Henrywood and Agarwal, Equation (12)

Average Error: 41.71% → 25.14%

Time: 58.1s

Precision: binary64

Cost: 33800

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := t_1 \cdot \frac{t_0}{\sqrt{-\ell}}\\ \mathbf{if}\;\ell \leq -2.4 \cdot 10^{+224}:\\ \;\;\;\;t_2 \cdot \left(1 + 0.5 \cdot \frac{\left(D \cdot \left(h \cdot \left(\frac{M}{d} \cdot \left(D \cdot M\right)\right)\right)\right) \cdot \frac{-0.25}{d}}{\ell}\right)\\ \mathbf{elif}\;\ell \leq -9.2 \cdot 10^{-35}:\\ \;\;\;\;\frac{t_0}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}, \frac{-0.5}{\frac{\ell}{h}}, 1\right)\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_2 \cdot \left(1 + -0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{-98}:\\ \;\;\;\;\left(t_1 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 + 0.5 \cdot \frac{\frac{0.25}{d} \cdot \left(M \cdot \left(\frac{D}{-d} \cdot \left(h \cdot \left(D \cdot M\right)\right)\right)\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{0.5 \cdot M}{d}\right)}^{2}\right)\right)\\ \end{array} \]

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 (sqrt (- d)))
        (t_1 (sqrt (/ d h)))
        (t_2 (* t_1 (/ t_0 (sqrt (- l))))))
   (if (<= l -2.4e+224)
     (*
      t_2
      (+ 1.0 (* 0.5 (/ (* (* D (* h (* (/ M d) (* D M)))) (/ -0.25 d)) l))))
     (if (<= l -9.2e-35)
       (*
        (/ t_0 (sqrt (- h)))
        (*
         (sqrt (/ d l))
         (fma (pow (* M (/ D (* d 2.0))) 2.0) (/ -0.5 (/ l h)) 1.0)))
       (if (<= l -5e-310)
         (* t_2 (+ 1.0 (* -0.5 (/ (* h (pow (* M (* 0.5 (/ D d))) 2.0)) l))))
         (if (<= l 3e-98)
           (*
            (* t_1 (/ (sqrt d) (sqrt l)))
            (+
             1.0
             (* 0.5 (/ (* (/ 0.25 d) (* M (* (/ D (- d)) (* h (* D M))))) l))))
           (*
            (/ d (* (sqrt l) (sqrt h)))
            (+ 1.0 (* -0.5 (* (/ h l) (pow (* D (/ (* 0.5 M) d)) 2.0)))))))))))

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 = sqrt((d / h));
	double t_2 = t_1 * (t_0 / sqrt(-l));
	double tmp;
	if (l <= -2.4e+224) {
		tmp = t_2 * (1.0 + (0.5 * (((D * (h * ((M / d) * (D * M)))) * (-0.25 / d)) / l)));
	} else if (l <= -9.2e-35) {
		tmp = (t_0 / sqrt(-h)) * (sqrt((d / l)) * fma(pow((M * (D / (d * 2.0))), 2.0), (-0.5 / (l / h)), 1.0));
	} else if (l <= -5e-310) {
		tmp = t_2 * (1.0 + (-0.5 * ((h * pow((M * (0.5 * (D / d))), 2.0)) / l)));
	} else if (l <= 3e-98) {
		tmp = (t_1 * (sqrt(d) / sqrt(l))) * (1.0 + (0.5 * (((0.25 / d) * (M * ((D / -d) * (h * (D * M))))) / l)));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + (-0.5 * ((h / l) * pow((D * ((0.5 * M) / d)), 2.0))));
	}
	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 = sqrt(Float64(d / h))
	t_2 = Float64(t_1 * Float64(t_0 / sqrt(Float64(-l))))
	tmp = 0.0
	if (l <= -2.4e+224)
		tmp = Float64(t_2 * Float64(1.0 + Float64(0.5 * Float64(Float64(Float64(D * Float64(h * Float64(Float64(M / d) * Float64(D * M)))) * Float64(-0.25 / d)) / l))));
	elseif (l <= -9.2e-35)
		tmp = Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(sqrt(Float64(d / l)) * fma((Float64(M * Float64(D / Float64(d * 2.0))) ^ 2.0), Float64(-0.5 / Float64(l / h)), 1.0)));
	elseif (l <= -5e-310)
		tmp = Float64(t_2 * Float64(1.0 + Float64(-0.5 * Float64(Float64(h * (Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0)) / l))));
	elseif (l <= 3e-98)
		tmp = Float64(Float64(t_1 * Float64(sqrt(d) / sqrt(l))) * Float64(1.0 + Float64(0.5 * Float64(Float64(Float64(0.25 / d) * Float64(M * Float64(Float64(D / Float64(-d)) * Float64(h * Float64(D * M))))) / l))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(D * Float64(Float64(0.5 * M) / d)) ^ 2.0)))));
	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[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.4e+224], N[(t$95$2 * N[(1.0 + N[(0.5 * N[(N[(N[(D * N[(h * N[(N[(M / d), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.25 / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -9.2e-35], N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(M * N[(D / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 / N[(l / h), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(t$95$2 * N[(1.0 + N[(-0.5 * N[(N[(h * N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3e-98], N[(N[(t$95$1 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(0.5 * N[(N[(N[(0.25 / d), $MachinePrecision] * N[(M * N[(N[(D / (-d)), $MachinePrecision] * N[(h * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(N[(0.5 * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -2.40000000000000001e224

