Henrywood and Agarwal, Equation (12)

Average Error: 26.5 → 13.5

Time: 49.3s

Precision: binary64

Cost: 33868

Math

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

↓

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ t_1 := 1 + 0.5 \cdot \frac{h \cdot \left(M \cdot \frac{-0.5}{\frac{d}{D}}\right)}{\frac{\ell}{M \cdot \frac{0.5}{\frac{d}{D}}}}\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;h \leq -2.4 \cdot 10^{-160}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{t_2}{\sqrt{-\ell}}\right) \cdot t_1\\ \mathbf{elif}\;h \leq -2.7 \cdot 10^{-302}:\\ \;\;\;\;t_1 \cdot \left(\frac{t_2}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;h \leq 1.15 \cdot 10^{-60}:\\ \;\;\;\;t_0 \cdot \left(1 + \left(\frac{D}{\frac{\ell}{M}} \cdot \left(\frac{h}{d} \cdot \frac{M}{d}\right)\right) \cdot \left(D \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{t_0 \cdot \left(1 + \frac{h}{\frac{\ell}{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}} \cdot -0.5\right)}\right)}^{3}\\ \end{array} \]

FPCore

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

↓

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ d (* (sqrt h) (sqrt l))))
        (t_1
         (+
          1.0
          (*
           0.5
           (/ (* h (* M (/ -0.5 (/ d D)))) (/ l (* M (/ 0.5 (/ d D))))))))
        (t_2 (sqrt (- d))))
   (if (<= h -2.4e-160)
     (* (* (sqrt (/ d h)) (/ t_2 (sqrt (- l)))) t_1)
     (if (<= h -2.7e-302)
       (* t_1 (* (/ t_2 (sqrt (- h))) (sqrt (/ d l))))
       (if (<= h 1.15e-60)
         (* t_0 (+ 1.0 (* (* (/ D (/ l M)) (* (/ h d) (/ M d))) (* D -0.125))))
         (pow
          (cbrt
           (*
            t_0
            (+ 1.0 (* (/ h (/ l (pow (* M (* D (/ 0.5 d))) 2.0))) -0.5))))
          3.0))))))

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 = d / (sqrt(h) * sqrt(l));
	double t_1 = 1.0 + (0.5 * ((h * (M * (-0.5 / (d / D)))) / (l / (M * (0.5 / (d / D))))));
	double t_2 = sqrt(-d);
	double tmp;
	if (h <= -2.4e-160) {
		tmp = (sqrt((d / h)) * (t_2 / sqrt(-l))) * t_1;
	} else if (h <= -2.7e-302) {
		tmp = t_1 * ((t_2 / sqrt(-h)) * sqrt((d / l)));
	} else if (h <= 1.15e-60) {
		tmp = t_0 * (1.0 + (((D / (l / M)) * ((h / d) * (M / d))) * (D * -0.125)));
	} else {
		tmp = pow(cbrt((t_0 * (1.0 + ((h / (l / pow((M * (D * (0.5 / d))), 2.0))) * -0.5)))), 3.0);
	}
	return tmp;
}

