Henrywood and Agarwal, Equation (12)

Average Error: 26.7 → 17.6

Time: 39.3s

Precision: binary64

Cost: 33992

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sqrt[3]{h}}{\ell}\\ \mathbf{if}\;d \leq -3 \cdot 10^{-121}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-271}:\\ \;\;\;\;\frac{D}{\frac{\frac{d}{M}}{M \cdot D}} \cdot \left(\left(\left|t_0\right| \cdot \sqrt{t_0}\right) \cdot 0.125\right) - d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq 5.2 \cdot 10^{+43}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

FPCore

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

↓

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (cbrt h) l)))
   (if (<= d -3e-121)
     (*
      (* (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ d l)))
      (+ 1.0 (* (* (pow (* (/ M 2.0) (/ D d)) 2.0) (/ h l)) -0.5)))
     (if (<= d 3.8e-271)
       (-
        (* (/ D (/ (/ d M) (* M D))) (* (* (fabs t_0) (sqrt t_0)) 0.125))
        (* d (sqrt (/ (/ 1.0 l) h))))
       (if (<= d 5.2e+43)
         (*
          (/ (/ d (sqrt h)) (sqrt l))
          (fma (pow (* 0.5 (* M (/ D d))) 2.0) (* (/ h l) -0.5) 1.0))
         (/ d (* (sqrt h) (sqrt l))))))))

C

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

↓

double code(double d, double h, double l, double M, double D) {
	double t_0 = cbrt(h) / l;
	double tmp;
	if (d <= -3e-121) {
		tmp = ((sqrt(-d) / sqrt(-h)) * sqrt((d / l))) * (1.0 + ((pow(((M / 2.0) * (D / d)), 2.0) * (h / l)) * -0.5));
	} else if (d <= 3.8e-271) {
		tmp = ((D / ((d / M) / (M * D))) * ((fabs(t_0) * sqrt(t_0)) * 0.125)) - (d * sqrt(((1.0 / l) / h)));
	} else if (d <= 5.2e+43) {
		tmp = ((d / sqrt(h)) / sqrt(l)) * fma(pow((0.5 * (M * (D / d))), 2.0), ((h / l) * -0.5), 1.0);
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Julia

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

↓

function code(d, h, l, M, D)
	t_0 = Float64(cbrt(h) / l)
	tmp = 0.0
	if (d <= -3e-121)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * sqrt(Float64(d / l))) * Float64(1.0 + Float64(Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * Float64(h / l)) * -0.5)));
	elseif (d <= 3.8e-271)
		tmp = Float64(Float64(Float64(D / Float64(Float64(d / M) / Float64(M * D))) * Float64(Float64(abs(t_0) * sqrt(t_0)) * 0.125)) - Float64(d * sqrt(Float64(Float64(1.0 / l) / h))));
	elseif (d <= 5.2e+43)
		tmp = Float64(Float64(Float64(d / sqrt(h)) / sqrt(l)) * fma((Float64(0.5 * Float64(M * Float64(D / d))) ^ 2.0), Float64(Float64(h / l) * -0.5), 1.0));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

Wolfram

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

↓

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Power[h, 1/3], $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[d, -3e-121], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.8e-271], N[(N[(N[(D / N[(N[(d / M), $MachinePrecision] / N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[t$95$0], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] - N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.2e+43], N[(N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(0.5 * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.9999999999999999e-121

   1. Initial program 22.4

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

   2. Simplified 22.4

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

      [Start] 22.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 22.4	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 22.4	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 22.4	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 22.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 22.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 22.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 22.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 14.4

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

   if -2.9999999999999999e-121 < d < 3.8000000000000001e-271

   1. Initial program 38.1

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

   2. Simplified 39.1

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

      [Start] 38.1	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 38.1	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 38.1	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 38.1	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 38.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 38.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 38.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 39.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 39.4

