Henrywood and Agarwal, Equation (12)

Average Error: 41.42% → 23.94%

Time: 56.8s

Precision: binary64

Cost: 33928

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := M \cdot \left(0.5 \cdot \frac{D}{d}\right)\\ t_2 := D \cdot \left(0.5 \cdot \frac{M}{d}\right)\\ t_3 := \mathsf{fma}\left(-0.5, \frac{t_2}{\frac{\ell}{h \cdot t_2}}, 1\right)\\ t_4 := 1 + {t_1}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\\ t_5 := \sqrt{-d}\\ t_6 := \frac{t_5}{\sqrt{-\ell}}\\ t_7 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -2.8 \cdot 10^{+263}:\\ \;\;\;\;\frac{t_6}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;\ell \leq -7200000:\\ \;\;\;\;\left(\frac{t_5}{\sqrt{-h}} \cdot t_7\right) \cdot \left(1 + {\left(t_1 \cdot \sqrt{\frac{h}{\ell}}\right)}^{2} \cdot -0.5\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_0 \cdot \left(t_6 \cdot t_3\right)\\ \mathbf{elif}\;\ell \leq 235:\\ \;\;\;\;t_0 \cdot \left(t_3 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+112}:\\ \;\;\;\;\frac{\left(t_7 \cdot \sqrt{d}\right) \cdot t_4}{\sqrt{h}}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+132}:\\ \;\;\;\;t_0 \cdot \left(t_7 \cdot \mathsf{fma}\left(-0.5, \frac{t_2}{2 \cdot \left(\frac{d}{M} \cdot \frac{\frac{\ell}{D}}{h}\right)}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_4 \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

FPCore

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

↓

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (* M (* 0.5 (/ D d))))
        (t_2 (* D (* 0.5 (/ M d))))
        (t_3 (fma -0.5 (/ t_2 (/ l (* h t_2))) 1.0))
        (t_4 (+ 1.0 (* (pow t_1 2.0) (* (/ h l) -0.5))))
        (t_5 (sqrt (- d)))
        (t_6 (/ t_5 (sqrt (- l))))
        (t_7 (sqrt (/ d l))))
   (if (<= l -2.8e+263)
     (/ t_6 (sqrt (/ h d)))
     (if (<= l -7200000.0)
       (*
        (* (/ t_5 (sqrt (- h))) t_7)
        (+ 1.0 (* (pow (* t_1 (sqrt (/ h l))) 2.0) -0.5)))
       (if (<= l -5e-310)
         (* t_0 (* t_6 t_3))
         (if (<= l 235.0)
           (* t_0 (* t_3 (/ (sqrt d) (sqrt l))))
           (if (<= l 1.6e+112)
             (/ (* (* t_7 (sqrt d)) t_4) (sqrt h))
             (if (<= l 5.8e+132)
               (*
                t_0
                (*
                 t_7
                 (fma -0.5 (/ t_2 (* 2.0 (* (/ d M) (/ (/ l D) h)))) 1.0)))
               (* t_4 (/ d (* (sqrt l) (sqrt h))))))))))))

C

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

↓

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = M * (0.5 * (D / d));
	double t_2 = D * (0.5 * (M / d));
	double t_3 = fma(-0.5, (t_2 / (l / (h * t_2))), 1.0);
	double t_4 = 1.0 + (pow(t_1, 2.0) * ((h / l) * -0.5));
	double t_5 = sqrt(-d);
	double t_6 = t_5 / sqrt(-l);
	double t_7 = sqrt((d / l));
	double tmp;
	if (l <= -2.8e+263) {
		tmp = t_6 / sqrt((h / d));
	} else if (l <= -7200000.0) {
		tmp = ((t_5 / sqrt(-h)) * t_7) * (1.0 + (pow((t_1 * sqrt((h / l))), 2.0) * -0.5));
	} else if (l <= -5e-310) {
		tmp = t_0 * (t_6 * t_3);
	} else if (l <= 235.0) {
		tmp = t_0 * (t_3 * (sqrt(d) / sqrt(l)));
	} else if (l <= 1.6e+112) {
		tmp = ((t_7 * sqrt(d)) * t_4) / sqrt(h);
	} else if (l <= 5.8e+132) {
		tmp = t_0 * (t_7 * fma(-0.5, (t_2 / (2.0 * ((d / M) * ((l / D) / h)))), 1.0));
	} else {
		tmp = t_4 * (d / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