Java

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

↓

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = d / (Math.sqrt(h) * Math.sqrt(l));
	double t_1 = 1.0 + (0.5 * ((h * (M * (-0.5 / (d / D)))) / (l / (M * (0.5 / (d / D))))));
	double t_2 = Math.sqrt(-d);
	double tmp;
	if (h <= -2.4e-160) {
		tmp = (Math.sqrt((d / h)) * (t_2 / Math.sqrt(-l))) * t_1;
	} else if (h <= -2.7e-302) {
		tmp = t_1 * ((t_2 / Math.sqrt(-h)) * Math.sqrt((d / l)));
	} else if (h <= 1.15e-60) {
		tmp = t_0 * (1.0 + (((D / (l / M)) * ((h / d) * (M / d))) * (D * -0.125)));
	} else {
		tmp = Math.pow(Math.cbrt((t_0 * (1.0 + ((h / (l / Math.pow((M * (D * (0.5 / d))), 2.0))) * -0.5)))), 3.0);
	}
	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 = Float64(d / Float64(sqrt(h) * sqrt(l)))
	t_1 = Float64(1.0 + Float64(0.5 * Float64(Float64(h * Float64(M * Float64(-0.5 / Float64(d / D)))) / Float64(l / Float64(M * Float64(0.5 / Float64(d / D)))))))
	t_2 = sqrt(Float64(-d))
	tmp = 0.0
	if (h <= -2.4e-160)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(t_2 / sqrt(Float64(-l)))) * t_1);
	elseif (h <= -2.7e-302)
		tmp = Float64(t_1 * Float64(Float64(t_2 / sqrt(Float64(-h))) * sqrt(Float64(d / l))));
	elseif (h <= 1.15e-60)
		tmp = Float64(t_0 * Float64(1.0 + Float64(Float64(Float64(D / Float64(l / M)) * Float64(Float64(h / d) * Float64(M / d))) * Float64(D * -0.125))));
	else
		tmp = cbrt(Float64(t_0 * Float64(1.0 + Float64(Float64(h / Float64(l / (Float64(M * Float64(D * Float64(0.5 / d))) ^ 2.0))) * -0.5)))) ^ 3.0;
	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[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(0.5 * N[(N[(h * N[(M * N[(-0.5 / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l / N[(M * N[(0.5 / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[h, -2.4e-160], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$2 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[h, -2.7e-302], N[(t$95$1 * N[(N[(t$95$2 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 1.15e-60], N[(t$95$0 * N[(1.0 + N[(N[(N[(D / N[(l / M), $MachinePrecision]), $MachinePrecision] * N[(N[(h / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(t$95$0 * N[(1.0 + N[(N[(h / N[(l / N[Power[N[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -2.39999999999999991e-160

   1. Initial program 25.3

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

   2. Simplified 25.3

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

      [Start] 25.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 25.3	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 25.3	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 25.3	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 25.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 25.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 25.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 25.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 25.8

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)} - 1\right)}\right) \]

   4. Simplified 23.6

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

      [Start] 25.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(e^{\mathsf{log1p}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)} - 1\right)\right) \]
      expm1-def [=>] 25.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]
      expm1-log1p [=>] 25.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      associate-*r/ [=>] 24.5	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}}\right) \]
      associate-*l/ [<=] 23.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot h\right)}\right) \]
      *-commutative [=>] 23.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)}\right) \]
      *-commutative [=>] 23.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(M \cdot \color{blue}{\left(\frac{D}{d} \cdot 0.5\right)}\right)}^{2}}{\ell}\right)\right) \]
      associate-*r* [=>] 23.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot 0.5\right)}}^{2}}{\ell}\right)\right) \]

   5. Applied egg-rr 22.3

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

   6. Applied egg-rr 15.2

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

   if -2.39999999999999991e-160 < h < -2.70000000000000006e-302

   1. Initial program 34.6

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

   2. Simplified 36.2

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

      [Start] 34.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 34.6	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 34.6	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 34.6	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 34.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 34.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 34.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 36.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 36.4

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)} - 1\right)}\right) \]

   4. Simplified 38.2

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

      [Start] 36.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(e^{\mathsf{log1p}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)} - 1\right)\right) \]
      expm1-def [=>] 36.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]
      expm1-log1p [=>] 36.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      associate-*r/ [=>] 36.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}}\right) \]
      associate-*l/ [<=] 38.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot h\right)}\right) \]
      *-commutative [=>] 38.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)}\right) \]
      *-commutative [=>] 38.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(M \cdot \color{blue}{\left(\frac{D}{d} \cdot 0.5\right)}\right)}^{2}}{\ell}\right)\right) \]
      associate-*r* [=>] 38.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot 0.5\right)}}^{2}}{\ell}\right)\right) \]

   5. Applied egg-rr 32.6

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

   6. Applied egg-rr 14.2

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

   if -2.70000000000000006e-302 < h < 1.1500000000000001e-60

   1. Initial program 29.7

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

   2. Simplified 30.4

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

      [Start] 29.7	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 29.7	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 29.7	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 29.7	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 29.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 29.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 29.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 30.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Taylor expanded in M around 0 44.3