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

   4. Taylor expanded in d around -inf 42.6

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

   5. Simplified 39.8

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

      [Start] 42.6	\[ 0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + -1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]
      mul-1-neg [=>] 42.6	\[ 0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
      unsub-neg [=>] 42.6	\[ \color{blue}{0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
      *-commutative [=>] 42.6	\[ \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot 0.125} - d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]
      associate-*l* [=>] 42.6	\[ \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right)} - d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]
      associate-/l* [=>] 44.6	\[ \color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]
      unpow2 [=>] 44.6	\[ \frac{\color{blue}{D \cdot D}}{\frac{d}{{M}^{2}}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]
      associate-/l* [=>] 43.6	\[ \color{blue}{\frac{D}{\frac{\frac{d}{{M}^{2}}}{D}}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]
      unpow2 [=>] 43.6	\[ \frac{D}{\frac{\frac{d}{\color{blue}{M \cdot M}}}{D}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]
      associate-/r* [=>] 42.4	\[ \frac{D}{\frac{\color{blue}{\frac{\frac{d}{M}}{M}}}{D}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]
      associate-/l/ [=>] 39.8	\[ \frac{D}{\color{blue}{\frac{\frac{d}{M}}{D \cdot M}}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]
      associate-/r* [=>] 39.8	\[ \frac{D}{\frac{\frac{d}{M}}{D \cdot M}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125\right) - d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Applied egg-rr 34.4

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

   7. Simplified 31.5

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

      [Start] 34.4	\[ \frac{D}{\frac{\frac{d}{M}}{D \cdot M}} \cdot \left(\left(\sqrt{{\left(\frac{\sqrt[3]{h}}{\ell}\right)}^{2}} \cdot \sqrt{\frac{\sqrt[3]{h}}{\ell}}\right) \cdot 0.125\right) - d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}} \]
      unpow2 [=>] 34.4	\[ \frac{D}{\frac{\frac{d}{M}}{D \cdot M}} \cdot \left(\left(\sqrt{\color{blue}{\frac{\sqrt[3]{h}}{\ell} \cdot \frac{\sqrt[3]{h}}{\ell}}} \cdot \sqrt{\frac{\sqrt[3]{h}}{\ell}}\right) \cdot 0.125\right) - d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}} \]
      rem-sqrt-square [=>] 31.5	\[ \frac{D}{\frac{\frac{d}{M}}{D \cdot M}} \cdot \left(\left(\color{blue}{\left|\frac{\sqrt[3]{h}}{\ell}\right|} \cdot \sqrt{\frac{\sqrt[3]{h}}{\ell}}\right) \cdot 0.125\right) - d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}} \]

   if 3.8000000000000001e-271 < d < 5.20000000000000042e43

   1. Initial program 27.1

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

   2. Simplified 28.1

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

      [Start] 27.1	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 27.1	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 27.1	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 27.1	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 27.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 27.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 27.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 28.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 21.9

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

   4. Simplified 22.0

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

      [Start] 21.9	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \]
      *-lft-identity [<=] 21.9	\[ \color{blue}{1 \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} + \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \]
      *-commutative [<=] 21.9	\[ 1 \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \color{blue}{\left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]
      distribute-rgt-in [<=] 21.9	\[ \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)} \]
      associate-/r* [=>] 22.0	\[ \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \cdot \left(1 + \left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \]
      +-commutative [=>] 22.0	\[ \frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \color{blue}{\left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} + 1\right)} \]
      *-commutative [=>] 22.0	\[ \frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \left(\color{blue}{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)} + 1\right) \]
      fma-def [=>] 22.0	\[ \frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \color{blue}{\mathsf{fma}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]
      *-commutative [=>] 22.0	\[ \frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \color{blue}{\left(\frac{D}{d} \cdot 0.5\right)}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right) \]
      associate-*r* [=>] 22.0	\[ \frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left({\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot 0.5\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right) \]
      *-commutative [=>] 22.0	\[ \frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left({\left(\left(M \cdot \frac{D}{d}\right) \cdot 0.5\right)}^{2}, \color{blue}{\frac{h}{\ell} \cdot -0.5}, 1\right) \]

   if 5.20000000000000042e43 < d

   1. Initial program 25.4

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

   2. Simplified 25.1

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

      [Start] 25.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 25.4	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 25.4	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 25.4	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 25.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 25.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 25.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 25.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Taylor expanded in d around inf 17.9

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

   4. Simplified 17.3

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

      [Start] 17.9	\[ \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]
      *-commutative [=>] 17.9	\[ \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
      associate-/r* [=>] 17.3	\[ d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   5. Applied egg-rr 10.1

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

   6. Simplified 7.9

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

      [Start] 10.1	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \]
      associate-/l/ [=>] 7.9	\[ \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 17.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3 \cdot 10^{-121}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-271}:\\ \;\;\;\;\frac{D}{\frac{\frac{d}{M}}{M \cdot D}} \cdot \left(\left(\left|\frac{\sqrt[3]{h}}{\ell}\right| \cdot \sqrt{\frac{\sqrt[3]{h}}{\ell}}\right) \cdot 0.125\right) - d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq 5.2 \cdot 10^{+43}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternatives

