Henrywood and Agarwal, Equation (12)

Average Error: 26.6 → 17.0

Time: 48.6s

Precision: binary64

Cost: 33668

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+103}:\\ \;\;\;\;\frac{t_2}{\sqrt{-h}} \cdot \left(t_0 \cdot \mathsf{fma}\left(-0.5, {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}, 1\right)\right)\\ \mathbf{elif}\;\ell \leq -2.7 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\frac{D}{\frac{\frac{d}{M}}{D \cdot M}} \cdot 0.125, \sqrt{\frac{h}{{\ell}^{3}}}, \left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -5.5 \cdot 10^{-108}:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(1 + -0.5 \cdot {\left(\frac{M}{\frac{2}{\frac{D}{d}}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(t_1 \cdot \frac{t_2}{\sqrt{-\ell}}\right) \cdot \left(1 + -0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}, 1\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

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 l))) (t_1 (sqrt (/ d h))) (t_2 (sqrt (- d))))
   (if (<= l -4.5e+103)
     (*
      (/ t_2 (sqrt (- h)))
      (* t_0 (fma -0.5 (* (pow (* D (/ (/ M d) 2.0)) 2.0) (/ h l)) 1.0)))
     (if (<= l -2.7e-11)
       (fma
        (* (/ D (/ (/ d M) (* D M))) 0.125)
        (sqrt (/ h (pow l 3.0)))
        (* (- d) (sqrt (/ (/ 1.0 l) h))))
       (if (<= l -5.5e-108)
         (*
          (* t_0 t_1)
          (+ 1.0 (* -0.5 (pow (* (/ M (/ 2.0 (/ D d))) (sqrt (/ h l))) 2.0))))
         (if (<= l -4e-310)
           (*
            (* t_1 (/ t_2 (sqrt (- l))))
            (+ 1.0 (* -0.5 (/ (* h (pow (* (/ D d) (/ M 2.0)) 2.0)) l))))
           (*
            (fma -0.5 (* (/ h l) (pow (* D (/ M (* d 2.0))) 2.0)) 1.0)
            (/ d (* (sqrt l) (sqrt h))))))))))

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 / l));
	double t_1 = sqrt((d / h));
	double t_2 = sqrt(-d);
	double tmp;
	if (l <= -4.5e+103) {
		tmp = (t_2 / sqrt(-h)) * (t_0 * fma(-0.5, (pow((D * ((M / d) / 2.0)), 2.0) * (h / l)), 1.0));
	} else if (l <= -2.7e-11) {
		tmp = fma(((D / ((d / M) / (D * M))) * 0.125), sqrt((h / pow(l, 3.0))), (-d * sqrt(((1.0 / l) / h))));
	} else if (l <= -5.5e-108) {
		tmp = (t_0 * t_1) * (1.0 + (-0.5 * pow(((M / (2.0 / (D / d))) * sqrt((h / l))), 2.0)));
	} else if (l <= -4e-310) {
		tmp = (t_1 * (t_2 / sqrt(-l))) * (1.0 + (-0.5 * ((h * pow(((D / d) * (M / 2.0)), 2.0)) / l)));
	} else {
		tmp = fma(-0.5, ((h / l) * pow((D * (M / (d * 2.0))), 2.0)), 1.0) * (d / (sqrt(l) * sqrt(h)));
	}
	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 / l))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -4.5e+103)
		tmp = Float64(Float64(t_2 / sqrt(Float64(-h))) * Float64(t_0 * fma(-0.5, Float64((Float64(D * Float64(Float64(M / d) / 2.0)) ^ 2.0) * Float64(h / l)), 1.0)));
	elseif (l <= -2.7e-11)
		tmp = fma(Float64(Float64(D / Float64(Float64(d / M) / Float64(D * M))) * 0.125), sqrt(Float64(h / (l ^ 3.0))), Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h))));
	elseif (l <= -5.5e-108)
		tmp = Float64(Float64(t_0 * t_1) * Float64(1.0 + Float64(-0.5 * (Float64(Float64(M / Float64(2.0 / Float64(D / d))) * sqrt(Float64(h / l))) ^ 2.0))));
	elseif (l <= -4e-310)
		tmp = Float64(Float64(t_1 * Float64(t_2 / sqrt(Float64(-l)))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h * (Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0)) / l))));
	else
		tmp = Float64(fma(-0.5, Float64(Float64(h / l) * (Float64(D * Float64(M / Float64(d * 2.0))) ^ 2.0)), 1.0) * Float64(d / Float64(sqrt(l) * sqrt(h))));
	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[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -4.5e+103], N[(N[(t$95$2 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(-0.5 * N[(N[Power[N[(D * N[(N[(M / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2.7e-11], N[(N[(N[(D / N[(N[(d / M), $MachinePrecision] / N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5.5e-108], N[(N[(t$95$0 * t$95$1), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[Power[N[(N[(M / N[(2.0 / N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -4e-310], N[(N[(t$95$1 * N[(t$95$2 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -4.50000000000000001e103

