Henrywood and Agarwal, Equation (12)

Average Accuracy: 58.8% → 74.7%

Time: 1.2min

Precision: binary64

Cost: 28116

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

Math

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

↓

\[\begin{array}{l} t_0 := D \cdot \left(\frac{M}{d} \cdot 0.5\right)\\ t_1 := \sqrt{-d}\\ t_2 := \mathsf{fma}\left(-0.5, \frac{t_0}{\frac{\ell}{h \cdot t_0}}, 1\right)\\ t_3 := \frac{t_1}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot t_2\right)\\ t_4 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -4 \cdot 10^{+107}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq -9.8 \cdot 10^{+57}:\\ \;\;\;\;t_4 \cdot \left(t_2 \cdot \frac{t_1}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-173}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-293}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 3.85 \cdot 10^{+93}:\\ \;\;\;\;t_4 \cdot \left(t_2 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

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 (* D (* (/ M d) 0.5)))
        (t_1 (sqrt (- d)))
        (t_2 (fma -0.5 (/ t_0 (/ l (* h t_0))) 1.0))
        (t_3 (* (/ t_1 (sqrt (- h))) (* (sqrt (/ d l)) t_2)))
        (t_4 (sqrt (/ d h))))
   (if (<= d -4e+107)
     t_3
     (if (<= d -9.8e+57)
       (* t_4 (* t_2 (/ t_1 (sqrt (- l)))))
       (if (<= d -1.45e-173)
         t_3
         (if (<= d 1.05e-293)
           (*
            (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
            (+ 1.0 (* (/ h l) (* -0.5 (pow (/ (* D M) (* d 2.0)) 2.0)))))
           (if (<= d 3.85e+93)
             (* t_4 (* t_2 (/ (sqrt d) (sqrt l))))
             (* d (* (pow l -0.5) (pow h -0.5))))))))))

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 = D * ((M / d) * 0.5);
	double t_1 = sqrt(-d);
	double t_2 = fma(-0.5, (t_0 / (l / (h * t_0))), 1.0);
	double t_3 = (t_1 / sqrt(-h)) * (sqrt((d / l)) * t_2);
	double t_4 = sqrt((d / h));
	double tmp;
	if (d <= -4e+107) {
		tmp = t_3;
	} else if (d <= -9.8e+57) {
		tmp = t_4 * (t_2 * (t_1 / sqrt(-l)));
	} else if (d <= -1.45e-173) {
		tmp = t_3;
	} else if (d <= 1.05e-293) {
		tmp = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 + ((h / l) * (-0.5 * pow(((D * M) / (d * 2.0)), 2.0))));
	} else if (d <= 3.85e+93) {
		tmp = t_4 * (t_2 * (sqrt(d) / sqrt(l)));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	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(D * Float64(Float64(M / d) * 0.5))
	t_1 = sqrt(Float64(-d))
	t_2 = fma(-0.5, Float64(t_0 / Float64(l / Float64(h * t_0))), 1.0)
	t_3 = Float64(Float64(t_1 / sqrt(Float64(-h))) * Float64(sqrt(Float64(d / l)) * t_2))
	t_4 = sqrt(Float64(d / h))
	tmp = 0.0
	if (d <= -4e+107)
		tmp = t_3;
	elseif (d <= -9.8e+57)
		tmp = Float64(t_4 * Float64(t_2 * Float64(t_1 / sqrt(Float64(-l)))));
	elseif (d <= -1.45e-173)
		tmp = t_3;
	elseif (d <= 1.05e-293)
		tmp = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(D * M) / Float64(d * 2.0)) ^ 2.0)))));
	elseif (d <= 3.85e+93)
		tmp = Float64(t_4 * Float64(t_2 * Float64(sqrt(d) / sqrt(l))));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	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[(D * N[(N[(M / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$2 = N[(-0.5 * N[(t$95$0 / N[(l / N[(h * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -4e+107], t$95$3, If[LessEqual[d, -9.8e+57], N[(t$95$4 * N[(t$95$2 * N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.45e-173], t$95$3, If[LessEqual[d, 1.05e-293], 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[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(D * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.85e+93], N[(t$95$4 * N[(t$95$2 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -3.9999999999999999e107 or -9.7999999999999998e57 < d < -1.4499999999999999e-173

