Henrywood and Agarwal, Equation (12)

Average Accuracy: 36.1% → 47.4%

Time: 34.4s

Precision: binary64

Cost: 27536

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := 0.5 \cdot \left(D \cdot \frac{M}{d}\right)\\ t_2 := 1 - h \cdot \left(t_1 \cdot \left(t_1 \cdot \frac{0.5}{\ell}\right)\right)\\ t_3 := {\left(\frac{d}{h}\right)}^{0.5}\\ \mathbf{if}\;h \leq -4.4 \cdot 10^{-178}:\\ \;\;\;\;\left(t_3 \cdot \frac{t_0}{\sqrt{-\ell}}\right) \cdot t_2\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{t_0}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq 3 \cdot 10^{-182}:\\ \;\;\;\;\left(1 + {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;h \leq 6 \cdot 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, \left(D \cdot M\right) \cdot \left(\frac{\frac{\sqrt{h}}{{\ell}^{1.5}}}{\frac{d}{-0.125}} \cdot \left(D \cdot M\right)\right)\right)\\ \mathbf{elif}\;h \leq 1.18 \cdot 10^{+151}:\\ \;\;\;\;t_2 \cdot \left(t_3 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot {h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

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 (sqrt (- d)))
        (t_1 (* 0.5 (* D (/ M d))))
        (t_2 (- 1.0 (* h (* t_1 (* t_1 (/ 0.5 l))))))
        (t_3 (pow (/ d h) 0.5)))
   (if (<= h -4.4e-178)
     (* (* t_3 (/ t_0 (sqrt (- l)))) t_2)
     (if (<= h -5e-310)
       (* (/ t_0 (sqrt (- h))) (sqrt (/ d l)))
       (if (<= h 3e-182)
         (*
          (+ 1.0 (* (pow (* (* 0.5 M) (/ D d)) 2.0) (* -0.5 (/ h l))))
          (/ d (* (sqrt h) (sqrt l))))
         (if (<= h 6e-80)
           (fma
            d
            (sqrt (/ 1.0 (* h l)))
            (* (* D M) (* (/ (/ (sqrt h) (pow l 1.5)) (/ d -0.125)) (* D M))))
           (if (<= h 1.18e+151)
             (* t_2 (* t_3 (/ (sqrt d) (sqrt l))))
             (/ (* d (pow h -0.5)) (sqrt 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 = sqrt(-d);
	double t_1 = 0.5 * (D * (M / d));
	double t_2 = 1.0 - (h * (t_1 * (t_1 * (0.5 / l))));
	double t_3 = pow((d / h), 0.5);
	double tmp;
	if (h <= -4.4e-178) {
		tmp = (t_3 * (t_0 / sqrt(-l))) * t_2;
	} else if (h <= -5e-310) {
		tmp = (t_0 / sqrt(-h)) * sqrt((d / l));
	} else if (h <= 3e-182) {
		tmp = (1.0 + (pow(((0.5 * M) * (D / d)), 2.0) * (-0.5 * (h / l)))) * (d / (sqrt(h) * sqrt(l)));
	} else if (h <= 6e-80) {
		tmp = fma(d, sqrt((1.0 / (h * l))), ((D * M) * (((sqrt(h) / pow(l, 1.5)) / (d / -0.125)) * (D * M))));
	} else if (h <= 1.18e+151) {
		tmp = t_2 * (t_3 * (sqrt(d) / sqrt(l)));
	} else {
		tmp = (d * pow(h, -0.5)) / sqrt(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 = sqrt(Float64(-d))
	t_1 = Float64(0.5 * Float64(D * Float64(M / d)))
	t_2 = Float64(1.0 - Float64(h * Float64(t_1 * Float64(t_1 * Float64(0.5 / l)))))
	t_3 = Float64(d / h) ^ 0.5
	tmp = 0.0
	if (h <= -4.4e-178)
		tmp = Float64(Float64(t_3 * Float64(t_0 / sqrt(Float64(-l)))) * t_2);
	elseif (h <= -5e-310)
		tmp = Float64(Float64(t_0 / sqrt(Float64(-h))) * sqrt(Float64(d / l)));
	elseif (h <= 3e-182)
		tmp = Float64(Float64(1.0 + Float64((Float64(Float64(0.5 * M) * Float64(D / d)) ^ 2.0) * Float64(-0.5 * Float64(h / l)))) * Float64(d / Float64(sqrt(h) * sqrt(l))));
	elseif (h <= 6e-80)
		tmp = fma(d, sqrt(Float64(1.0 / Float64(h * l))), Float64(Float64(D * M) * Float64(Float64(Float64(sqrt(h) / (l ^ 1.5)) / Float64(d / -0.125)) * Float64(D * M))));
	elseif (h <= 1.18e+151)
		tmp = Float64(t_2 * Float64(t_3 * Float64(sqrt(d) / sqrt(l))));
	else
		tmp = Float64(Float64(d * (h ^ -0.5)) / sqrt(l));
	end
	return 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[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(h * N[(t$95$1 * N[(t$95$1 * N[(0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision]}, If[LessEqual[h, -4.4e-178], N[(N[(t$95$3 * N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[h, -5e-310], N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 3e-182], N[(N[(1.0 + N[(N[Power[N[(N[(0.5 * M), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 6e-80], N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(D * M), $MachinePrecision] * N[(N[(N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision] / N[(d / -0.125), $MachinePrecision]), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 1.18e+151], N[(t$95$2 * N[(t$95$3 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if h < -4.4000000000000002e-178

