Henrywood and Agarwal, Equation (9a)

Average Accuracy: 78.3% → 86.0%

Time: 23.0s

Precision: binary64

Cost: 14472

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

Math

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

↓

\[\begin{array}{l} t_0 := M \cdot \frac{D}{d}\\ t_1 := 0.5 \cdot t_0\\ \mathbf{if}\;M \cdot D \leq -1 \cdot 10^{+164}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}{\frac{\ell}{h} \cdot \frac{2}{t_0}}}\\ \mathbf{elif}\;M \cdot D \leq -5 \cdot 10^{-111}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \frac{{\left(M \cdot D\right)}^{2}}{d \cdot \left(\ell \cdot 4\right)}}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(t_1 \cdot \frac{t_1}{\ell}\right)}\\ \end{array} \]

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 (* M (/ D d))) (t_1 (* 0.5 t_0)))
   (if (<= (* M D) -1e+164)
     (* w0 (sqrt (- 1.0 (/ (* M (* 0.5 (/ D d))) (* (/ l h) (/ 2.0 t_0))))))
     (if (<= (* M D) -5e-111)
       (* w0 (sqrt (- 1.0 (/ (* h (/ (pow (* M D) 2.0) (* d (* l 4.0)))) d))))
       (* w0 (sqrt (- 1.0 (* h (* t_1 (/ t_1 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 t_0 = M * (D / d);
	double t_1 = 0.5 * t_0;
	double tmp;
	if ((M * D) <= -1e+164) {
		tmp = w0 * sqrt((1.0 - ((M * (0.5 * (D / d))) / ((l / h) * (2.0 / t_0)))));
	} else if ((M * D) <= -5e-111) {
		tmp = w0 * sqrt((1.0 - ((h * (pow((M * D), 2.0) / (d * (l * 4.0)))) / d)));
	} else {
		tmp = w0 * sqrt((1.0 - (h * (t_1 * (t_1 / 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = m * (d / d_1)
    t_1 = 0.5d0 * t_0
    if ((m * d) <= (-1d+164)) then
        tmp = w0 * sqrt((1.0d0 - ((m * (0.5d0 * (d / d_1))) / ((l / h) * (2.0d0 / t_0)))))
    else if ((m * d) <= (-5d-111)) then
        tmp = w0 * sqrt((1.0d0 - ((h * (((m * d) ** 2.0d0) / (d_1 * (l * 4.0d0)))) / d_1)))
    else
        tmp = w0 * sqrt((1.0d0 - (h * (t_1 * (t_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 t_0 = M * (D / d);
	double t_1 = 0.5 * t_0;
	double tmp;
	if ((M * D) <= -1e+164) {
		tmp = w0 * Math.sqrt((1.0 - ((M * (0.5 * (D / d))) / ((l / h) * (2.0 / t_0)))));
	} else if ((M * D) <= -5e-111) {
		tmp = w0 * Math.sqrt((1.0 - ((h * (Math.pow((M * D), 2.0) / (d * (l * 4.0)))) / d)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (h * (t_1 * (t_1 / 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):
	t_0 = M * (D / d)
	t_1 = 0.5 * t_0
	tmp = 0
	if (M * D) <= -1e+164:
		tmp = w0 * math.sqrt((1.0 - ((M * (0.5 * (D / d))) / ((l / h) * (2.0 / t_0)))))
	elif (M * D) <= -5e-111:
		tmp = w0 * math.sqrt((1.0 - ((h * (math.pow((M * D), 2.0) / (d * (l * 4.0)))) / d)))
	else:
		tmp = w0 * math.sqrt((1.0 - (h * (t_1 * (t_1 / 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)
	t_0 = Float64(M * Float64(D / d))
	t_1 = Float64(0.5 * t_0)
	tmp = 0.0
	if (Float64(M * D) <= -1e+164)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(M * Float64(0.5 * Float64(D / d))) / Float64(Float64(l / h) * Float64(2.0 / t_0))))));
	elseif (Float64(M * D) <= -5e-111)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * Float64((Float64(M * D) ^ 2.0) / Float64(d * Float64(l * 4.0)))) / d))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(t_1 * Float64(t_1 / 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)
	t_0 = M * (D / d);
	t_1 = 0.5 * t_0;
	tmp = 0.0;
	if ((M * D) <= -1e+164)
		tmp = w0 * sqrt((1.0 - ((M * (0.5 * (D / d))) / ((l / h) * (2.0 / t_0)))));
	elseif ((M * D) <= -5e-111)
		tmp = w0 * sqrt((1.0 - ((h * (((M * D) ^ 2.0) / (d * (l * 4.0)))) / d)));
	else
		tmp = w0 * sqrt((1.0 - (h * (t_1 * (t_1 / 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_] := Block[{t$95$0 = N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * t$95$0), $MachinePrecision]}, If[LessEqual[N[(M * D), $MachinePrecision], -1e+164], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(l / h), $MachinePrecision] * N[(2.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(M * D), $MachinePrecision], -5e-111], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[(N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision] / N[(d * N[(l * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(t$95$1 * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < -1e164