   1. Initial program 28.8

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

   2. Simplified 29.1

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

      [Start] 28.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 28.8	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 28.8	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 28.8	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 28.8	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 28.8	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      sub-neg [=>] 28.8	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]
      +-commutative [=>] 28.8	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]
      associate-*l* [=>] 28.8	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) + 1\right)\right) \]
      distribute-lft-neg-in [=>] 28.8	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(-\frac{1}{2}\right) \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)} + 1\right)\right) \]
      fma-def [=>] 28.8	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(-\frac{1}{2}, {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}, 1\right)}\right) \]

   3. Applied egg-rr 20.7

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

   if -4.50000000000000001e103 < l < -2.70000000000000005e-11

   1. Initial program 18.5

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

   2. Simplified 19.3

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

      [Start] 18.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 18.5	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 18.5	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 18.5	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 18.5	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 18.5	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 18.5	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 19.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 18.4

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

   4. Taylor expanded in d around -inf 25.7

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

   5. Simplified 10.3

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

      [Start] 25.7	\[ 0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + -1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]
      associate-*r* [=>] 25.7	\[ \color{blue}{\left(0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} + -1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]
      fma-def [=>] 25.7	\[ \color{blue}{\mathsf{fma}\left(0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}, \sqrt{\frac{h}{{\ell}^{3}}}, -1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]
      *-commutative [=>] 25.7	\[ \mathsf{fma}\left(\color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot 0.125}, \sqrt{\frac{h}{{\ell}^{3}}}, -1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right) \]
      associate-/l* [=>] 25.5	\[ \mathsf{fma}\left(\color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot 0.125, \sqrt{\frac{h}{{\ell}^{3}}}, -1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right) \]
      unpow2 [=>] 25.5	\[ \mathsf{fma}\left(\frac{\color{blue}{D \cdot D}}{\frac{d}{{M}^{2}}} \cdot 0.125, \sqrt{\frac{h}{{\ell}^{3}}}, -1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right) \]
      associate-/l* [=>] 19.9	\[ \mathsf{fma}\left(\color{blue}{\frac{D}{\frac{\frac{d}{{M}^{2}}}{D}}} \cdot 0.125, \sqrt{\frac{h}{{\ell}^{3}}}, -1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right) \]
      unpow2 [=>] 19.9	\[ \mathsf{fma}\left(\frac{D}{\frac{\frac{d}{\color{blue}{M \cdot M}}}{D}} \cdot 0.125, \sqrt{\frac{h}{{\ell}^{3}}}, -1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right) \]
      associate-/r* [=>] 15.4	\[ \mathsf{fma}\left(\frac{D}{\frac{\color{blue}{\frac{\frac{d}{M}}{M}}}{D}} \cdot 0.125, \sqrt{\frac{h}{{\ell}^{3}}}, -1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right) \]
      associate-/l/ [=>] 10.7	\[ \mathsf{fma}\left(\frac{D}{\color{blue}{\frac{\frac{d}{M}}{D \cdot M}}} \cdot 0.125, \sqrt{\frac{h}{{\ell}^{3}}}, -1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right) \]
      *-commutative [=>] 10.7	\[ \mathsf{fma}\left(\frac{D}{\frac{\frac{d}{M}}{D \cdot M}} \cdot 0.125, \sqrt{\frac{h}{{\ell}^{3}}}, -1 \cdot \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)}\right) \]
      mul-1-neg [=>] 10.7	\[ \mathsf{fma}\left(\frac{D}{\frac{\frac{d}{M}}{D \cdot M}} \cdot 0.125, \sqrt{\frac{h}{{\ell}^{3}}}, \color{blue}{-\sqrt{\frac{1}{\ell \cdot h}} \cdot d}\right) \]
      *-commutative [<=] 10.7	\[ \mathsf{fma}\left(\frac{D}{\frac{\frac{d}{M}}{D \cdot M}} \cdot 0.125, \sqrt{\frac{h}{{\ell}^{3}}}, -\color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}}\right) \]
      distribute-rgt-neg-in [=>] 10.7	\[ \mathsf{fma}\left(\frac{D}{\frac{\frac{d}{M}}{D \cdot M}} \cdot 0.125, \sqrt{\frac{h}{{\ell}^{3}}}, \color{blue}{d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)}\right) \]
      associate-/r* [=>] 10.3	\[ \mathsf{fma}\left(\frac{D}{\frac{\frac{d}{M}}{D \cdot M}} \cdot 0.125, \sqrt{\frac{h}{{\ell}^{3}}}, d \cdot \left(-\sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right)\right) \]