   1. Initial program 38.9%

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

   2. Applied egg-rr 38.9%

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

      [Start] 38.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      expm1-log1p-u [=>] 38.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]
      expm1-udef [=>] 38.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} - 1\right)}\right) \]
      log1p-udef [=>] 38.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(e^{\color{blue}{\log \left(1 + \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}} - 1\right)\right) \]
      add-exp-log [<=] 38.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(1 + \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} - 1\right)\right) \]
      associate-*l* [=>] 38.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) - 1\right)\right) \]
      metadata-eval [=>] 38.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]
      *-un-lft-identity [=>] 38.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + 0.5 \cdot \left({\left(\frac{\color{blue}{1 \cdot \left(M \cdot D\right)}}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]
      times-frac [=>] 38.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + 0.5 \cdot \left({\color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]
      metadata-eval [=>] 38.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + 0.5 \cdot \left({\left(\color{blue}{0.5} \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]

   3. Simplified 38.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(h \cdot \frac{{\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2}}{\frac{\ell}{0.5}} + 0\right)}\right) \]
      Proof

      [Start] 38.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + 0.5 \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]
      +-commutative [=>] 38.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(0.5 \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) + 1\right)} - 1\right)\right) \]
      associate--l+ [=>] 38.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) + \left(1 - 1\right)\right)}\right) \]
      associate-*r* [=>] 38.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(0.5 \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}\right) \cdot \frac{h}{\ell}} + \left(1 - 1\right)\right)\right) \]
      *-commutative [=>] 38.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{h}{\ell} \cdot \left(0.5 \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}\right)} + \left(1 - 1\right)\right)\right) \]
      associate-*l/ [=>] 38.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{h \cdot \left(0.5 \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}\right)}{\ell}} + \left(1 - 1\right)\right)\right) \]
      associate-*r/ [<=] 41.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{h \cdot \frac{0.5 \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}}{\ell}} + \left(1 - 1\right)\right)\right) \]
      *-commutative [=>] 41.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{\color{blue}{{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot 0.5}}{\ell} + \left(1 - 1\right)\right)\right) \]
      associate-/l* [=>] 41.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \color{blue}{\frac{{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}}{\frac{\ell}{0.5}}} + \left(1 - 1\right)\right)\right) \]
      *-commutative [=>] 41.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{{\left(0.5 \cdot \frac{\color{blue}{D \cdot M}}{d}\right)}^{2}}{\frac{\ell}{0.5}} + \left(1 - 1\right)\right)\right) \]
      associate-/l* [=>] 38.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{{\left(0.5 \cdot \color{blue}{\frac{D}{\frac{d}{M}}}\right)}^{2}}{\frac{\ell}{0.5}} + \left(1 - 1\right)\right)\right) \]
      metadata-eval [=>] 38.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{{\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2}}{\frac{\ell}{0.5}} + \color{blue}{0}\right)\right) \]