   1. Initial program 63.4%

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

   2. Simplified 62.6%

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

      [Start] 63.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 63.1	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 63.1	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 63.1	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 63.1	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 63.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      sub-neg [=>] 63.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]
      +-commutative [=>] 63.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]
      associate-*l* [=>] 63.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) + 1\right)\right) \]
      distribute-lft-neg-in [=>] 63.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(-\frac{1}{2}\right) \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)} + 1\right)\right) \]
      fma-def [=>] 63.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(-\frac{1}{2}, {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}, 1\right)}\right) \]

   3. Applied egg-rr 68.3%

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

      [Start] 62.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}, 1\right)\right) \]
      associate-*r/ [=>] 63.9	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \color{blue}{\frac{{\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot h}{\ell}}, 1\right)\right) \]
      unpow2 [=>] 63.9	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{\color{blue}{\left(\left(D \cdot \frac{\frac{M}{d}}{2}\right) \cdot \left(D \cdot \frac{\frac{M}{d}}{2}\right)\right)} \cdot h}{\ell}, 1\right)\right) \]
      associate-*l* [=>] 66.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{\color{blue}{\left(D \cdot \frac{\frac{M}{d}}{2}\right) \cdot \left(\left(D \cdot \frac{\frac{M}{d}}{2}\right) \cdot h\right)}}{\ell}, 1\right)\right) \]
      associate-/l* [=>] 68.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \color{blue}{\frac{D \cdot \frac{\frac{M}{d}}{2}}{\frac{\ell}{\left(D \cdot \frac{\frac{M}{d}}{2}\right) \cdot h}}}, 1\right)\right) \]
      div-inv [=>] 68.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{1}{2}\right)}}{\frac{\ell}{\left(D \cdot \frac{\frac{M}{d}}{2}\right) \cdot h}}, 1\right)\right) \]
      metadata-eval [=>] 68.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot \color{blue}{0.5}\right)}{\frac{\ell}{\left(D \cdot \frac{\frac{M}{d}}{2}\right) \cdot h}}, 1\right)\right) \]
      div-inv [=>] 68.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\left(D \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{1}{2}\right)}\right) \cdot h}}, 1\right)\right) \]
      metadata-eval [=>] 68.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\left(D \cdot \left(\frac{M}{d} \cdot \color{blue}{0.5}\right)\right) \cdot h}}, 1\right)\right) \]

   4. Applied egg-rr 79.9%

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

      [Start] 68.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right) \cdot h}}, 1\right)\right) \]
      frac-2neg [=>] 68.3	\[ \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right) \cdot h}}, 1\right)\right) \]
      sqrt-div [=>] 79.9	\[ \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right) \cdot h}}, 1\right)\right) \]

   if -3.9999999999999999e107 < d < -9.7999999999999998e57

   1. Initial program 71.5%

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

   2. Simplified 70.8%

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

      [Start] 71.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 71.5	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 71.5	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 71.5	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 71.5	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 71.5	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      sub-neg [=>] 71.5	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]
      +-commutative [=>] 71.5	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]
      associate-*l* [=>] 71.5	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) + 1\right)\right) \]
      distribute-lft-neg-in [=>] 71.5	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(-\frac{1}{2}\right) \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)} + 1\right)\right) \]
      fma-def [=>] 71.5	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(-\frac{1}{2}, {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}, 1\right)}\right) \]