   if -2.70000000000000005e-11 < l < -5.50000000000000031e-108

   1. Initial program 20.4

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

   2. Simplified 21.1

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

      [Start] 20.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 20.4	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 20.4	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 20.4	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 20.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 20.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 20.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 21.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 20.4

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

   4. Simplified 20.5

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

      [Start] 20.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]
      associate-/r/ [<=] 20.5	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{M}{\frac{2}{\frac{D}{d}}}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   if -5.50000000000000031e-108 < l < -3.999999999999988e-310

   1. Initial program 32.7

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

   2. Simplified 33.3

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

      [Start] 32.7	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 32.7	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 32.7	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 32.7	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 32.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 32.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 32.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 33.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 26.7

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

   4. Applied egg-rr 13.1

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

   if -3.999999999999988e-310 < l

   1. Initial program 27.0

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

   2. Simplified 27.3

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

      [Start] 27.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 27.1	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 27.1	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 27.1	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 27.1	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 27.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      sub-neg [=>] 27.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]
      +-commutative [=>] 27.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]
      associate-*l* [=>] 27.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) + 1\right)\right) \]
      distribute-lft-neg-in [=>] 27.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(-\frac{1}{2}\right) \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)} + 1\right)\right) \]
      fma-def [=>] 27.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(-\frac{1}{2}, {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}, 1\right)}\right) \]

   3. Applied egg-rr 16.8

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

   4. Simplified 16.8

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

      [Start] 16.8	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \left(\left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]
      distribute-rgt1-in [=>] 16.8	\[ \color{blue}{\left(\left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5 + 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]
      *-commutative [=>] 16.8	\[ \left(\color{blue}{-0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)} + 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]
      fma-udef [<=] 16.8	\[ \color{blue}{\mathsf{fma}\left(-0.5, {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}, 1\right)} \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]
      *-commutative [<=] 16.8	\[ \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(-0.5, {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}, 1\right)} \]
      *-commutative [=>] 16.8	\[ \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \mathsf{fma}\left(-0.5, {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}, 1\right) \]
      associate-/l/ [=>] 16.8	\[ \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \mathsf{fma}\left(-0.5, {\left(D \cdot \color{blue}{\frac{M}{2 \cdot d}}\right)}^{2} \cdot \frac{h}{\ell}, 1\right) \]