   1. Initial program 36.7%

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

   2. Simplified 41.5%

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

      [Start] 36.7	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      associate-/l* [=>] 41.5	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}}^{2} \cdot \frac{h}{\ell}} \]

   3. Applied egg-rr 48.7%

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

      [Start] 41.5	\[ w0 \cdot \sqrt{1 - {\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot \frac{h}{\ell}} \]
      associate-*r/ [=>] 40.9	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot h}{\ell}}} \]
      associate-/l* [=>] 41.5	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2}}{\frac{\ell}{h}}}} \]
      unpow2 [=>] 41.5	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{M}{\frac{2 \cdot d}{D}} \cdot \frac{M}{\frac{2 \cdot d}{D}}}}{\frac{\ell}{h}}} \]
      clear-num [=>] 41.5	\[ w0 \cdot \sqrt{1 - \frac{\frac{M}{\frac{2 \cdot d}{D}} \cdot \color{blue}{\frac{1}{\frac{\frac{2 \cdot d}{D}}{M}}}}{\frac{\ell}{h}}} \]
      un-div-inv [=>] 41.5	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{\frac{M}{\frac{2 \cdot d}{D}}}{\frac{\frac{2 \cdot d}{D}}{M}}}}{\frac{\ell}{h}}} \]
      associate-/l/ [=>] 48.8	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{M}{\frac{2 \cdot d}{D}}}{\frac{\ell}{h} \cdot \frac{\frac{2 \cdot d}{D}}{M}}}} \]
      div-inv [=>] 48.8	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{M \cdot \frac{1}{\frac{2 \cdot d}{D}}}}{\frac{\ell}{h} \cdot \frac{\frac{2 \cdot d}{D}}{M}}} \]
      associate-/l* [=>] 48.7	\[ w0 \cdot \sqrt{1 - \frac{M \cdot \frac{1}{\color{blue}{\frac{2}{\frac{D}{d}}}}}{\frac{\ell}{h} \cdot \frac{\frac{2 \cdot d}{D}}{M}}} \]
      associate-/r/ [=>] 48.7	\[ w0 \cdot \sqrt{1 - \frac{M \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{D}{d}\right)}}{\frac{\ell}{h} \cdot \frac{\frac{2 \cdot d}{D}}{M}}} \]
      metadata-eval [=>] 48.7	\[ w0 \cdot \sqrt{1 - \frac{M \cdot \left(\color{blue}{0.5} \cdot \frac{D}{d}\right)}{\frac{\ell}{h} \cdot \frac{\frac{2 \cdot d}{D}}{M}}} \]
      associate-/l* [=>] 48.7	\[ w0 \cdot \sqrt{1 - \frac{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}{\frac{\ell}{h} \cdot \frac{\color{blue}{\frac{2}{\frac{D}{d}}}}{M}}} \]
      associate-/l/ [=>] 48.7	\[ w0 \cdot \sqrt{1 - \frac{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}{\frac{\ell}{h} \cdot \color{blue}{\frac{2}{M \cdot \frac{D}{d}}}}} \]

   if -1e164 < (*.f64 M D) < -5.0000000000000003e-111

   1. Initial program 78.3%

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

   2. Simplified 76.7%

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

      [Start] 78.3	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      associate-*l/ [<=] 76.7	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2 \cdot d} \cdot D\right)}}^{2} \cdot \frac{h}{\ell}} \]
      *-commutative [=>] 76.7	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(D \cdot \frac{M}{2 \cdot d}\right)}}^{2} \cdot \frac{h}{\ell}} \]

