Henrywood and Agarwal, Equation (12)

Average Error: 27.0 → 16.8

Time: 35.9s

Precision: binary64

Cost: 110608

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \frac{d}{\sqrt{h \cdot \ell}}\\ t_3 := \sqrt{\frac{d}{h}}\\ t_4 := M \cdot \frac{D}{d}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.125, \frac{\left(h \cdot M\right) \cdot \left(D \cdot D\right)}{\frac{\ell}{M} \cdot \left(d \cdot d\right)}, 1\right)\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-120}:\\ \;\;\;\;t_3 \cdot \left(t_0 \cdot \mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}, 1\right)\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\left|t_2\right| \cdot \mathsf{fma}\left(-0.125, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(h \cdot M\right)}{\ell}, 1\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+255}:\\ \;\;\;\;t_3 \cdot \left(t_0 \cdot \left(1 + \left(t_4 \cdot t_4\right) \cdot \frac{-0.125}{\frac{\ell}{h}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|t_2 \cdot \mathsf{fma}\left(-0.125, \left(h \cdot M\right) \cdot \frac{{\left(\frac{D}{d}\right)}^{2}}{\frac{\ell}{M}}, 1\right)\right|\\ \end{array} \]

FPCore

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

↓

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (- 1.0 (* (* 0.5 (pow (/ (* M D) (* d 2.0)) 2.0)) (/ h l)))))
        (t_2 (/ d (sqrt (* h l))))
        (t_3 (sqrt (/ d h)))
        (t_4 (* M (/ D d))))
   (if (<= t_1 (- INFINITY))
     (*
      (sqrt (* (/ d h) (/ d l)))
      (fma -0.125 (/ (* (* h M) (* D D)) (* (/ l M) (* d d))) 1.0))
     (if (<= t_1 -2e-120)
       (*
        t_3
        (* t_0 (fma -0.5 (* (/ h l) (pow (* D (/ (/ M d) 2.0)) 2.0)) 1.0)))
       (if (<= t_1 0.0)
         (*
          (fabs t_2)
          (fma -0.125 (* (* (/ D d) (/ D d)) (/ (* M (* h M)) l)) 1.0))
         (if (<= t_1 2e+255)
           (* t_3 (* t_0 (+ 1.0 (* (* t_4 t_4) (/ -0.125 (/ l h))))))
           (fabs
            (*
             t_2
             (fma -0.125 (* (* h M) (/ (pow (/ D d) 2.0) (/ l M))) 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 / l));
	double t_1 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((0.5 * pow(((M * D) / (d * 2.0)), 2.0)) * (h / l)));
	double t_2 = d / sqrt((h * l));
	double t_3 = sqrt((d / h));
	double t_4 = M * (D / d);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = sqrt(((d / h) * (d / l))) * fma(-0.125, (((h * M) * (D * D)) / ((l / M) * (d * d))), 1.0);
	} else if (t_1 <= -2e-120) {
		tmp = t_3 * (t_0 * fma(-0.5, ((h / l) * pow((D * ((M / d) / 2.0)), 2.0)), 1.0));
	} else if (t_1 <= 0.0) {
		tmp = fabs(t_2) * fma(-0.125, (((D / d) * (D / d)) * ((M * (h * M)) / l)), 1.0);
	} else if (t_1 <= 2e+255) {
		tmp = t_3 * (t_0 * (1.0 + ((t_4 * t_4) * (-0.125 / (l / h)))));
	} else {
		tmp = fabs((t_2 * fma(-0.125, ((h * M) * (pow((D / d), 2.0) / (l / M))), 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 / l))
	t_1 = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(0.5 * (Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0)) * Float64(h / l))))
	t_2 = Float64(d / sqrt(Float64(h * l)))
	t_3 = sqrt(Float64(d / h))
	t_4 = Float64(M * Float64(D / d))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * fma(-0.125, Float64(Float64(Float64(h * M) * Float64(D * D)) / Float64(Float64(l / M) * Float64(d * d))), 1.0));
	elseif (t_1 <= -2e-120)
		tmp = Float64(t_3 * Float64(t_0 * fma(-0.5, Float64(Float64(h / l) * (Float64(D * Float64(Float64(M / d) / 2.0)) ^ 2.0)), 1.0)));
	elseif (t_1 <= 0.0)
		tmp = Float64(abs(t_2) * fma(-0.125, Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(M * Float64(h * M)) / l)), 1.0));
	elseif (t_1 <= 2e+255)
		tmp = Float64(t_3 * Float64(t_0 * Float64(1.0 + Float64(Float64(t_4 * t_4) * Float64(-0.125 / Float64(l / h))))));
	else
		tmp = abs(Float64(t_2 * fma(-0.125, Float64(Float64(h * M) * Float64((Float64(D / d) ^ 2.0) / Float64(l / M))), 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[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(0.5 * N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(N[(N[(h * M), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision] / N[(N[(l / M), $MachinePrecision] * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e-120], N[(t$95$3 * N[(t$95$0 * N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(N[(M / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[Abs[t$95$2], $MachinePrecision] * N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+255], N[(t$95$3 * N[(t$95$0 * N[(1.0 + N[(N[(t$95$4 * t$95$4), $MachinePrecision] * N[(-0.125 / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(t$95$2 * N[(-0.125 * N[(N[(h * M), $MachinePrecision] * N[(N[Power[N[(D / d), $MachinePrecision], 2.0], $MachinePrecision] / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 61.2