   3. Applied egg-rr 76.1%

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

      [Start] 70.8	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}, 1\right)\right) \]
      associate-*r/ [=>] 72.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \color{blue}{\frac{{\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot h}{\ell}}, 1\right)\right) \]
      unpow2 [=>] 72.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{\color{blue}{\left(\left(D \cdot \frac{\frac{M}{d}}{2}\right) \cdot \left(D \cdot \frac{\frac{M}{d}}{2}\right)\right)} \cdot h}{\ell}, 1\right)\right) \]
      associate-*l* [=>] 74.5	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{\color{blue}{\left(D \cdot \frac{\frac{M}{d}}{2}\right) \cdot \left(\left(D \cdot \frac{\frac{M}{d}}{2}\right) \cdot h\right)}}{\ell}, 1\right)\right) \]
      associate-/l* [=>] 76.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \color{blue}{\frac{D \cdot \frac{\frac{M}{d}}{2}}{\frac{\ell}{\left(D \cdot \frac{\frac{M}{d}}{2}\right) \cdot h}}}, 1\right)\right) \]
      div-inv [=>] 76.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{1}{2}\right)}}{\frac{\ell}{\left(D \cdot \frac{\frac{M}{d}}{2}\right) \cdot h}}, 1\right)\right) \]
      metadata-eval [=>] 76.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot \color{blue}{0.5}\right)}{\frac{\ell}{\left(D \cdot \frac{\frac{M}{d}}{2}\right) \cdot h}}, 1\right)\right) \]
      div-inv [=>] 76.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\left(D \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{1}{2}\right)}\right) \cdot h}}, 1\right)\right) \]
      metadata-eval [=>] 76.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\left(D \cdot \left(\frac{M}{d} \cdot \color{blue}{0.5}\right)\right) \cdot h}}, 1\right)\right) \]

   4. Applied egg-rr 81.6%

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

      [Start] 76.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right) \cdot h}}, 1\right)\right) \]
      frac-2neg [=>] 76.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\color{blue}{\frac{-d}{-\ell}}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right) \cdot h}}, 1\right)\right) \]
      sqrt-div [=>] 81.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right) \cdot h}}, 1\right)\right) \]

   if -1.4499999999999999e-173 < d < 1.05000000000000003e-293

   1. Initial program 33.9%

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

   if 1.05000000000000003e-293 < d < 3.85000000000000002e93

   1. Initial program 61.1%

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

   2. Simplified 59.5%

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

      [Start] 61.1	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 60.7	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 60.7	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 60.7	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 60.7	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 60.7	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      sub-neg [=>] 60.7	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]
      +-commutative [=>] 60.7	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]
      associate-*l* [=>] 60.7	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) + 1\right)\right) \]
      distribute-lft-neg-in [=>] 60.7	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(-\frac{1}{2}\right) \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)} + 1\right)\right) \]
      fma-def [=>] 60.7	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(-\frac{1}{2}, {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}, 1\right)}\right) \]

   3. Applied egg-rr 65.8%

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

      [Start] 59.5	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}, 1\right)\right) \]
      associate-*r/ [=>] 60.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \color{blue}{\frac{{\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot h}{\ell}}, 1\right)\right) \]
      unpow2 [=>] 60.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{\color{blue}{\left(\left(D \cdot \frac{\frac{M}{d}}{2}\right) \cdot \left(D \cdot \frac{\frac{M}{d}}{2}\right)\right)} \cdot h}{\ell}, 1\right)\right) \]
      associate-*l* [=>] 63.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{\color{blue}{\left(D \cdot \frac{\frac{M}{d}}{2}\right) \cdot \left(\left(D \cdot \frac{\frac{M}{d}}{2}\right) \cdot h\right)}}{\ell}, 1\right)\right) \]
      associate-/l* [=>] 65.8	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \color{blue}{\frac{D \cdot \frac{\frac{M}{d}}{2}}{\frac{\ell}{\left(D \cdot \frac{\frac{M}{d}}{2}\right) \cdot h}}}, 1\right)\right) \]
      div-inv [=>] 65.8	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{1}{2}\right)}}{\frac{\ell}{\left(D \cdot \frac{\frac{M}{d}}{2}\right) \cdot h}}, 1\right)\right) \]
      metadata-eval [=>] 65.8	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot \color{blue}{0.5}\right)}{\frac{\ell}{\left(D \cdot \frac{\frac{M}{d}}{2}\right) \cdot h}}, 1\right)\right) \]
      div-inv [=>] 65.8	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\left(D \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{1}{2}\right)}\right) \cdot h}}, 1\right)\right) \]
      metadata-eval [=>] 65.8	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\left(D \cdot \left(\frac{M}{d} \cdot \color{blue}{0.5}\right)\right) \cdot h}}, 1\right)\right) \]

