Henrywood and Agarwal, Equation (12)

Average Error: 26.8 → 14.5

Time: 47.4s

Precision: binary64

Cost: 27984

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := D \cdot \left(\frac{M}{d} \cdot 0.5\right)\\ t_3 := \mathsf{fma}\left(-0.5, \frac{t_2}{\frac{\ell}{h \cdot t_2}}, 1\right)\\ t_4 := \frac{t_0}{\sqrt{-\ell}}\\ \mathbf{if}\;\ell \leq -1.46 \cdot 10^{+262}:\\ \;\;\;\;t_1 \cdot \left(t_4 \cdot \mathsf{fma}\left(-0.5, \frac{\frac{M}{d}}{\frac{4}{D}} \cdot \left(\left(\frac{M}{d} \cdot D\right) \cdot \frac{h}{\ell}\right), 1\right)\right)\\ \mathbf{elif}\;\ell \leq -6.2 \cdot 10^{-73}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{-h}}{t_0}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot t_3\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_1 \cdot \left(t_4 \cdot t_3\right)\\ \mathbf{elif}\;\ell \leq 3.45 \cdot 10^{-37}:\\ \;\;\;\;t_1 \cdot \left(t_3 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (- d)))
        (t_1 (sqrt (/ d h)))
        (t_2 (* D (* (/ M d) 0.5)))
        (t_3 (fma -0.5 (/ t_2 (/ l (* h t_2))) 1.0))
        (t_4 (/ t_0 (sqrt (- l)))))
   (if (<= l -1.46e+262)
     (*
      t_1
      (*
       t_4
       (fma -0.5 (* (/ (/ M d) (/ 4.0 D)) (* (* (/ M d) D) (/ h l))) 1.0)))
     (if (<= l -6.2e-73)
       (* (/ 1.0 (/ (sqrt (- h)) t_0)) (* (sqrt (/ d l)) t_3))
       (if (<= l -2e-310)
         (* t_1 (* t_4 t_3))
         (if (<= l 3.45e-37)
           (* t_1 (* t_3 (/ (sqrt d) (sqrt l))))
           (*
            (/ d (* (sqrt l) (sqrt h)))
            (+ 1.0 (* (pow (* M (* 0.5 (/ D d))) 2.0) (* -0.5 (/ h l)))))))))))

C

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

↓

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(-d);
	double t_1 = sqrt((d / h));
	double t_2 = D * ((M / d) * 0.5);
	double t_3 = fma(-0.5, (t_2 / (l / (h * t_2))), 1.0);
	double t_4 = t_0 / sqrt(-l);
	double tmp;
	if (l <= -1.46e+262) {
		tmp = t_1 * (t_4 * fma(-0.5, (((M / d) / (4.0 / D)) * (((M / d) * D) * (h / l))), 1.0));
	} else if (l <= -6.2e-73) {
		tmp = (1.0 / (sqrt(-h) / t_0)) * (sqrt((d / l)) * t_3);
	} else if (l <= -2e-310) {
		tmp = t_1 * (t_4 * t_3);
	} else if (l <= 3.45e-37) {
		tmp = t_1 * (t_3 * (sqrt(d) / sqrt(l)));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + (pow((M * (0.5 * (D / d))), 2.0) * (-0.5 * (h / l))));
	}
	return tmp;
}

Julia

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

↓

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(-d))
	t_1 = sqrt(Float64(d / h))
	t_2 = Float64(D * Float64(Float64(M / d) * 0.5))
	t_3 = fma(-0.5, Float64(t_2 / Float64(l / Float64(h * t_2))), 1.0)
	t_4 = Float64(t_0 / sqrt(Float64(-l)))
	tmp = 0.0
	if (l <= -1.46e+262)
		tmp = Float64(t_1 * Float64(t_4 * fma(-0.5, Float64(Float64(Float64(M / d) / Float64(4.0 / D)) * Float64(Float64(Float64(M / d) * D) * Float64(h / l))), 1.0)));
	elseif (l <= -6.2e-73)
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(-h)) / t_0)) * Float64(sqrt(Float64(d / l)) * t_3));
	elseif (l <= -2e-310)
		tmp = Float64(t_1 * Float64(t_4 * t_3));
	elseif (l <= 3.45e-37)
		tmp = Float64(t_1 * Float64(t_3 * Float64(sqrt(d) / sqrt(l))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 + Float64((Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0) * Float64(-0.5 * Float64(h / l)))));
	end
	return tmp
end

