Henrywood and Agarwal, Equation (12)

Average Accuracy: 57.1% → 78.7%

Time: 56.7s

Precision: binary64

Cost: 104593

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

Math

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

↓

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+296}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\left(\frac{M}{d} \cdot \frac{h \cdot D}{d}\right) \cdot \left(M \cdot \frac{D}{\ell}\right)\right)\right)\\ \mathbf{elif}\;t_0 \leq -5 \cdot 10^{-181} \lor \neg \left(t_0 \leq 0\right) \land t_0 \leq 2 \cdot 10^{+233}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (*
  (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
  (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))

↓

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (- 1.0 (* (* 0.5 (pow (/ (* M D) (* d 2.0)) 2.0)) (/ h l))))))
   (if (<= t_0 -1e+296)
     (*
      (* (sqrt (/ d h)) (sqrt (/ d l)))
      (- 1.0 (* 0.125 (* (* (/ M d) (/ (* h D) d)) (* M (/ D l))))))
     (if (or (<= t_0 -5e-181) (and (not (<= t_0 0.0)) (<= t_0 2e+233)))
       t_0
       (fabs (/ d (sqrt (* h l))))))))

C

double code(double d, double h, double l, double M, double D) {
	return (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

↓

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((0.5 * pow(((M * D) / (d * 2.0)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -1e+296) {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - (0.125 * (((M / d) * ((h * D) / d)) * (M * (D / l)))));
	} else if ((t_0 <= -5e-181) || (!(t_0 <= 0.0) && (t_0 <= 2e+233))) {
		tmp = t_0;
	} else {
		tmp = fabs((d / sqrt((h * l))));
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d / h) ** 0.5d0) * ((d / l) ** 0.5d0)) * (1.0d0 - ((0.5d0 * (((m * d_1) / (d * 2.0d0)) ** 2.0d0)) * (h / l)))
    if (t_0 <= (-1d+296)) then
        tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0d0 - (0.125d0 * (((m / d) * ((h * d_1) / d)) * (m * (d_1 / l)))))
    else if ((t_0 <= (-5d-181)) .or. (.not. (t_0 <= 0.0d0)) .and. (t_0 <= 2d+233)) then
        tmp = t_0
    else
        tmp = abs((d / sqrt((h * l))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

↓

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((0.5 * Math.pow(((M * D) / (d * 2.0)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -1e+296) {
		tmp = (Math.sqrt((d / h)) * Math.sqrt((d / l))) * (1.0 - (0.125 * (((M / d) * ((h * D) / d)) * (M * (D / l)))));
	} else if ((t_0 <= -5e-181) || (!(t_0 <= 0.0) && (t_0 <= 2e+233))) {
		tmp = t_0;
	} else {
		tmp = Math.abs((d / Math.sqrt((h * l))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	return (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))

↓

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 - ((0.5 * math.pow(((M * D) / (d * 2.0)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= -1e+296:
		tmp = (math.sqrt((d / h)) * math.sqrt((d / l))) * (1.0 - (0.125 * (((M / d) * ((h * D) / d)) * (M * (D / l)))))
	elif (t_0 <= -5e-181) or (not (t_0 <= 0.0) and (t_0 <= 2e+233)):
		tmp = t_0
	else:
		tmp = math.fabs((d / math.sqrt((h * l))))
	return tmp

Julia

function code(d, h, l, M, D)
	return Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
end

↓

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(0.5 * (Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= -1e+296)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(0.125 * Float64(Float64(Float64(M / d) * Float64(Float64(h * D) / d)) * Float64(M * Float64(D / l))))));
	elseif ((t_0 <= -5e-181) || (!(t_0 <= 0.0) && (t_0 <= 2e+233)))
		tmp = t_0;
	else
		tmp = abs(Float64(d / sqrt(Float64(h * l))));
	end
	return tmp
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
end

↓

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((0.5 * (((M * D) / (d * 2.0)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= -1e+296)
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - (0.125 * (((M / d) * ((h * D) / d)) * (M * (D / l)))));
	elseif ((t_0 <= -5e-181) || (~((t_0 <= 0.0)) && (t_0 <= 2e+233)))
		tmp = t_0;
	else
		tmp = abs((d / sqrt((h * l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(0.5 * N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+296], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.125 * N[(N[(N[(M / d), $MachinePrecision] * N[(N[(h * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(M * N[(D / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -5e-181], And[N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision], LessEqual[t$95$0, 2e+233]]], t$95$0, N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < -9.99999999999999981e295