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

      [Start] 64.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 64.0	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 64.0	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 64.0	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 64.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 64.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 64.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 61.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 61.3

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

   4. Simplified 61.5

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

      [Start] 61.3	\[ \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} + \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5\right) \]
      *-lft-identity [<=] 61.3	\[ \color{blue}{1 \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} + \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5\right) \]
      *-commutative [<=] 61.3	\[ 1 \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} + \color{blue}{\left(\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]
      distribute-rgt-in [<=] 61.3	\[ \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5\right)} \]
      associate-/r* [=>] 61.5	\[ \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \cdot \left(1 + \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5\right) \]
      +-commutative [=>] 61.5	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \color{blue}{\left(\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5 + 1\right)} \]
      associate-*l* [=>] 61.5	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(\color{blue}{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)} + 1\right) \]
      fma-def [=>] 61.5	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \color{blue}{\mathsf{fma}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)} \]
      associate-*r/ [=>] 61.5	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \mathsf{fma}\left({\left(M \cdot \color{blue}{\frac{0.5 \cdot D}{d}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \]
      associate-/l* [=>] 61.5	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \mathsf{fma}\left({\left(M \cdot \color{blue}{\frac{0.5}{\frac{d}{D}}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \]

   5. Taylor expanded in M around 0 61.5

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

   6. Simplified 57.2

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

      [Start] 61.5	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right) \]
      +-commutative [=>] 61.5	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} + 1\right)} \]
      fma-def [=>] 61.5	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}, 1\right)} \]
      times-frac [=>] 61.9	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \mathsf{fma}\left(-0.125, \color{blue}{\frac{{D}^{2}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}}, 1\right) \]
      unpow2 [=>] 61.9	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \mathsf{fma}\left(-0.125, \frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}, 1\right) \]
      unpow2 [=>] 61.9	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \mathsf{fma}\left(-0.125, \frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \frac{{M}^{2} \cdot h}{\ell}, 1\right) \]
      times-frac [=>] 60.0	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \mathsf{fma}\left(-0.125, \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \frac{{M}^{2} \cdot h}{\ell}, 1\right) \]
      *-commutative [=>] 60.0	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \mathsf{fma}\left(-0.125, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{\color{blue}{h \cdot {M}^{2}}}{\ell}, 1\right) \]
      unpow2 [=>] 60.0	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \mathsf{fma}\left(-0.125, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{\ell}, 1\right) \]
      associate-*r* [=>] 57.2	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \mathsf{fma}\left(-0.125, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{\ell}, 1\right) \]