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

↓

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = Float64(M * Float64(0.5 * Float64(D / d)))
	t_2 = Float64(D * Float64(0.5 * Float64(M / d)))
	t_3 = fma(-0.5, Float64(t_2 / Float64(l / Float64(h * t_2))), 1.0)
	t_4 = Float64(1.0 + Float64((t_1 ^ 2.0) * Float64(Float64(h / l) * -0.5)))
	t_5 = sqrt(Float64(-d))
	t_6 = Float64(t_5 / sqrt(Float64(-l)))
	t_7 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -2.8e+263)
		tmp = Float64(t_6 / sqrt(Float64(h / d)));
	elseif (l <= -7200000.0)
		tmp = Float64(Float64(Float64(t_5 / sqrt(Float64(-h))) * t_7) * Float64(1.0 + Float64((Float64(t_1 * sqrt(Float64(h / l))) ^ 2.0) * -0.5)));
	elseif (l <= -5e-310)
		tmp = Float64(t_0 * Float64(t_6 * t_3));
	elseif (l <= 235.0)
		tmp = Float64(t_0 * Float64(t_3 * Float64(sqrt(d) / sqrt(l))));
	elseif (l <= 1.6e+112)
		tmp = Float64(Float64(Float64(t_7 * sqrt(d)) * t_4) / sqrt(h));
	elseif (l <= 5.8e+132)
		tmp = Float64(t_0 * Float64(t_7 * fma(-0.5, Float64(t_2 / Float64(2.0 * Float64(Float64(d / M) * Float64(Float64(l / D) / h)))), 1.0)));
	else
		tmp = Float64(t_4 * Float64(d / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

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

↓

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(D * N[(0.5 * N[(M / d), $MachinePrecision]), $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[(1.0 + N[(N[Power[t$95$1, 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -2.8e+263], N[(t$95$6 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -7200000.0], N[(N[(N[(t$95$5 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$7), $MachinePrecision] * N[(1.0 + N[(N[Power[N[(t$95$1 * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(t$95$0 * N[(t$95$6 * t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 235.0], N[(t$95$0 * N[(t$95$3 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.6e+112], N[(N[(N[(t$95$7 * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.8e+132], N[(t$95$0 * N[(t$95$7 * N[(-0.5 * N[(t$95$2 / N[(2.0 * N[(N[(d / M), $MachinePrecision] * N[(N[(l / D), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$4 * N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if l < -2.7999999999999998e263

   1. Initial program 46.99

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

   2. Simplified 47.81

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

      [Start] 46.99	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 46.99	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 46.99	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 46.99	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 46.99	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 46.99	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      cancel-sign-sub-inv [=>] 46.99	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]
      +-commutative [=>] 46.99	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell} + 1\right)}\right) \]
      *-commutative [=>] 46.99	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}\right) \cdot \frac{h}{\ell} + 1\right)\right) \]
      distribute-rgt-neg-in [=>] 46.99	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(-\frac{1}{2}\right)\right)} \cdot \frac{h}{\ell} + 1\right)\right) \]
      associate-*l* [=>] 46.99	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\left(-\frac{1}{2}\right) \cdot \frac{h}{\ell}\right)} + 1\right)\right) \]
      fma-def [=>] 46.99	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}, \left(-\frac{1}{2}\right) \cdot \frac{h}{\ell}, 1\right)}\right) \]

   3. Taylor expanded in M around 0 48.78

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

   4. Applied egg-rr 47.85

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

   5. Applied egg-rr 34.57

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

   if -2.7999999999999998e263 < l < -7.2e6

   1. Initial program 38.22

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

   2. Simplified 38.49

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

      [Start] 38.22	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 38.22	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 38.22	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 38.22	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 38.22	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 38.22	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 38.22	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 38.49	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 27.01