   4. Applied egg-rr 74.4%

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

      [Start] 65.8	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right) \cdot h}}, 1\right)\right) \]
      sqrt-div [=>] 74.4	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right) \cdot h}}, 1\right)\right) \]
      div-inv [=>] 74.4	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\left(\sqrt{d} \cdot \frac{1}{\sqrt{\ell}}\right)} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right) \cdot h}}, 1\right)\right) \]

   5. Simplified 74.4%

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

      [Start] 74.4	\[ \sqrt{\frac{d}{h}} \cdot \left(\left(\sqrt{d} \cdot \frac{1}{\sqrt{\ell}}\right) \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right) \cdot h}}, 1\right)\right) \]
      associate-*r/ [=>] 74.4	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{\sqrt{d} \cdot 1}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right) \cdot h}}, 1\right)\right) \]
      *-rgt-identity [=>] 74.4	\[ \sqrt{\frac{d}{h}} \cdot \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right) \cdot h}}, 1\right)\right) \]

   if 3.85000000000000002e93 < d

   1. Initial program 57.4%

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

   2. Taylor expanded in d around inf 74.2%

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

   3. Applied egg-rr 33.0%

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

      [Start] 74.2	\[ \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]
      expm1-log1p-u [=>] 72.4	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot d \]
      expm1-udef [=>] 32.6	\[ \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)} - 1\right)} \cdot d \]
      inv-pow [=>] 32.6	\[ \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}}\right)} - 1\right) \cdot d \]
      metadata-eval [<=] 32.6	\[ \left(e^{\mathsf{log1p}\left(\sqrt{{\left(\ell \cdot h\right)}^{\color{blue}{\left(-1\right)}}}\right)} - 1\right) \cdot d \]
      sqrt-pow1 [=>] 33.0	\[ \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \cdot d \]
      metadata-eval [=>] 33.0	\[ \left(e^{\mathsf{log1p}\left({\left(\ell \cdot h\right)}^{\left(\frac{\color{blue}{-1}}{2}\right)}\right)} - 1\right) \cdot d \]
      metadata-eval [=>] 33.0	\[ \left(e^{\mathsf{log1p}\left({\left(\ell \cdot h\right)}^{\color{blue}{-0.5}}\right)} - 1\right) \cdot d \]

   4. Simplified 74.5%

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

      [Start] 33.0	\[ \left(e^{\mathsf{log1p}\left({\left(\ell \cdot h\right)}^{-0.5}\right)} - 1\right) \cdot d \]
      expm1-def [=>] 72.8	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\ell \cdot h\right)}^{-0.5}\right)\right)} \cdot d \]
      expm1-log1p [=>] 74.5	\[ \color{blue}{{\left(\ell \cdot h\right)}^{-0.5}} \cdot d \]

   5. Applied egg-rr 88.5%

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

      [Start] 74.5	\[ {\left(\ell \cdot h\right)}^{-0.5} \cdot d \]
      unpow-prod-down [=>] 88.5	\[ \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \cdot d \]

3. Recombined 5 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{+107}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{h \cdot \left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}}, 1\right)\right)\\ \mathbf{elif}\;d \leq -9.8 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{h \cdot \left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}}, 1\right) \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-173}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{h \cdot \left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}}, 1\right)\right)\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-293}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 3.85 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{h \cdot \left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}}, 1\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	70.0%
Cost	28248