   7. Applied egg-rr 57.5

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

   8. Simplified 53.8

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

      [Start] 57.5	\[ \sqrt{\frac{\frac{d \cdot d}{\ell}}{h}} \cdot \mathsf{fma}\left(-0.125, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{\left(h \cdot M\right) \cdot M}{\ell}, 1\right) \]
      associate-/l/ [=>] 56.3	\[ \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \mathsf{fma}\left(-0.125, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{\left(h \cdot M\right) \cdot M}{\ell}, 1\right) \]
      *-commutative [=>] 56.3	\[ \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot \mathsf{fma}\left(-0.125, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{\left(h \cdot M\right) \cdot M}{\ell}, 1\right) \]
      times-frac [=>] 53.8	\[ \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \mathsf{fma}\left(-0.125, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{\left(h \cdot M\right) \cdot M}{\ell}, 1\right) \]

   9. Applied egg-rr 56.6

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

   if -inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < -1.99999999999999996e-120

   1. Initial program 1.5

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

   2. Simplified 7.2

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

      [Start] 1.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 3.1	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 3.1	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 3.1	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 3.1	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 3.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      sub-neg [=>] 3.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]
      +-commutative [=>] 3.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]
      associate-*l* [=>] 3.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) + 1\right)\right) \]
      distribute-lft-neg-in [=>] 3.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(-\frac{1}{2}\right) \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)} + 1\right)\right) \]
      fma-def [=>] 3.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(-\frac{1}{2}, {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}, 1\right)}\right) \]

   if -1.99999999999999996e-120 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 0.0

   1. Initial program 39.0

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

   2. Simplified 40.3

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

      [Start] 39.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 39.0	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 39.0	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 39.0	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 39.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 39.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 39.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 40.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 36.4

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

   4. Simplified 40.1

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

      [Start] 36.4	\[ \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} + \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5\right) \]
      *-lft-identity [<=] 36.4	\[ \color{blue}{1 \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} + \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5\right) \]
      *-commutative [<=] 36.4	\[ 1 \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} + \color{blue}{\left(\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]
      distribute-rgt-in [<=] 36.4	\[ \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5\right)} \]
      associate-/r* [=>] 40.1	\[ \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \cdot \left(1 + \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5\right) \]
      +-commutative [=>] 40.1	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \color{blue}{\left(\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5 + 1\right)} \]
      associate-*l* [=>] 40.1	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(\color{blue}{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)} + 1\right) \]
      fma-def [=>] 40.1	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \color{blue}{\mathsf{fma}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)} \]
      associate-*r/ [=>] 40.1	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \mathsf{fma}\left({\left(M \cdot \color{blue}{\frac{0.5 \cdot D}{d}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \]
      associate-/l* [=>] 40.1	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \mathsf{fma}\left({\left(M \cdot \color{blue}{\frac{0.5}{\frac{d}{D}}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \]

   5. Taylor expanded in M around 0 55.8

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

   6. Simplified 46.9

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

      [Start] 55.8	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right) \]
      +-commutative [=>] 55.8	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} + 1\right)} \]
      fma-def [=>] 55.8	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}, 1\right)} \]
      times-frac [=>] 56.8	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \mathsf{fma}\left(-0.125, \color{blue}{\frac{{D}^{2}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}}, 1\right) \]
      unpow2 [=>] 56.8	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \mathsf{fma}\left(-0.125, \frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}, 1\right) \]
      unpow2 [=>] 56.8	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \mathsf{fma}\left(-0.125, \frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \frac{{M}^{2} \cdot h}{\ell}, 1\right) \]
      times-frac [=>] 47.2	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \mathsf{fma}\left(-0.125, \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \frac{{M}^{2} \cdot h}{\ell}, 1\right) \]
      *-commutative [=>] 47.2	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \mathsf{fma}\left(-0.125, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{\color{blue}{h \cdot {M}^{2}}}{\ell}, 1\right) \]
      unpow2 [=>] 47.2	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \mathsf{fma}\left(-0.125, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{\ell}, 1\right) \]
      associate-*r* [=>] 46.9	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \mathsf{fma}\left(-0.125, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{\ell}, 1\right) \]