   1. Initial program 2.7%

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

   2. Simplified 6.1%

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

      [Start] 2.7	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 2.7	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 2.7	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 2.7	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 2.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      *-commutative [=>] 2.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 2.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]
      times-frac [=>] 6.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 6.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Taylor expanded in M around 0 7.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]

   4. Simplified 14.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{D \cdot \left(D \cdot h\right)}{d \cdot \ell}\right)}\right) \]
      Proof

      [Start] 7.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right) \]
      associate-*r/ [=>] 7.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right) \]
      *-commutative [=>] 7.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right) \]
      associate-*r/ [<=] 7.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
      associate-*r* [=>] 6.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}\right) \]
      *-commutative [=>] 6.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{\left({M}^{2} \cdot {D}^{2}\right)} \cdot h}{{d}^{2} \cdot \ell}\right) \]
      associate-*l* [=>] 7.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{{M}^{2} \cdot \left({D}^{2} \cdot h\right)}}{{d}^{2} \cdot \ell}\right) \]
      unpow2 [=>] 7.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{{M}^{2} \cdot \left({D}^{2} \cdot h\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]
      associate-*l* [=>] 8.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{{M}^{2} \cdot \left({D}^{2} \cdot h\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]
      times-frac [=>] 11.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\left(\frac{{M}^{2}}{d} \cdot \frac{{D}^{2} \cdot h}{d \cdot \ell}\right)}\right) \]
      unpow2 [=>] 11.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{\color{blue}{M \cdot M}}{d} \cdot \frac{{D}^{2} \cdot h}{d \cdot \ell}\right)\right) \]
      unpow2 [=>] 11.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot h}{d \cdot \ell}\right)\right) \]
      associate-*l* [=>] 14.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{\color{blue}{D \cdot \left(D \cdot h\right)}}{d \cdot \ell}\right)\right) \]

   5. Applied egg-rr 24.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{\frac{M}{d} \cdot M}{\frac{d}{D \cdot h} \cdot \frac{\ell}{D}}}\right) \]
      Proof

      [Start] 14.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{M \cdot M}{d} \cdot \frac{D \cdot \left(D \cdot h\right)}{d \cdot \ell}\right)\right) \]
      clear-num [=>] 14.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{M \cdot M}{d} \cdot \color{blue}{\frac{1}{\frac{d \cdot \ell}{D \cdot \left(D \cdot h\right)}}}\right)\right) \]
      un-div-inv [=>] 14.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{\frac{M \cdot M}{d}}{\frac{d \cdot \ell}{D \cdot \left(D \cdot h\right)}}}\right) \]
      associate-/l* [=>] 16.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{\frac{M}{\frac{d}{M}}}}{\frac{d \cdot \ell}{D \cdot \left(D \cdot h\right)}}\right) \]
      associate-/r/ [=>] 16.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{\frac{M}{d} \cdot M}}{\frac{d \cdot \ell}{D \cdot \left(D \cdot h\right)}}\right) \]
      *-commutative [=>] 16.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{\frac{M}{d} \cdot M}{\frac{d \cdot \ell}{\color{blue}{\left(D \cdot h\right) \cdot D}}}\right) \]
      times-frac [=>] 24.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{\frac{M}{d} \cdot M}{\color{blue}{\frac{d}{D \cdot h} \cdot \frac{\ell}{D}}}\right) \]

   6. Applied egg-rr 34.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\left(\left(\frac{M}{d} \cdot \frac{D \cdot h}{d}\right) \cdot \left(M \cdot \frac{D}{\ell}\right)\right)}\right) \]
      Proof

      [Start] 24.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{\frac{M}{d} \cdot M}{\frac{d}{D \cdot h} \cdot \frac{\ell}{D}}\right) \]
      times-frac [=>] 34.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\left(\frac{\frac{M}{d}}{\frac{d}{D \cdot h}} \cdot \frac{M}{\frac{\ell}{D}}\right)}\right) \]
      div-inv [=>] 34.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\color{blue}{\left(\frac{M}{d} \cdot \frac{1}{\frac{d}{D \cdot h}}\right)} \cdot \frac{M}{\frac{\ell}{D}}\right)\right) \]
      clear-num [<=] 34.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\left(\frac{M}{d} \cdot \color{blue}{\frac{D \cdot h}{d}}\right) \cdot \frac{M}{\frac{\ell}{D}}\right)\right) \]
      div-inv [=>] 34.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\left(\frac{M}{d} \cdot \frac{D \cdot h}{d}\right) \cdot \color{blue}{\left(M \cdot \frac{1}{\frac{\ell}{D}}\right)}\right)\right) \]
      clear-num [<=] 34.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\left(\frac{M}{d} \cdot \frac{D \cdot h}{d}\right) \cdot \left(M \cdot \color{blue}{\frac{D}{\ell}}\right)\right)\right) \]

   if -9.99999999999999981e295 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < -5.0000000000000001e-181 or -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 1.99999999999999995e233

