Henrywood and Agarwal, Equation (12)

Average Error: 26.7 → 14.1

Time: 49.7s

Precision: binary64

Cost: 27984

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := D \cdot \left(\frac{M}{d} \cdot 0.5\right)\\ t_2 := \mathsf{fma}\left(-0.5, \frac{t_1}{\frac{\ell}{h \cdot t_1}}, 1\right)\\ t_3 := \sqrt{\frac{d}{h}}\\ t_4 := t_3 \cdot \left(\frac{t_0}{\sqrt{-\ell}} \cdot t_2\right)\\ \mathbf{if}\;\ell \leq -2.2 \cdot 10^{+204}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\ell \leq -1.6 \cdot 10^{-29}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \frac{t_0}{\sqrt{-h}}\right) \cdot \left(1 + -0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{-65}:\\ \;\;\;\;t_3 \cdot \left(t_2 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{+199}:\\ \;\;\;\;\mathsf{fma}\left(d, \sqrt{\frac{1}{\ell \cdot h}}, \frac{D \cdot M}{\frac{d}{\left(D \cdot M\right) \cdot \frac{\sqrt{h} \cdot -0.125}{{\ell}^{1.5}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}, h \cdot \frac{-0.5}{\ell}, 1\right)\\ \end{array} \]

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 (* D (* (/ M d) 0.5)))
        (t_2 (fma -0.5 (/ t_1 (/ l (* h t_1))) 1.0))
        (t_3 (sqrt (/ d h)))
        (t_4 (* t_3 (* (/ t_0 (sqrt (- l))) t_2))))
   (if (<= l -2.2e+204)
     t_4
     (if (<= l -1.6e-29)
       (*
        (* (sqrt (/ d l)) (/ t_0 (sqrt (- h))))
        (+ 1.0 (* -0.5 (* (pow (* (/ M 2.0) (/ D d)) 2.0) (/ h l)))))
       (if (<= l -2e-310)
         t_4
         (if (<= l 8e-65)
           (* t_3 (* t_2 (/ (sqrt d) (sqrt l))))
           (if (<= l 7.2e+199)
             (fma
              d
              (sqrt (/ 1.0 (* l h)))
              (/
               (* D M)
               (/ d (* (* D M) (/ (* (sqrt h) -0.125) (pow l 1.5))))))
             (*
              (/ (/ d (sqrt h)) (sqrt l))
              (fma
               (pow (* M (* D (/ 0.5 d))) 2.0)
               (* h (/ -0.5 l))
               1.0)))))))))

C

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

↓

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(-d);
	double t_1 = D * ((M / d) * 0.5);
	double t_2 = fma(-0.5, (t_1 / (l / (h * t_1))), 1.0);
	double t_3 = sqrt((d / h));
	double t_4 = t_3 * ((t_0 / sqrt(-l)) * t_2);
	double tmp;
	if (l <= -2.2e+204) {
		tmp = t_4;
	} else if (l <= -1.6e-29) {
		tmp = (sqrt((d / l)) * (t_0 / sqrt(-h))) * (1.0 + (-0.5 * (pow(((M / 2.0) * (D / d)), 2.0) * (h / l))));
	} else if (l <= -2e-310) {
		tmp = t_4;
	} else if (l <= 8e-65) {
		tmp = t_3 * (t_2 * (sqrt(d) / sqrt(l)));
	} else if (l <= 7.2e+199) {
		tmp = fma(d, sqrt((1.0 / (l * h))), ((D * M) / (d / ((D * M) * ((sqrt(h) * -0.125) / pow(l, 1.5))))));
	} else {
		tmp = ((d / sqrt(h)) / sqrt(l)) * fma(pow((M * (D * (0.5 / d))), 2.0), (h * (-0.5 / l)), 1.0);
	}
	return tmp;
}