   3. Applied egg-rr 75.8%

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

      [Start] 76.7	\[ w0 \cdot \sqrt{1 - {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      associate-*r/ [=>] 78.9	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot h}{\ell}}} \]
      associate-/l* [=>] 76.9	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}} \]
      unpow2 [=>] 76.9	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(D \cdot \frac{M}{2 \cdot d}\right) \cdot \left(D \cdot \frac{M}{2 \cdot d}\right)}}{\frac{\ell}{h}}} \]
      associate-*r/ [=>] 76.9	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{D \cdot M}{2 \cdot d}} \cdot \left(D \cdot \frac{M}{2 \cdot d}\right)}{\frac{\ell}{h}}} \]
      associate-*r/ [=>] 78.6	\[ w0 \cdot \sqrt{1 - \frac{\frac{D \cdot M}{2 \cdot d} \cdot \color{blue}{\frac{D \cdot M}{2 \cdot d}}}{\frac{\ell}{h}}} \]
      frac-times [=>] 74.7	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)}}}{\frac{\ell}{h}}} \]
      associate-/l/ [=>] 75.8	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\frac{\ell}{h} \cdot \left(\left(2 \cdot d\right) \cdot \left(2 \cdot d\right)\right)}}} \]
      *-commutative [=>] 75.8	\[ w0 \cdot \sqrt{1 - \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\frac{\ell}{h} \cdot \left(\color{blue}{\left(d \cdot 2\right)} \cdot \left(2 \cdot d\right)\right)}} \]
      *-commutative [=>] 75.8	\[ w0 \cdot \sqrt{1 - \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\frac{\ell}{h} \cdot \left(\left(d \cdot 2\right) \cdot \color{blue}{\left(d \cdot 2\right)}\right)}} \]
      swap-sqr [=>] 75.8	\[ w0 \cdot \sqrt{1 - \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\frac{\ell}{h} \cdot \color{blue}{\left(\left(d \cdot d\right) \cdot \left(2 \cdot 2\right)\right)}}} \]
      metadata-eval [=>] 75.8	\[ w0 \cdot \sqrt{1 - \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\frac{\ell}{h} \cdot \left(\left(d \cdot d\right) \cdot \color{blue}{4}\right)}} \]

   4. Simplified 77.4%

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

      [Start] 75.8	\[ w0 \cdot \sqrt{1 - \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\frac{\ell}{h} \cdot \left(\left(d \cdot d\right) \cdot 4\right)}} \]
      associate-/l* [=>] 77.4	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{D \cdot M}{\frac{\frac{\ell}{h} \cdot \left(\left(d \cdot d\right) \cdot 4\right)}{D \cdot M}}}} \]
      associate-/r/ [=>] 77.4	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{D \cdot M}{\frac{\ell}{h} \cdot \left(\left(d \cdot d\right) \cdot 4\right)} \cdot \left(D \cdot M\right)}} \]
      *-commutative [=>] 77.4	\[ w0 \cdot \sqrt{1 - \frac{D \cdot M}{\frac{\ell}{h} \cdot \color{blue}{\left(4 \cdot \left(d \cdot d\right)\right)}} \cdot \left(D \cdot M\right)} \]
      unpow2 [<=] 77.4	\[ w0 \cdot \sqrt{1 - \frac{D \cdot M}{\frac{\ell}{h} \cdot \left(4 \cdot \color{blue}{{d}^{2}}\right)} \cdot \left(D \cdot M\right)} \]
      associate-*r* [=>] 77.4	\[ w0 \cdot \sqrt{1 - \frac{D \cdot M}{\color{blue}{\left(\frac{\ell}{h} \cdot 4\right) \cdot {d}^{2}}} \cdot \left(D \cdot M\right)} \]
      unpow2 [=>] 77.4	\[ w0 \cdot \sqrt{1 - \frac{D \cdot M}{\left(\frac{\ell}{h} \cdot 4\right) \cdot \color{blue}{\left(d \cdot d\right)}} \cdot \left(D \cdot M\right)} \]