   4. Applied egg-rr 40.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \color{blue}{\left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \frac{0.5}{\ell}\right)\right)} + 0\right)\right) \]
      Proof

      [Start] 38.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{{\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2}}{\frac{\ell}{0.5}} + 0\right)\right) \]
      div-inv [=>] 38.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \color{blue}{\left({\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot \frac{1}{\frac{\ell}{0.5}}\right)} + 0\right)\right) \]
      metadata-eval [<=] 38.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \left({\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot \frac{1}{\frac{\ell}{\color{blue}{\frac{1}{2}}}}\right) + 0\right)\right) \]
      clear-num [<=] 38.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \left({\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot \color{blue}{\frac{\frac{1}{2}}{\ell}}\right) + 0\right)\right) \]
      unpow2 [=>] 38.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \left(\color{blue}{\left(\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right) \cdot \left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)\right)} \cdot \frac{\frac{1}{2}}{\ell}\right) + 0\right)\right) \]
      associate-*l* [=>] 40.1	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \color{blue}{\left(\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right) \cdot \left(\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right) \cdot \frac{\frac{1}{2}}{\ell}\right)\right)} + 0\right)\right) \]
      div-inv [=>] 40.1	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \left(\left(0.5 \cdot \color{blue}{\left(D \cdot \frac{1}{\frac{d}{M}}\right)}\right) \cdot \left(\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right) \cdot \frac{\frac{1}{2}}{\ell}\right)\right) + 0\right)\right) \]
      clear-num [<=] 40.1	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \left(\left(0.5 \cdot \left(D \cdot \color{blue}{\frac{M}{d}}\right)\right) \cdot \left(\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right) \cdot \frac{\frac{1}{2}}{\ell}\right)\right) + 0\right)\right) \]
      div-inv [=>] 40.1	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\left(0.5 \cdot \color{blue}{\left(D \cdot \frac{1}{\frac{d}{M}}\right)}\right) \cdot \frac{\frac{1}{2}}{\ell}\right)\right) + 0\right)\right) \]
      clear-num [<=] 40.1	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\left(0.5 \cdot \left(D \cdot \color{blue}{\frac{M}{d}}\right)\right) \cdot \frac{\frac{1}{2}}{\ell}\right)\right) + 0\right)\right) \]
      metadata-eval [=>] 40.1	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \frac{\color{blue}{0.5}}{\ell}\right)\right) + 0\right)\right) \]

   5. Applied egg-rr 46.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}}\right) \cdot \left(1 - \left(h \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \frac{0.5}{\ell}\right)\right) + 0\right)\right) \]
      Proof

      [Start] 40.1	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \frac{0.5}{\ell}\right)\right) + 0\right)\right) \]
      metadata-eval [=>] 40.1	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(h \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \frac{0.5}{\ell}\right)\right) + 0\right)\right) \]
      unpow1/2 [=>] 40.1	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(h \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \frac{0.5}{\ell}\right)\right) + 0\right)\right) \]
      frac-2neg [=>] 40.1	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{-d}{-\ell}}}\right) \cdot \left(1 - \left(h \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \frac{0.5}{\ell}\right)\right) + 0\right)\right) \]
      sqrt-div [=>] 46.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}}\right) \cdot \left(1 - \left(h \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \frac{0.5}{\ell}\right)\right) + 0\right)\right) \]

   if -4.4000000000000002e-178 < h < -4.999999999999985e-310

   1. Initial program 22.2%

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

   2. Simplified 18.5%

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

      [Start] 22.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 22.2	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 22.2	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 22.2	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 22.2	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 22.2	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      sub-neg [=>] 22.2	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]
      +-commutative [=>] 22.2	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]
      *-commutative [=>] 22.2	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) + 1\right)\right) \]
      distribute-rgt-neg-in [=>] 22.2	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{h}{\ell} \cdot \left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} + 1\right)\right) \]
      fma-def [=>] 22.2	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, -\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}, 1\right)}\right) \]