   7. Applied egg-rr 49.1

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

   8. Simplified 49.0

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

      [Start] 49.1	\[ \sqrt{\frac{\frac{d \cdot d}{\ell}}{h}} \cdot \mathsf{fma}\left(-0.125, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{\left(h \cdot M\right) \cdot M}{\ell}, 1\right) \]
      associate-/l/ [=>] 46.7	\[ \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \mathsf{fma}\left(-0.125, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{\left(h \cdot M\right) \cdot M}{\ell}, 1\right) \]
      *-commutative [=>] 46.7	\[ \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot \mathsf{fma}\left(-0.125, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{\left(h \cdot M\right) \cdot M}{\ell}, 1\right) \]
      times-frac [=>] 49.0	\[ \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \mathsf{fma}\left(-0.125, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{\left(h \cdot M\right) \cdot M}{\ell}, 1\right) \]

   9. Applied egg-rr 28.9

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 1.99999999999999998e255

   1. Initial program 1.0

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

   2. Simplified 1.2

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

      [Start] 1.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 1.0	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 1.0	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 1.0	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 1.0	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 1.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      cancel-sign-sub-inv [=>] 1.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]
      +-commutative [=>] 1.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell} + 1\right)}\right) \]
      *-commutative [=>] 1.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}\right) \cdot \frac{h}{\ell} + 1\right)\right) \]
      distribute-rgt-neg-in [=>] 1.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(-\frac{1}{2}\right)\right)} \cdot \frac{h}{\ell} + 1\right)\right) \]
      associate-*l* [=>] 1.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\left(-\frac{1}{2}\right) \cdot \frac{h}{\ell}\right)} + 1\right)\right) \]
      fma-def [=>] 1.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}, \left(-\frac{1}{2}\right) \cdot \frac{h}{\ell}, 1\right)}\right) \]