\[\begin{array}{l} t_0 := D \cdot \left(\frac{M}{d} \cdot 0.5\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{-d}\\ t_3 := \mathsf{fma}\left(-0.5, \frac{t_0}{\frac{\ell}{h \cdot t_0}}, 1\right)\\ t_4 := \sqrt{\frac{d}{\ell}}\\ t_5 := t_4 \cdot t_1\\ \mathbf{if}\;h \leq -1.1 \cdot 10^{-135}:\\ \;\;\;\;t_1 \cdot \left(t_3 \cdot \frac{t_2}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;h \leq -1.5 \cdot 10^{-193}:\\ \;\;\;\;\left(\frac{t_2}{\sqrt{-h}} \cdot t_4\right) \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\right)\\ \mathbf{elif}\;h \leq -5.2 \cdot 10^{-258}:\\ \;\;\;\;t_5 \cdot \left(1 + 0.5 \cdot \left(\frac{M \cdot \frac{D}{d}}{\frac{d}{D} \cdot \frac{\ell}{h \cdot M}} \cdot -0.25\right)\right)\\ \mathbf{elif}\;h \leq -3.3 \cdot 10^{-304}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;h \leq 7.4 \cdot 10^{-308}:\\ \;\;\;\;t_5 \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \frac{h \cdot \left(\frac{D}{d} \cdot \left(M \cdot \frac{M}{\ell}\right)\right)}{\frac{d}{D}}\right)\right)\\ \mathbf{elif}\;h \leq 8 \cdot 10^{+218}:\\ \;\;\;\;t_1 \cdot \left(t_3 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{h} \cdot \left(\left(D \cdot M\right) \cdot \left(M \cdot \frac{D \cdot -0.125}{d}\right)\right)}{{\ell}^{0.75}}}{{\ell}^{0.75}}\\ \end{array} \]

Alternative 2
Accuracy	72.9%
Cost	27984

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := D \cdot \left(\frac{M}{d} \cdot 0.5\right)\\ t_2 := \mathsf{fma}\left(-0.5, \frac{t_1}{\frac{\ell}{h \cdot t_1}}, 1\right)\\ t_3 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -2.35 \cdot 10^{-94}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t_0\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -3.4 \cdot 10^{-173}:\\ \;\;\;\;\left(t_0 \cdot t_2\right) \cdot t_3\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-295}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{+92}:\\ \;\;\;\;t_3 \cdot \left(t_2 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 3
Accuracy	70.6%
Cost	27540

\[\begin{array}{l} t_0 := D \cdot \left(\frac{M}{d} \cdot 0.5\right)\\ t_1 := \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{t_0}{\frac{\ell}{h \cdot t_0}}, 1\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -1.1 \cdot 10^{+96}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-173}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{-293}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{-221}:\\ \;\;\;\;M \cdot \frac{\left(D \cdot \left(D \cdot M\right)\right) \cdot \left(\sqrt{h} \cdot \left(-0.125 \cdot {\ell}^{-1.5}\right)\right)}{d}\\ \mathbf{elif}\;d \leq 5 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h}}\\ \mathbf{elif}\;d \leq 1.32 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 4
Accuracy	70.5%
Cost	27540

\[\begin{array}{l} t_0 := D \cdot \left(\frac{M}{d} \cdot 0.5\right)\\ t_1 := \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{t_0}{\frac{\ell}{h \cdot t_0}}, 1\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -2.2 \cdot 10^{+95}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-173}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 1.3 \cdot 10^{-294}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{-223}:\\ \;\;\;\;M \cdot \frac{\left(D \cdot \left(D \cdot M\right)\right) \cdot \left(\sqrt{h} \cdot \left(-0.125 \cdot {\ell}^{-1.5}\right)\right)}{d}\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{d \cdot h}{\frac{\ell}{-0.5}}, {\left(\frac{D}{\frac{d}{M \cdot 0.5}}\right)}^{2}, d\right)}{\sqrt{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;d \leq 9.2 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 5
Accuracy	69.9%
Cost	27408

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := D \cdot \left(\frac{M}{d} \cdot 0.5\right)\\ \mathbf{if}\;d \leq -1.4 \cdot 10^{-95}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t_0\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-173}:\\ \;\;\;\;\left(t_0 \cdot \mathsf{fma}\left(-0.5, \frac{t_1}{\frac{\ell}{h \cdot t_1}}, 1\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{-305}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{-226}:\\ \;\;\;\;M \cdot \frac{\left(D \cdot \left(D \cdot M\right)\right) \cdot \left(\sqrt{h} \cdot \left(-0.125 \cdot {\ell}^{-1.5}\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{h}}\\ \end{array} \]