   1. Initial program 98.5%

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

   if -5.0000000000000001e-181 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < -0.0 or 1.99999999999999995e233 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))

   1. Initial program 13.3%

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

   2. Taylor expanded in d around inf 36.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   3. Simplified 36.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
      Proof

      [Start] 36.1	\[ \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]
      *-commutative [=>] 36.1	\[ \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
      associate-/r* [=>] 36.1	\[ d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   4. Applied egg-rr 35.9%

      \[\leadsto d \cdot \color{blue}{{\left(\sqrt[3]{\frac{\frac{1}{\ell}}{h}}\right)}^{1.5}} \]
      Proof

      [Start] 36.1	\[ d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}} \]
      pow1/2 [=>] 36.1	\[ d \cdot \color{blue}{{\left(\frac{\frac{1}{\ell}}{h}\right)}^{0.5}} \]
      add-cube-cbrt [=>] 35.9	\[ d \cdot {\color{blue}{\left(\left(\sqrt[3]{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt[3]{\frac{\frac{1}{\ell}}{h}}\right) \cdot \sqrt[3]{\frac{\frac{1}{\ell}}{h}}\right)}}^{0.5} \]
      pow3 [=>] 35.9	\[ d \cdot {\color{blue}{\left({\left(\sqrt[3]{\frac{\frac{1}{\ell}}{h}}\right)}^{3}\right)}}^{0.5} \]
      pow-pow [=>] 35.9	\[ d \cdot \color{blue}{{\left(\sqrt[3]{\frac{\frac{1}{\ell}}{h}}\right)}^{\left(3 \cdot 0.5\right)}} \]
      metadata-eval [=>] 35.9	\[ d \cdot {\left(\sqrt[3]{\frac{\frac{1}{\ell}}{h}}\right)}^{\color{blue}{1.5}} \]

   5. Applied egg-rr 35.9%

      \[\leadsto d \cdot {\color{blue}{\left(0 + \sqrt[3]{\frac{\frac{1}{\ell}}{h}}\right)}}^{1.5} \]
      Proof

      [Start] 35.9	\[ d \cdot {\left(\sqrt[3]{\frac{\frac{1}{\ell}}{h}}\right)}^{1.5} \]
      add-log-exp [=>] 12.9	\[ d \cdot {\color{blue}{\log \left(e^{\sqrt[3]{\frac{\frac{1}{\ell}}{h}}}\right)}}^{1.5} \]
      *-un-lft-identity [=>] 12.9	\[ d \cdot {\log \color{blue}{\left(1 \cdot e^{\sqrt[3]{\frac{\frac{1}{\ell}}{h}}}\right)}}^{1.5} \]
      log-prod [=>] 12.9	\[ d \cdot {\color{blue}{\left(\log 1 + \log \left(e^{\sqrt[3]{\frac{\frac{1}{\ell}}{h}}}\right)\right)}}^{1.5} \]
      metadata-eval [=>] 12.9	\[ d \cdot {\left(\color{blue}{0} + \log \left(e^{\sqrt[3]{\frac{\frac{1}{\ell}}{h}}}\right)\right)}^{1.5} \]
      add-log-exp [<=] 35.9	\[ d \cdot {\left(0 + \color{blue}{\sqrt[3]{\frac{\frac{1}{\ell}}{h}}}\right)}^{1.5} \]

   6. Simplified 35.9%

      \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{\sqrt[3]{\ell \cdot h}}\right)}}^{1.5} \]
      Proof