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

   4. Applied egg-rr 23.81

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

   if -7.2e6 < l < -4.999999999999985e-310

   1. Initial program 43.12

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

   2. Simplified 44.46

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

      [Start] 43.12	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 43.96	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 43.96	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 43.96	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 43.96	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 43.96	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      sub-neg [=>] 43.96	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]
      +-commutative [=>] 43.96	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]
      associate-*l* [=>] 43.96	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) + 1\right)\right) \]
      distribute-lft-neg-in [=>] 43.96	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(-\frac{1}{2}\right) \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)} + 1\right)\right) \]
      fma-def [=>] 43.96	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(-\frac{1}{2}, {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}, 1\right)}\right) \]

   3. Applied egg-rr 36.77

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

   4. Applied egg-rr 21.93

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

   if -4.999999999999985e-310 < l < 235

   1. Initial program 40.04

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

   2. Simplified 41.66

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

      [Start] 40.04	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 40.63	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 40.63	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 40.63	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 40.63	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 40.63	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      sub-neg [=>] 40.63	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]
      +-commutative [=>] 40.63	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]
      associate-*l* [=>] 40.63	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) + 1\right)\right) \]
      distribute-lft-neg-in [=>] 40.63	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(-\frac{1}{2}\right) \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)} + 1\right)\right) \]
      fma-def [=>] 40.63	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(-\frac{1}{2}, {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}, 1\right)}\right) \]

   3. Applied egg-rr 33.52

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

   4. Applied egg-rr 19.46

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

   5. Simplified 19.42

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

      [Start] 19.46	\[ \sqrt{\frac{d}{h}} \cdot \left(\left(\sqrt{d} \cdot \frac{1}{\sqrt{\ell}}\right) \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right) \cdot h}}, 1\right)\right) \]
      associate-*r/ [=>] 19.42	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{\sqrt{d} \cdot 1}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right) \cdot h}}, 1\right)\right) \]
      *-rgt-identity [=>] 19.42	\[ \sqrt{\frac{d}{h}} \cdot \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right) \cdot h}}, 1\right)\right) \]

   if 235 < l < 1.59999999999999993e112

   1. Initial program 32.61

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

   2. Simplified 33.22

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

      [Start] 32.61	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 32.61	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 32.61	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 32.61	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 32.61	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 32.61	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 32.61	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 33.22	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 22.95

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

   4. Simplified 22.95

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

      [Start] 22.95	\[ \frac{\sqrt{d} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)}{\sqrt{h}} \]
      associate-*r* [=>] 22.95	\[ \frac{\color{blue}{\left(\sqrt{d} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + \left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)}}{\sqrt{h}} \]
      *-commutative [=>] 22.95	\[ \frac{\left(\sqrt{d} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + \color{blue}{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)}\right)}{\sqrt{h}} \]
      *-commutative [=>] 22.95	\[ \frac{\left(\sqrt{d} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \color{blue}{\left(\frac{h}{\ell} \cdot -0.5\right)}\right)}{\sqrt{h}} \]

   if 1.59999999999999993e112 < l < 5.7999999999999997e132

   1. Initial program 37.4

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

   2. Simplified 38.05

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

      [Start] 37.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 37.4	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 37.4	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 37.41	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 37.41	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 37.41	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      sub-neg [=>] 37.41	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]
      +-commutative [=>] 37.41	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]
      associate-*l* [=>] 37.41	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) + 1\right)\right) \]
      distribute-lft-neg-in [=>] 37.41	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(-\frac{1}{2}\right) \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)} + 1\right)\right) \]
      fma-def [=>] 37.41	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(-\frac{1}{2}, {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}, 1\right)}\right) \]