Alternative 1
Error	21.0
Cost	104464

\[\begin{array}{l} t_0 := d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ t_1 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_2 := 1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\\ t_3 := \left(t_1 \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot t_2\\ t_4 := \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;t_4 \cdot \left(1 + 0.5 \cdot \left(\frac{D}{\frac{\ell}{D} \cdot \frac{d}{\frac{M \cdot \left(h \cdot M\right)}{d}}} \cdot -0.25\right)\right)\\ \mathbf{elif}\;t_3 \leq -500000000000:\\ \;\;\;\;t_2 \cdot \left(t_1 \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;\frac{M \cdot D}{\frac{d}{M \cdot D}} \cdot \left(0.125 \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) - t_0\\ \mathbf{elif}\;t_3 \leq 4 \cdot 10^{+298}:\\ \;\;\;\;\left(1 + \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5\right) \cdot t_4\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	17.0
Cost	33796

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ t_1 := M \cdot \left(0.5 \cdot \frac{D}{d}\right)\\ \mathbf{if}\;\ell \leq -4.1 \cdot 10^{+111}:\\ \;\;\;\;t_0 \cdot \left(1 - 0.5 \cdot {\left(t_1 \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{elif}\;\ell \leq -2.8 \cdot 10^{-44}:\\ \;\;\;\;\frac{M \cdot D}{\frac{d}{M \cdot D}} \cdot \left(0.125 \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) - d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -6.1 \cdot 10^{-127}:\\ \;\;\;\;t_0 \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \left(0.5 \cdot D\right)}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -2.7 \cdot 10^{-307}:\\ \;\;\;\;t_0 \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{D}{\frac{\ell \cdot \frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}{D}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {t_1}^{2}\right)\\ \end{array} \]

Alternative 3
Error	20.8
Cost	21316

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -300000000000:\\ \;\;\;\;\left(t_1 \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{D}{\frac{\ell \cdot \frac{\frac{d}{M} \cdot \frac{d}{M}}{h}}{D}}\right)\right)\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-128}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot t_1\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{D}{\frac{\frac{d \cdot d}{M \cdot \left(h \cdot \frac{M}{\ell}\right)}}{D}}\right)\right)\\ \mathbf{elif}\;d \leq -1.9 \cdot 10^{-204}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-271}:\\ \;\;\;\;\frac{M \cdot D}{\frac{d}{M \cdot D}} \cdot \left(0.125 \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) - d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+42}:\\ \;\;\;\;t_0 \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	21.4
Cost	21136

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{D}{d}\\ t_1 := \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{if}\;d \leq -3.9 \cdot 10^{+202}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -2.35 \cdot 10^{-121}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{M}{\frac{\ell}{h}} \cdot \frac{\left(M \cdot \frac{D}{d}\right) \cdot t_0}{2}\right)\right)}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-271}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq 7 \cdot 10^{+44}:\\ \;\;\;\;t_1 \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(M \cdot t_0\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	21.1
Cost	21136

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{D}{d}\\ t_1 := \sqrt{\frac{\frac{1}{\ell}}{h}}\\ t_2 := \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{if}\;d \leq -3.8 \cdot 10^{+202}:\\ \;\;\;\;\left(-d\right) \cdot t_1\\ \mathbf{elif}\;d \leq -2.85 \cdot 10^{-123}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{M}{\frac{\ell}{h}} \cdot \frac{\left(M \cdot \frac{D}{d}\right) \cdot t_0}{2}\right)\right)}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-271}:\\ \;\;\;\;\frac{D}{\frac{\frac{d}{M}}{M \cdot D}} \cdot \left(0.125 \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) - d \cdot t_1\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{+41}:\\ \;\;\;\;t_2 \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(M \cdot t_0\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	20.9
Cost	21136