      [Start] 35.9	\[ d \cdot {\left(0 + \sqrt[3]{\frac{\frac{1}{\ell}}{h}}\right)}^{1.5} \]
      +-lft-identity [=>] 35.9	\[ d \cdot {\color{blue}{\left(\sqrt[3]{\frac{\frac{1}{\ell}}{h}}\right)}}^{1.5} \]
      unpow1/3 [<=] 34.2	\[ d \cdot {\color{blue}{\left({\left(\frac{\frac{1}{\ell}}{h}\right)}^{0.3333333333333333}\right)}}^{1.5} \]
      exp-to-pow [<=] 34.3	\[ d \cdot {\color{blue}{\left(e^{\log \left(\frac{\frac{1}{\ell}}{h}\right) \cdot 0.3333333333333333}\right)}}^{1.5} \]
      associate-/r* [<=] 34.2	\[ d \cdot {\left(e^{\log \color{blue}{\left(\frac{1}{\ell \cdot h}\right)} \cdot 0.3333333333333333}\right)}^{1.5} \]
      log-rec [=>] 34.3	\[ d \cdot {\left(e^{\color{blue}{\left(-\log \left(\ell \cdot h\right)\right)} \cdot 0.3333333333333333}\right)}^{1.5} \]
      *-rgt-identity [<=] 34.3	\[ d \cdot {\left(e^{\color{blue}{\left(\left(-\log \left(\ell \cdot h\right)\right) \cdot 1\right)} \cdot 0.3333333333333333}\right)}^{1.5} \]
      *-rgt-identity [=>] 34.3	\[ d \cdot {\left(e^{\color{blue}{\left(-\log \left(\ell \cdot h\right)\right)} \cdot 0.3333333333333333}\right)}^{1.5} \]
      distribute-lft-neg-out [=>] 34.3	\[ d \cdot {\left(e^{\color{blue}{-\log \left(\ell \cdot h\right) \cdot 0.3333333333333333}}\right)}^{1.5} \]
      exp-neg [=>] 34.3	\[ d \cdot {\color{blue}{\left(\frac{1}{e^{\log \left(\ell \cdot h\right) \cdot 0.3333333333333333}}\right)}}^{1.5} \]
      exp-to-pow [=>] 34.2	\[ d \cdot {\left(\frac{1}{\color{blue}{{\left(\ell \cdot h\right)}^{0.3333333333333333}}}\right)}^{1.5} \]
      unpow1/3 [=>] 35.9	\[ d \cdot {\left(\frac{1}{\color{blue}{\sqrt[3]{\ell \cdot h}}}\right)}^{1.5} \]

   7. Applied egg-rr 19.7%

      \[\leadsto \color{blue}{{\left(\left(d \cdot d\right) \cdot {\left(\ell \cdot h\right)}^{-1}\right)}^{0.5}} \]
      Proof

