3 \cdot 10^{+136}:\\ \;\;\;\;w0 \cdot \sqrt{1 + M \cdot \frac{\frac{D}{-4} \cdot \frac{M}{\frac{d}{h}}}{\frac{d}{\frac{D}{\ell}}}}\\ \mathbf{elif}\;D \leq 3.6 \cdot 10^{+274}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(\frac{h}{\ell} \cdot \left(M \cdot D\right)\right) \cdot \left(-0.25 \cdot \frac{M \cdot D}{d \cdot d}\right)}\\ \end{array} \]

Alternative 4
Accuracy	85.4%
Cost	8392

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot D}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)} \cdot \left(h \cdot \left(M \cdot D\right)\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -5 \cdot 10^{-258}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{M \cdot \left(\frac{D}{d} \cdot -0.5\right)}{\frac{\ell}{h} \cdot \left(\frac{2}{D} \cdot \frac{d}{M}\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 5
Accuracy	80.4%
Cost	8272

\[\begin{array}{l} t_0 := w0 \cdot \sqrt{1 + \left(\frac{h}{\ell} \cdot \left(M \cdot D\right)\right) \cdot \left(-0.25 \cdot \frac{M \cdot D}{d \cdot d}\right)}\\ \mathbf{if}\;d \leq -8.5 \cdot 10^{+82}:\\ \;\;\;\;w0\\ \mathbf{elif}\;d \leq -8 \cdot 10^{-127}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{-161}:\\ \;\;\;\;w0\\ \mathbf{elif}\;d \leq 10^{+82}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + M \cdot \frac{\frac{D}{-4} \cdot \frac{M}{\frac{d}{h}}}{\frac{d}{\frac{D}{\ell}}}}\\ \end{array} \]

Alternative 6
Accuracy	83.9%
Cost	8264

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot D}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)} \cdot \left(h \cdot \left(M \cdot D\right)\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -5 \cdot 10^{-258}:\\ \;\;\;\;w0 \cdot \sqrt{1 - D \cdot \left(\frac{h}{\ell} \cdot \frac{M}{\frac{d}{\frac{M}{d} \cdot \left(D \cdot 0.25\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 7
Accuracy	79.2%
Cost	8140

\[\begin{array}{l} t_0 := w0 \cdot \sqrt{1 + M \cdot \frac{\frac{D}{-4} \cdot \frac{M}{\frac{d}{h}}}{\frac{d}{\frac{D}{\ell}}}}\\ \mathbf{if}\;D \leq -4 \cdot 10^{-188}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;D \leq 1.65 \cdot 10^{-43}:\\ \;\;\;\;w0\\ \mathbf{elif}\;D \leq 4.5 \cdot 10^{+63}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{M}{\ell} \cdot \left(\left(\frac{M}{d} \cdot \frac{D \cdot -0.25}{d}\right) \cdot \left(D \cdot h\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Accuracy	79.4%
Cost	8140

\[\begin{array}{l} t_0 := h \cdot \left(M \cdot D\right)\\ t_1 := \left(d \cdot -4\right) \cdot \frac{d}{D}\\ \mathbf{if}\;D \leq -2 \cdot 10^{-25}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M}{\frac{\frac{\ell}{h}}{-D}} \cdot \frac{M}{t_1}}\\ \mathbf{elif}\;D \leq 10^{-244}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{M}{\frac{t_1 \cdot \frac{\ell}{M \cdot D}}{h}}}\\ \mathbf{elif}\;D \leq 2 \cdot 10^{+73}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{M}{d \cdot \left(\frac{d \cdot -4}{D} \cdot \frac{\ell}{t_0}\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot D}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)} \cdot t_0}\\ \end{array} \]

Alternative 9
Accuracy	79.7%
Cost	8140

\[\begin{array}{l} t_0 := h \cdot \left(M \cdot D\right)\\ \mathbf{if}\;D \leq -1 \cdot 10^{-43}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{M \cdot \frac{D \cdot \frac{M}{d}}{d \cdot -4}}{\frac{\ell}{D \cdot h}}}\\ \mathbf{elif}\;D \leq 10^{-244}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{M}{\frac{\left(\left(d \cdot -4\right) \cdot \frac{d}{D}\right) \cdot \frac{\ell}{M \cdot D}}{h}}}\\ \mathbf{elif}\;D \leq 5 \cdot 10^{+74}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{M}{d \cdot \left(\frac{d \cdot -4}{D} \cdot \frac{\ell}{t_0}\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot D}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)} \cdot t_0}\\ \end{array} \]

Alternative 10
Accuracy	81.9%
Cost	8008

\[\begin{array}{l} t_0 := h \cdot \left(M \cdot D\right)\\ \mathbf{if}\;d \leq -1.4 \cdot 10^{-113}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot D}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)} \cdot t_0}\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{+34}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{M}{d \cdot \left(\frac{d \cdot -4}{D} \cdot \frac{\ell}{t_0}\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + M \cdot \frac{\frac{D}{-4} \cdot \frac{M}{\frac{d}{h}}}{\frac{d}{\frac{D}{\ell}}}}\\ \end{array} \]

Alternative 11
Accuracy	80.8%
Cost	7876

\[\begin{array}{l} \mathbf{if}\;M \leq -2.35 \cdot 10^{-97}:\\ \;\;\;\;w0 \cdot \sqrt{1 + M \cdot \frac{\frac{D}{-4} \cdot \frac{M}{\frac{d}{h}}}{\frac{d}{\frac{D}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 12
Accuracy	78.9%
Cost	64

\[w0 \]

Reproduce

herbie shell --seed 2023140 
(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))))))