Julia

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

↓

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(-d))
	t_1 = Float64(D * Float64(Float64(M / d) * 0.5))
	t_2 = fma(-0.5, Float64(t_1 / Float64(l / Float64(h * t_1))), 1.0)
	t_3 = sqrt(Float64(d / h))
	t_4 = Float64(t_3 * Float64(Float64(t_0 / sqrt(Float64(-l))) * t_2))
	tmp = 0.0
	if (l <= -2.2e+204)
		tmp = t_4;
	elseif (l <= -1.6e-29)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * Float64(t_0 / sqrt(Float64(-h)))) * Float64(1.0 + Float64(-0.5 * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * Float64(h / l)))));
	elseif (l <= -2e-310)
		tmp = t_4;
	elseif (l <= 8e-65)
		tmp = Float64(t_3 * Float64(t_2 * Float64(sqrt(d) / sqrt(l))));
	elseif (l <= 7.2e+199)
		tmp = fma(d, sqrt(Float64(1.0 / Float64(l * h))), Float64(Float64(D * M) / Float64(d / Float64(Float64(D * M) * Float64(Float64(sqrt(h) * -0.125) / (l ^ 1.5))))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(h)) / sqrt(l)) * fma((Float64(M * Float64(D * Float64(0.5 / d))) ^ 2.0), Float64(h * Float64(-0.5 / l)), 1.0));
	end
	return tmp
end

Wolfram

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

↓

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$1 = N[(D * N[(N[(M / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-0.5 * N[(t$95$1 / N[(l / N[(h * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.2e+204], t$95$4, If[LessEqual[l, -1.6e-29], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 / N[Sqrt[(-h)], $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[l, -2e-310], t$95$4, If[LessEqual[l, 8e-65], N[(t$95$3 * N[(t$95$2 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.2e+199], N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(D * M), $MachinePrecision] / N[(d / N[(N[(D * M), $MachinePrecision] * N[(N[(N[Sqrt[h], $MachinePrecision] * -0.125), $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * N[(-0.5 / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -2.20000000000000011e204 or -1.6e-29 < l < -1.999999999999994e-310

   1. Initial program 30.4

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

   2. Simplified 30.6

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

      [Start] 30.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 30.6	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 30.6	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 30.6	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 30.6	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 30.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      sub-neg [=>] 30.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]
      +-commutative [=>] 30.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]
      associate-*l* [=>] 30.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) + 1\right)\right) \]
      distribute-lft-neg-in [=>] 30.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(-\frac{1}{2}\right) \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)} + 1\right)\right) \]
      fma-def [=>] 30.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(-\frac{1}{2}, {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}, 1\right)}\right) \]

   3. Applied egg-rr 26.6

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

   4. Applied egg-rr 16.7

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

   if -2.20000000000000011e204 < l < -1.6e-29

   1. Initial program 23.0

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

   2. Simplified 23.1

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

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

   3. Applied egg-rr 15.1

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

   if -1.999999999999994e-310 < l < 7.99999999999999939e-65

   1. Initial program 30.1

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

   2. Simplified 30.9

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

      [Start] 30.1	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 30.6	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 30.6	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 30.6	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 30.6	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 30.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      sub-neg [=>] 30.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]
      +-commutative [=>] 30.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]
      associate-*l* [=>] 30.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) + 1\right)\right) \]
      distribute-lft-neg-in [=>] 30.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(-\frac{1}{2}\right) \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)} + 1\right)\right) \]
      fma-def [=>] 30.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(-\frac{1}{2}, {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}, 1\right)}\right) \]

   3. Applied egg-rr 24.8

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

   4. Applied egg-rr 12.7

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

   5. Simplified 12.7

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

      [Start] 12.7	\[ \sqrt{\frac{d}{h}} \cdot \left(\left(\sqrt{d} \cdot \frac{1}{\sqrt{\ell}}\right) \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right) \cdot h}}, 1\right)\right) \]
      associate-*r/ [=>] 12.7	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{\sqrt{d} \cdot 1}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right) \cdot h}}, 1\right)\right) \]
      *-rgt-identity [=>] 12.7	\[ \sqrt{\frac{d}{h}} \cdot \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right) \cdot h}}, 1\right)\right) \]

   if 7.99999999999999939e-65 < l < 7.20000000000000002e199

   1. Initial program 21.9

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

   2. Simplified 23.0

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

      [Start] 21.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 21.9	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 21.9	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 21.9	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 21.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 21.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 21.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 23.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Taylor expanded in d around 0 31.9