   5. Applied egg-rr 83.2%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{{\left(D \cdot M\right)}^{2}}{\left(\ell \cdot 4\right) \cdot d} \cdot h}{d}}} \]
      Proof

      [Start] 77.4	\[ w0 \cdot \sqrt{1 - \frac{D \cdot M}{\left(\frac{\ell}{h} \cdot 4\right) \cdot \left(d \cdot d\right)} \cdot \left(D \cdot M\right)} \]
      associate-*l/ [=>] 75.8	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\frac{\ell}{h} \cdot 4\right) \cdot \left(d \cdot d\right)}}} \]
      associate-*r* [=>] 79.4	\[ w0 \cdot \sqrt{1 - \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\color{blue}{\left(\left(\frac{\ell}{h} \cdot 4\right) \cdot d\right) \cdot d}}} \]
      associate-/r* [=>] 79.9	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\frac{\ell}{h} \cdot 4\right) \cdot d}}{d}}} \]
      associate-*l/ [=>] 80.0	\[ w0 \cdot \sqrt{1 - \frac{\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\color{blue}{\frac{\ell \cdot 4}{h}} \cdot d}}{d}} \]
      associate-*l/ [=>] 83.8	\[ w0 \cdot \sqrt{1 - \frac{\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\color{blue}{\frac{\left(\ell \cdot 4\right) \cdot d}{h}}}}{d}} \]
      associate-/r/ [=>] 83.2	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\ell \cdot 4\right) \cdot d} \cdot h}}{d}} \]
      pow2 [=>] 83.2	\[ w0 \cdot \sqrt{1 - \frac{\frac{\color{blue}{{\left(D \cdot M\right)}^{2}}}{\left(\ell \cdot 4\right) \cdot d} \cdot h}{d}} \]

   if -5.0000000000000003e-111 < (*.f64 M D)

   1. Initial program 81.8%

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

   2. Simplified 81.8%

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

      [Start] 81.8	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      associate-/l* [=>] 81.8	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}}^{2} \cdot \frac{h}{\ell}} \]

   3. Applied egg-rr 84.1%

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

      [Start] 81.8	\[ w0 \cdot \sqrt{1 - {\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot \frac{h}{\ell}} \]
      associate-*r/ [=>] 88.3	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot h}{\ell}}} \]
      associate-/l* [=>] 82.7	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2}}{\frac{\ell}{h}}}} \]
      unpow2 [=>] 82.7	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{M}{\frac{2 \cdot d}{D}} \cdot \frac{M}{\frac{2 \cdot d}{D}}}}{\frac{\ell}{h}}} \]
      clear-num [=>] 82.7	\[ w0 \cdot \sqrt{1 - \frac{\frac{M}{\frac{2 \cdot d}{D}} \cdot \color{blue}{\frac{1}{\frac{\frac{2 \cdot d}{D}}{M}}}}{\frac{\ell}{h}}} \]
      un-div-inv [=>] 82.7	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{\frac{M}{\frac{2 \cdot d}{D}}}{\frac{\frac{2 \cdot d}{D}}{M}}}}{\frac{\ell}{h}}} \]
      associate-/l/ [=>] 84.3	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{M}{\frac{2 \cdot d}{D}}}{\frac{\ell}{h} \cdot \frac{\frac{2 \cdot d}{D}}{M}}}} \]
      div-inv [=>] 84.1	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{M \cdot \frac{1}{\frac{2 \cdot d}{D}}}}{\frac{\ell}{h} \cdot \frac{\frac{2 \cdot d}{D}}{M}}} \]
      associate-/l* [=>] 84.1	\[ w0 \cdot \sqrt{1 - \frac{M \cdot \frac{1}{\color{blue}{\frac{2}{\frac{D}{d}}}}}{\frac{\ell}{h} \cdot \frac{\frac{2 \cdot d}{D}}{M}}} \]
      associate-/r/ [=>] 84.1	\[ w0 \cdot \sqrt{1 - \frac{M \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{D}{d}\right)}}{\frac{\ell}{h} \cdot \frac{\frac{2 \cdot d}{D}}{M}}} \]
      metadata-eval [=>] 84.1	\[ w0 \cdot \sqrt{1 - \frac{M \cdot \left(\color{blue}{0.5} \cdot \frac{D}{d}\right)}{\frac{\ell}{h} \cdot \frac{\frac{2 \cdot d}{D}}{M}}} \]
      associate-/l* [=>] 84.1	\[ w0 \cdot \sqrt{1 - \frac{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}{\frac{\ell}{h} \cdot \frac{\color{blue}{\frac{2}{\frac{D}{d}}}}{M}}} \]
      associate-/l/ [=>] 84.1	\[ w0 \cdot \sqrt{1 - \frac{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}{\frac{\ell}{h} \cdot \color{blue}{\frac{2}{M \cdot \frac{D}{d}}}}} \]