Wolfram

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

↓

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(D * N[(N[(M / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-0.5 * N[(t$95$2 / N[(l / N[(h * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.46e+262], N[(t$95$1 * N[(t$95$4 * N[(-0.5 * N[(N[(N[(M / d), $MachinePrecision] / N[(4.0 / D), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -6.2e-73], N[(N[(1.0 / N[(N[Sqrt[(-h)], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(t$95$1 * N[(t$95$4 * t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.45e-37], N[(t$95$1 * N[(t$95$3 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -1.4599999999999999e262

   1. Initial program 29.2

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

   2. Simplified 28.7

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

      [Start] 29.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 29.2	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 29.2	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 29.2	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 29.2	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 29.2	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      sub-neg [=>] 29.2	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]
      +-commutative [=>] 29.2	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]
      associate-*l* [=>] 29.2	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) + 1\right)\right) \]
      distribute-lft-neg-in [=>] 29.2	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(-\frac{1}{2}\right) \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)} + 1\right)\right) \]
      fma-def [=>] 29.2	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(-\frac{1}{2}, {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}, 1\right)}\right) \]

   3. Applied egg-rr 28.7

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\frac{\ell}{h} \cdot 4}}, 1\right)\right) \]

   4. Applied egg-rr 27.7

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \color{blue}{\frac{\frac{M}{d}}{\frac{4}{D}} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \frac{h}{\ell}\right)}, 1\right)\right) \]

   5. Applied egg-rr 18.4

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}} \cdot \mathsf{fma}\left(-0.5, \frac{\frac{M}{d}}{\frac{4}{D}} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \frac{h}{\ell}\right), 1\right)\right) \]

   if -1.4599999999999999e262 < l < -6.19999999999999938e-73

   1. Initial program 24.8

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

   2. Simplified 25.3

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

      [Start] 24.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 24.9	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 24.9	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 24.9	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 24.9	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 24.9	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      sub-neg [=>] 24.9	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]
      +-commutative [=>] 24.9	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]
      associate-*l* [=>] 24.9	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) + 1\right)\right) \]
      distribute-lft-neg-in [=>] 24.9	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(-\frac{1}{2}\right) \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)} + 1\right)\right) \]
      fma-def [=>] 24.9	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(-\frac{1}{2}, {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}, 1\right)}\right) \]

   3. Applied egg-rr 22.8

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

   4. Applied egg-rr 22.7

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

   5. Applied egg-rr 15.1

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

   if -6.19999999999999938e-73 < l < -1.999999999999994e-310

   1. Initial program 31.2

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

   2. Simplified 31.9

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

      [Start] 31.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 31.6	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 31.6	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 31.6	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 31.6	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 31.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      sub-neg [=>] 31.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]
      +-commutative [=>] 31.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]
      associate-*l* [=>] 31.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) + 1\right)\right) \]
      distribute-lft-neg-in [=>] 31.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(-\frac{1}{2}\right) \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)} + 1\right)\right) \]
      fma-def [=>] 31.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(-\frac{1}{2}, {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}, 1\right)}\right) \]

   3. Applied egg-rr 25.9

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

   4. Applied egg-rr 12.5

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

   if -1.999999999999994e-310 < l < 3.4499999999999999e-37

   1. Initial program 28.5

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

   2. Simplified 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] 28.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 28.6	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 28.6	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 28.6	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 28.6	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 28.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      sub-neg [=>] 28.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]
      +-commutative [=>] 28.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]
      associate-*l* [=>] 28.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) + 1\right)\right) \]
      distribute-lft-neg-in [=>] 28.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(-\frac{1}{2}\right) \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)} + 1\right)\right) \]
      fma-def [=>] 28.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(-\frac{1}{2}, {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}, 1\right)}\right) \]