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

   4. Simplified 16.4

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

      [Start] 31.9	\[ \sqrt{\frac{1}{\ell \cdot h}} \cdot d + -0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]
      *-commutative [=>] 31.9	\[ \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} + -0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]
      fma-def [=>] 31.9	\[ \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{\ell \cdot h}}, -0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]
      *-commutative [=>] 31.9	\[ \mathsf{fma}\left(d, \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}, -0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) \]
      *-commutative [=>] 31.9	\[ \mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot -0.125}\right) \]
      associate-*l* [=>] 31.9	\[ \mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)}\right) \]
      *-commutative [=>] 31.9	\[ \mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)\right) \]
      unpow2 [=>] 31.9	\[ \mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, \frac{\color{blue}{\left(M \cdot M\right)} \cdot {D}^{2}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)\right) \]
      unpow2 [=>] 31.9	\[ \mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, \frac{\left(M \cdot M\right) \cdot \color{blue}{\left(D \cdot D\right)}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)\right) \]
      unswap-sqr [=>] 20.2	\[ \mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, \frac{\color{blue}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}}{d} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)\right) \]
      associate-/l* [=>] 16.4	\[ \mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, \color{blue}{\frac{M \cdot D}{\frac{d}{M \cdot D}}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)\right) \]

   5. Applied egg-rr 7.4

      \[\leadsto \mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, \color{blue}{\frac{M \cdot D}{\frac{d}{\left(M \cdot D\right) \cdot \frac{\sqrt{h} \cdot -0.125}{{\ell}^{1.5}}}}}\right) \]

   if 7.20000000000000002e199 < l

   1. Initial program 30.7

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

   2. Simplified 30.4

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

      [Start] 30.7	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 30.7	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 30.7	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 30.7	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 30.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 30.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 30.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 30.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 16.2

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

   4. Simplified 20.0

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

      [Start] 16.2	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \]
      *-rgt-identity [<=] 16.2	\[ \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot 1} + \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \]
      distribute-lft-in [<=] 16.2	\[ \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)} \]
      associate-/r* [=>] 20.0	\[ \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \cdot \left(1 + \left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \]
      +-commutative [=>] 20.0	\[ \frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \color{blue}{\left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} + 1\right)} \]
      *-commutative [=>] 20.0	\[ \frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \left(\color{blue}{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)} + 1\right) \]
      fma-def [=>] 20.0	\[ \frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \color{blue}{\mathsf{fma}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]
      associate-*r/ [=>] 20.0	\[ \frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \color{blue}{\frac{0.5 \cdot D}{d}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right) \]
      associate-/l* [=>] 20.0	\[ \frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \color{blue}{\frac{0.5}{\frac{d}{D}}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right) \]
      associate-/r/ [=>] 20.0	\[ \frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \color{blue}{\left(\frac{0.5}{d} \cdot D\right)}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right) \]
      associate-*r/ [=>] 20.0	\[ \frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \left(\frac{0.5}{d} \cdot D\right)\right)}^{2}, \color{blue}{\frac{-0.5 \cdot h}{\ell}}, 1\right) \]
      associate-/l* [=>] 20.0	\[ \frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \left(\frac{0.5}{d} \cdot D\right)\right)}^{2}, \color{blue}{\frac{-0.5}{\frac{\ell}{h}}}, 1\right) \]
      associate-/r/ [=>] 20.0	\[ \frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \left(\frac{0.5}{d} \cdot D\right)\right)}^{2}, \color{blue}{\frac{-0.5}{\ell} \cdot h}, 1\right) \]