   4. Applied egg-rr 82.2%

      \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{{\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2}}{\frac{\ell}{h}}}\right)} - 1\right)} \]
      Proof

      [Start] 84.1	\[ w0 \cdot \sqrt{1 - \frac{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}{\frac{\ell}{h} \cdot \frac{2}{M \cdot \frac{D}{d}}}} \]
      expm1-log1p-u [=>] 83.7	\[ w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}{\frac{\ell}{h} \cdot \frac{2}{M \cdot \frac{D}{d}}}}\right)\right)} \]
      expm1-udef [=>] 83.7	\[ w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}{\frac{\ell}{h} \cdot \frac{2}{M \cdot \frac{D}{d}}}}\right)} - 1\right)} \]

   5. Simplified 88.4%

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

      [Start] 82.2	\[ w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{{\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2}}{\frac{\ell}{h}}}\right)} - 1\right) \]
      expm1-def [=>] 82.2	\[ w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{{\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2}}{\frac{\ell}{h}}}\right)\right)} \]
      expm1-log1p [=>] 82.5	\[ w0 \cdot \color{blue}{\sqrt{1 - \frac{{\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2}}{\frac{\ell}{h}}}} \]
      associate-/r/ [=>] 88.4	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2}}{\ell} \cdot h}} \]
      *-commutative [=>] 88.4	\[ w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \color{blue}{\left(0.5 \cdot \frac{D}{d}\right)}\right)}^{2}}{\ell} \cdot h} \]

   6. Applied egg-rr 90.0%

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

      [Start] 88.4	\[ w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot h} \]
      unpow2 [=>] 88.4	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}{\ell} \cdot h} \]
      associate-/l* [=>] 90.0	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}{\frac{\ell}{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}}} \cdot h} \]
      associate-/r/ [=>] 90.0	\[ w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}{\ell} \cdot \left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)\right)} \cdot h} \]
      *-commutative [=>] 90.0	\[ w0 \cdot \sqrt{1 - \left(\frac{M \cdot \color{blue}{\left(\frac{D}{d} \cdot 0.5\right)}}{\ell} \cdot \left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)\right) \cdot h} \]
      associate-*r* [=>] 90.0	\[ w0 \cdot \sqrt{1 - \left(\frac{\color{blue}{\left(M \cdot \frac{D}{d}\right) \cdot 0.5}}{\ell} \cdot \left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)\right) \cdot h} \]
      *-commutative [=>] 90.0	\[ w0 \cdot \sqrt{1 - \left(\frac{\left(M \cdot \frac{D}{d}\right) \cdot 0.5}{\ell} \cdot \left(M \cdot \color{blue}{\left(\frac{D}{d} \cdot 0.5\right)}\right)\right) \cdot h} \]
      associate-*r* [=>] 90.0	\[ w0 \cdot \sqrt{1 - \left(\frac{\left(M \cdot \frac{D}{d}\right) \cdot 0.5}{\ell} \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot 0.5\right)}\right) \cdot h} \]