   3. Applied egg-rr 23.4

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

   4. Applied egg-rr 12.9

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

   5. Simplified 12.8

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

      [Start] 12.9	\[ \sqrt{\frac{d}{h}} \cdot \left(\left(\sqrt{d} \cdot \frac{1}{\sqrt{\ell}}\right) \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right) \cdot h}}, 1\right)\right) \]
      associate-*r/ [=>] 12.8	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{\sqrt{d} \cdot 1}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right) \cdot h}}, 1\right)\right) \]
      *-rgt-identity [=>] 12.8	\[ \sqrt{\frac{d}{h}} \cdot \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right) \cdot h}}, 1\right)\right) \]

   if 3.4499999999999999e-37 < l

   1. Initial program 26.1

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

   2. Simplified 26.2

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

      [Start] 26.1	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 26.1	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 26.1	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 26.1	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 26.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 26.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 26.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 26.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 14.7

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

   4. Simplified 14.7

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

      [Start] 14.7	\[ \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} + \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \]
      *-rgt-identity [<=] 14.7	\[ \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1} + \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \]
      distribute-lft-in [<=] 14.7	\[ \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)} \]
      *-commutative [=>] 14.7	\[ \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \color{blue}{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)}\right) \]
      *-commutative [=>] 14.7	\[ \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \color{blue}{\left(\frac{h}{\ell} \cdot -0.5\right)}\right) \]

3. Recombined 5 regimes into one program.

4. Final simplification 14.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.46 \cdot 10^{+262}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{\frac{M}{d}}{\frac{4}{D}} \cdot \left(\left(\frac{M}{d} \cdot D\right) \cdot \frac{h}{\ell}\right), 1\right)\right)\\ \mathbf{elif}\;\ell \leq -6.2 \cdot 10^{-73}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{-h}}{\sqrt{-d}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{h \cdot \left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}}, 1\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{h \cdot \left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}}, 1\right)\right)\\ \mathbf{elif}\;\ell \leq 3.45 \cdot 10^{-37}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{h \cdot \left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}}, 1\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	20.8
Cost	83724

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := D \cdot \left(\frac{M}{d} \cdot 0.5\right)\\ t_2 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right)\\ t_3 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ t_4 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{-123}:\\ \;\;\;\;t_4 \cdot \left(t_0 \cdot \mathsf{fma}\left(-0.5, \frac{t_1}{\frac{\frac{\ell}{h}}{t_1}}, 1\right)\right)\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{-235}:\\ \;\;\;\;t_3 \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+260}:\\ \;\;\;\;t_4 \cdot \left(\mathsf{fma}\left(-0.5, \frac{\frac{M}{d}}{\frac{4}{D}} \cdot \left(\left(\frac{M}{d} \cdot D\right) \cdot \frac{h}{\ell}\right), 1\right) \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 2
Error	17.2
Cost	27984

\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := t_1 \cdot \left(\frac{t_0}{\sqrt{-\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{\frac{M}{d}}{\frac{4}{D}} \cdot \left(\left(\frac{M}{d} \cdot D\right) \cdot \frac{h}{\ell}\right), 1\right)\right)\\ t_3 := D \cdot \left(\frac{M}{d} \cdot 0.5\right)\\ \mathbf{if}\;\ell \leq -5.8 \cdot 10^{+262}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -1.35 \cdot 10^{-72}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \frac{t_0}{\sqrt{-h}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 10^{-35}:\\ \;\;\;\;t_1 \cdot \left(\mathsf{fma}\left(-0.5, \frac{t_3}{\frac{\ell}{h \cdot t_3}}, 1\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \end{array} \]