3. Recombined 5 regimes into one program.

4. Final simplification 14.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.2 \cdot 10^{+204}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{h \cdot \left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}}, 1\right)\right)\\ \mathbf{elif}\;\ell \leq -1.6 \cdot 10^{-29}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right) \cdot \left(1 + -0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{h \cdot \left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}}, 1\right)\right)\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{h \cdot \left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}}, 1\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{+199}:\\ \;\;\;\;\mathsf{fma}\left(d, \sqrt{\frac{1}{\ell \cdot h}}, \frac{D \cdot M}{\frac{d}{\left(D \cdot M\right) \cdot \frac{\sqrt{h} \cdot -0.125}{{\ell}^{1.5}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}, h \cdot \frac{-0.5}{\ell}, 1\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	14.4
Cost	27976

\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := D \cdot \left(\frac{M}{d} \cdot 0.5\right)\\ t_2 := \mathsf{fma}\left(-0.5, \frac{t_1}{\frac{\ell}{h \cdot t_1}}, 1\right)\\ t_3 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;\ell \leq -3.6 \cdot 10^{+220}:\\ \;\;\;\;t_3 \cdot \left(\frac{t_0}{\sqrt{-\ell}} \cdot t_2\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{-h}}{t_0}} \cdot \left(t_2 \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{-66}:\\ \;\;\;\;t_3 \cdot \left(t_2 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{+200}:\\ \;\;\;\;\mathsf{fma}\left(d, \sqrt{\frac{1}{\ell \cdot h}}, \frac{D \cdot M}{\frac{d}{\left(D \cdot M\right) \cdot \frac{\sqrt{h} \cdot -0.125}{{\ell}^{1.5}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}, h \cdot \frac{-0.5}{\ell}, 1\right)\\ \end{array} \]

Alternative 2
Error	14.4
Cost	27852

\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := D \cdot \left(\frac{M}{d} \cdot 0.5\right)\\ t_2 := \mathsf{fma}\left(-0.5, \frac{t_1}{\frac{\ell}{h \cdot t_1}}, 1\right)\\ t_3 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+220}:\\ \;\;\;\;t_3 \cdot \left(\frac{t_0}{\sqrt{-\ell}} \cdot t_2\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(t_2 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{t_0}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{-64}:\\ \;\;\;\;t_3 \cdot \left(t_2 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+202}:\\ \;\;\;\;\mathsf{fma}\left(d, \sqrt{\frac{1}{\ell \cdot h}}, \frac{D \cdot M}{\frac{d}{\left(D \cdot M\right) \cdot \frac{\sqrt{h} \cdot -0.125}{{\ell}^{1.5}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}, h \cdot \frac{-0.5}{\ell}, 1\right)\\ \end{array} \]

Alternative 3
Error	18.9
Cost	27536

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -1.62 \cdot 10^{+230}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}} \cdot t_1}{\sqrt{-\ell}}\\ \mathbf{elif}\;\ell \leq -3.2 \cdot 10^{+64}:\\ \;\;\;\;\frac{t_1}{\frac{\sqrt{-h}}{t_0}}\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{-65}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(t_0 \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{h \cdot \left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}}, 1\right)\right)\\ \mathbf{elif}\;\ell \leq 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(d, \sqrt{\frac{1}{\ell \cdot h}}, \frac{D \cdot M}{\frac{d}{\left(D \cdot M\right) \cdot \frac{\sqrt{h} \cdot -0.125}{{\ell}^{1.5}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}, h \cdot \frac{-0.5}{\ell}, 1\right)\\ \end{array} \]

Alternative 4
Error	17.6
Cost	27536

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -3.05 \cdot 10^{+237}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}} \cdot t_1}{\sqrt{-\ell}}\\ \mathbf{elif}\;\ell \leq -1.1 \cdot 10^{-40}:\\ \;\;\;\;\left(t_0 \cdot \frac{t_1}{\sqrt{-h}}\right) \cdot \left(1 + -0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(t_0 \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{h \cdot \left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}}, 1\right)\right)\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+199}:\\ \;\;\;\;\mathsf{fma}\left(d, \sqrt{\frac{1}{\ell \cdot h}}, \frac{D \cdot M}{\frac{d}{\left(D \cdot M\right) \cdot \frac{\sqrt{h} \cdot -0.125}{{\ell}^{1.5}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}, h \cdot \frac{-0.5}{\ell}, 1\right)\\ \end{array} \]