   3. Taylor expanded in h around 0 22.7%

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

   4. Applied egg-rr 37.1%

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

      [Start] 22.7	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right) \]
      frac-2neg [=>] 22.7	\[ \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right) \]
      sqrt-div [=>] 37.1	\[ \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right) \]

   if -4.999999999999985e-310 < h < 3.0000000000000001e-182

   1. Initial program 35.5%

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

   2. Simplified 35.5%

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

      [Start] 35.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 35.5	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 35.5	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 35.5	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 35.5	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      *-commutative [=>] 35.5	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 35.5	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]
      times-frac [=>] 35.5	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 35.5	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 67.4%

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

      [Start] 35.5	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      sub-neg [=>] 35.5	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\left(1 + \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} \]
      distribute-lft-in [=>] 35.5	\[ \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot 1 + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
      *-commutative [<=] 35.5	\[ \color{blue}{1 \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      *-un-lft-identity [<=] 35.5	\[ \color{blue}{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      sqrt-div [=>] 35.3	\[ \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      sqrt-div [=>] 41.6	\[ \frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      frac-times [=>] 41.5	\[ \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      add-sqr-sqrt [<=] 41.7	\[ \frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}} + \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      sqrt-div [=>] 67.4	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      sqrt-div [=>] 67.3	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      frac-times [=>] 67.3	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}} \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      add-sqr-sqrt [<=] 67.4	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(-{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      distribute-rgt-neg-in [=>] 67.4	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Simplified 67.4%

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

      [Start] 67.4	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right) \]
      *-commutative [<=] 67.4	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \color{blue}{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]
      distribute-rgt1-in [=>] 67.4	\[ \color{blue}{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right) + 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]
      distribute-lft-neg-in [=>] 67.4	\[ \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \color{blue}{\left(\left(-0.5\right) \cdot \frac{h}{\ell}\right)} + 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]
      metadata-eval [=>] 67.4	\[ \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{-0.5} \cdot \frac{h}{\ell}\right) + 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]

   if 3.0000000000000001e-182 < h < 6.00000000000000014e-80

   1. Initial program 37.3%

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

   2. Simplified 37.6%

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

      [Start] 37.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 37.3	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 37.3	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 37.3	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 37.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      *-commutative [=>] 37.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 37.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]
      times-frac [=>] 37.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 37.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Taylor expanded in d around -inf 0.0%

      \[\leadsto \color{blue}{-1 \cdot \left(\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) + 0.125 \cdot \left(\frac{{\left(\sqrt{-1}\right)}^{2} \cdot \left({D}^{2} \cdot {M}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]

   4. Simplified 31.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, \frac{-0.125 \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{\frac{d}{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}}\right)} \]
      Proof