   1. Initial program 52.91

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

   2. Simplified 51.73

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

      [Start] 52.91	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 52.91	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 52.91	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 52.91	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 52.91	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 52.91	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 52.91	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 51.73	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 53.89

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

   4. Taylor expanded in M around 0 73.24

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}}{\ell}\right) \]

   5. Simplified 62.39

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{\frac{0.25}{d} \cdot \frac{\left(\left(D \cdot \left(D \cdot h\right)\right) \cdot M\right) \cdot M}{d}}}{\ell}\right) \]
      Proof

      [Start] 73.24	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\ell}\right) \]
      associate-*r/ [=>] 73.24	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{\frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}}}{\ell}\right) \]
      unpow2 [=>] 73.24	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{d \cdot d}}}{\ell}\right) \]
      times-frac [=>] 68.45	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{\frac{0.25}{d} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d}}}{\ell}\right) \]
      *-commutative [=>] 68.45	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \frac{{D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{d}}{\ell}\right) \]
      associate-*r* [=>] 67.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \frac{\color{blue}{\left({D}^{2} \cdot h\right) \cdot {M}^{2}}}{d}}{\ell}\right) \]
      unpow2 [=>] 67.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \frac{\left({D}^{2} \cdot h\right) \cdot \color{blue}{\left(M \cdot M\right)}}{d}}{\ell}\right) \]
      associate-*r* [=>] 63.25	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \frac{\color{blue}{\left(\left({D}^{2} \cdot h\right) \cdot M\right) \cdot M}}{d}}{\ell}\right) \]
      unpow2 [=>] 63.25	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \frac{\left(\left(\color{blue}{\left(D \cdot D\right)} \cdot h\right) \cdot M\right) \cdot M}{d}}{\ell}\right) \]
      associate-*l* [=>] 62.39	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \frac{\left(\color{blue}{\left(D \cdot \left(D \cdot h\right)\right)} \cdot M\right) \cdot M}{d}}{\ell}\right) \]

   6. Taylor expanded in D around 0 68.45

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \color{blue}{\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{d}}}{\ell}\right) \]

   7. Simplified 57.72

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \color{blue}{\left(D \cdot \left(h \cdot \left(\frac{M}{d} \cdot \left(M \cdot D\right)\right)\right)\right)}}{\ell}\right) \]
      Proof