   3. Applied egg-rr 1.2

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

   4. Taylor expanded in M around 0 22.9

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

   5. Simplified 1.2

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

      [Start] 22.9	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} + 1\right)\right) \]
      associate-/l* [=>] 22.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \color{blue}{\frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{{M}^{2} \cdot h}}} + 1\right)\right) \]
      associate-*r/ [=>] 22.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{-0.125 \cdot {D}^{2}}{\frac{\ell \cdot {d}^{2}}{{M}^{2} \cdot h}}} + 1\right)\right) \]
      *-commutative [<=] 22.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\color{blue}{{D}^{2} \cdot -0.125}}{\frac{\ell \cdot {d}^{2}}{{M}^{2} \cdot h}} + 1\right)\right) \]
      *-commutative [=>] 22.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{{D}^{2} \cdot -0.125}{\frac{\color{blue}{{d}^{2} \cdot \ell}}{{M}^{2} \cdot h}} + 1\right)\right) \]
      times-frac [=>] 18.7	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{{D}^{2} \cdot -0.125}{\color{blue}{\frac{{d}^{2}}{{M}^{2}} \cdot \frac{\ell}{h}}} + 1\right)\right) \]
      times-frac [=>] 18.4	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{{M}^{2}}} \cdot \frac{-0.125}{\frac{\ell}{h}}} + 1\right)\right) \]
      associate-/l* [<=] 18.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}} \cdot \frac{-0.125}{\frac{\ell}{h}} + 1\right)\right) \]
      *-commutative [=>] 18.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}} \cdot \frac{-0.125}{\frac{\ell}{h}} + 1\right)\right) \]
      unpow2 [=>] 18.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\color{blue}{\left(M \cdot M\right)} \cdot {D}^{2}}{{d}^{2}} \cdot \frac{-0.125}{\frac{\ell}{h}} + 1\right)\right) \]
      unpow2 [=>] 18.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\left(M \cdot M\right) \cdot \color{blue}{\left(D \cdot D\right)}}{{d}^{2}} \cdot \frac{-0.125}{\frac{\ell}{h}} + 1\right)\right) \]
      swap-sqr [<=] 7.4	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}}{{d}^{2}} \cdot \frac{-0.125}{\frac{\ell}{h}} + 1\right)\right) \]
      unpow2 [=>] 7.4	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{d \cdot d}} \cdot \frac{-0.125}{\frac{\ell}{h}} + 1\right)\right) \]
      times-frac [=>] 1.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d}\right)} \cdot \frac{-0.125}{\frac{\ell}{h}} + 1\right)\right) \]
      *-commutative [<=] 1.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{\color{blue}{D \cdot M}}{d} \cdot \frac{M \cdot D}{d}\right) \cdot \frac{-0.125}{\frac{\ell}{h}} + 1\right)\right) \]
      associate-*l/ [<=] 1.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\color{blue}{\left(\frac{D}{d} \cdot M\right)} \cdot \frac{M \cdot D}{d}\right) \cdot \frac{-0.125}{\frac{\ell}{h}} + 1\right)\right) \]
      *-commutative [=>] 1.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\color{blue}{\left(M \cdot \frac{D}{d}\right)} \cdot \frac{M \cdot D}{d}\right) \cdot \frac{-0.125}{\frac{\ell}{h}} + 1\right)\right) \]
      *-commutative [<=] 1.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{\color{blue}{D \cdot M}}{d}\right) \cdot \frac{-0.125}{\frac{\ell}{h}} + 1\right)\right) \]
      associate-*l/ [<=] 1.2	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\left(M \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot M\right)}\right) \cdot \frac{-0.125}{\frac{\ell}{h}} + 1\right)\right) \]
      *-commutative [=>] 1.2	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\left(M \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(M \cdot \frac{D}{d}\right)}\right) \cdot \frac{-0.125}{\frac{\ell}{h}} + 1\right)\right) \]

   if 1.99999999999999998e255 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))

   1. Initial program 62.2

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

   2. Simplified 62.2

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

      [Start] 62.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 62.2	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 62.2	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 62.2	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 62.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 62.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 62.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 62.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 51.4

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

   4. Simplified 52.1

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

      [Start] 51.4	\[ \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} + \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5\right) \]
      *-lft-identity [<=] 51.4	\[ \color{blue}{1 \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} + \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5\right) \]
      *-commutative [<=] 51.4	\[ 1 \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} + \color{blue}{\left(\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]
      distribute-rgt-in [<=] 51.4	\[ \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5\right)} \]
      associate-/r* [=>] 52.1	\[ \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \cdot \left(1 + \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5\right) \]
      +-commutative [=>] 52.1	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \color{blue}{\left(\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5 + 1\right)} \]
      associate-*l* [=>] 52.1	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(\color{blue}{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)} + 1\right) \]
      fma-def [=>] 52.1	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \color{blue}{\mathsf{fma}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)} \]
      associate-*r/ [=>] 52.1	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \mathsf{fma}\left({\left(M \cdot \color{blue}{\frac{0.5 \cdot D}{d}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \]
      associate-/l* [=>] 52.1	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \mathsf{fma}\left({\left(M \cdot \color{blue}{\frac{0.5}{\frac{d}{D}}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \]