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

   4. Simplified 36.3

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

      [Start] 44.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)\right) \]
      associate-/r* [=>] 42.5	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{\frac{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}}\right)\right) \]
      associate-/l/ [=>] 44.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right)\right) \]
      associate-*r* [=>] 44.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}\right)\right) \]
      *-commutative [<=] 44.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}{\color{blue}{\ell \cdot {d}^{2}}}\right)\right) \]
      times-frac [=>] 42.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{\ell} \cdot \frac{h}{{d}^{2}}\right)}\right)\right) \]
      unpow2 [=>] 42.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{{D}^{2} \cdot \color{blue}{\left(M \cdot M\right)}}{\ell} \cdot \frac{h}{{d}^{2}}\right)\right)\right) \]
      associate-*r* [=>] 40.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\color{blue}{\left({D}^{2} \cdot M\right) \cdot M}}{\ell} \cdot \frac{h}{{d}^{2}}\right)\right)\right) \]
      unpow2 [=>] 40.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot M\right) \cdot M}{\ell} \cdot \frac{h}{{d}^{2}}\right)\right)\right) \]
      unpow2 [=>] 40.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}{\ell} \cdot \frac{h}{\color{blue}{d \cdot d}}\right)\right)\right) \]
      associate-/r* [=>] 36.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}{\ell} \cdot \color{blue}{\frac{\frac{h}{d}}{d}}\right)\right)\right) \]

   5. Applied egg-rr 32.3

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

   6. Applied egg-rr 29.3

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

   7. Applied egg-rr 11.1

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

   if 1.1500000000000001e-60 < h

   1. Initial program 22.9

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

   2. Simplified 23.1

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

      [Start] 22.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 22.9	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 22.9	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 22.9	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 22.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 23.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 23.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 23.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 23.7

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)} - 1\right)}\right) \]

   4. Simplified 20.4

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

      [Start] 23.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(e^{\mathsf{log1p}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)} - 1\right)\right) \]
      expm1-def [=>] 23.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]
      expm1-log1p [=>] 23.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      associate-*r/ [=>] 21.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}}\right) \]
      associate-*l/ [<=] 20.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot h\right)}\right) \]
      *-commutative [=>] 20.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)}\right) \]
      *-commutative [=>] 20.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(M \cdot \color{blue}{\left(\frac{D}{d} \cdot 0.5\right)}\right)}^{2}}{\ell}\right)\right) \]
      associate-*r* [=>] 20.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot 0.5\right)}}^{2}}{\ell}\right)\right) \]

   5. Applied egg-rr 20.1

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

   6. Applied egg-rr 12.8

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \frac{h}{\frac{\ell}{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}} \cdot -0.5\right)}\right)}^{3}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 13.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.4 \cdot 10^{-160}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 + 0.5 \cdot \frac{h \cdot \left(M \cdot \frac{-0.5}{\frac{d}{D}}\right)}{\frac{\ell}{M \cdot \frac{0.5}{\frac{d}{D}}}}\right)\\ \mathbf{elif}\;h \leq -2.7 \cdot 10^{-302}:\\ \;\;\;\;\left(1 + 0.5 \cdot \frac{h \cdot \left(M \cdot \frac{-0.5}{\frac{d}{D}}\right)}{\frac{\ell}{M \cdot \frac{0.5}{\frac{d}{D}}}}\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;h \leq 1.15 \cdot 10^{-60}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(\frac{D}{\frac{\ell}{M}} \cdot \left(\frac{h}{d} \cdot \frac{M}{d}\right)\right) \cdot \left(D \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \frac{h}{\frac{\ell}{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}} \cdot -0.5\right)}\right)}^{3}\\ \end{array} \]