      [Start] 35.9	\[ d \cdot {\left(\frac{1}{\sqrt[3]{\ell \cdot h}}\right)}^{1.5} \]
      add-sqr-sqrt [=>] 34.9	\[ \color{blue}{\sqrt{d \cdot {\left(\frac{1}{\sqrt[3]{\ell \cdot h}}\right)}^{1.5}} \cdot \sqrt{d \cdot {\left(\frac{1}{\sqrt[3]{\ell \cdot h}}\right)}^{1.5}}} \]
      sqrt-unprod [=>] 22.3	\[ \color{blue}{\sqrt{\left(d \cdot {\left(\frac{1}{\sqrt[3]{\ell \cdot h}}\right)}^{1.5}\right) \cdot \left(d \cdot {\left(\frac{1}{\sqrt[3]{\ell \cdot h}}\right)}^{1.5}\right)}} \]
      pow1/2 [=>] 22.3	\[ \color{blue}{{\left(\left(d \cdot {\left(\frac{1}{\sqrt[3]{\ell \cdot h}}\right)}^{1.5}\right) \cdot \left(d \cdot {\left(\frac{1}{\sqrt[3]{\ell \cdot h}}\right)}^{1.5}\right)\right)}^{0.5}} \]
      swap-sqr [=>] 19.5	\[ {\color{blue}{\left(\left(d \cdot d\right) \cdot \left({\left(\frac{1}{\sqrt[3]{\ell \cdot h}}\right)}^{1.5} \cdot {\left(\frac{1}{\sqrt[3]{\ell \cdot h}}\right)}^{1.5}\right)\right)}}^{0.5} \]
      pow-prod-up [=>] 19.5	\[ {\left(\left(d \cdot d\right) \cdot \color{blue}{{\left(\frac{1}{\sqrt[3]{\ell \cdot h}}\right)}^{\left(1.5 + 1.5\right)}}\right)}^{0.5} \]
      inv-pow [=>] 19.5	\[ {\left(\left(d \cdot d\right) \cdot {\color{blue}{\left({\left(\sqrt[3]{\ell \cdot h}\right)}^{-1}\right)}}^{\left(1.5 + 1.5\right)}\right)}^{0.5} \]
      pow1/3 [=>] 19.0	\[ {\left(\left(d \cdot d\right) \cdot {\left({\color{blue}{\left({\left(\ell \cdot h\right)}^{0.3333333333333333}\right)}}^{-1}\right)}^{\left(1.5 + 1.5\right)}\right)}^{0.5} \]
      pow-pow [=>] 19.0	\[ {\left(\left(d \cdot d\right) \cdot {\color{blue}{\left({\left(\ell \cdot h\right)}^{\left(0.3333333333333333 \cdot -1\right)}\right)}}^{\left(1.5 + 1.5\right)}\right)}^{0.5} \]
      pow-pow [=>] 19.7	\[ {\left(\left(d \cdot d\right) \cdot \color{blue}{{\left(\ell \cdot h\right)}^{\left(\left(0.3333333333333333 \cdot -1\right) \cdot \left(1.5 + 1.5\right)\right)}}\right)}^{0.5} \]
      metadata-eval [=>] 19.7	\[ {\left(\left(d \cdot d\right) \cdot {\left(\ell \cdot h\right)}^{\left(\color{blue}{-0.3333333333333333} \cdot \left(1.5 + 1.5\right)\right)}\right)}^{0.5} \]
      metadata-eval [=>] 19.7	\[ {\left(\left(d \cdot d\right) \cdot {\left(\ell \cdot h\right)}^{\left(-0.3333333333333333 \cdot \color{blue}{3}\right)}\right)}^{0.5} \]
      metadata-eval [=>] 19.7	\[ {\left(\left(d \cdot d\right) \cdot {\left(\ell \cdot h\right)}^{\color{blue}{-1}}\right)}^{0.5} \]