   5. Taylor expanded in M around 0 53.1

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

   6. Simplified 48.0

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

      [Start] 53.1	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right) \]
      +-commutative [=>] 53.1	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} + 1\right)} \]
      fma-def [=>] 53.1	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}, 1\right)} \]
      times-frac [=>] 53.2	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \mathsf{fma}\left(-0.125, \color{blue}{\frac{{D}^{2}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}}, 1\right) \]
      unpow2 [=>] 53.2	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \mathsf{fma}\left(-0.125, \frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}, 1\right) \]
      unpow2 [=>] 53.2	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \mathsf{fma}\left(-0.125, \frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \frac{{M}^{2} \cdot h}{\ell}, 1\right) \]
      times-frac [=>] 50.3	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \mathsf{fma}\left(-0.125, \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \frac{{M}^{2} \cdot h}{\ell}, 1\right) \]
      *-commutative [=>] 50.3	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \mathsf{fma}\left(-0.125, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{\color{blue}{h \cdot {M}^{2}}}{\ell}, 1\right) \]
      unpow2 [=>] 50.3	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \mathsf{fma}\left(-0.125, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{\ell}, 1\right) \]
      associate-*r* [=>] 48.0	\[ \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \mathsf{fma}\left(-0.125, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{\ell}, 1\right) \]

   7. Applied egg-rr 47.9

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

   8. Applied egg-rr 55.1

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

   9. Simplified 29.4

      \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(-0.125, \left(M \cdot h\right) \cdot \frac{{\left(\frac{D}{d}\right)}^{2}}{\frac{\ell}{M}}, 1\right)\right|} \]
      Proof

      [Start] 55.1	\[ \sqrt{{\left(\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(-0.125, \left(\left(M \cdot h\right) \cdot \frac{M}{\ell}\right) \cdot {\left(\frac{D}{d}\right)}^{2}, 1\right)\right)}^{2}} \]
      unpow2 [=>] 55.1	\[ \sqrt{\color{blue}{\left(\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(-0.125, \left(\left(M \cdot h\right) \cdot \frac{M}{\ell}\right) \cdot {\left(\frac{D}{d}\right)}^{2}, 1\right)\right) \cdot \left(\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(-0.125, \left(\left(M \cdot h\right) \cdot \frac{M}{\ell}\right) \cdot {\left(\frac{D}{d}\right)}^{2}, 1\right)\right)}} \]
      rem-sqrt-square [=>] 31.8	\[ \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(-0.125, \left(\left(M \cdot h\right) \cdot \frac{M}{\ell}\right) \cdot {\left(\frac{D}{d}\right)}^{2}, 1\right)\right|} \]

3. Recombined 5 regimes into one program.

4. Final simplification 16.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -\infty:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.125, \frac{\left(h \cdot M\right) \cdot \left(D \cdot D\right)}{\frac{\ell}{M} \cdot \left(d \cdot d\right)}, 1\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -2 \cdot 10^{-120}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}, 1\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \mathsf{fma}\left(-0.125, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(h \cdot M\right)}{\ell}, 1\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{+255}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right) \cdot \frac{-0.125}{\frac{\ell}{h}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(-0.125, \left(h \cdot M\right) \cdot \frac{{\left(\frac{D}{d}\right)}^{2}}{\frac{\ell}{M}}, 1\right)\right|\\ \end{array} \]

Alternatives

Alternative 1
Error	16.8
Cost	110608

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{h \cdot \ell}}\\ t_1 := M \cdot \frac{D}{d}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_3 := \sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(t_1 \cdot t_1\right) \cdot \frac{-0.125}{\frac{\ell}{h}}\right)\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.125, \frac{\left(h \cdot M\right) \cdot \left(D \cdot D\right)}{\frac{\ell}{M} \cdot \left(d \cdot d\right)}, 1\right)\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-120}:\\ \;\;\;\;t_3 \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\left|t_0\right| \cdot \mathsf{fma}\left(-0.125, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(h \cdot M\right)}{\ell}, 1\right)\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+255}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot t_3\\ \mathbf{else}:\\ \;\;\;\;\left|t_0 \cdot \mathsf{fma}\left(-0.125, \left(h \cdot M\right) \cdot \frac{{\left(\frac{D}{d}\right)}^{2}}{\frac{\ell}{M}}, 1\right)\right|\\ \end{array} \]