      [Start] 68.45	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{d}}{\ell}\right) \]
      associate-*r* [=>] 67.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \frac{\color{blue}{\left({D}^{2} \cdot h\right) \cdot {M}^{2}}}{d}}{\ell}\right) \]
      unpow2 [=>] 67.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot h\right) \cdot {M}^{2}}{d}}{\ell}\right) \]
      associate-*r* [<=] 66.68	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \frac{\color{blue}{\left(D \cdot \left(D \cdot h\right)\right)} \cdot {M}^{2}}{d}}{\ell}\right) \]
      unpow2 [=>] 66.68	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \frac{\left(D \cdot \left(D \cdot h\right)\right) \cdot \color{blue}{\left(M \cdot M\right)}}{d}}{\ell}\right) \]
      associate-*r/ [<=] 67.21	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \color{blue}{\left(\left(D \cdot \left(D \cdot h\right)\right) \cdot \frac{M \cdot M}{d}\right)}}{\ell}\right) \]
      associate-*l* [=>] 63.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \color{blue}{\left(D \cdot \left(\left(D \cdot h\right) \cdot \frac{M \cdot M}{d}\right)\right)}}{\ell}\right) \]
      *-commutative [=>] 63.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \left(D \cdot \left(\color{blue}{\left(h \cdot D\right)} \cdot \frac{M \cdot M}{d}\right)\right)}{\ell}\right) \]
      associate-*l* [=>] 61.03	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \left(D \cdot \color{blue}{\left(h \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right)}\right)}{\ell}\right) \]
      *-commutative [<=] 61.03	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \left(D \cdot \left(h \cdot \color{blue}{\left(\frac{M \cdot M}{d} \cdot D\right)}\right)\right)}{\ell}\right) \]
      associate-/l* [=>] 58.86	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \left(D \cdot \left(h \cdot \left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot D\right)\right)\right)}{\ell}\right) \]
      associate-/r/ [=>] 58.86	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \left(D \cdot \left(h \cdot \left(\color{blue}{\left(\frac{M}{d} \cdot M\right)} \cdot D\right)\right)\right)}{\ell}\right) \]
      associate-*l* [=>] 57.72	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \left(D \cdot \left(h \cdot \color{blue}{\left(\frac{M}{d} \cdot \left(M \cdot D\right)\right)}\right)\right)}{\ell}\right) \]

   8. Applied egg-rr 45.74

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \left(D \cdot \left(h \cdot \left(\frac{M}{d} \cdot \left(M \cdot D\right)\right)\right)\right)}{\ell}\right) \]

   if -2.40000000000000001e224 < l < -9.1999999999999996e-35

   1. Initial program 36.19

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

   2. Simplified 37.01

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

      [Start] 36.19	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 36.34	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 36.34	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 36.35	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 36.35	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 36.35	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      cancel-sign-sub-inv [=>] 36.35	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]
      +-commutative [=>] 36.35	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell} + 1\right)}\right) \]
      *-commutative [=>] 36.35	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}\right) \cdot \frac{h}{\ell} + 1\right)\right) \]
      distribute-rgt-neg-in [=>] 36.35	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(-\frac{1}{2}\right)\right)} \cdot \frac{h}{\ell} + 1\right)\right) \]
      associate-*l* [=>] 36.35	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\left(-\frac{1}{2}\right) \cdot \frac{h}{\ell}\right)} + 1\right)\right) \]
      fma-def [=>] 36.35	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}, \left(-\frac{1}{2}\right) \cdot \frac{h}{\ell}, 1\right)}\right) \]

   3. Applied egg-rr 25.54

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

   if -9.1999999999999996e-35 < l < -4.999999999999985e-310

   1. Initial program 43.31

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

   2. Simplified 44.45

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

      [Start] 43.31	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 43.31	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 43.31	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 43.31	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 43.31	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 43.31	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 43.31	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 44.45	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 35

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

   4. Applied egg-rr 20.17

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

   if -4.999999999999985e-310 < l < 3e-98

   1. Initial program 51.03

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

   2. Simplified 51.9

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

      [Start] 51.03	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 51.03	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 51.03	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 51.03	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 51.03	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 51.03	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 51.03	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 51.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 43.58

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

   4. Taylor expanded in M around 0 66.61

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}}{\ell}\right) \]

   5. Simplified 55.89

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{\frac{0.25}{d} \cdot \frac{\left(\left(D \cdot \left(D \cdot h\right)\right) \cdot M\right) \cdot M}{d}}}{\ell}\right) \]
      Proof