Alternatives

Alternative 1
Error	18.1
Cost	21716

\[\begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{\ell}}{h}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \left(1 + 0.5 \cdot \frac{h \cdot \left(M \cdot \frac{-0.5}{\frac{d}{D}}\right)}{\frac{\ell}{M \cdot \frac{0.5}{\frac{d}{D}}}}\right) \cdot \left(t_1 \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{if}\;d \leq -5 \cdot 10^{+132}:\\ \;\;\;\;d \cdot \left(-t_0\right)\\ \mathbf{elif}\;d \leq -1.65 \cdot 10^{-117}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -3.9 \cdot 10^{-242}:\\ \;\;\;\;\frac{D}{\frac{\frac{d}{M}}{M \cdot D}} \cdot \left(0.125 \cdot \sqrt{\frac{1}{\ell} \cdot \frac{h}{\ell \cdot \ell}}\right) - d \cdot t_0\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{-209}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{-75}:\\ \;\;\;\;\left(t_1 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(\frac{D}{\frac{\left(d \cdot \frac{d}{h}\right) \cdot \frac{\ell}{M \cdot D}}{M}} \cdot -0.25\right)\right)\\ \mathbf{elif}\;d \leq 1400000000000:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(\frac{D}{\frac{\ell}{M}} \cdot \left(\frac{h}{d} \cdot \frac{M}{d}\right)\right) \cdot \left(D \cdot -0.125\right)\right)\\ \end{array} \]

Alternative 2
Error	19.1
Cost	21580

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \left(t_1 \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(\frac{M \cdot D}{\left(d \cdot \frac{d}{h}\right) \cdot \frac{\ell}{M \cdot D}} \cdot -0.25\right)\right)\\ t_3 := \frac{h}{d} \cdot \frac{M}{d}\\ \mathbf{if}\;\ell \leq -3 \cdot 10^{+254}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -9 \cdot 10^{+188}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -3.2 \cdot 10^{-294}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{-110}:\\ \;\;\;\;t_0 \cdot \left(1 + \left(\frac{D}{\frac{\ell}{M}} \cdot t_3\right) \cdot \left(D \cdot -0.125\right)\right)\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{+156}:\\ \;\;\;\;t_0 \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+246}:\\ \;\;\;\;t_0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(t_3 \cdot \left(M \cdot \frac{D}{\ell}\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 6.8 \cdot 10^{+304}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{D \cdot D}{\frac{\ell}{h}} \cdot \frac{-0.125}{\frac{d}{M} \cdot \frac{d}{M}}\right)\right)\\ \end{array} \]

Alternative 3
Error	19.0
Cost	21580

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.2 \cdot 10^{-294}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(\frac{M \cdot D}{\left(d \cdot \frac{d}{h}\right) \cdot \frac{\ell}{M \cdot D}} \cdot -0.25\right)\right)\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{-104}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(\frac{D}{\frac{\ell}{M}} \cdot \left(\frac{h}{d} \cdot \frac{M}{d}\right)\right) \cdot \left(D \cdot -0.125\right)\right)\\ \mathbf{elif}\;\ell \leq 1.02 \cdot 10^{+217}:\\ \;\;\;\;\left(1 + 0.5 \cdot \frac{h \cdot \left(M \cdot \frac{-0.5}{\frac{d}{D}}\right)}{\frac{\ell}{M \cdot \frac{0.5}{\frac{d}{D}}}}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+245}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{\mathsf{fma}\left(-0.125, D \cdot \left(\frac{M}{d} \cdot \left(\frac{h}{d} \cdot \left(M \cdot \frac{D}{\ell}\right)\right)\right), 1\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 4
Error	16.3
Cost	21580

\[\begin{array}{l} t_0 := 1 + 0.5 \cdot \frac{h \cdot \left(M \cdot \frac{-0.5}{\frac{d}{D}}\right)}{\frac{\ell}{M \cdot \frac{0.5}{\frac{d}{D}}}}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot t_0\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{-102}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(\frac{D}{\frac{\ell}{M}} \cdot \left(\frac{h}{d} \cdot \frac{M}{d}\right)\right) \cdot \left(D \cdot -0.125\right)\right)\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+215}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+244}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{\mathsf{fma}\left(-0.125, D \cdot \left(\frac{M}{d} \cdot \left(\frac{h}{d} \cdot \left(M \cdot \frac{D}{\ell}\right)\right)\right), 1\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 5
Error	17.8
Cost	21400