Alternative 6
Accuracy	69.6%
Cost	21584

\[\begin{array}{l} t_0 := D \cdot \left(\frac{M}{d} \cdot 0.5\right)\\ t_1 := \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{t_0}{\frac{\ell}{h \cdot t_0}}, 1\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -1.3 \cdot 10^{+96}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq 1.32 \cdot 10^{-301}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-225}:\\ \;\;\;\;M \cdot \frac{\left(D \cdot \left(D \cdot M\right)\right) \cdot \left(\sqrt{h} \cdot \left(-0.125 \cdot {\ell}^{-1.5}\right)\right)}{d}\\ \mathbf{elif}\;d \leq 4.9 \cdot 10^{+73}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 7
Accuracy	69.7%
Cost	21584

\[\begin{array}{l} t_0 := D \cdot \left(\frac{M}{d} \cdot 0.5\right)\\ t_1 := \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{t_0}{\frac{\ell}{h \cdot t_0}}, 1\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -4.2 \cdot 10^{+95}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq -1.9 \cdot 10^{-173}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{-295}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{+79}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 8
Accuracy	65.5%
Cost	21400

\[\begin{array}{l} t_0 := d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ t_1 := -0.5 \cdot \frac{h}{\ell}\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := \sqrt{\frac{d}{\ell}}\\ t_4 := t_2 \cdot \left(t_3 \cdot \left(1 + t_1 \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{if}\;h \leq -3.8 \cdot 10^{+258}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;h \leq -9 \cdot 10^{+127}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq -1.6 \cdot 10^{-78}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;h \leq -3.3 \cdot 10^{-193}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq -8.1 \cdot 10^{-258}:\\ \;\;\;\;\left(t_3 \cdot t_2\right) \cdot \left(1 + 0.5 \cdot \left(\frac{M \cdot \frac{D}{d}}{\frac{d}{D} \cdot \frac{\ell}{h \cdot M}} \cdot -0.25\right)\right)\\ \mathbf{elif}\;h \leq -3.3 \cdot 10^{-304}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot t_1\right)\\ \end{array} \]

Alternative 9
Accuracy	67.0%
Cost	21268

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{1}{h \cdot \ell}}\\ t_3 := {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\\ \mathbf{if}\;d \leq -1.65 \cdot 10^{+46}:\\ \;\;\;\;d \cdot \left(-t_2\right)\\ \mathbf{elif}\;d \leq -2.76 \cdot 10^{-70}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \mathsf{fma}\left(-0.125, \left(M \cdot \left(h \cdot M\right)\right) \cdot \left(\frac{D}{d \cdot d} \cdot \frac{D}{\ell}\right), 1\right)\right)\\ \mathbf{elif}\;d \leq -3.1 \cdot 10^{-264}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{0.125}{\frac{d}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}} - d \cdot t_2\\ \mathbf{elif}\;d \leq 1.25 \cdot 10^{-307}:\\ \;\;\;\;\left(t_1 \cdot t_0\right) \cdot \left(1 + -0.5 \cdot \frac{t_3}{\frac{\ell}{h}}\right)\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-225}:\\ \;\;\;\;M \cdot \frac{\left(D \cdot \left(D \cdot M\right)\right) \cdot \left(\sqrt{h} \cdot \left(-0.125 \cdot {\ell}^{-1.5}\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + t_3 \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \end{array} \]

Alternative 10
Accuracy	68.5%
Cost	21004

\[\begin{array}{l} \mathbf{if}\;d \leq -2.05 \cdot 10^{+45}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq 7.8 \cdot 10^{-292}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + 0.5 \cdot \left(\frac{M \cdot \frac{D}{d}}{\frac{d}{D} \cdot \frac{\ell}{h \cdot M}} \cdot -0.25\right)\right)\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{-225}:\\ \;\;\;\;M \cdot \frac{\left(D \cdot \left(D \cdot M\right)\right) \cdot \left(\sqrt{h} \cdot \left(-0.125 \cdot {\ell}^{-1.5}\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \end{array} \]