   3. Applied egg-rr 36.51

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

   4. Applied egg-rr 68.89

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

   5. Simplified 36.93

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

      [Start] 68.89	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{e^{\mathsf{log1p}\left(\frac{\ell}{\left(D \cdot \frac{M}{d}\right) \cdot \left(0.5 \cdot h\right)}\right)} - 1}, 1\right)\right) \]
      expm1-def [=>] 61.28	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell}{\left(D \cdot \frac{M}{d}\right) \cdot \left(0.5 \cdot h\right)}\right)\right)}}, 1\right)\right) \]
      expm1-log1p [=>] 36.51	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\color{blue}{\frac{\ell}{\left(D \cdot \frac{M}{d}\right) \cdot \left(0.5 \cdot h\right)}}}, 1\right)\right) \]
      *-rgt-identity [<=] 36.51	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\color{blue}{\frac{\ell}{\left(D \cdot \frac{M}{d}\right) \cdot \left(0.5 \cdot h\right)} \cdot 1}}, 1\right)\right) \]
      associate-*l/ [=>] 36.51	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\color{blue}{\frac{\ell \cdot 1}{\left(D \cdot \frac{M}{d}\right) \cdot \left(0.5 \cdot h\right)}}}, 1\right)\right) \]
      associate-*r* [=>] 36.51	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell \cdot 1}{\color{blue}{\left(\left(D \cdot \frac{M}{d}\right) \cdot 0.5\right) \cdot h}}}, 1\right)\right) \]
      associate-*r* [<=] 36.51	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell \cdot 1}{\color{blue}{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)} \cdot h}}, 1\right)\right) \]
      times-frac [=>] 35.15	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\color{blue}{\frac{\ell}{D \cdot \left(\frac{M}{d} \cdot 0.5\right)} \cdot \frac{1}{h}}}, 1\right)\right) \]
      /-rgt-identity [<=] 35.15	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\color{blue}{\frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{1}}} \cdot \frac{1}{h}}, 1\right)\right) \]
      associate-/l* [=>] 35.16	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\color{blue}{\frac{D}{\frac{1}{\frac{M}{d} \cdot 0.5}}}} \cdot \frac{1}{h}}, 1\right)\right) \]
      *-commutative [=>] 35.16	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\frac{D}{\frac{1}{\color{blue}{0.5 \cdot \frac{M}{d}}}}} \cdot \frac{1}{h}}, 1\right)\right) \]
      associate-/r* [=>] 35.16	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\frac{D}{\color{blue}{\frac{\frac{1}{0.5}}{\frac{M}{d}}}}} \cdot \frac{1}{h}}, 1\right)\right) \]
      metadata-eval [=>] 35.16	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\frac{D}{\frac{\color{blue}{2}}{\frac{M}{d}}}} \cdot \frac{1}{h}}, 1\right)\right) \]
      associate-/l* [<=] 35.16	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\frac{D}{\color{blue}{\frac{2 \cdot d}{M}}}} \cdot \frac{1}{h}}, 1\right)\right) \]
      associate-*r/ [<=] 35.16	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\frac{\ell}{\frac{D}{\color{blue}{2 \cdot \frac{d}{M}}}} \cdot \frac{1}{h}}, 1\right)\right) \]
      associate-/r/ [=>] 35.16	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\color{blue}{\left(\frac{\ell}{D} \cdot \left(2 \cdot \frac{d}{M}\right)\right)} \cdot \frac{1}{h}}, 1\right)\right) \]
      *-commutative [<=] 35.16	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\color{blue}{\left(\left(2 \cdot \frac{d}{M}\right) \cdot \frac{\ell}{D}\right)} \cdot \frac{1}{h}}, 1\right)\right) \]
      associate-*l* [=>] 36.94	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\color{blue}{\left(2 \cdot \frac{d}{M}\right) \cdot \left(\frac{\ell}{D} \cdot \frac{1}{h}\right)}}, 1\right)\right) \]
      associate-*l* [=>] 36.94	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{\color{blue}{2 \cdot \left(\frac{d}{M} \cdot \left(\frac{\ell}{D} \cdot \frac{1}{h}\right)\right)}}, 1\right)\right) \]
      associate-*r/ [=>] 36.93	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(\frac{M}{d} \cdot 0.5\right)}{2 \cdot \left(\frac{d}{M} \cdot \color{blue}{\frac{\frac{\ell}{D} \cdot 1}{h}}\right)}, 1\right)\right) \]

   if 5.7999999999999997e132 < l

   1. Initial program 48.62

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

   2. Simplified 48.48

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

      [Start] 48.62	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 48.62	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 48.62	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 48.62	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 48.62	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 48.62	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 48.62	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 48.48	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 26.4