      [Start] 66.61	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\ell}\right) \]
      associate-*r/ [=>] 66.61	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{\frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}}}{\ell}\right) \]
      unpow2 [=>] 66.61	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{d \cdot d}}}{\ell}\right) \]
      times-frac [=>] 60.59	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{\frac{0.25}{d} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d}}}{\ell}\right) \]
      *-commutative [=>] 60.59	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \frac{{D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{d}}{\ell}\right) \]
      associate-*r* [=>] 60.42	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \frac{\color{blue}{\left({D}^{2} \cdot h\right) \cdot {M}^{2}}}{d}}{\ell}\right) \]
      unpow2 [=>] 60.42	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \frac{\left({D}^{2} \cdot h\right) \cdot \color{blue}{\left(M \cdot M\right)}}{d}}{\ell}\right) \]
      associate-*r* [=>] 58.32	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \frac{\color{blue}{\left(\left({D}^{2} \cdot h\right) \cdot M\right) \cdot M}}{d}}{\ell}\right) \]
      unpow2 [=>] 58.32	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \frac{\left(\left(\color{blue}{\left(D \cdot D\right)} \cdot h\right) \cdot M\right) \cdot M}{d}}{\ell}\right) \]
      associate-*l* [=>] 55.89	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \frac{\left(\color{blue}{\left(D \cdot \left(D \cdot h\right)\right)} \cdot M\right) \cdot M}{d}}{\ell}\right) \]

   6. Applied egg-rr 34.14

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left(\sqrt{d} \cdot \frac{1}{\sqrt{\ell}}\right)}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \frac{\left(\left(D \cdot \left(D \cdot h\right)\right) \cdot M\right) \cdot M}{d}}{\ell}\right) \]

   7. Simplified 34.11

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \frac{\left(\left(D \cdot \left(D \cdot h\right)\right) \cdot M\right) \cdot M}{d}}{\ell}\right) \]
      Proof

      [Start] 34.14	\[ \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot \frac{1}{\sqrt{\ell}}\right)\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \frac{\left(\left(D \cdot \left(D \cdot h\right)\right) \cdot M\right) \cdot M}{d}}{\ell}\right) \]
      associate-*r/ [=>] 34.11	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{d} \cdot 1}{\sqrt{\ell}}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \frac{\left(\left(D \cdot \left(D \cdot h\right)\right) \cdot M\right) \cdot M}{d}}{\ell}\right) \]
      *-rgt-identity [=>] 34.11	\[ \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \frac{\left(\left(D \cdot \left(D \cdot h\right)\right) \cdot M\right) \cdot M}{d}}{\ell}\right) \]

   8. Applied egg-rr 29.93

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \color{blue}{\left(\frac{D \cdot \left(D \cdot \left(h \cdot M\right)\right)}{-d} \cdot \left(-M\right)\right)}}{\ell}\right) \]

   9. Simplified 22.1

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \color{blue}{\left(\left(\frac{D}{-d} \cdot \left(h \cdot \left(M \cdot D\right)\right)\right) \cdot \left(-M\right)\right)}}{\ell}\right) \]
      Proof

      [Start] 29.93	\[ \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \left(\frac{D \cdot \left(D \cdot \left(h \cdot M\right)\right)}{-d} \cdot \left(-M\right)\right)}{\ell}\right) \]
      associate-/l* [=>] 24.64	\[ \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \left(\color{blue}{\frac{D}{\frac{-d}{D \cdot \left(h \cdot M\right)}}} \cdot \left(-M\right)\right)}{\ell}\right) \]
      associate-/r/ [=>] 24.91	\[ \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \left(\color{blue}{\left(\frac{D}{-d} \cdot \left(D \cdot \left(h \cdot M\right)\right)\right)} \cdot \left(-M\right)\right)}{\ell}\right) \]
      *-commutative [=>] 24.91	\[ \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \left(\left(\frac{D}{-d} \cdot \color{blue}{\left(\left(h \cdot M\right) \cdot D\right)}\right) \cdot \left(-M\right)\right)}{\ell}\right) \]
      associate-*l* [=>] 22.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\frac{0.25}{d} \cdot \left(\left(\frac{D}{-d} \cdot \color{blue}{\left(h \cdot \left(M \cdot D\right)\right)}\right) \cdot \left(-M\right)\right)}{\ell}\right) \]

   if 3e-98 < l

   1. Initial program 39.45

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

   2. Simplified 39.94

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

      [Start] 39.45	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 39.45	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 39.45	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 39.45	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 39.45	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 39.45	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 39.45	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 39.94	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 41.35