Alternative 11
Accuracy	63.0%
Cost	15184

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ t_1 := d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{if}\;d \leq -2.2 \cdot 10^{+44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -6.5 \cdot 10^{-68}:\\ \;\;\;\;t_0 \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \frac{D \cdot D}{d \cdot \left(d \cdot \frac{\ell}{M \cdot \left(h \cdot M\right)}\right)}\right)\right)\\ \mathbf{elif}\;d \leq -1.02 \cdot 10^{-144}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 12
Accuracy	63.3%
Cost	15184

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ t_1 := d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{if}\;d \leq -9.5 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -2.1 \cdot 10^{-67}:\\ \;\;\;\;t_0 \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \frac{D \cdot D}{d \cdot \left(d \cdot \frac{\ell}{M \cdot \left(h \cdot M\right)}\right)}\right)\right)\\ \mathbf{elif}\;d \leq -1.5 \cdot 10^{-141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -2.6 \cdot 10^{-300}:\\ \;\;\;\;t_0 \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \frac{M \cdot M}{\frac{d}{D} \cdot \left(\frac{\ell}{h} \cdot \frac{d}{D}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 13
Accuracy	66.2%
Cost	15184

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -2.8 \cdot 10^{+41}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-291}:\\ \;\;\;\;t_0 \cdot \left(1 + 0.5 \cdot \left(\frac{M \cdot \frac{D}{d}}{\frac{d}{D} \cdot \frac{\ell}{h \cdot M}} \cdot -0.25\right)\right)\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{-225}:\\ \;\;\;\;M \cdot \frac{\left(D \cdot \left(D \cdot M\right)\right) \cdot \left(\sqrt{h} \cdot \left(-0.125 \cdot {\ell}^{-1.5}\right)\right)}{d}\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{+71}:\\ \;\;\;\;t_0 \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \frac{h \cdot \left(\frac{D}{d} \cdot \left(M \cdot \frac{M}{\ell}\right)\right)}{\frac{d}{D}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 14
Accuracy	62.5%
Cost	14920

\[\begin{array}{l} \mathbf{if}\;d \leq -1.05 \cdot 10^{+41}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq -2.3 \cdot 10^{-32}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell \cdot \frac{h}{d}}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 15
Accuracy	62.5%
Cost	14920

\[\begin{array}{l} \mathbf{if}\;d \leq -1.9 \cdot 10^{+39}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq -1.12 \cdot 10^{-39}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)\right)\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell \cdot \frac{h}{d}}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 16
Accuracy	64.7%
Cost	14920

\[\begin{array}{l} \mathbf{if}\;d \leq -1.55 \cdot 10^{+46}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{+71}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \frac{h \cdot \left(\frac{D}{d} \cdot \left(M \cdot \frac{M}{\ell}\right)\right)}{\frac{d}{D}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 17
Accuracy	67.4%
Cost	14920

\[\begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{+41}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq 5.4 \cdot 10^{+29}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + 0.5 \cdot \left(\frac{M \cdot \frac{D}{d}}{\frac{d}{D} \cdot \frac{\ell}{h \cdot M}} \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 18
Accuracy	61.5%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;\ell \leq 9.2 \cdot 10^{-241}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 19
Accuracy	61.5%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;\ell \leq 9.2 \cdot 10^{-241}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 20
Accuracy	56.0%
Cost	7108

\[\begin{array}{l} \mathbf{if}\;d \leq -1.5 \cdot 10^{-202}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell} \cdot \frac{1}{h}}\\ \end{array} \]

Alternative 21
Accuracy	56.0%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;d \leq -1.5 \cdot 10^{-202}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]

Alternative 22
Accuracy	46.5%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\ell \leq -8.5 \cdot 10^{-271}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 23
Accuracy	47.8%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{-167}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 24
Accuracy	47.8%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{-167}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]

Alternative 25
Accuracy	32.0%
Cost	6720

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

Reproduce

herbie shell --seed 2023133 
(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)))))