Alternative 2
Error	16.9
Cost	104401

\[\begin{array}{l} t_0 := M \cdot \frac{D}{d}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(t_0 \cdot t_0\right) \cdot \frac{-0.125}{\frac{\ell}{h}}\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.125, \frac{\left(h \cdot M\right) \cdot \left(D \cdot D\right)}{\frac{\ell}{M} \cdot \left(d \cdot d\right)}, 1\right)\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-120}:\\ \;\;\;\;t_2 \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;t_1 \leq 0 \lor \neg \left(t_1 \leq 2 \cdot 10^{+255}\right):\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \mathsf{fma}\left(-0.125, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(h \cdot M\right)}{\ell}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot t_2\\ \end{array} \]

Alternative 3
Error	21.4
Cost	21004

\[\begin{array}{l} t_0 := M \cdot \frac{D}{d}\\ \mathbf{if}\;d \leq -3.5 \cdot 10^{+92}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -6.2 \cdot 10^{-40}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(t_0 \cdot t_0\right) \cdot \frac{-0.125}{\frac{\ell}{h}}\right)\right) \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq -8.2 \cdot 10^{-304}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(h \cdot M\right)}{\ell}, 1\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 4
Error	22.7
Cost	14920

\[\begin{array}{l} t_0 := M \cdot \frac{D}{d}\\ \mathbf{if}\;d \leq -2.1 \cdot 10^{+92}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -1.85 \cdot 10^{-39}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(t_0 \cdot t_0\right) \cdot \frac{-0.125}{\frac{\ell}{h}}\right)\right) \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq -8.2 \cdot 10^{-304}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(h \cdot M\right)}{\ell}, 1\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 5
Error	23.5
Cost	14860

\[\begin{array}{l} t_0 := d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{if}\;d \leq -2.2 \cdot 10^{+57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.95 \cdot 10^{-35}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -8.2 \cdot 10^{-304}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(h \cdot M\right)}{\ell}, 1\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 6
Error	22.8
Cost	14860

\[\begin{array}{l} t_0 := M \cdot \frac{D}{d}\\ \mathbf{if}\;d \leq -1.8 \cdot 10^{+92}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -2.3 \cdot 10^{-41}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(t_0 \cdot t_0\right) \cdot \frac{-0.125}{\frac{\ell}{h}}\right)\right)\\ \mathbf{elif}\;d \leq -8.2 \cdot 10^{-304}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(h \cdot M\right)}{\ell}, 1\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 7
Error	23.8
Cost	14664

\[\begin{array}{l} t_0 := d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{if}\;d \leq -1.12 \cdot 10^{+55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -2.2 \cdot 10^{-48}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.125, \frac{\left(h \cdot M\right) \cdot \left(D \cdot D\right)}{\frac{\ell}{M} \cdot \left(d \cdot d\right)}, 1\right)\\ \mathbf{elif}\;d \leq 1.44 \cdot 10^{-300}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 8
Error	23.2
Cost	13516

\[\begin{array}{l} t_0 := d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{if}\;d \leq -7.2 \cdot 10^{+62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -5.5 \cdot 10^{-57}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-298}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 9
Error	23.9
Cost	13252

\[\begin{array}{l} \mathbf{if}\;d \leq 7 \cdot 10^{-298}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 10
Error	34.5
Cost	7112

\[\begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq 1.95 \cdot 10^{+167}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \end{array} \]

Alternative 11
Error	27.6
Cost	7112

\[\begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;h \leq 1.95 \cdot 10^{+167}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \end{array} \]

Alternative 12
Error	34.8
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq -3.1 \cdot 10^{-252}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 13
Error	44.2
Cost	6784

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

Alternative 14
Error	44.2
Cost	6720

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

Reproduce

herbie shell --seed 2023056 
(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)))))