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

   4. Applied egg-rr 21.67

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

   5. Simplified 22.61

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

      [Start] 21.67	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(\frac{h}{\frac{\ell}{{\left(\frac{0.5 \cdot M}{\frac{d}{D}}\right)}^{2}}} \cdot -0.5\right) \]
      *-lft-identity [<=] 21.67	\[ \color{blue}{1 \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} + \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(\frac{h}{\frac{\ell}{{\left(\frac{0.5 \cdot M}{\frac{d}{D}}\right)}^{2}}} \cdot -0.5\right) \]
      *-commutative [<=] 21.67	\[ 1 \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \color{blue}{\left(\frac{h}{\frac{\ell}{{\left(\frac{0.5 \cdot M}{\frac{d}{D}}\right)}^{2}}} \cdot -0.5\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]
      distribute-rgt-in [<=] 21.67	\[ \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \frac{h}{\frac{\ell}{{\left(\frac{0.5 \cdot M}{\frac{d}{D}}\right)}^{2}}} \cdot -0.5\right)} \]
      associate-/r/ [=>] 22.6	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \color{blue}{\left(\frac{h}{\ell} \cdot {\left(\frac{0.5 \cdot M}{\frac{d}{D}}\right)}^{2}\right)} \cdot -0.5\right) \]
      associate-/r/ [=>] 22.61	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot {\color{blue}{\left(\frac{0.5 \cdot M}{d} \cdot D\right)}}^{2}\right) \cdot -0.5\right) \]
      *-commutative [=>] 22.61	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot {\left(\frac{\color{blue}{M \cdot 0.5}}{d} \cdot D\right)}^{2}\right) \cdot -0.5\right) \]

3. Recombined 5 regimes into one program.

4. Final simplification 25.14

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{+224}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 + 0.5 \cdot \frac{\left(D \cdot \left(h \cdot \left(\frac{M}{d} \cdot \left(D \cdot M\right)\right)\right)\right) \cdot \frac{-0.25}{d}}{\ell}\right)\\ \mathbf{elif}\;\ell \leq -9.2 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}, \frac{-0.5}{\frac{\ell}{h}}, 1\right)\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 + -0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{-98}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 + 0.5 \cdot \frac{\frac{0.25}{d} \cdot \left(M \cdot \left(\frac{D}{-d} \cdot \left(h \cdot \left(D \cdot M\right)\right)\right)\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{0.5 \cdot M}{d}\right)}^{2}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	25.19%
Cost	27660

\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := t_1 \cdot \frac{t_0}{\sqrt{-\ell}}\\ \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+224}:\\ \;\;\;\;t_2 \cdot \left(1 + 0.5 \cdot \frac{\left(D \cdot \left(h \cdot \left(\frac{M}{d} \cdot \left(D \cdot M\right)\right)\right)\right) \cdot \frac{-0.25}{d}}{\ell}\right)\\ \mathbf{elif}\;\ell \leq -1.45 \cdot 10^{-34}:\\ \;\;\;\;\left(\frac{t_0}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_2 \cdot \left(1 + -0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 1.38 \cdot 10^{-98}:\\ \;\;\;\;\left(t_1 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 + 0.5 \cdot \frac{\frac{0.25}{d} \cdot \left(M \cdot \left(\frac{D}{-d} \cdot \left(h \cdot \left(D \cdot M\right)\right)\right)\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{0.5 \cdot M}{d}\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 2
Error	27.28%
Cost	27528

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;\ell \leq -1.05 \cdot 10^{-33}:\\ \;\;\;\;\frac{D}{\frac{\frac{d}{M}}{D \cdot M}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(t_0 \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 + -0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 1.42 \cdot 10^{-98}:\\ \;\;\;\;\left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 + 0.5 \cdot \frac{\frac{0.25}{d} \cdot \left(M \cdot \left(\frac{D}{-d} \cdot \left(h \cdot \left(D \cdot M\right)\right)\right)\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{0.5 \cdot M}{d}\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 3
Error	31.26%
Cost	21648

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \left(t_0 \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 + 0.5 \cdot \frac{\left(D \cdot \left(h \cdot \left(\frac{M}{d} \cdot \left(D \cdot M\right)\right)\right)\right) \cdot \frac{-0.25}{d}}{\ell}\right)\\ \mathbf{if}\;h \leq -4.6 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;h \leq -2.4 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d}\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;h \leq 1.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \frac{-0.5}{\ell} \cdot \frac{D}{\frac{d}{\frac{0.25 \cdot M}{\frac{d}{M \cdot \left(h \cdot D\right)}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 + 0.5 \cdot \frac{\frac{0.25}{d} \cdot \left(M \cdot \left(\frac{D}{-d} \cdot \left(h \cdot \left(D \cdot M\right)\right)\right)\right)}{\ell}\right)\\ \end{array} \]