   8. Simplified 61.7%

      \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \]
      Proof

      [Start] 19.7	\[ {\left(\left(d \cdot d\right) \cdot {\left(\ell \cdot h\right)}^{-1}\right)}^{0.5} \]
      unpow1/2 [=>] 19.7	\[ \color{blue}{\sqrt{\left(d \cdot d\right) \cdot {\left(\ell \cdot h\right)}^{-1}}} \]
      *-commutative [=>] 19.7	\[ \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1} \cdot \left(d \cdot d\right)}} \]
      metadata-eval [<=] 19.7	\[ \sqrt{{\left(\ell \cdot h\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}} \cdot \left(d \cdot d\right)} \]
      pow-sqr [<=] 19.6	\[ \sqrt{\color{blue}{\left({\left(\ell \cdot h\right)}^{-0.5} \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)} \cdot \left(d \cdot d\right)} \]
      swap-sqr [<=] 22.4	\[ \sqrt{\color{blue}{\left({\left(\ell \cdot h\right)}^{-0.5} \cdot d\right) \cdot \left({\left(\ell \cdot h\right)}^{-0.5} \cdot d\right)}} \]
      rem-sqrt-square [=>] 61.7	\[ \color{blue}{\left|{\left(\ell \cdot h\right)}^{-0.5} \cdot d\right|} \]
      *-commutative [=>] 61.7	\[ \left|\color{blue}{d \cdot {\left(\ell \cdot h\right)}^{-0.5}}\right| \]
      rem-exp-log [<=] 58.5	\[ \left|d \cdot \color{blue}{e^{\log \left({\left(\ell \cdot h\right)}^{-0.5}\right)}}\right| \]
      log-pow [=>] 58.5	\[ \left|d \cdot e^{\color{blue}{-0.5 \cdot \log \left(\ell \cdot h\right)}}\right| \]
      metadata-eval [<=] 58.5	\[ \left|d \cdot e^{\color{blue}{\left(1.5 \cdot -0.3333333333333333\right)} \cdot \log \left(\ell \cdot h\right)}\right| \]
      associate-*r* [<=] 58.0	\[ \left|d \cdot e^{\color{blue}{1.5 \cdot \left(-0.3333333333333333 \cdot \log \left(\ell \cdot h\right)\right)}}\right| \]
      *-commutative [<=] 58.0	\[ \left|d \cdot e^{\color{blue}{\left(-0.3333333333333333 \cdot \log \left(\ell \cdot h\right)\right) \cdot 1.5}}\right| \]
      metadata-eval [<=] 58.0	\[ \left|d \cdot e^{\left(-0.3333333333333333 \cdot \log \left(\ell \cdot h\right)\right) \cdot \color{blue}{\left(0.75 + 0.75\right)}}\right| \]
      distribute-lft-out [<=] 58.0	\[ \left|d \cdot e^{\color{blue}{\left(-0.3333333333333333 \cdot \log \left(\ell \cdot h\right)\right) \cdot 0.75 + \left(-0.3333333333333333 \cdot \log \left(\ell \cdot h\right)\right) \cdot 0.75}}\right| \]
      distribute-rgt-out [=>] 58.0	\[ \left|d \cdot e^{\color{blue}{0.75 \cdot \left(-0.3333333333333333 \cdot \log \left(\ell \cdot h\right) + -0.3333333333333333 \cdot \log \left(\ell \cdot h\right)\right)}}\right| \]
      exp-prod [=>] 58.3	\[ \left|d \cdot \color{blue}{{\left(e^{0.75}\right)}^{\left(-0.3333333333333333 \cdot \log \left(\ell \cdot h\right) + -0.3333333333333333 \cdot \log \left(\ell \cdot h\right)\right)}}\right| \]
      distribute-rgt-out [=>] 58.3	\[ \left|d \cdot {\left(e^{0.75}\right)}^{\color{blue}{\left(\log \left(\ell \cdot h\right) \cdot \left(-0.3333333333333333 + -0.3333333333333333\right)\right)}}\right| \]
      metadata-eval [=>] 58.3	\[ \left|d \cdot {\left(e^{0.75}\right)}^{\left(\log \left(\ell \cdot h\right) \cdot \color{blue}{-0.6666666666666666}\right)}\right| \]
      metadata-eval [<=] 58.3	\[ \left|d \cdot {\left(e^{0.75}\right)}^{\left(\log \left(\ell \cdot h\right) \cdot \color{blue}{\left(-0.6666666666666666\right)}\right)}\right| \]
      distribute-rgt-neg-in [<=] 58.3	\[ \left|d \cdot {\left(e^{0.75}\right)}^{\color{blue}{\left(-\log \left(\ell \cdot h\right) \cdot 0.6666666666666666\right)}}\right| \]
      *-commutative [<=] 58.3	\[ \left|d \cdot {\left(e^{0.75}\right)}^{\left(-\color{blue}{0.6666666666666666 \cdot \log \left(\ell \cdot h\right)}\right)}\right| \]

3. Recombined 3 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{+296}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\left(\frac{M}{d} \cdot \frac{h \cdot D}{d}\right) \cdot \left(M \cdot \frac{D}{\ell}\right)\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-181} \lor \neg \left(\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0\right) \land \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{+233}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	77.8%
Cost	83661

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-181}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5 \cdot \frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{elif}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 2 \cdot 10^{+233}\right):\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Accuracy	77.6%
Cost	83661

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-181}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot {\left(\frac{M}{\frac{d \cdot 2}{D}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 2 \cdot 10^{+233}\right):\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Accuracy	68.1%
Cost	21400

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;d \leq -5.2 \cdot 10^{+100}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -2.7 \cdot 10^{+76}:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(1 - 0.125 \cdot \left(\frac{M}{d} \cdot \frac{M}{\frac{d}{h} \cdot \frac{\frac{\ell}{D}}{D}}\right)\right)\\ \mathbf{elif}\;d \leq -7.4 \cdot 10^{+48}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -0.001:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - 0.125 \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\frac{d}{\frac{h}{d \cdot \ell}}}\right)\\ \mathbf{elif}\;d \leq -4.5 \cdot 10^{-164}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;d \leq 3.1 \cdot 10^{-285}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 4
Accuracy	68.1%
Cost	21400

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;d \leq -3.6 \cdot 10^{+100}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -2.7 \cdot 10^{+76}:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(1 - 0.125 \cdot \left(\frac{M}{d} \cdot \frac{M}{\frac{d}{h} \cdot \frac{\frac{\ell}{D}}{D}}\right)\right)\\ \mathbf{elif}\;d \leq -6.5 \cdot 10^{+47}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -0.001:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - 0.125 \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\frac{d}{\frac{h}{d \cdot \ell}}}\right)\\ \mathbf{elif}\;d \leq -4.4 \cdot 10^{-164}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \frac{{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2}}{\frac{\ell}{h}}\right)\right)\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-283}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 5
Accuracy	66.1%
Cost	21268

