Henrywood and Agarwal, Equation (12)

Average Error: 26.7 → 17.2

Time: 50.4s

Precision: binary64

Cost: 40200

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{if}\;d \leq -980000000:\\ \;\;\;\;\left(\frac{t_0}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -2.95 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{t_0}{\sqrt{-\ell}} \cdot \mathsf{fma}\left(-0.5, {\left(\frac{\frac{M}{2} \cdot D}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}, 1\right)\right)\\ \mathbf{elif}\;d \leq 0.0054:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}, 1\right) \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (- d))) (t_1 (/ d (* (sqrt h) (sqrt l)))))
   (if (<= d -980000000.0)
     (*
      (* (/ t_0 (sqrt (- h))) (sqrt (/ d l)))
      (- 1.0 (* 0.5 (* (pow (* (/ M 2.0) (/ D d)) 2.0) (/ h l)))))
     (if (<= d -2.95e-288)
       (*
        (sqrt (/ d h))
        (*
         (/ t_0 (sqrt (- l)))
         (fma -0.5 (pow (* (/ (* (/ M 2.0) D) d) (sqrt (/ h l))) 2.0) 1.0)))
       (if (<= d 0.0054)
         (* (fma (/ h l) (* -0.5 (pow (* (/ 0.5 d) (* M D)) 2.0)) 1.0) t_1)
         t_1)))))

C

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

↓

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(-d);
	double t_1 = d / (sqrt(h) * sqrt(l));
	double tmp;
	if (d <= -980000000.0) {
		tmp = ((t_0 / sqrt(-h)) * sqrt((d / l))) * (1.0 - (0.5 * (pow(((M / 2.0) * (D / d)), 2.0) * (h / l))));
	} else if (d <= -2.95e-288) {
		tmp = sqrt((d / h)) * ((t_0 / sqrt(-l)) * fma(-0.5, pow(((((M / 2.0) * D) / d) * sqrt((h / l))), 2.0), 1.0));
	} else if (d <= 0.0054) {
		tmp = fma((h / l), (-0.5 * pow(((0.5 / d) * (M * D)), 2.0)), 1.0) * t_1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

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

↓

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(-d))
	t_1 = Float64(d / Float64(sqrt(h) * sqrt(l)))
	tmp = 0.0
	if (d <= -980000000.0)
		tmp = Float64(Float64(Float64(t_0 / sqrt(Float64(-h))) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(0.5 * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * Float64(h / l)))));
	elseif (d <= -2.95e-288)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(t_0 / sqrt(Float64(-l))) * fma(-0.5, (Float64(Float64(Float64(Float64(M / 2.0) * D) / d) * sqrt(Float64(h / l))) ^ 2.0), 1.0)));
	elseif (d <= 0.0054)
		tmp = Float64(fma(Float64(h / l), Float64(-0.5 * (Float64(Float64(0.5 / d) * Float64(M * D)) ^ 2.0)), 1.0) * t_1);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

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

↓

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$1 = N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -980000000.0], N[(N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2.95e-288], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(N[(N[(M / 2.0), $MachinePrecision] * D), $MachinePrecision] / d), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 0.0054], N[(N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(0.5 / d), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -9.8e8

   1. Initial program 22.8

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

   2. Simplified 22.6

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

      [Start] 22.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 22.8	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 22.8	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 22.8	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 22.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 22.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 22.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 22.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 12.9

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

   if -9.8e8 < d < -2.95000000000000007e-288

   1. Initial program 28.2

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

   2. Simplified 29.6

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

      [Start] 28.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 28.5	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 28.5	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 28.5	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 28.5	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 28.5	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      sub-neg [=>] 28.5	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]
      +-commutative [=>] 28.5	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]
      associate-*l* [=>] 28.5	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) + 1\right)\right) \]
      distribute-lft-neg-in [=>] 28.5	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(-\frac{1}{2}\right) \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)} + 1\right)\right) \]
      fma-def [=>] 28.5	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(-\frac{1}{2}, {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}, 1\right)}\right) \]

   3. Applied egg-rr 27.6

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

   4. Applied egg-rr 25.7

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

   5. Applied egg-rr 19.3

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

   if -2.95000000000000007e-288 < d < 0.0054000000000000003

   1. Initial program 31.6

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

   2. Simplified 32.8

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

      [Start] 31.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 31.6	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 31.6	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 31.6	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 31.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 31.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 31.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 32.8	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 30.8