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{D}{d}\\ t_1 := \sqrt{\frac{\frac{1}{\ell}}{h}}\\ t_2 := \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{if}\;d \leq -1.85 \cdot 10^{+214}:\\ \;\;\;\;\left(-d\right) \cdot t_1\\ \mathbf{elif}\;d \leq -6.6 \cdot 10^{-122}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{M}{\frac{\ell}{h}} \cdot \frac{\left(M \cdot \frac{D}{d}\right) \cdot t_0}{2}\right)\right)}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-271}:\\ \;\;\;\;\frac{M \cdot D}{\frac{d}{M \cdot D}} \cdot \left(0.125 \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) - d \cdot t_1\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+44}:\\ \;\;\;\;t_2 \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(M \cdot t_0\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	22.0
Cost	15317

\[\begin{array}{l} t_0 := \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + 0.5 \cdot \left(\frac{D}{\frac{\ell}{D} \cdot \frac{d}{\frac{M \cdot \left(h \cdot M\right)}{d}}} \cdot -0.25\right)\right)\\ \mathbf{if}\;d \leq -3.3 \cdot 10^{+92}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -1.2 \cdot 10^{-128}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{-306}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq 1.25 \cdot 10^{-49} \lor \neg \left(d \leq 4.4 \cdot 10^{+21}\right):\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	21.8
Cost	15317

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -5.2 \cdot 10^{+92}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -2.3 \cdot 10^{-130}:\\ \;\;\;\;t_0 \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{D}{\frac{\frac{d \cdot d}{M \cdot \left(h \cdot \frac{M}{\ell}\right)}}{D}}\right)\right)\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{-306}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{-49} \lor \neg \left(d \leq 6.2 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(1 + 0.5 \cdot \left(\frac{D}{\frac{\ell}{D} \cdot \frac{d}{\frac{M \cdot \left(h \cdot M\right)}{d}}} \cdot -0.25\right)\right)\\ \end{array} \]

Alternative 9
Error	22.5
Cost	15317

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -5.8 \cdot 10^{+206}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -1.02 \cdot 10^{-122}:\\ \;\;\;\;\frac{t_0 \cdot \left(1 + -0.5 \cdot \left(\frac{M}{\frac{\ell}{h}} \cdot \frac{\left(M \cdot \frac{D}{d}\right) \cdot \left(0.5 \cdot \frac{D}{d}\right)}{2}\right)\right)}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{-306}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-50} \lor \neg \left(d \leq 1.6 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot t_0\right) \cdot \left(1 + 0.5 \cdot \left(\frac{D}{\frac{\ell}{D} \cdot \frac{d}{\frac{M \cdot \left(h \cdot M\right)}{d}}} \cdot -0.25\right)\right)\\ \end{array} \]

Alternative 10
Error	23.4
Cost	15316

\[\begin{array}{l} t_0 := M \cdot \frac{D}{d}\\ t_1 := 0.5 \cdot \frac{D}{d}\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := \sqrt{\frac{\ell}{d}}\\ \mathbf{if}\;d \leq -3.5 \cdot 10^{+202}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -5.8 \cdot 10^{-124}:\\ \;\;\;\;\frac{t_2 \cdot \left(1 + -0.5 \cdot \left(\frac{M}{\frac{\ell}{h}} \cdot \frac{t_0 \cdot t_1}{2}\right)\right)}{t_3}\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{-271}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq 4.3 \cdot 10^{-105}:\\ \;\;\;\;\frac{t_2 \cdot \left(1 + -0.5 \cdot \frac{0.5}{\frac{\frac{\ell}{h}}{t_0 \cdot \left(M \cdot t_1\right)}}\right)}{t_3}\\ \mathbf{elif}\;d \leq 3.1 \cdot 10^{+22}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot t_2\right) \cdot \left(1 + 0.5 \cdot \left(\frac{D}{\frac{\ell}{D} \cdot \frac{d}{\frac{M \cdot \left(h \cdot M\right)}{d}}} \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 11
Error	22.9
Cost	14920

\[\begin{array}{l} \mathbf{if}\;d \leq -2.35 \cdot 10^{+58}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -3.15 \cdot 10^{-121}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}{\ell} \cdot \frac{\frac{h}{d}}{d}\right)\right)\right)\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{-306}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 12
Error	24.3
Cost	13384

\[\begin{array}{l} \mathbf{if}\;h \leq -1.85 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;h \leq -9.5 \cdot 10^{-308}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 13
Error	23.9
Cost	13252

\[\begin{array}{l} \mathbf{if}\;d \leq 9.5 \cdot 10^{-306}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 14
Error	28.2
Cost	7044

\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.9 \cdot 10^{-260}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 15
Error	28.0
Cost	7044

\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.9 \cdot 10^{-260}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 16
Error	33.7
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\ell \leq 2.45 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 17
Error	44.5
Cost	6720

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

Reproduce

herbie shell --seed 2023067 
(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)))))