Alternative 4
Error	29.23%
Cost	21648

\[\begin{array}{l} t_0 := 1 + 0.5 \cdot \frac{\left(D \cdot \left(h \cdot \left(\frac{M}{d} \cdot \left(D \cdot M\right)\right)\right)\right) \cdot \frac{-0.25}{d}}{\ell}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;h \leq -4.1 \cdot 10^{-41}:\\ \;\;\;\;\left(t_1 \cdot \frac{t_2}{\sqrt{-\ell}}\right) \cdot t_0\\ \mathbf{elif}\;h \leq -9.6 \cdot 10^{-136}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d}\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t_0 \cdot \left(\frac{t_2}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;h \leq 2.25 \cdot 10^{-54}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \frac{-0.5}{\ell} \cdot \frac{D}{\frac{d}{\frac{0.25 \cdot M}{\frac{d}{M \cdot \left(h \cdot D\right)}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 + 0.5 \cdot \frac{\frac{0.25}{d} \cdot \left(M \cdot \left(\frac{D}{-d} \cdot \left(h \cdot \left(D \cdot M\right)\right)\right)\right)}{\ell}\right)\\ \end{array} \]

Alternative 5
Error	32.16%
Cost	21532

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ t_1 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ t_2 := \left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{if}\;d \leq -1.4 \cdot 10^{+122}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -2.2 \cdot 10^{-43}:\\ \;\;\;\;t_0 \cdot \left(1 + 0.5 \cdot \left(\frac{D \cdot M}{\left(d \cdot \frac{d}{h}\right) \cdot \frac{\ell}{D \cdot M}} \cdot -0.25\right)\right)\\ \mathbf{elif}\;d \leq -1.3 \cdot 10^{-163}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -1.95 \cdot 10^{-303}:\\ \;\;\;\;t_0 \cdot \left(1 + 0.5 \cdot \left(\frac{\frac{\frac{h}{d}}{\frac{d}{D \cdot \left(D \cdot M\right)}}}{\frac{\ell}{M}} \cdot -0.25\right)\right)\\ \mathbf{elif}\;d \leq 9.4 \cdot 10^{-298}:\\ \;\;\;\;t_0 \cdot \left(1 + 0.5 \cdot \frac{\left(D \cdot \frac{h \cdot \left(M \cdot \left(D \cdot M\right)\right)}{d}\right) \cdot \frac{-0.25}{d}}{\ell}\right)\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{-174}:\\ \;\;\;\;t_1 \cdot \left(1 + \frac{-0.5}{d \cdot \frac{\ell}{\left(\left(D \cdot M\right) \cdot \left(0.25 \cdot D\right)\right) \cdot \left(h \cdot \frac{M}{d}\right)}}\right)\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{+133}:\\ \;\;\;\;t_1 \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{0.5 \cdot M}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 6
Error	32.16%
Cost	21532

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := t_0 \cdot t_1\\ t_3 := \left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ t_4 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{if}\;d \leq -9.5 \cdot 10^{+121}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq -2.2 \cdot 10^{-43}:\\ \;\;\;\;t_2 \cdot \left(1 + 0.5 \cdot \left(\frac{D \cdot M}{\left(d \cdot \frac{d}{h}\right) \cdot \frac{\ell}{D \cdot M}} \cdot -0.25\right)\right)\\ \mathbf{elif}\;d \leq -1.05 \cdot 10^{-163}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq -1.8 \cdot 10^{-303}:\\ \;\;\;\;t_2 \cdot \left(1 + 0.5 \cdot \left(\frac{\frac{\frac{h}{d}}{\frac{d}{D \cdot \left(D \cdot M\right)}}}{\frac{\ell}{M}} \cdot -0.25\right)\right)\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{-298}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 + {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \left(h \cdot \frac{-0.5}{\ell}\right)\right)\right)\\ \mathbf{elif}\;d \leq 2.55 \cdot 10^{-174}:\\ \;\;\;\;t_4 \cdot \left(1 + \frac{-0.5}{d \cdot \frac{\ell}{\left(\left(D \cdot M\right) \cdot \left(0.25 \cdot D\right)\right) \cdot \left(h \cdot \frac{M}{d}\right)}}\right)\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{+135}:\\ \;\;\;\;t_4 \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{0.5 \cdot M}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 7
Error	32.17%
Cost	21392