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := -0.5 \cdot \frac{h}{\ell}\\ t_3 := \sqrt{\frac{\frac{1}{\ell}}{h}}\\ t_4 := \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{if}\;d \leq -3.8 \cdot 10^{+132}:\\ \;\;\;\;d \cdot \left(-t_3\right)\\ \mathbf{elif}\;d \leq -9.6 \cdot 10^{-119}:\\ \;\;\;\;\left(1 + 0.5 \cdot \frac{h \cdot \left(M \cdot \frac{-0.5}{\frac{d}{D}}\right)}{\frac{\ell}{M \cdot \frac{0.5}{\frac{d}{D}}}}\right) \cdot \left(t_1 \cdot t_0\right)\\ \mathbf{elif}\;d \leq -2.5 \cdot 10^{-242}:\\ \;\;\;\;\frac{D}{\frac{\frac{d}{M}}{M \cdot D}} \cdot \left(0.125 \cdot \sqrt{\frac{1}{\ell} \cdot \frac{h}{\ell \cdot \ell}}\right) - d \cdot t_3\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{-301}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \left(1 + {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot t_2\right)\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-143}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{\mathsf{fma}\left(-0.125, D \cdot \left(\frac{M}{d} \cdot \left(\frac{h}{d} \cdot \left(M \cdot \frac{D}{\ell}\right)\right)\right), 1\right)}{\sqrt{\ell}}\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{+48}:\\ \;\;\;\;t_4 \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot t_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_4 \cdot \left(1 + \left(\frac{D}{\frac{\ell}{M}} \cdot \left(\frac{h}{d} \cdot \frac{M}{d}\right)\right) \cdot \left(D \cdot -0.125\right)\right)\\ \end{array} \]

Alternative 6
Error	18.6
Cost	21332

\[\begin{array}{l} t_0 := \left(1 + 0.5 \cdot \frac{h \cdot \left(M \cdot \frac{-0.5}{\frac{d}{D}}\right)}{\frac{\ell}{M \cdot \frac{0.5}{\frac{d}{D}}}}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ t_1 := \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{if}\;d \leq -3.1 \cdot 10^{+132}:\\ \;\;\;\;d \cdot \left(-t_1\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-118}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -6.5 \cdot 10^{-242}:\\ \;\;\;\;\frac{D}{\frac{\frac{d}{M}}{M \cdot D}} \cdot \left(0.125 \cdot \sqrt{\frac{1}{\ell} \cdot \frac{h}{\ell \cdot \ell}}\right) - d \cdot t_1\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{-295}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{\mathsf{fma}\left(-0.125, D \cdot \left(\frac{M}{d} \cdot \left(\frac{h}{d} \cdot \left(M \cdot \frac{D}{\ell}\right)\right)\right), 1\right)}{\sqrt{\ell}}\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{+19}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 7
Error	18.6
Cost	21332

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{\frac{1}{\ell}}{h}}\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := \left(1 + 0.5 \cdot \frac{h \cdot \left(M \cdot \frac{-0.5}{\frac{d}{D}}\right)}{\frac{\ell}{M \cdot \frac{0.5}{\frac{d}{D}}}}\right) \cdot \left(t_2 \cdot t_0\right)\\ \mathbf{if}\;d \leq -6.2 \cdot 10^{+132}:\\ \;\;\;\;d \cdot \left(-t_1\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-118}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq -2.5 \cdot 10^{-242}:\\ \;\;\;\;\frac{D}{\frac{\frac{d}{M}}{M \cdot D}} \cdot \left(0.125 \cdot \sqrt{\frac{1}{\ell} \cdot \frac{h}{\ell \cdot \ell}}\right) - d \cdot t_1\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-291}:\\ \;\;\;\;t_2 \cdot \left(t_0 \cdot \left(1 + {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{-98}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{\mathsf{fma}\left(-0.125, D \cdot \left(\frac{M}{d} \cdot \left(\frac{h}{d} \cdot \left(M \cdot \frac{D}{\ell}\right)\right)\right), 1\right)}{\sqrt{\ell}}\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{+19}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 8
Error	20.0
Cost	21004