3. Recombined 5 regimes into one program.

4. Final simplification 17.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+103}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}, 1\right)\right)\\ \mathbf{elif}\;\ell \leq -2.7 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\frac{D}{\frac{\frac{d}{M}}{D \cdot M}} \cdot 0.125, \sqrt{\frac{h}{{\ell}^{3}}}, \left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -5.5 \cdot 10^{-108}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + -0.5 \cdot {\left(\frac{M}{\frac{2}{\frac{D}{d}}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 + -0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}, 1\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternatives

Alternative 1
Error	17.2
Cost	27792

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\\ t_3 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -4.4 \cdot 10^{+122}:\\ \;\;\;\;\left(\frac{t_3}{\sqrt{-h}} \cdot t_0\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot t_2\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\frac{D}{\frac{\frac{d}{M}}{D \cdot M}} \cdot 0.125, \sqrt{\frac{h}{{\ell}^{3}}}, \left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -5.5 \cdot 10^{-108}:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(1 + -0.5 \cdot {\left(\frac{M}{\frac{2}{\frac{D}{d}}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(t_1 \cdot \frac{t_3}{\sqrt{-\ell}}\right) \cdot \left(1 + -0.5 \cdot \frac{h \cdot t_2}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}, 1\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 2
Error	17.3
Cost	27660

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ t_1 := {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\\ \mathbf{if}\;\ell \leq -5.7 \cdot 10^{+122}:\\ \;\;\;\;t_0 \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot t_1\right)\right)\\ \mathbf{elif}\;\ell \leq -8 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\frac{D}{\frac{\frac{d}{M}}{D \cdot M}} \cdot 0.125, \sqrt{\frac{h}{{\ell}^{3}}}, \left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t_0 \cdot \left(1 + -0.5 \cdot \frac{h \cdot t_1}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}, 1\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 3
Error	19.6
Cost	27532

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+109}:\\ \;\;\;\;\frac{t_0}{\frac{\sqrt{-h}}{\sqrt{-d}}}\\ \mathbf{elif}\;\ell \leq -8.4 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\frac{D}{\frac{\frac{d}{M}}{D \cdot M}} \cdot 0.125, \sqrt{\frac{h}{{\ell}^{3}}}, \left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -9 \cdot 10^{-290}:\\ \;\;\;\;\left(t_0 \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + -0.5 \cdot {\left(\frac{M}{\frac{2}{\frac{D}{d}}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}, 1\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 4
Error	20.0
Cost	27532

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;\ell \leq -4.4 \cdot 10^{+122}:\\ \;\;\;\;\left(t_0 \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -2.05 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\frac{D}{\frac{\frac{d}{M}}{D \cdot M}} \cdot 0.125, \sqrt{\frac{h}{{\ell}^{3}}}, \left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -9 \cdot 10^{-290}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot t_0\right) \cdot \left(1 + -0.5 \cdot {\left(\frac{M}{\frac{2}{\frac{D}{d}}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}, 1\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 5
Error	20.3
Cost	27408

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;\ell \leq -2.45 \cdot 10^{+87}:\\ \;\;\;\;\frac{t_0}{\frac{\sqrt{-h}}{\sqrt{-d}}}\\ \mathbf{elif}\;\ell \leq -1.2 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \frac{D}{\frac{\frac{d}{M \cdot M}}{D}}\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq -4.4 \cdot 10^{-40}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \mathsf{fma}\left(-0.125, \left(M \cdot \left(h \cdot M\right)\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{\ell \cdot d}\right), 1\right)\right)\\ \mathbf{elif}\;\ell \leq -9 \cdot 10^{-290}:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{M}{\frac{2}{\frac{D}{d}}}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}, 1\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 6
Error	19.2
Cost	27336