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

   4. Applied egg-rr 27.6

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

   5. Simplified 26.1

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

      [Start] 27.6	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(\left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5\right) \]
      *-rgt-identity [<=] 27.6	\[ \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot 1} + \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(\left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5\right) \]
      distribute-lft-in [<=] 27.6	\[ \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5\right)} \]
      +-commutative [=>] 27.6	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \color{blue}{\left(\left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}\right) \cdot -0.5 + 1\right)} \]
      associate-*l* [=>] 27.6	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(\color{blue}{\frac{h}{\ell} \cdot \left({\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2} \cdot -0.5\right)} + 1\right) \]
      fma-def [=>] 27.6	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, {\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2} \cdot -0.5, 1\right)} \]
      *-commutative [=>] 27.6	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\color{blue}{\left(\frac{0.5}{\frac{d}{D}} \cdot M\right)}}^{2} \cdot -0.5, 1\right) \]
      associate-/r/ [=>] 27.6	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\color{blue}{\left(\frac{0.5}{d} \cdot D\right)} \cdot M\right)}^{2} \cdot -0.5, 1\right) \]
      associate-*l* [=>] 26.1	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\color{blue}{\left(\frac{0.5}{d} \cdot \left(D \cdot M\right)\right)}}^{2} \cdot -0.5, 1\right) \]

   if 0.0054000000000000003 < d

   1. Initial program 24.2

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

   2. Taylor expanded in d around inf 21.3

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

   3. Simplified 20.8

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

      [Start] 21.3	\[ \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]
      *-commutative [=>] 21.3	\[ \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
      associate-/l/ [<=] 20.8	\[ d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   4. Applied egg-rr 14.7

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

   5. Simplified 10.5

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

      [Start] 14.7	\[ \frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \]
      associate-/l/ [=>] 10.5	\[ \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 17.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -980000000:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -2.95 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \mathsf{fma}\left(-0.5, {\left(\frac{\frac{M}{2} \cdot D}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}, 1\right)\right)\\ \mathbf{elif}\;d \leq 0.0054:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}, 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternatives

Alternative 1
Error	20.2
Cost	62728

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{M \cdot D}{d} \cdot \frac{0.5 \cdot \frac{\frac{D}{2}}{d}}{\frac{\ell}{h \cdot M}}, 1\right)\right)\\ \mathbf{elif}\;t_0 \leq 10^{+267}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{h \cdot \ell}}{d}}\\ \end{array} \]

Alternative 2
Error	20.7
Cost	48068

\[\begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq 10^{+267}:\\ \;\;\;\;\left(1 + -0.5 \cdot {\left(\sqrt{\frac{h}{\ell}} \cdot \left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)\right)}^{2}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{h \cdot \ell}}{d}}\\ \end{array} \]

Alternative 3
Error	17.9
Cost	33928

\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{if}\;d \leq -1.32 \cdot 10^{+60}:\\ \;\;\;\;\left(\frac{t_0}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -2.95 \cdot 10^{-288}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{t_0}{\sqrt{-\ell}}\right) \cdot \left(1 + -0.5 \cdot {\left(\sqrt{\frac{h}{\ell}} \cdot \left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)\right)}^{2}\right)\\ \mathbf{elif}\;d \leq 0.62:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}, 1\right) \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	18.5
Cost	27848

\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{if}\;d \leq -1.9 \cdot 10^{-129}:\\ \;\;\;\;\left(\frac{t_0}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -2.95 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{t_0}{\sqrt{-\ell}} \cdot \mathsf{fma}\left(-0.5, D \cdot \left(\left(\left(M \cdot \frac{0.5}{d}\right) \cdot \left(h \cdot \frac{0.5}{\ell}\right)\right) \cdot \frac{M \cdot D}{d}\right), 1\right)\right)\\ \mathbf{elif}\;d \leq 0.7:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}, 1\right) \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	18.6
Cost	27528

\[\begin{array}{l} t_0 := 1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;d \leq -3.25 \cdot 10^{-70}:\\ \;\;\;\;\left(\frac{t_2}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot t_0\\ \mathbf{elif}\;d \leq -2.95 \cdot 10^{-288}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{\frac{d}{h}} \cdot \frac{t_2}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;d \leq 0.011:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}, 1\right) \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	20.2
Cost	27396

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{if}\;d \leq -2.95 \cdot 10^{-288}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;d \leq 0.00062:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}, 1\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	21.3
Cost	21456