      [Start] 0.0	\[ -1 \cdot \left(\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) + 0.125 \cdot \left(\frac{{\left(\sqrt{-1}\right)}^{2} \cdot \left({D}^{2} \cdot {M}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]
      associate-*r* [=>] 0.0	\[ \color{blue}{\left(-1 \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} + 0.125 \cdot \left(\frac{{\left(\sqrt{-1}\right)}^{2} \cdot \left({D}^{2} \cdot {M}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]
      fma-def [=>] 0.0	\[ \color{blue}{\mathsf{fma}\left(-1 \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot d\right), \sqrt{\frac{1}{\ell \cdot h}}, 0.125 \cdot \left(\frac{{\left(\sqrt{-1}\right)}^{2} \cdot \left({D}^{2} \cdot {M}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]
      *-commutative [=>] 0.0	\[ \mathsf{fma}\left(-1 \cdot \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)}, \sqrt{\frac{1}{\ell \cdot h}}, 0.125 \cdot \left(\frac{{\left(\sqrt{-1}\right)}^{2} \cdot \left({D}^{2} \cdot {M}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) \]
      unpow2 [=>] 0.0	\[ \mathsf{fma}\left(-1 \cdot \left(d \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}\right), \sqrt{\frac{1}{\ell \cdot h}}, 0.125 \cdot \left(\frac{{\left(\sqrt{-1}\right)}^{2} \cdot \left({D}^{2} \cdot {M}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) \]
      rem-square-sqrt [=>] 0.0	\[ \mathsf{fma}\left(-1 \cdot \left(d \cdot \color{blue}{-1}\right), \sqrt{\frac{1}{\ell \cdot h}}, 0.125 \cdot \left(\frac{{\left(\sqrt{-1}\right)}^{2} \cdot \left({D}^{2} \cdot {M}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) \]
      *-commutative [=>] 0.0	\[ \mathsf{fma}\left(-1 \cdot \color{blue}{\left(-1 \cdot d\right)}, \sqrt{\frac{1}{\ell \cdot h}}, 0.125 \cdot \left(\frac{{\left(\sqrt{-1}\right)}^{2} \cdot \left({D}^{2} \cdot {M}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) \]
      associate-*r* [=>] 0.0	\[ \mathsf{fma}\left(\color{blue}{\left(-1 \cdot -1\right) \cdot d}, \sqrt{\frac{1}{\ell \cdot h}}, 0.125 \cdot \left(\frac{{\left(\sqrt{-1}\right)}^{2} \cdot \left({D}^{2} \cdot {M}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) \]
      metadata-eval [=>] 0.0	\[ \mathsf{fma}\left(\color{blue}{1} \cdot d, \sqrt{\frac{1}{\ell \cdot h}}, 0.125 \cdot \left(\frac{{\left(\sqrt{-1}\right)}^{2} \cdot \left({D}^{2} \cdot {M}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) \]
      *-lft-identity [=>] 0.0	\[ \mathsf{fma}\left(\color{blue}{d}, \sqrt{\frac{1}{\ell \cdot h}}, 0.125 \cdot \left(\frac{{\left(\sqrt{-1}\right)}^{2} \cdot \left({D}^{2} \cdot {M}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) \]
      *-commutative [=>] 0.0	\[ \mathsf{fma}\left(d, \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}, 0.125 \cdot \left(\frac{{\left(\sqrt{-1}\right)}^{2} \cdot \left({D}^{2} \cdot {M}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) \]
      associate-*r* [=>] 0.0	\[ \mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, \color{blue}{\left(0.125 \cdot \frac{{\left(\sqrt{-1}\right)}^{2} \cdot \left({D}^{2} \cdot {M}^{2}\right)}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}\right) \]

   5. Applied egg-rr 74.1%

      \[\leadsto \mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, \color{blue}{\left(\frac{\frac{\sqrt{h}}{{\ell}^{1.5}}}{\frac{d}{-0.125}} \cdot \left(D \cdot M\right)\right) \cdot \left(D \cdot M\right)}\right) \]
      Proof

      [Start] 31.6	\[ \mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, \frac{-0.125 \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{\frac{d}{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}}\right) \]
      associate-/r/ [=>] 33.0	\[ \mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, \color{blue}{\frac{-0.125 \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d} \cdot \left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right)}\right) \]
      add-sqr-sqrt [=>] 33.0	\[ \mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, \frac{-0.125 \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d} \cdot \color{blue}{\left(\sqrt{D \cdot \left(D \cdot \left(M \cdot M\right)\right)} \cdot \sqrt{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}\right)}\right) \]
      associate-*r* [=>] 33.0	\[ \mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, \color{blue}{\left(\frac{-0.125 \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d} \cdot \sqrt{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}\right) \cdot \sqrt{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}}\right) \]