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -1.65 \cdot 10^{+109}:\\ \;\;\;\;\frac{t_0}{\frac{\sqrt{-h}}{\sqrt{-d}}}\\ \mathbf{elif}\;\ell \leq -7.8 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\frac{D}{\frac{\frac{d}{M}}{D \cdot M}} \cdot 0.125, \sqrt{\frac{h}{{\ell}^{3}}}, \left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -9 \cdot 10^{-290}:\\ \;\;\;\;\left(t_0 \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(\frac{M}{\frac{2}{\frac{D}{d}}}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}, 1\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 7
Error	21.1
Cost	21516

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := t_0 \cdot t_1\\ \mathbf{if}\;d \leq -2 \cdot 10^{+129}:\\ \;\;\;\;\frac{\sqrt{-d}}{\frac{\sqrt{-h}}{t_0}}\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{-158}:\\ \;\;\;\;t_2 \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 1.28 \cdot 10^{-5}:\\ \;\;\;\;\left(t_1 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 + -0.5 \cdot \left(\left(M \cdot \left(D \cdot \left(D \cdot \frac{\frac{M}{d}}{\ell \cdot \frac{-d}{h}}\right)\right)\right) \cdot -0.25\right)\right)\\ \mathbf{elif}\;d \leq 1.16 \cdot 10^{+47}:\\ \;\;\;\;t_2 \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 8
Error	22.0
Cost	20868

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -1.7 \cdot 10^{-140}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 + {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{-0.5}{\frac{\ell}{h}}\right)\right)\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{t_1}{\frac{\sqrt{-h}}{\sqrt{-d}}}\\ \mathbf{elif}\;h \leq 1.02 \cdot 10^{-172}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;h \leq 1.9 \cdot 10^{+37}:\\ \;\;\;\;\left(t_1 \cdot t_0\right) \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \left(D \cdot \frac{\frac{h}{\frac{d}{M}} \cdot \frac{D}{\frac{d}{M}}}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{-0.5}}{\frac{\sqrt{h}}{d}}\\ \end{array} \]

Alternative 9
Error	21.3
Cost	20868

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := t_0 \cdot \sqrt{\frac{d}{h}}\\ \mathbf{if}\;h \leq -2.1 \cdot 10^{-141}:\\ \;\;\;\;t_1 \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(\frac{M}{\frac{2}{\frac{D}{d}}}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{t_0}{\frac{\sqrt{-h}}{\sqrt{-d}}}\\ \mathbf{elif}\;h \leq 1.15 \cdot 10^{-172}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;h \leq 1.65 \cdot 10^{+39}:\\ \;\;\;\;t_1 \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \left(D \cdot \frac{\frac{h}{\frac{d}{M}} \cdot \frac{D}{\frac{d}{M}}}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{-0.5}}{\frac{\sqrt{h}}{d}}\\ \end{array} \]

Alternative 10
Error	21.9
Cost	20172

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{-d}\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := \left(t_0 \cdot t_2\right) \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \left(D \cdot \frac{\frac{h}{\frac{d}{M}} \cdot \frac{D}{\frac{d}{M}}}{\ell}\right)\right)\right)\\ \mathbf{if}\;h \leq -1 \cdot 10^{+208}:\\ \;\;\;\;\frac{t_1 \cdot t_2}{\sqrt{-\ell}}\\ \mathbf{elif}\;h \leq -2.45 \cdot 10^{-142}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{t_0}{\frac{\sqrt{-h}}{t_1}}\\ \mathbf{elif}\;h \leq 1.12 \cdot 10^{-172}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;h \leq 7.5 \cdot 10^{+38}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{-0.5}}{\frac{\sqrt{h}}{d}}\\ \end{array} \]

Alternative 11
Error	22.0
Cost	20040

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \left(t_0 \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \left(D \cdot \frac{\frac{h}{\frac{d}{M}} \cdot \frac{D}{\frac{d}{M}}}{\ell}\right)\right)\right)\\ \mathbf{if}\;h \leq -6.8 \cdot 10^{-62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\frac{\sqrt{-h}}{t_0}}\\ \mathbf{elif}\;h \leq 1.15 \cdot 10^{-172}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;h \leq 7.5 \cdot 10^{+38}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{-0.5}}{\frac{\sqrt{h}}{d}}\\ \end{array} \]