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

   4. Simplified 26.4

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

      [Start] 26.4	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \]
      *-lft-identity [<=] 26.4	\[ \color{blue}{1 \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} + \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \]
      *-commutative [<=] 26.4	\[ 1 \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \color{blue}{\left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]
      distribute-rgt-in [<=] 26.4	\[ \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)} \]
      *-commutative [=>] 26.4	\[ \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 + \left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \]
      *-commutative [=>] 26.4	\[ \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \color{blue}{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)}\right) \]
      *-commutative [=>] 26.4	\[ \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \color{blue}{\left(\frac{h}{\ell} \cdot -0.5\right)}\right) \]

3. Recombined 7 regimes into one program.

4. Final simplification 23.94

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.8 \cdot 10^{+263}:\\ \;\;\;\;\frac{\frac{\sqrt{-d}}{\sqrt{-\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;\ell \leq -7200000:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + {\left(\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2} \cdot -0.5\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{\ell}{h \cdot \left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}}, 1\right)\right)\\ \mathbf{elif}\;\ell \leq 235:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\mathsf{fma}\left(-0.5, \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{\ell}{h \cdot \left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}}, 1\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+112}:\\ \;\;\;\;\frac{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{d}\right) \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)}{\sqrt{h}}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{2 \cdot \left(\frac{d}{M} \cdot \frac{\frac{\ell}{D}}{h}\right)}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternatives

Alternative 1
Error	24.45%
Cost	33800

\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := D \cdot \left(0.5 \cdot \frac{M}{d}\right)\\ t_2 := \mathsf{fma}\left(-0.5, \frac{t_1}{2 \cdot \left(\frac{d}{M} \cdot \frac{\frac{\ell}{D}}{h}\right)}, 1\right)\\ t_3 := \frac{t_0}{\sqrt{-\ell}}\\ t_4 := \sqrt{\frac{d}{h}}\\ t_5 := \mathsf{fma}\left(-0.5, \frac{t_1}{\frac{\ell}{h \cdot t_1}}, 1\right)\\ t_6 := \sqrt{\frac{d}{\ell}}\\ t_7 := 1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\\ \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+221}:\\ \;\;\;\;t_4 \cdot \left(t_3 \cdot t_2\right)\\ \mathbf{elif}\;\ell \leq -120000:\\ \;\;\;\;\frac{t_0}{\sqrt{-h}} \cdot \left(t_6 \cdot \mathsf{fma}\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}, \frac{-0.5}{\frac{\ell}{h}}, 1\right)\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_4 \cdot \left(t_3 \cdot t_5\right)\\ \mathbf{elif}\;\ell \leq 96:\\ \;\;\;\;t_4 \cdot \left(t_5 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+111}:\\ \;\;\;\;\frac{\left(t_6 \cdot \sqrt{d}\right) \cdot t_7}{\sqrt{h}}\\ \mathbf{elif}\;\ell \leq 1.76 \cdot 10^{+133}:\\ \;\;\;\;t_4 \cdot \left(t_6 \cdot t_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_7 \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 2
Error	27.96%
Cost	28116

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := D \cdot \left(0.5 \cdot \frac{M}{d}\right)\\ t_2 := {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\\ t_3 := 1 + t_2 \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\\ t_4 := \sqrt{-d}\\ t_5 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -3.6 \cdot 10^{+253}:\\ \;\;\;\;\frac{\frac{t_4}{\sqrt{-\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;\ell \leq -2.8 \cdot 10^{-15}:\\ \;\;\;\;\left(\frac{t_4}{\sqrt{-h}} \cdot t_5\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -6.4 \cdot 10^{-103}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-308}:\\ \;\;\;\;\left(t_5 \cdot t_0\right) \cdot \left(1 + -0.5 \cdot \frac{h \cdot t_2}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 100:\\ \;\;\;\;t_0 \cdot \left(\mathsf{fma}\left(-0.5, \frac{t_1}{\frac{\ell}{h \cdot t_1}}, 1\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 3.1 \cdot 10^{+112}:\\ \;\;\;\;\frac{\left(t_5 \cdot \sqrt{d}\right) \cdot t_3}{\sqrt{h}}\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{+133}:\\ \;\;\;\;t_0 \cdot \left(t_5 \cdot \mathsf{fma}\left(-0.5, \frac{t_1}{2 \cdot \left(\frac{d}{M} \cdot \frac{\frac{\ell}{D}}{h}\right)}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 3
Error	25.06%
Cost	27984