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{if}\;d \leq -1.9 \cdot 10^{+123}:\\ \;\;\;\;\left(-d\right) \cdot t_1\\ \mathbf{elif}\;d \leq -2.2 \cdot 10^{-43}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot t_0\right) \cdot \left(1 + 0.5 \cdot \left(\frac{D \cdot M}{\left(d \cdot \frac{d}{h}\right) \cdot \frac{\ell}{D \cdot M}} \cdot -0.25\right)\right)\\ \mathbf{elif}\;d \leq -7.5 \cdot 10^{-284}:\\ \;\;\;\;\frac{D}{\frac{\frac{d}{M}}{D \cdot M}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot t_1\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{-295}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \left(t_0 \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \frac{-0.5}{d \cdot \frac{\ell}{\left(\left(D \cdot M\right) \cdot \left(0.25 \cdot D\right)\right) \cdot \left(h \cdot \frac{M}{d}\right)}}\right)\\ \end{array} \]

Alternative 8
Error	32.45%
Cost	21264

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ t_1 := \left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{if}\;d \leq -9.5 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -2.2 \cdot 10^{-43}:\\ \;\;\;\;t_0 \cdot \left(1 + 0.5 \cdot \left(\frac{D \cdot M}{\left(d \cdot \frac{d}{h}\right) \cdot \frac{\ell}{D \cdot M}} \cdot -0.25\right)\right)\\ \mathbf{elif}\;d \leq -1.75 \cdot 10^{-163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-296}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \frac{-0.5}{d \cdot \frac{\ell}{\left(\left(D \cdot M\right) \cdot \left(0.25 \cdot D\right)\right) \cdot \left(h \cdot \frac{M}{d}\right)}}\right)\\ \end{array} \]

Alternative 9
Error	31.32%
Cost	21264

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \left(t_0 \cdot t_1\right) \cdot \left(1 + 0.5 \cdot \left(\frac{D \cdot M}{\left(d \cdot \frac{d}{h}\right) \cdot \frac{\ell}{D \cdot M}} \cdot -0.25\right)\right)\\ t_3 := \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{if}\;d \leq -1.45 \cdot 10^{+122}:\\ \;\;\;\;\left(-d\right) \cdot t_3\\ \mathbf{elif}\;d \leq -2.2 \cdot 10^{-43}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -2.1 \cdot 10^{-282}:\\ \;\;\;\;\frac{D}{\frac{\frac{d}{M}}{D \cdot M}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot t_3\\ \mathbf{elif}\;d \leq 8.8 \cdot 10^{-296}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 + {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \left(h \cdot \frac{-0.5}{\ell}\right)\right)\right)\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{-32} \lor \neg \left(d \leq 4.6 \cdot 10^{+57}\right):\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \frac{-0.5}{d \cdot \frac{\ell}{\left(\left(D \cdot M\right) \cdot \left(0.25 \cdot D\right)\right) \cdot \left(h \cdot \frac{M}{d}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Error	32.78%
Cost	15581

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ t_1 := t_0 \cdot \left(1 + 0.5 \cdot \left(\frac{D \cdot M}{\left(d \cdot \frac{d}{h}\right) \cdot \frac{\ell}{D \cdot M}} \cdot -0.25\right)\right)\\ t_2 := \left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{if}\;d \leq -1.5 \cdot 10^{+122}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -2.2 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -1.15 \cdot 10^{-163}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -2.15 \cdot 10^{-304}:\\ \;\;\;\;t_0 \cdot \left(1 + 0.5 \cdot \left(\frac{\frac{\frac{h}{d}}{\frac{d}{D \cdot \left(D \cdot M\right)}}}{\frac{\ell}{M}} \cdot -0.25\right)\right)\\ \mathbf{elif}\;d \leq 4.25 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{\frac{d}{\ell \cdot \frac{h}{d}}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 4.9 \cdot 10^{-33} \lor \neg \left(d \leq 9.4 \cdot 10^{+55}\right):\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \frac{-0.5}{d \cdot \frac{\ell}{\left(\left(D \cdot M\right) \cdot \left(0.25 \cdot D\right)\right) \cdot \left(h \cdot \frac{M}{d}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	32.71%
Cost	15581