Alternative 12
Error	22.0
Cost	20040

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \left(t_0 \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \left(D \cdot \frac{\frac{h}{\frac{d}{M}} \cdot \frac{D}{\frac{d}{M}}}{\ell}\right)\right)\right)\\ \mathbf{if}\;h \leq -5.8 \cdot 10^{-142}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{t_0}{\frac{\sqrt{-h}}{\sqrt{-d}}}\\ \mathbf{elif}\;h \leq 1.02 \cdot 10^{-172}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;h \leq 1.5 \cdot 10^{+40}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{-0.5}}{\frac{\sqrt{h}}{d}}\\ \end{array} \]

Alternative 13
Error	22.8
Cost	15316

\[\begin{array}{l} t_0 := \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \left(D \cdot \frac{\frac{h}{\frac{d}{M}} \cdot \frac{D}{\frac{d}{M}}}{\ell}\right)\right)\right)\\ \mathbf{if}\;h \leq -4.05 \cdot 10^{-26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq -8.5 \cdot 10^{-285}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq 1.15 \cdot 10^{-172}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;h \leq 7 \cdot 10^{+38}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{-0.5}}{\frac{\sqrt{h}}{d}}\\ \end{array} \]

Alternative 14
Error	23.0
Cost	15316

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ t_1 := t_0 \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \left(D \cdot \frac{\frac{h}{\frac{d}{M}} \cdot \frac{D}{\frac{d}{M}}}{\ell}\right)\right)\right)\\ \mathbf{if}\;h \leq -7.2 \cdot 10^{-26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;h \leq -8 \cdot 10^{-285}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;h \leq 6 \cdot 10^{-307}:\\ \;\;\;\;t_0 \cdot \left(1 + -0.5 \cdot \left(\left(M \cdot \left(\left(D \cdot D\right) \cdot \frac{\frac{\frac{M}{d}}{\ell}}{\frac{-d}{h}}\right)\right) \cdot -0.25\right)\right)\\ \mathbf{elif}\;h \leq 6.5 \cdot 10^{-173}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;h \leq 6.7 \cdot 10^{+38}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{-0.5}}{\frac{\sqrt{h}}{d}}\\ \end{array} \]

Alternative 15
Error	25.0
Cost	13648

\[\begin{array}{l} t_0 := \frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{if}\;\ell \leq -9.2 \cdot 10^{+122}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -1.3 \cdot 10^{-11}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -2.2 \cdot 10^{-230}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{-296}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 16
Error	23.1
Cost	13648

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{if}\;h \leq -5.6 \cdot 10^{+112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq -8.2 \cdot 10^{-285}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;h \leq -6.5 \cdot 10^{-296}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h \cdot \frac{\ell}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 17
Error	23.5
Cost	13512

\[\begin{array}{l} \mathbf{if}\;d \leq -4.6 \cdot 10^{+129}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;d \leq 1.42 \cdot 10^{-298}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 18
Error	23.7
Cost	13384

\[\begin{array}{l} \mathbf{if}\;h \leq -7.4 \cdot 10^{+112}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;h \leq -1.4 \cdot 10^{-303}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 19
Error	27.5
Cost	7244

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{if}\;h \leq -1.7 \cdot 10^{+114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq -1.4 \cdot 10^{-303}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;h \leq 4.6 \cdot 10^{+60}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 20
Error	27.6
Cost	7244

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{if}\;h \leq -1.05 \cdot 10^{+114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq -1.4 \cdot 10^{-303}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;h \leq 4.8 \cdot 10^{+60}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 21
Error	33.4
Cost	7113

\[\begin{array}{l} \mathbf{if}\;h \leq 6 \cdot 10^{-307} \lor \neg \left(h \leq 3.7 \cdot 10^{+59}\right):\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]

Alternative 22
Error	44.1
Cost	6784

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

Alternative 23
Error	44.1
Cost	6720

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

Reproduce

herbie shell --seed 2023011 
(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)))))