\[\begin{array}{l} t_0 := D \cdot \left(0.5 \cdot \frac{M}{d}\right)\\ t_1 := \sqrt{-d}\\ t_2 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -5.5 \cdot 10^{+179}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -7 \cdot 10^{+39}:\\ \;\;\;\;\left(\frac{t_1}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -1.3 \cdot 10^{-295}:\\ \;\;\;\;t_2 \cdot \left(\frac{t_1}{\sqrt{-\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{t_0}{2 \cdot \left(\frac{d}{M} \cdot \frac{\frac{\ell}{D}}{h}\right)}, 1\right)\right)\\ \mathbf{elif}\;d \leq 4.7 \cdot 10^{+85}:\\ \;\;\;\;t_2 \cdot \left(\mathsf{fma}\left(-0.5, \frac{t_0}{\frac{\ell}{h \cdot t_0}}, 1\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 4
Error	24.57%
Cost	27984

\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := D \cdot \left(0.5 \cdot \frac{M}{d}\right)\\ t_2 := \mathsf{fma}\left(-0.5, \frac{t_1}{2 \cdot \left(\frac{d}{M} \cdot \frac{\frac{\ell}{D}}{h}\right)}, 1\right)\\ t_3 := \sqrt{\frac{d}{h}}\\ t_4 := \mathsf{fma}\left(-0.5, \frac{t_1}{\frac{\ell}{h \cdot t_1}}, 1\right)\\ t_5 := \sqrt{\frac{d}{\ell}}\\ t_6 := 1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\\ t_7 := \frac{t_0}{\sqrt{-\ell}}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+221}:\\ \;\;\;\;t_3 \cdot \left(t_7 \cdot t_2\right)\\ \mathbf{elif}\;\ell \leq -510000000:\\ \;\;\;\;\left(\frac{t_0}{\sqrt{-h}} \cdot t_5\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_3 \cdot \left(t_7 \cdot t_4\right)\\ \mathbf{elif}\;\ell \leq 125:\\ \;\;\;\;t_3 \cdot \left(t_4 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 1.42 \cdot 10^{+112}:\\ \;\;\;\;\frac{\left(t_5 \cdot \sqrt{d}\right) \cdot t_6}{\sqrt{h}}\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+132}:\\ \;\;\;\;t_3 \cdot \left(t_5 \cdot t_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_6 \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 5
Error	28.99%
Cost	27528

\[\begin{array}{l} t_0 := D \cdot \left(0.5 \cdot \frac{M}{d}\right)\\ t_1 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ t_2 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -4.8 \cdot 10^{+160}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -5.2 \cdot 10^{-166}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot t_2\right)\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-96}:\\ \;\;\;\;\left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot t_1\\ \mathbf{elif}\;d \leq 1.46 \cdot 10^{+81}:\\ \;\;\;\;t_2 \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{t_0}{\frac{\ell}{h \cdot t_0}}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	32.08%
Cost	21584

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := D \cdot \left(0.5 \cdot \frac{M}{d}\right)\\ \mathbf{if}\;d \leq -6.8 \cdot 10^{+160}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -3.8 \cdot 10^{-118}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \mathsf{fma}\left(-0.5, \frac{t_2}{\frac{\ell}{h \cdot t_2}}, 1\right)\right)\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{-211}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;d \leq 8.2 \cdot 10^{+80}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \mathsf{fma}\left(-0.5, \frac{t_2}{2 \cdot \left(\frac{d}{M} \cdot \frac{\frac{\ell}{D}}{h}\right)}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 7
Error	32.01%
Cost	21396

\[\begin{array}{l} t_0 := {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\\ t_1 := \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + -0.5 \cdot \frac{h \cdot t_0}{\ell}\right)\\ \mathbf{if}\;\ell \leq -2.6 \cdot 10^{+125}:\\ \;\;\;\;\frac{\frac{\sqrt{-d}}{\sqrt{-\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;\ell \leq -1.4 \cdot 10^{-103}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-309}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 8.6 \cdot 10^{-193}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \leq 9.6 \cdot 10^{-128}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t_0 \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 8
Error	31.61%
Cost	21268