Alternative 5
Error	19.2
Cost	21580

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -2.65 \cdot 10^{+228}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}} \cdot t_1}{\sqrt{-\ell}}\\ \mathbf{elif}\;\ell \leq -7.8 \cdot 10^{+64}:\\ \;\;\;\;\frac{t_1}{\frac{\sqrt{-h}}{t_0}}\\ \mathbf{elif}\;\ell \leq 1.18 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(t_0 \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{h \cdot \left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}}, 1\right)\right)\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(d, \sqrt{\frac{1}{\ell \cdot h}}, \frac{D \cdot M}{\frac{d}{D \cdot M}} \cdot \left(-0.125 \cdot \sqrt{\frac{h}{\ell} \cdot \frac{1}{\ell \cdot \ell}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 6
Error	19.9
Cost	21456

\[\begin{array}{l} t_0 := {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -8 \cdot 10^{+227}:\\ \;\;\;\;\frac{t_2 \cdot t_3}{\sqrt{-\ell}}\\ \mathbf{elif}\;\ell \leq -1.6 \cdot 10^{+64}:\\ \;\;\;\;\frac{t_3}{\frac{\sqrt{-h}}{t_1}}\\ \mathbf{elif}\;\ell \leq 2.05 \cdot 10^{-106}:\\ \;\;\;\;\left(t_2 \cdot t_1\right) \cdot \left(1 + -0.5 \cdot \left(\frac{1}{\ell} \cdot \left(h \cdot t_0\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(d, \sqrt{\frac{1}{\ell \cdot h}}, \frac{D \cdot M}{\frac{d}{D \cdot M}} \cdot \left(-0.125 \cdot \sqrt{\frac{h}{\ell} \cdot \frac{1}{\ell \cdot \ell}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t_0 \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 7
Error	19.3
Cost	21456

\[\begin{array}{l} t_0 := D \cdot \left(\frac{M}{d} \cdot 0.5\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+228}:\\ \;\;\;\;\frac{t_2 \cdot t_3}{\sqrt{-\ell}}\\ \mathbf{elif}\;\ell \leq -1.5 \cdot 10^{+67}:\\ \;\;\;\;\frac{t_3}{\frac{\sqrt{-h}}{t_1}}\\ \mathbf{elif}\;\ell \leq 5.9 \cdot 10^{-27}:\\ \;\;\;\;t_2 \cdot \left(\mathsf{fma}\left(-0.5, \frac{t_0}{\frac{\ell}{h \cdot t_0}}, 1\right) \cdot t_1\right)\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(d, \sqrt{\frac{1}{\ell \cdot h}}, \frac{D \cdot M}{\frac{d}{D \cdot M}} \cdot \left(-0.125 \cdot \sqrt{\frac{h}{\ell} \cdot \frac{1}{\ell \cdot \ell}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 8
Error	19.5
Cost	21260

\[\begin{array}{l} t_0 := {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+230}:\\ \;\;\;\;\frac{t_2 \cdot t_3}{\sqrt{-\ell}}\\ \mathbf{elif}\;\ell \leq -3.3 \cdot 10^{+63}:\\ \;\;\;\;\frac{t_3}{\frac{\sqrt{-h}}{t_1}}\\ \mathbf{elif}\;\ell \leq 1.32 \cdot 10^{-232}:\\ \;\;\;\;\left(t_2 \cdot t_1\right) \cdot \left(1 + -0.5 \cdot \left(\frac{1}{\ell} \cdot \left(h \cdot t_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t_0 \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 9
Error	20.1
Cost	21136

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -3.8 \cdot 10^{+227}:\\ \;\;\;\;\frac{t_2 \cdot t_3}{\sqrt{-\ell}}\\ \mathbf{elif}\;\ell \leq -9.5 \cdot 10^{+66}:\\ \;\;\;\;\frac{t_3}{\frac{\sqrt{-h}}{t_1}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \left(t_2 \cdot t_1\right)\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-137}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right) \cdot t_0\\ \end{array} \]