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -3.8 \cdot 10^{+56}:\\ \;\;\;\;\left(t_1 \cdot t_2\right) \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{-269}:\\ \;\;\;\;t_2 \cdot \left(t_1 \cdot \mathsf{fma}\left(-0.5, \frac{M \cdot D}{d} \cdot \frac{0.5 \cdot \frac{\frac{D}{2}}{d}}{\frac{\ell}{h \cdot M}}, 1\right)\right)\\ \mathbf{elif}\;d \leq 5.6 \cdot 10^{-68}:\\ \;\;\;\;t_0 \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{elif}\;d \leq 1.65 \cdot 10^{+87}:\\ \;\;\;\;t_2 \cdot \left(t_1 \cdot \mathsf{fma}\left(-0.5, D \cdot \left(M \cdot \frac{\left(0.25 \cdot \frac{M}{\ell}\right) \cdot \frac{h}{d}}{\frac{d}{D}}\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	21.8
Cost	21264

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \left(t_1 \cdot t_0\right) \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{-25}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \mathsf{fma}\left(-0.5, D \cdot \left(\left(\left(M \cdot \frac{0.5}{d}\right) \cdot \left(h \cdot \frac{0.5}{\ell}\right)\right) \cdot \frac{M \cdot D}{d}\right), 1\right)\right)\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{-256}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 2.02 \cdot 10^{-224}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{-64}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 9
Error	21.7
Cost	21064

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;h \leq -2.7 \cdot 10^{-43}:\\ \;\;\;\;\left(t_1 \cdot t_2\right) \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq 7.2 \cdot 10^{-273}:\\ \;\;\;\;t_2 \cdot \left(t_1 \cdot \mathsf{fma}\left(-0.125, \frac{\frac{D}{d} \cdot \left(M \cdot \left(h \cdot M\right)\right)}{d \cdot \frac{\ell}{D}}, 1\right)\right)\\ \mathbf{elif}\;h \leq 2.5 \cdot 10^{+112}:\\ \;\;\;\;t_0 \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	24.9
Cost	20872

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{if}\;d \leq -2.95 \cdot 10^{-288}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - 0.125 \cdot \left(D \cdot \frac{D}{\frac{\ell}{h} \cdot \frac{d}{\frac{M}{\frac{d}{M}}}}\right)\right)\\ \mathbf{elif}\;d \leq 0.08:\\ \;\;\;\;t_0 \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	22.5
Cost	20872

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ t_1 := \frac{h}{\ell} \cdot -0.5\\ \mathbf{if}\;d \leq -2.95 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + t_1 \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 0.39:\\ \;\;\;\;t_0 \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	21.9
Cost	20872

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{if}\;d \leq -2.95 \cdot 10^{-288}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 0.28:\\ \;\;\;\;t_0 \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	26.6
Cost	15052

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq 1.26 \cdot 10^{-307}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 2.75 \cdot 10^{-158}:\\ \;\;\;\;D \cdot \frac{M}{\frac{d}{\left(M \cdot D\right) \cdot -0.125} \cdot \frac{{\ell}^{1.5}}{\sqrt{h}}}\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{+52}:\\ \;\;\;\;t_0 \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}{\ell} \cdot \frac{\frac{h}{d}}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 14
Error	27.3
Cost	15052

\[\begin{array}{l} \mathbf{if}\;d \leq -2.95 \cdot 10^{-288}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - 0.125 \cdot \left(D \cdot \frac{D}{\frac{\ell}{h} \cdot \frac{d}{\frac{M}{\frac{d}{M}}}}\right)\right)\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{-159}:\\ \;\;\;\;D \cdot \frac{M}{\frac{d}{\left(M \cdot D\right) \cdot -0.125} \cdot \frac{{\ell}^{1.5}}{\sqrt{h}}}\\ \mathbf{elif}\;d \leq 4.1 \cdot 10^{+52}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}{\ell} \cdot \frac{\frac{h}{d}}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 15
Error	27.4
Cost	14352

\[\begin{array}{l} t_0 := D \cdot \frac{M}{\frac{d}{\left(M \cdot D\right) \cdot -0.125} \cdot \frac{{\ell}^{1.5}}{\sqrt{h}}}\\ \mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-111}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-68}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;d \leq 8.2 \cdot 10^{-33}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 16
Error	25.4
Cost	13380

\[\begin{array}{l} \mathbf{if}\;d \leq 1.36 \cdot 10^{-250}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 17
Error	31.2
Cost	13252

\[\begin{array}{l} \mathbf{if}\;d \leq 2.4 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 18
Error	34.9
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq -1.3 \cdot 10^{-249}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 19
Error	44.0
Cost	6720

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

Alternative 20
Error	59.8
Cost	192

\[d \cdot 0 \]

Reproduce

herbie shell --seed 2023076 
(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)))))