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ t_1 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -8.2 \cdot 10^{+161}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-111}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot t_1\right)\\ \mathbf{elif}\;d \leq -1.1 \cdot 10^{-143}:\\ \;\;\;\;\frac{\sqrt{-d} \cdot t_1}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{-277}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{+36}:\\ \;\;\;\;\left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	32.02%
Cost	21136

\[\begin{array}{l} t_0 := \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{if}\;\ell \leq -1.85 \cdot 10^{+126}:\\ \;\;\;\;\frac{t_0}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;\ell \leq -7 \cdot 10^{-152}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_0 \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{-212}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 10
Error	33.49%
Cost	19908

\[\begin{array}{l} \mathbf{if}\;h \leq -1.32 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{-d}}{\frac{\sqrt{-\ell}}{\sqrt{\frac{d}{h}}}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 11
Error	33.5%
Cost	19908

\[\begin{array}{l} \mathbf{if}\;h \leq -2 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{-d} \cdot \sqrt{\frac{d}{h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 12
Error	33.49%
Cost	19908

\[\begin{array}{l} \mathbf{if}\;h \leq -1.05 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 13
Error	38.18%
Cost	14920

\[\begin{array}{l} \mathbf{if}\;d \leq -1.9 \cdot 10^{+89}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -3.3 \cdot 10^{+62}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + 0.5 \cdot \left(\left(\frac{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}{\ell} \cdot \frac{\frac{h}{d}}{d}\right) \cdot -0.25\right)\right)\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{-211}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 14
Error	35.44%
Cost	14920

\[\begin{array}{l} \mathbf{if}\;h \leq -2.45 \cdot 10^{+117}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;h \leq -1.75 \cdot 10^{+26}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \left(D \cdot \frac{D}{\frac{\frac{\frac{d}{\frac{h}{\ell}}}{\frac{M}{d}}}{M}}\right) \cdot -0.125\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 15
Error	38.73%
Cost	14600

\[\begin{array}{l} \mathbf{if}\;d \leq -9.8 \cdot 10^{+123}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -5000000:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{-211}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 16
Error	38.14%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;d \leq 4.2 \cdot 10^{-211}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 17
Error	52.27%
Cost	7244

\[\begin{array}{l} t_0 := \sqrt{d \cdot \frac{\frac{d}{\ell}}{h}}\\ \mathbf{if}\;h \leq 3 \cdot 10^{-304}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq 1.18 \cdot 10^{+160}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;h \leq 7.6 \cdot 10^{+259}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]

Alternative 18
Error	52.36%
Cost	7244

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{if}\;h \leq 7.8 \cdot 10^{-304}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq 1.55 \cdot 10^{+162}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;h \leq 5.4 \cdot 10^{+259}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]

Alternative 19
Error	52.2%
Cost	7244

\[\begin{array}{l} \mathbf{if}\;h \leq 1.5 \cdot 10^{-302}:\\ \;\;\;\;\sqrt{\frac{d}{\ell \cdot \frac{h}{d}}}\\ \mathbf{elif}\;h \leq 7.5 \cdot 10^{+161}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;h \leq 1.5 \cdot 10^{+260}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]

Alternative 20
Error	52.23%
Cost	7244

\[\begin{array}{l} \mathbf{if}\;h \leq 1.55 \cdot 10^{-302}:\\ \;\;\;\;\sqrt{\frac{d}{\ell \cdot \frac{h}{d}}}\\ \mathbf{elif}\;h \leq 4.5 \cdot 10^{+155}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;h \leq 4.7 \cdot 10^{+259}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]

Alternative 21
Error	43.43%
Cost	7244

\[\begin{array}{l} \mathbf{if}\;h \leq -6.2 \cdot 10^{-306}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;h \leq 3.8 \cdot 10^{+157}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;h \leq 2.45 \cdot 10^{+260}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]

Alternative 22
Error	43.24%
Cost	7244

\[\begin{array}{l} \mathbf{if}\;h \leq -6.2 \cdot 10^{-306}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;h \leq 1.25 \cdot 10^{+160}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;h \leq 3.5 \cdot 10^{+259}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]

Alternative 23
Error	54.45%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;h \leq -6.2 \cdot 10^{-306}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]

Alternative 24
Error	68.72%
Cost	6720

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

Reproduce

herbie shell --seed 2023115 
(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)))))