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(1 + 0.5 \cdot \frac{h \cdot \left(M \cdot \frac{-0.5}{\frac{d}{D}}\right)}{\frac{\ell}{M \cdot \frac{0.5}{\frac{d}{D}}}}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 9.2 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+158}:\\ \;\;\;\;t_0 \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+246}:\\ \;\;\;\;t_0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\left(\frac{h}{d} \cdot \frac{M}{d}\right) \cdot \left(M \cdot \frac{D}{\ell}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 9
Error	19.2
Cost	15312

\[\begin{array}{l} t_0 := \left(1 + 0.5 \cdot \frac{h \cdot \left(M \cdot \frac{-0.5}{\frac{d}{D}}\right)}{\frac{\ell}{M \cdot \frac{0.5}{\frac{d}{D}}}}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ t_1 := \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{if}\;d \leq -2.75 \cdot 10^{+132}:\\ \;\;\;\;d \cdot \left(-t_1\right)\\ \mathbf{elif}\;d \leq -4.6 \cdot 10^{-117}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -2.5 \cdot 10^{-242}:\\ \;\;\;\;\frac{D}{\frac{\frac{d}{M}}{M \cdot D}} \cdot \left(0.125 \cdot \sqrt{\frac{1}{\ell} \cdot \frac{h}{\ell \cdot \ell}}\right) - d \cdot t_1\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{+19}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 10
Error	19.7
Cost	15312

\[\begin{array}{l} t_0 := \frac{\ell}{M \cdot \frac{0.5}{\frac{d}{D}}}\\ t_1 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{if}\;d \leq -3.1 \cdot 10^{+132}:\\ \;\;\;\;d \cdot \left(-t_2\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-118}:\\ \;\;\;\;\left(1 + 0.5 \cdot \frac{h \cdot \left(M \cdot \frac{-0.5}{\frac{d}{D}}\right)}{t_0}\right) \cdot t_1\\ \mathbf{elif}\;d \leq -4.6 \cdot 10^{-241}:\\ \;\;\;\;\frac{D}{\frac{\frac{d}{M}}{M \cdot D}} \cdot \left(0.125 \cdot \sqrt{\frac{1}{\ell} \cdot \frac{h}{\ell \cdot \ell}}\right) - d \cdot t_2\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{+19}:\\ \;\;\;\;t_1 \cdot \left(1 + -0.5 \cdot \frac{\frac{D}{d} \cdot \frac{h}{\frac{2}{M}}}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 11
Error	22.7
Cost	15192

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-95}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-201}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d \cdot \frac{2}{M}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -2.6 \cdot 10^{-282}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{-130}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \leq 4.1 \cdot 10^{+245}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\left(\frac{h}{d} \cdot \frac{M}{d}\right) \cdot \left(M \cdot \frac{D}{\ell}\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 6.8 \cdot 10^{+304}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{D \cdot D}{\frac{\ell}{h}} \cdot \frac{-0.125}{\frac{d}{M} \cdot \frac{d}{M}}\right)\right)\\ \end{array} \]

Alternative 12
Error	22.0
Cost	15184

\[\begin{array}{l} t_0 := \frac{\ell}{M \cdot D}\\ t_1 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ t_2 := \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\left(\frac{h}{d} \cdot \frac{M}{d}\right) \cdot \left(M \cdot \frac{D}{\ell}\right)\right)\right)\right)\\ \mathbf{if}\;d \leq -4.9 \cdot 10^{+132}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;d \leq -5.5 \cdot 10^{-253}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(\frac{M \cdot D}{\left(d \cdot \frac{d}{h}\right) \cdot t_0} \cdot -0.25\right)\right) \cdot t_1\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{-143}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{+19}:\\ \;\;\;\;t_1 \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \frac{D}{\frac{t_0 \cdot \left(d \cdot d\right)}{h \cdot M}}\right)\right)\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{+197}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 13
Error	19.9
Cost	15061

\[\begin{array}{l} \mathbf{if}\;d \leq -2.75 \cdot 10^{+132}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;d \leq -4.1 \cdot 10^{-166}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(\frac{D}{\frac{\left(d \cdot \frac{d}{h}\right) \cdot \frac{\ell}{M \cdot D}}{M}} \cdot -0.25\right)\right)\\ \mathbf{elif}\;d \leq 1.12 \cdot 10^{-278}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{-98} \lor \neg \left(d \leq 2.9 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\left(\frac{h}{d} \cdot \frac{M}{d}\right) \cdot \left(M \cdot \frac{D}{\ell}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 14
Error	24.2
Cost	15060