   if 6.00000000000000014e-80 < h < 1.18000000000000005e151

   1. Initial program 51.5%

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

   2. Applied egg-rr 51.5%

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

      [Start] 51.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      expm1-log1p-u [=>] 50.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]
      expm1-udef [=>] 50.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} - 1\right)}\right) \]
      log1p-udef [=>] 50.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(e^{\color{blue}{\log \left(1 + \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}} - 1\right)\right) \]
      add-exp-log [<=] 51.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(1 + \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} - 1\right)\right) \]
      associate-*l* [=>] 51.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) - 1\right)\right) \]
      metadata-eval [=>] 51.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]
      *-un-lft-identity [=>] 51.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + 0.5 \cdot \left({\left(\frac{\color{blue}{1 \cdot \left(M \cdot D\right)}}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]
      times-frac [=>] 51.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + 0.5 \cdot \left({\color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]
      metadata-eval [=>] 51.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + 0.5 \cdot \left({\left(\color{blue}{0.5} \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]

   3. Simplified 51.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(h \cdot \frac{{\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2}}{\frac{\ell}{0.5}} + 0\right)}\right) \]
      Proof

      [Start] 51.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + 0.5 \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]
      +-commutative [=>] 51.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(0.5 \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) + 1\right)} - 1\right)\right) \]
      associate--l+ [=>] 51.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) + \left(1 - 1\right)\right)}\right) \]
      associate-*r* [=>] 51.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(0.5 \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}\right) \cdot \frac{h}{\ell}} + \left(1 - 1\right)\right)\right) \]
      *-commutative [=>] 51.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{h}{\ell} \cdot \left(0.5 \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}\right)} + \left(1 - 1\right)\right)\right) \]
      associate-*l/ [=>] 51.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{h \cdot \left(0.5 \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}\right)}{\ell}} + \left(1 - 1\right)\right)\right) \]
      associate-*r/ [<=] 51.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{h \cdot \frac{0.5 \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}}{\ell}} + \left(1 - 1\right)\right)\right) \]
      *-commutative [=>] 51.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{\color{blue}{{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot 0.5}}{\ell} + \left(1 - 1\right)\right)\right) \]
      associate-/l* [=>] 51.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \color{blue}{\frac{{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}}{\frac{\ell}{0.5}}} + \left(1 - 1\right)\right)\right) \]
      *-commutative [=>] 51.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{{\left(0.5 \cdot \frac{\color{blue}{D \cdot M}}{d}\right)}^{2}}{\frac{\ell}{0.5}} + \left(1 - 1\right)\right)\right) \]
      associate-/l* [=>] 51.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{{\left(0.5 \cdot \color{blue}{\frac{D}{\frac{d}{M}}}\right)}^{2}}{\frac{\ell}{0.5}} + \left(1 - 1\right)\right)\right) \]
      metadata-eval [=>] 51.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{{\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2}}{\frac{\ell}{0.5}} + \color{blue}{0}\right)\right) \]

   4. Applied egg-rr 53.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \color{blue}{\left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \frac{0.5}{\ell}\right)\right)} + 0\right)\right) \]
      Proof