Alternative 3
Error	15.6
Cost	27984

\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := \frac{t_0}{\sqrt{-\ell}}\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := D \cdot \left(\frac{M}{d} \cdot 0.5\right)\\ t_4 := \mathsf{fma}\left(-0.5, \frac{t_3}{\frac{\ell}{h \cdot t_3}}, 1\right)\\ \mathbf{if}\;\ell \leq -1.42 \cdot 10^{+262}:\\ \;\;\;\;t_2 \cdot \left(t_1 \cdot \mathsf{fma}\left(-0.5, \frac{\frac{M}{d}}{\frac{4}{D}} \cdot \left(\left(\frac{M}{d} \cdot D\right) \cdot \frac{h}{\ell}\right), 1\right)\right)\\ \mathbf{elif}\;\ell \leq -4.4 \cdot 10^{-72}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \frac{t_0}{\sqrt{-h}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_2 \cdot \left(t_1 \cdot t_4\right)\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{-38}:\\ \;\;\;\;t_2 \cdot \left(t_4 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \end{array} \]

Alternative 4
Error	17.5
Cost	27852

\[\begin{array}{l} t_0 := D \cdot \left(\frac{M}{d} \cdot 0.5\right)\\ t_1 := \mathsf{fma}\left(-0.5, \frac{t_0}{\frac{\ell}{h \cdot t_0}}, 1\right)\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -8 \cdot 10^{+115}:\\ \;\;\;\;\left(t_2 \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{-287}:\\ \;\;\;\;\left(t_2 \cdot t_1\right) \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq 8.2 \cdot 10^{+60}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t_1 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 5
Error	18.3
Cost	27724

\[\begin{array}{l} t_0 := D \cdot \left(\frac{M}{d} \cdot 0.5\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -2.7 \cdot 10^{+114}:\\ \;\;\;\;\left(t_1 \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-287}:\\ \;\;\;\;\left(t_1 \cdot \mathsf{fma}\left(-0.5, \frac{t_0}{\frac{\ell}{h \cdot t_0}}, 1\right)\right) \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{+60}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\frac{M}{d}}{\frac{4}{D}} \cdot \left(\left(\frac{M}{d} \cdot D\right) \cdot \frac{h}{\ell}\right), 1\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 6
Error	18.7
Cost	27396

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ t_1 := D \cdot \left(\frac{M}{d} \cdot 0.5\right)\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -7 \cdot 10^{+115}:\\ \;\;\;\;\left(t_2 \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 5.6 \cdot 10^{-202}:\\ \;\;\;\;\left(t_2 \cdot \mathsf{fma}\left(-0.5, \frac{t_1}{\frac{\ell}{h \cdot t_1}}, 1\right)\right) \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{+106}:\\ \;\;\;\;t_0 \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	21.8
Cost	21324

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ t_1 := \sqrt{\frac{d}{h}} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\frac{M}{d}}{\frac{4}{D}} \cdot \left(\left(\frac{M}{d} \cdot D\right) \cdot \frac{h}{\ell}\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{if}\;d \leq 5.6 \cdot 10^{-202}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{-113}:\\ \;\;\;\;t_0 \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{+57}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	21.5
Cost	21324

\[\begin{array}{l} t_0 := \frac{\frac{M}{d}}{\frac{4}{D}}\\ t_1 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq 1.15 \cdot 10^{-201}:\\ \;\;\;\;t_2 \cdot \left(\mathsf{fma}\left(-0.5, t_0 \cdot \left(\left(\frac{M}{d} \cdot D\right) \cdot \frac{h}{\ell}\right), 1\right) \cdot t_3\right)\\ \mathbf{elif}\;d \leq 2.25 \cdot 10^{-113}:\\ \;\;\;\;t_1 \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{+58}:\\ \;\;\;\;t_2 \cdot \left(t_3 \cdot \mathsf{fma}\left(-0.5, t_0 \cdot \frac{h \cdot \left(M \cdot D\right)}{\ell \cdot d}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	20.7
Cost	21324