\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_1 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -8.8 \cdot 10^{+207}:\\ \;\;\;\;t_1 \cdot \left(1 - 0.125 \cdot \frac{M \cdot D}{\frac{d}{M} \cdot \left(\frac{d}{h} \cdot \frac{\ell}{D}\right)}\right)\\ \mathbf{elif}\;\ell \leq -2.5 \cdot 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -2.4 \cdot 10^{-288}:\\ \;\;\;\;t_1 \cdot \left(1 - 0.125 \cdot \left(\frac{\frac{M}{d}}{\frac{\ell}{D}} \cdot \frac{M}{\frac{d}{h \cdot D}}\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{-223}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \left(1 + \frac{-0.125}{\frac{d}{h}} \cdot \frac{M \cdot \frac{M}{d}}{\frac{\frac{\ell}{D}}{D}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 6
Accuracy	68.7%
Cost	20872

\[\begin{array}{l} t_0 := {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}\\ \mathbf{if}\;h \leq -270000000:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{t_0}{\frac{\ell}{0.5}}\right)\\ \mathbf{elif}\;h \leq 1.8 \cdot 10^{-288}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t_0 \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 7
Accuracy	61.3%
Cost	15456

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{M \cdot D}{\frac{d}{0.5}}\right)}^{2}\right)\\ t_1 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_2 := \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{if}\;d \leq -2.3 \cdot 10^{+48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -4.5 \cdot 10^{-164}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.2 \cdot 10^{-302}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-239}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{-219}:\\ \;\;\;\;-0.125 \cdot \frac{\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-193}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\left(D \cdot D\right) \cdot \frac{M \cdot \frac{h \cdot M}{d}}{d}\right) \cdot \frac{-0.125}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 5.3 \cdot 10^{-86}:\\ \;\;\;\;t_2 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Accuracy	67.9%
Cost	15448

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ t_1 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;d \leq -3.1 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -2.7 \cdot 10^{+76}:\\ \;\;\;\;t_0 \cdot \left(1 - 0.125 \cdot \left(\frac{M}{d} \cdot \frac{M}{\frac{d}{h} \cdot \frac{\frac{\ell}{D}}{D}}\right)\right)\\ \mathbf{elif}\;d \leq -1.56 \cdot 10^{+52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -0.0006:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - 0.125 \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\frac{d}{\frac{h}{d \cdot \ell}}}\right)\\ \mathbf{elif}\;d \leq -4.6 \cdot 10^{-273}:\\ \;\;\;\;t_0 \cdot \left(1 - 0.125 \cdot \left(\frac{\frac{M}{d}}{\frac{\ell}{D}} \cdot \frac{M}{\frac{d}{h \cdot D}}\right)\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{+70}:\\ \;\;\;\;\left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot \frac{\left(M \cdot \frac{0.5}{\frac{d}{D}}\right) \cdot \left(M \cdot \left(0.5 \cdot D\right)\right)}{d}\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]

Alternative 9
Accuracy	61.7%
Cost	15324

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{M \cdot D}{\frac{d}{0.5}}\right)}^{2}\right)\\ t_1 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;d \leq -7.1 \cdot 10^{+47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-164}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 1.18 \cdot 10^{-240}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 3.75 \cdot 10^{-221}:\\ \;\;\;\;-0.125 \cdot \frac{\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}\\ \mathbf{elif}\;d \leq 5.4 \cdot 10^{-193}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\left(D \cdot D\right) \cdot \frac{M \cdot \frac{h \cdot M}{d}}{d}\right) \cdot \frac{-0.125}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 10^{-149}:\\ \;\;\;\;\frac{d}{{\left({\left(h \cdot \ell\right)}^{0.25}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 10
Accuracy	62.1%
Cost	15320