      [Start] 51.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{{\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2}}{\frac{\ell}{0.5}} + 0\right)\right) \]
      div-inv [=>] 51.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \color{blue}{\left({\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot \frac{1}{\frac{\ell}{0.5}}\right)} + 0\right)\right) \]
      metadata-eval [<=] 51.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \left({\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot \frac{1}{\frac{\ell}{\color{blue}{\frac{1}{2}}}}\right) + 0\right)\right) \]
      clear-num [<=] 51.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \left({\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot \color{blue}{\frac{\frac{1}{2}}{\ell}}\right) + 0\right)\right) \]
      unpow2 [=>] 51.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \left(\color{blue}{\left(\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right) \cdot \left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)\right)} \cdot \frac{\frac{1}{2}}{\ell}\right) + 0\right)\right) \]
      associate-*l* [=>] 53.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \color{blue}{\left(\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right) \cdot \left(\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right) \cdot \frac{\frac{1}{2}}{\ell}\right)\right)} + 0\right)\right) \]
      div-inv [=>] 53.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \left(\left(0.5 \cdot \color{blue}{\left(D \cdot \frac{1}{\frac{d}{M}}\right)}\right) \cdot \left(\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right) \cdot \frac{\frac{1}{2}}{\ell}\right)\right) + 0\right)\right) \]
      clear-num [<=] 53.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \left(\left(0.5 \cdot \left(D \cdot \color{blue}{\frac{M}{d}}\right)\right) \cdot \left(\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right) \cdot \frac{\frac{1}{2}}{\ell}\right)\right) + 0\right)\right) \]
      div-inv [=>] 53.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\left(0.5 \cdot \color{blue}{\left(D \cdot \frac{1}{\frac{d}{M}}\right)}\right) \cdot \frac{\frac{1}{2}}{\ell}\right)\right) + 0\right)\right) \]
      clear-num [<=] 53.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\left(0.5 \cdot \left(D \cdot \color{blue}{\frac{M}{d}}\right)\right) \cdot \frac{\frac{1}{2}}{\ell}\right)\right) + 0\right)\right) \]
      metadata-eval [=>] 53.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \frac{\color{blue}{0.5}}{\ell}\right)\right) + 0\right)\right) \]

   5. Applied egg-rr 63.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(h \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \frac{0.5}{\ell}\right)\right) + 0\right)\right) \]
      Proof

      [Start] 53.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \frac{0.5}{\ell}\right)\right) + 0\right)\right) \]
      metadata-eval [=>] 53.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(h \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \frac{0.5}{\ell}\right)\right) + 0\right)\right) \]
      unpow1/2 [=>] 53.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(h \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \frac{0.5}{\ell}\right)\right) + 0\right)\right) \]
      sqrt-div [=>] 63.1	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(h \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \frac{0.5}{\ell}\right)\right) + 0\right)\right) \]

   if 1.18000000000000005e151 < h

   1. Initial program 32.0%

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

   2. Taylor expanded in d around inf 16.0%

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

   3. Simplified 15.9%

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

      [Start] 16.0	\[ \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]
      *-commutative [=>] 16.0	\[ \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
      *-commutative [=>] 16.0	\[ d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]
      associate-/r* [=>] 15.9	\[ d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   4. Applied egg-rr 40.7%

      \[\leadsto \color{blue}{\frac{{h}^{-0.5} \cdot d}{\sqrt{\ell}}} \]
      Proof

      [Start] 15.9	\[ d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \]
      *-commutative [=>] 15.9	\[ \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot d} \]
      sqrt-div [=>] 40.7	\[ \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \cdot d \]
      associate-*l/ [=>] 40.7	\[ \color{blue}{\frac{\sqrt{\frac{1}{h}} \cdot d}{\sqrt{\ell}}} \]
      inv-pow [=>] 40.7	\[ \frac{\sqrt{\color{blue}{{h}^{-1}}} \cdot d}{\sqrt{\ell}} \]
      sqrt-pow1 [=>] 40.7	\[ \frac{\color{blue}{{h}^{\left(\frac{-1}{2}\right)}} \cdot d}{\sqrt{\ell}} \]
      metadata-eval [=>] 40.7	\[ \frac{{h}^{\color{blue}{-0.5}} \cdot d}{\sqrt{\ell}} \]

3. Recombined 6 regimes into one program.

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -4.4 \cdot 10^{-178}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - h \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \frac{0.5}{\ell}\right)\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq 3 \cdot 10^{-182}:\\ \;\;\;\;\left(1 + {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;h \leq 6 \cdot 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, \left(D \cdot M\right) \cdot \left(\frac{\frac{\sqrt{h}}{{\ell}^{1.5}}}{\frac{d}{-0.125}} \cdot \left(D \cdot M\right)\right)\right)\\ \mathbf{elif}\;h \leq 1.18 \cdot 10^{+151}:\\ \;\;\;\;\left(1 - h \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \frac{0.5}{\ell}\right)\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot {h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	44.6%
Cost	21644

\[\begin{array}{l} t_0 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_1 := 0.5 \cdot \left(D \cdot \frac{M}{d}\right)\\ t_2 := 1 - h \cdot \left(t_1 \cdot \left(t_1 \cdot \frac{0.5}{\ell}\right)\right)\\ \mathbf{if}\;h \leq -1.65 \cdot 10^{-177}:\\ \;\;\;\;t_2 \cdot \left(t_0 \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq 5.6 \cdot 10^{-182}:\\ \;\;\;\;\left(1 + {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \]