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ t_1 := D \cdot \left(\frac{M}{d} \cdot 0.5\right)\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq 5.6 \cdot 10^{-202}:\\ \;\;\;\;t_2 \cdot \left(t_3 \cdot \mathsf{fma}\left(-0.5, \frac{t_1}{\frac{\ell}{h \cdot t_1}}, 1\right)\right)\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-113}:\\ \;\;\;\;t_0 \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{+59}:\\ \;\;\;\;t_2 \cdot \left(t_3 \cdot \mathsf{fma}\left(-0.5, \frac{\frac{M}{d}}{\frac{4}{D}} \cdot \frac{h \cdot \left(M \cdot D\right)}{\ell \cdot d}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	23.4
Cost	21136

\[\begin{array}{l} t_0 := 1 + -0.5 \cdot \left(0.25 \cdot \frac{M}{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\ell}{h \cdot M}}\right)\\ t_1 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := \sqrt{\frac{d}{h}} \cdot t_2\\ \mathbf{if}\;d \leq -4.1 \cdot 10^{-5}:\\ \;\;\;\;\left(t_2 \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right) \cdot t_0\\ \mathbf{elif}\;d \leq -5.2 \cdot 10^{-137}:\\ \;\;\;\;t_3 \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{\frac{D \cdot \left(D \cdot \left(M \cdot \left(h \cdot M\right)\right)\right)}{\ell \cdot d}}{d}\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-310}:\\ \;\;\;\;t_3 \cdot t_0\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{+106}:\\ \;\;\;\;t_1 \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	21.6
Cost	20996

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{if}\;d \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right)\\ \mathbf{elif}\;d \leq 3.45 \cdot 10^{+106}:\\ \;\;\;\;t_0 \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	21.7
Cost	20872

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ t_1 := -0.5 \cdot \frac{h}{\ell}\\ \mathbf{if}\;d \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot t_1\right)\right)\\ \mathbf{elif}\;d \leq 4 \cdot 10^{+106}:\\ \;\;\;\;t_0 \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	21.8
Cost	20872

\[\begin{array}{l} t_0 := {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -3.5 \cdot 10^{-11}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot t_1\right) \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{t_0}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\left(t_1 \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right) \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \frac{M}{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\ell}{h \cdot M}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + t_0 \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \end{array} \]

Alternative 14
Error	25.2
Cost	15052

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ t_1 := t_0 \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \frac{M}{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\ell}{h \cdot M}}\right)\right)\\ \mathbf{if}\;d \leq -8 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -3.7 \cdot 10^{-137}:\\ \;\;\;\;t_0 \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{\frac{D \cdot \left(D \cdot \left(M \cdot \left(h \cdot M\right)\right)\right)}{\ell \cdot d}}{d}\right)\right)\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-287}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 15
Error	25.1
Cost	15052

\[\begin{array}{l} t_0 := 1 + -0.5 \cdot \left(0.25 \cdot \frac{M}{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\ell}{h \cdot M}}\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{d}{h}} \cdot t_1\\ \mathbf{if}\;d \leq -0.000145:\\ \;\;\;\;\left(t_1 \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right) \cdot t_0\\ \mathbf{elif}\;d \leq -1.05 \cdot 10^{-136}:\\ \;\;\;\;t_2 \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{\frac{D \cdot \left(D \cdot \left(M \cdot \left(h \cdot M\right)\right)\right)}{\ell \cdot d}}{d}\right)\right)\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-287}:\\ \;\;\;\;t_2 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 16
Error	25.3
Cost	14920

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -1250000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_0 \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 17
Error	25.4
Cost	14788

\[\begin{array}{l} \mathbf{if}\;d \leq 1.45 \cdot 10^{-287}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \frac{M}{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\ell}{h \cdot M}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 18
Error	25.3
Cost	13380

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 19
Error	28.8
Cost	13252

\[\begin{array}{l} \mathbf{if}\;d \leq 1.55 \cdot 10^{-286}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 20
Error	34.6
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq -7.6 \cdot 10^{-200}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]

Alternative 21
Error	32.8
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\ell \leq 4.1 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]

Alternative 22
Error	43.8
Cost	6784

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

Alternative 23
Error	43.8
Cost	6720

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

Reproduce

herbie shell --seed 2023012 
(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)))))