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := t_0 \cdot t_1\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-114}:\\ \;\;\;\;t_2 \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \frac{D}{\frac{\frac{\ell}{M \cdot D} \cdot \left(d \cdot d\right)}{h \cdot M}}\right)\right)\\ \mathbf{elif}\;\ell \leq -3.2 \cdot 10^{-294}:\\ \;\;\;\;t_2 \cdot \left(1 + 0.5 \cdot \left(\left(\left(\left(M \cdot D\right) \cdot \frac{M \cdot D}{\ell}\right) \cdot \frac{\frac{h}{d}}{d}\right) \cdot -0.25\right)\right)\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{-130}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+245}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\left(\frac{h}{d} \cdot \frac{M}{d}\right) \cdot \left(M \cdot \frac{D}{\ell}\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 6.8 \cdot 10^{+304}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(\frac{D \cdot D}{\frac{\ell}{h}} \cdot \frac{-0.125}{\frac{d}{M} \cdot \frac{d}{M}}\right)\right)\\ \end{array} \]

Alternative 15
Error	18.8
Cost	15052

\[\begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{if}\;d \leq -6.5 \cdot 10^{+132}:\\ \;\;\;\;d \cdot \left(-t_0\right)\\ \mathbf{elif}\;d \leq -2.3 \cdot 10^{-173}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(\frac{M \cdot D}{\left(d \cdot \frac{d}{h}\right) \cdot \frac{\ell}{M \cdot D}} \cdot -0.25\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;d \leq 1.12 \cdot 10^{-287}:\\ \;\;\;\;\frac{D}{\frac{\frac{d}{M}}{M \cdot D}} \cdot \left(0.125 \cdot \sqrt{\frac{1}{\ell} \cdot \frac{h}{\ell \cdot \ell}}\right) - d \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\left(\frac{h}{d} \cdot \frac{M}{d}\right) \cdot \left(M \cdot \frac{D}{\ell}\right)\right)\right)\right)\\ \end{array} \]

Alternative 16
Error	19.6
Cost	14928

\[\begin{array}{l} \mathbf{if}\;d \leq -3.8 \cdot 10^{+132}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;d \leq -2.6 \cdot 10^{-97}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(\left(\left(\left(M \cdot D\right) \cdot \frac{M \cdot D}{\ell}\right) \cdot \frac{\frac{h}{d}}{d}\right) \cdot -0.25\right)\right)\\ \mathbf{elif}\;d \leq -2.5 \cdot 10^{-242}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{-271}:\\ \;\;\;\;\sqrt{\frac{d}{\ell \cdot \frac{h}{d}}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\left(\frac{h}{d} \cdot \frac{M}{d}\right) \cdot \left(M \cdot \frac{D}{\ell}\right)\right)\right)\right)\\ \end{array} \]

Alternative 17
Error	25.5
Cost	14600

\[\begin{array}{l} \mathbf{if}\;d \leq -2.75 \cdot 10^{+132}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{-135}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d \cdot \frac{2}{M}}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 18
Error	25.6
Cost	14600

\[\begin{array}{l} \mathbf{if}\;d \leq -3.8 \cdot 10^{+132}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;d \leq 3 \cdot 10^{-135}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot \left(0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 19
Error	23.5
Cost	13380

\[\begin{array}{l} \mathbf{if}\;d \leq 1.15 \cdot 10^{-289}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 20
Error	23.5
Cost	13252

\[\begin{array}{l} \mathbf{if}\;d \leq 10^{-289}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 21
Error	28.5
Cost	7044

\[\begin{array}{l} \mathbf{if}\;h \leq -4.2 \cdot 10^{-277}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 22
Error	28.3
Cost	7044

\[\begin{array}{l} \mathbf{if}\;h \leq -4.2 \cdot 10^{-277}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 23
Error	35.4
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq -3.2 \cdot 10^{-128}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 24
Error	34.1
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq 2.4 \cdot 10^{-139}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 25
Error	43.3
Cost	6784

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

Alternative 26
Error	43.3
Cost	6720

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

Reproduce

herbie shell --seed 2023017 
(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)))))