3. Recombined 3 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq -1 \cdot 10^{+164}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}{\frac{\ell}{h} \cdot \frac{2}{M \cdot \frac{D}{d}}}}\\ \mathbf{elif}\;M \cdot D \leq -5 \cdot 10^{-111}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \frac{{\left(M \cdot D\right)}^{2}}{d \cdot \left(\ell \cdot 4\right)}}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right) \cdot \frac{0.5 \cdot \left(M \cdot \frac{D}{d}\right)}{\ell}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	87.1%
Cost	28297

\[\begin{array}{l} t_0 := 0.5 \cdot \left(M \cdot \frac{D}{d}\right)\\ t_1 := {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+301} \lor \neg \left(t_1 \leq 10^{-119}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(t_0 \cdot \frac{t_0}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_1}\\ \end{array} \]

Alternative 2
Accuracy	82.6%
Cost	8524

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -4 \cdot 10^{+77}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \left(h \cdot \frac{\frac{M \cdot D}{d \cdot d}}{\ell \cdot 4}\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-176}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \left(\frac{D}{\frac{\ell}{h} \cdot \left(d \cdot 4\right)} \cdot \frac{M}{d}\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -2 \cdot 10^{-298}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(M \cdot D\right) \cdot \left(h \cdot \frac{\frac{D}{d} \cdot \frac{M}{d}}{\ell \cdot -4}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 3
Accuracy	82.6%
Cost	8264

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \left(\frac{D}{d} \cdot \left(h \cdot \frac{M}{d \cdot \left(\ell \cdot 4\right)}\right)\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-206}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \left(\frac{D}{\frac{\ell}{h} \cdot \left(d \cdot 4\right)} \cdot \frac{M}{d}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 4
Accuracy	82.8%
Cost	8264

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -4 \cdot 10^{+77}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \left(h \cdot \frac{\frac{M \cdot D}{d \cdot d}}{\ell \cdot 4}\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-206}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \left(\frac{D}{\frac{\ell}{h} \cdot \left(d \cdot 4\right)} \cdot \frac{M}{d}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 5
Accuracy	84.3%
Cost	8264

\[\begin{array}{l} t_0 := 0.5 \cdot \left(M \cdot \frac{D}{d}\right)\\ \mathbf{if}\;D \leq 2 \cdot 10^{+56}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(t_0 \cdot \frac{t_0}{\ell}\right)}\\ \mathbf{elif}\;D \leq 10^{+188}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \left(\frac{D}{\frac{\ell}{h} \cdot \left(d \cdot 4\right)} \cdot \frac{M}{d}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + h \cdot \left(t_0 \cdot \left(M \cdot \left(\left(0.5 \cdot \frac{D}{d}\right) \cdot \frac{-1}{\ell}\right)\right)\right)}\\ \end{array} \]

Alternative 6
Accuracy	84.9%
Cost	8137

\[\begin{array}{l} t_0 := 0.5 \cdot \left(M \cdot \frac{D}{d}\right)\\ \mathbf{if}\;D \leq 5 \cdot 10^{+55} \lor \neg \left(D \leq 3.2 \cdot 10^{+188}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(t_0 \cdot \frac{t_0}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \left(\frac{D}{\frac{\ell}{h} \cdot \left(d \cdot 4\right)} \cdot \frac{M}{d}\right)}\\ \end{array} \]

Alternative 7
Accuracy	78.6%
Cost	8008

\[\begin{array}{l} \mathbf{if}\;D \leq 3.1 \cdot 10^{-24}:\\ \;\;\;\;w0\\ \mathbf{elif}\;D \leq 2.85 \cdot 10^{+22}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\ell \cdot \left(d \cdot d\right)} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 8
Accuracy	81.3%
Cost	8004

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-206}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot D\right) \cdot \left(\frac{D}{d} \cdot \left(h \cdot \frac{M}{d \cdot \left(\ell \cdot 4\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 9
Accuracy	79.2%
Cost	64

\[w0 \]

Reproduce

herbie shell --seed 2023135 
(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))))))