Alternative 10
Error	19.5
Cost	21132

\[\begin{array}{l} t_0 := {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -1.7 \cdot 10^{+228}:\\ \;\;\;\;\frac{t_2 \cdot t_3}{\sqrt{-\ell}}\\ \mathbf{elif}\;\ell \leq -1.15 \cdot 10^{+73}:\\ \;\;\;\;\frac{t_3}{\frac{\sqrt{-h}}{t_1}}\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{-222}:\\ \;\;\;\;\left(t_2 \cdot t_1\right) \cdot \left(1 + -0.5 \cdot \frac{h \cdot t_0}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t_0 \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 11
Error	20.7
Cost	21004

\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{if}\;\ell \leq -1.85 \cdot 10^{+228}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}} \cdot t_0}{\sqrt{-\ell}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{t_0 \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq 7.6 \cdot 10^{-135}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right) \cdot t_1\\ \end{array} \]

Alternative 12
Error	22.2
Cost	20040

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{-d}\\ t_2 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -15200000000000:\\ \;\;\;\;\frac{t_1}{\frac{\sqrt{-h}}{t_0}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{t_1}{\frac{\sqrt{-\ell}}{t_2}}\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-142}:\\ \;\;\;\;\frac{D \cdot M}{\frac{d}{D \cdot M}} \cdot \left(-0.125 \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\\ \mathbf{elif}\;d \leq 11000000:\\ \;\;\;\;\left(t_2 \cdot t_0\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{h}{\ell} \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 13
Error	22.1
Cost	20040

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;d \leq -1.3 \cdot 10^{-29}:\\ \;\;\;\;\frac{t_2}{\frac{\sqrt{-h}}{t_0}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{t_1 \cdot t_2}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{-142}:\\ \;\;\;\;\frac{D \cdot M}{\frac{d}{D \cdot M}} \cdot \left(-0.125 \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\\ \mathbf{elif}\;d \leq 30000000:\\ \;\;\;\;\left(t_1 \cdot t_0\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{h}{\ell} \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 14
Error	21.3
Cost	19908

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -7.5 \cdot 10^{+36}:\\ \;\;\;\;\frac{\sqrt{-d}}{\frac{\sqrt{-h}}{t_0}}\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{-147}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot t_0\right) \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \frac{D \cdot \left(\frac{D}{d} \cdot \left(M \cdot \left(h \cdot \frac{M}{\ell}\right)\right)\right)}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 15
Error	24.9
Cost	15052

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}} \cdot t_0\\ \mathbf{if}\;\ell \leq -1.25 \cdot 10^{+93}:\\ \;\;\;\;t_0 \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;\ell \leq -200000:\\ \;\;\;\;t_1 \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{h}{\ell} \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot d}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_1 \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M}{\frac{\ell}{h \cdot M}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 16
Error	24.9
Cost	15052

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ t_1 := t_0 \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \frac{D \cdot \left(\frac{D}{d} \cdot \left(M \cdot \left(h \cdot \frac{M}{\ell}\right)\right)\right)}{d}\right)\right)\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{+237}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -5000000:\\ \;\;\;\;t_0 \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{h}{\ell} \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot d}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 17
Error	25.2
Cost	14600

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -2.4 \cdot 10^{+165}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot t_0\\ \mathbf{elif}\;h \leq -4 \cdot 10^{+110}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{t_0}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 18
Error	25.1
Cost	13508

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 19
Error	25.4
Cost	13380

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

Alternative 20
Error	25.1
Cost	13380

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

Alternative 21
Error	29.0
Cost	13252

\[\begin{array}{l} \mathbf{if}\;d \leq 2.1 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 22
Error	33.0
Cost	7113

\[\begin{array}{l} \mathbf{if}\;h \leq -1.8 \cdot 10^{-289} \lor \neg \left(h \leq 1.75 \cdot 10^{+197}\right):\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]

Alternative 23
Error	44.1
Cost	6720

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

Reproduce

herbie shell --seed 2023046 
(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)))))