Alternative 2
Accuracy	47.5%
Cost	21644

\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := 0.5 \cdot \left(D \cdot \frac{M}{d}\right)\\ t_2 := 1 - h \cdot \left(t_1 \cdot \left(t_1 \cdot \frac{0.5}{\ell}\right)\right)\\ t_3 := {\left(\frac{d}{h}\right)}^{0.5}\\ \mathbf{if}\;h \leq -2.2 \cdot 10^{-178}:\\ \;\;\;\;\left(t_3 \cdot \frac{t_0}{\sqrt{-\ell}}\right) \cdot t_2\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{t_0}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq 4.3 \cdot 10^{-182}:\\ \;\;\;\;\left(1 + {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(t_3 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \]

Alternative 3
Accuracy	43.7%
Cost	20872

\[\begin{array}{l} t_0 := 0.5 \cdot \left(D \cdot \frac{M}{d}\right)\\ \mathbf{if}\;h \leq -1.35 \cdot 10^{-179}:\\ \;\;\;\;\left(1 - h \cdot \left(t_0 \cdot \left(t_0 \cdot \frac{0.5}{\ell}\right)\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 4
Accuracy	40.7%
Cost	15244

\[\begin{array}{l} t_0 := 0.5 \cdot \left(D \cdot \frac{M}{d}\right)\\ t_1 := \left(1 - h \cdot \left(t_0 \cdot \left(t_0 \cdot \frac{0.5}{\ell}\right)\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{if}\;\ell \leq 7.2 \cdot 10^{-284}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.55 \cdot 10^{-68}:\\ \;\;\;\;\frac{d \cdot {h}^{-0.5}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{+68}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{{h}^{-0.5}}{\frac{\sqrt{\ell}}{d}}\\ \end{array} \]

Alternative 5
Accuracy	40.7%
Cost	15244

\[\begin{array}{l} t_0 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_1 := 0.5 \cdot \left(D \cdot \frac{M}{d}\right)\\ t_2 := 1 - h \cdot \left(t_1 \cdot \left(t_1 \cdot \frac{0.5}{\ell}\right)\right)\\ \mathbf{if}\;\ell \leq 1.4 \cdot 10^{-283}:\\ \;\;\;\;t_2 \cdot \left(t_0 \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{-73}:\\ \;\;\;\;\frac{d \cdot {h}^{-0.5}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{+70}:\\ \;\;\;\;t_2 \cdot \left(t_0 \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{h}^{-0.5}}{\frac{\sqrt{\ell}}{d}}\\ \end{array} \]

Alternative 6
Accuracy	37.9%
Cost	13508

\[\begin{array}{l} \mathbf{if}\;\ell \leq 6.2 \cdot 10^{-284}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 7
Accuracy	38.2%
Cost	13508

\[\begin{array}{l} \mathbf{if}\;\ell \leq 5.5 \cdot 10^{-284}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 8
Accuracy	37.9%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;\ell \leq 5.5 \cdot 10^{-284}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 9
Accuracy	34.4%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;d \leq 1.02 \cdot 10^{-254}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 10
Accuracy	30.3%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;h \leq -1.15 \cdot 10^{-239} \lor \neg \left(h \leq 5.5 \cdot 10^{+182}\right):\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 11
Accuracy	28.6%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;h \leq -1.15 \cdot 10^{-239}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 12
Accuracy	20.8%
Cost	6720

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

Reproduce

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