\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_1 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ t_2 := t_1 \cdot \left(1 - 0.125 \cdot \left(\frac{\frac{M}{d}}{\frac{\ell}{D}} \cdot \frac{M}{\frac{d}{h \cdot D}}\right)\right)\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+208}:\\ \;\;\;\;t_1 \cdot \left(1 - 0.125 \cdot \frac{M \cdot D}{\frac{d}{M} \cdot \left(\frac{d}{h} \cdot \frac{\ell}{D}\right)}\right)\\ \mathbf{elif}\;\ell \leq -5.8 \cdot 10^{-53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -9 \cdot 10^{-290}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{-217}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \left(1 + \frac{-0.125}{\frac{d}{h}} \cdot \frac{M \cdot \frac{M}{d}}{\frac{\frac{\ell}{D}}{D}}\right)\\ \mathbf{elif}\;\ell \leq 1.06 \cdot 10^{+92}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 11
Accuracy	65.0%
Cost	15316

\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_1 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -9.5 \cdot 10^{+207}:\\ \;\;\;\;t_1 \cdot \left(1 - 0.125 \cdot \frac{M \cdot D}{\frac{d}{M} \cdot \left(\frac{d}{h} \cdot \frac{\ell}{D}\right)}\right)\\ \mathbf{elif}\;\ell \leq -9.2 \cdot 10^{-52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -9.5 \cdot 10^{-282}:\\ \;\;\;\;t_1 \cdot \left(1 - 0.125 \cdot \left(\frac{\frac{M}{d}}{\frac{\ell}{D}} \cdot \frac{M}{\frac{d}{h \cdot D}}\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{-230}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \left(1 + \frac{-0.125}{\frac{d}{h}} \cdot \frac{M \cdot \frac{M}{d}}{\frac{\frac{\ell}{D}}{D}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot \frac{\left(M \cdot \frac{0.5}{\frac{d}{D}}\right) \cdot \left(M \cdot \left(0.5 \cdot D\right)\right)}{d}\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 12
Accuracy	66.2%
Cost	15056

\[\begin{array}{l} t_0 := \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\left(\frac{M}{d} \cdot \frac{h \cdot D}{d}\right) \cdot \left(M \cdot \frac{D}{\ell}\right)\right)\right)\\ t_1 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;d \leq -5.5 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-164}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{-221}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 13
Accuracy	66.5%
Cost	15056

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ t_1 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;d \leq -3 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -3.3 \cdot 10^{-164}:\\ \;\;\;\;t_0 \cdot \left(1 - 0.125 \cdot \left(\left(\frac{M}{d} \cdot \frac{h \cdot D}{d}\right) \cdot \left(M \cdot \frac{D}{\ell}\right)\right)\right)\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{-221}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{-7}:\\ \;\;\;\;t_0 \cdot \left(1 - 0.125 \cdot \left(\frac{\frac{M}{d}}{\frac{\ell}{D}} \cdot \frac{M}{\frac{d}{h \cdot D}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 14
Accuracy	62.3%
Cost	14996

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{M \cdot D}{\frac{d}{0.5}}\right)}^{2}\right)\\ t_1 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;d \leq -7.1 \cdot 10^{+47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -4.5 \cdot 10^{-164}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.92 \cdot 10^{-277}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 1.3 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \left(1 + \frac{-0.125}{\frac{d}{h}} \cdot \frac{M \cdot \frac{M}{d}}{\frac{\frac{\ell}{D}}{D}}\right)\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-80}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 15
Accuracy	61.8%
Cost	14864

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ t_1 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;d \leq -1.6 \cdot 10^{-42}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -4.5 \cdot 10^{-164}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{-277}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 3.1 \cdot 10^{-163}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 16
Accuracy	62.3%
Cost	14864

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{M \cdot D}{\frac{d}{0.5}}\right)}^{2}\right)\\ t_1 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;d \leq -5.3 \cdot 10^{+47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -2.3 \cdot 10^{-164}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-302}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{-163}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 17
Accuracy	64.2%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;h \leq -2.5 \cdot 10^{+120}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 18
Accuracy	63.0%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 19
Accuracy	56.8%
Cost	13120

\[\left|\frac{d}{\sqrt{h \cdot \ell}}\right| \]

Alternative 20
Accuracy	45.5%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq -4.7 \cdot 10^{-268}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 21
Accuracy	47.5%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq 4.9 \cdot 10^{-270}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 22
Accuracy	47.6%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq 4.8 \cdot 10^{-273}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]

Alternative 23
Accuracy	30.7%
Cost	6784

\[d \cdot {\left(h \cdot \ell\right)}^{-0.5} \]

Alternative 24
Accuracy	30.7%
Cost	6720

\[\frac{d}{\sqrt{h \cdot \ell}} \]

Reproduce

herbie shell --seed 2023152 
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  :precision binary64
  (* (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))) (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))