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ t_1 := t_0 \cdot \left(1 + 0.5 \cdot \left(\frac{D \cdot M}{\left(d \cdot \frac{d}{h}\right) \cdot \frac{\ell}{D \cdot M}} \cdot -0.25\right)\right)\\ t_2 := \left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{if}\;d \leq -1.05 \cdot 10^{+122}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -2.2 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -4.1 \cdot 10^{-163}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -1.95 \cdot 10^{-303}:\\ \;\;\;\;t_0 \cdot \left(1 + 0.5 \cdot \left(\frac{\frac{\frac{h}{d}}{\frac{d}{D \cdot \left(D \cdot M\right)}}}{\frac{\ell}{M}} \cdot -0.25\right)\right)\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{-298}:\\ \;\;\;\;t_0 \cdot \left(1 + 0.5 \cdot \frac{\left(D \cdot \frac{h \cdot \left(M \cdot \left(D \cdot M\right)\right)}{d}\right) \cdot \frac{-0.25}{d}}{\ell}\right)\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{-32} \lor \neg \left(d \leq 1.6 \cdot 10^{+59}\right):\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \frac{-0.5}{d \cdot \frac{\ell}{\left(\left(D \cdot M\right) \cdot \left(0.25 \cdot D\right)\right) \cdot \left(h \cdot \frac{M}{d}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	31.74%
Cost	15317

\[\begin{array}{l} t_0 := \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(\frac{D \cdot M}{\left(d \cdot \frac{d}{h}\right) \cdot \frac{\ell}{D \cdot M}} \cdot -0.25\right)\right)\\ t_1 := \left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{if}\;d \leq -2 \cdot 10^{+123}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -2.2 \cdot 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 4.3 \cdot 10^{-299}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 5.1 \cdot 10^{-33} \lor \neg \left(d \leq 8.4 \cdot 10^{+58}\right):\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \frac{-0.5}{d \cdot \frac{\ell}{\left(\left(D \cdot M\right) \cdot \left(0.25 \cdot D\right)\right) \cdot \left(h \cdot \frac{M}{d}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	34.53%
Cost	14792

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-308}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq 4.6 \cdot 10^{+149}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \frac{-0.5}{\ell} \cdot \frac{D}{\frac{d}{\frac{0.25 \cdot M}{\frac{d}{M \cdot \left(h \cdot D\right)}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \end{array} \]

Alternative 14
Error	38.41%
Cost	14732

\[\begin{array}{l} \mathbf{if}\;d \leq 1.75 \cdot 10^{-298}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{-118}:\\ \;\;\;\;\frac{d}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{-36}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + {\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 15
Error	36.13%
Cost	14660

\[\begin{array}{l} \mathbf{if}\;d \leq 4.3 \cdot 10^{-299}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \frac{-0.5}{d \cdot \frac{\ell}{\left(\left(D \cdot M\right) \cdot \left(0.25 \cdot D\right)\right) \cdot \left(h \cdot \frac{M}{d}\right)}}\right)\\ \end{array} \]

Alternative 16
Error	37.85%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;d \leq 4.25 \cdot 10^{-299}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 17
Error	37.81%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;d \leq 4.25 \cdot 10^{-299}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 18
Error	53.82%
Cost	7244

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq 2.3 \cdot 10^{-194}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{+145}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{+228}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]

Alternative 19
Error	53.01%
Cost	7244

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.6 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{+145}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+228}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]

Alternative 20
Error	52.99%
Cost	7244

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.7 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+144}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+225}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\sqrt{\ell \cdot h}}\\ \end{array} \]

Alternative 21
Error	44.95%
Cost	7244

\[\begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{+144}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+226}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\sqrt{\ell \cdot h}}\\ \end{array} \]

Alternative 22
Error	44.65%
Cost	7244

\[\begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{-309}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{+145}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{+228}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\sqrt{\ell \cdot h}}\\ \end{array} \]

Alternative 23
Error	54.5%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq -1.15 \cdot 10^{-240}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]

Alternative 24
Error	68.58%
Cost	6784

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

Alternative 25
Error	68.57%
Cost	6720

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

Reproduce

herbie shell --seed 2023090 
(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)))))