Henrywood and Agarwal, Equation (12)

Average Accuracy: 58.1% → 80.8%

Time: 42.9s

Precision: binary64

Cost: 104465

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ t_3 := M \cdot \frac{D}{d}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(1 - 0.125 \cdot \left(\frac{t_3}{\frac{1}{h}} \cdot \frac{t_3}{\ell}\right)\right)\\ \mathbf{elif}\;t_2 \leq -5 \cdot 10^{-213} \lor \neg \left(t_2 \leq 0\right) \land t_2 \leq 5 \cdot 10^{+281}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \left(0.5 \cdot D\right)}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]

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 h)))
        (t_1 (sqrt (/ d l)))
        (t_2
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (+ 1.0 (* (/ h l) (* (pow (/ (* M D) (* d 2.0)) 2.0) -0.5)))))
        (t_3 (* M (/ D d))))
   (if (<= t_2 (- INFINITY))
     (* (* t_0 t_1) (- 1.0 (* 0.125 (* (/ t_3 (/ 1.0 h)) (/ t_3 l)))))
     (if (or (<= t_2 -5e-213) (and (not (<= t_2 0.0)) (<= t_2 5e+281)))
       (*
        t_0
        (* t_1 (- 1.0 (* 0.5 (* (/ h l) (pow (/ (* M (* 0.5 D)) d) 2.0))))))
       (fabs (/ d (sqrt (* h l))))))))

C

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

↓

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = sqrt((d / l));
	double t_2 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 + ((h / l) * (pow(((M * D) / (d * 2.0)), 2.0) * -0.5)));
	double t_3 = M * (D / d);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (t_0 * t_1) * (1.0 - (0.125 * ((t_3 / (1.0 / h)) * (t_3 / l))));
	} else if ((t_2 <= -5e-213) || (!(t_2 <= 0.0) && (t_2 <= 5e+281))) {
		tmp = t_0 * (t_1 * (1.0 - (0.5 * ((h / l) * pow(((M * (0.5 * D)) / d), 2.0)))));
	} else {
		tmp = fabs((d / sqrt((h * l))));
	}
	return tmp;
}

Java

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

↓

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / h));
	double t_1 = Math.sqrt((d / l));
	double t_2 = (Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 + ((h / l) * (Math.pow(((M * D) / (d * 2.0)), 2.0) * -0.5)));
	double t_3 = M * (D / d);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = (t_0 * t_1) * (1.0 - (0.125 * ((t_3 / (1.0 / h)) * (t_3 / l))));
	} else if ((t_2 <= -5e-213) || (!(t_2 <= 0.0) && (t_2 <= 5e+281))) {
		tmp = t_0 * (t_1 * (1.0 - (0.5 * ((h / l) * Math.pow(((M * (0.5 * D)) / d), 2.0)))));
	} else {
		tmp = Math.abs((d / Math.sqrt((h * l))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	return (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))

↓

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / h))
	t_1 = math.sqrt((d / l))
	t_2 = (math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 + ((h / l) * (math.pow(((M * D) / (d * 2.0)), 2.0) * -0.5)))
	t_3 = M * (D / d)
	tmp = 0
	if t_2 <= -math.inf:
		tmp = (t_0 * t_1) * (1.0 - (0.125 * ((t_3 / (1.0 / h)) * (t_3 / l))))
	elif (t_2 <= -5e-213) or (not (t_2 <= 0.0) and (t_2 <= 5e+281)):
		tmp = t_0 * (t_1 * (1.0 - (0.5 * ((h / l) * math.pow(((M * (0.5 * D)) / d), 2.0)))))
	else:
		tmp = math.fabs((d / math.sqrt((h * l))))
	return tmp

Julia

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

↓

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(d / l))
	t_2 = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * -0.5))))
	t_3 = Float64(M * Float64(D / d))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(t_0 * t_1) * Float64(1.0 - Float64(0.125 * Float64(Float64(t_3 / Float64(1.0 / h)) * Float64(t_3 / l)))));
	elseif ((t_2 <= -5e-213) || (!(t_2 <= 0.0) && (t_2 <= 5e+281)))
		tmp = Float64(t_0 * Float64(t_1 * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M * Float64(0.5 * D)) / d) ^ 2.0))))));
	else
		tmp = abs(Float64(d / sqrt(Float64(h * l))));
	end
	return tmp
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
end

↓

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / h));
	t_1 = sqrt((d / l));
	t_2 = (((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 + ((h / l) * ((((M * D) / (d * 2.0)) ^ 2.0) * -0.5)));
	t_3 = M * (D / d);
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = (t_0 * t_1) * (1.0 - (0.125 * ((t_3 / (1.0 / h)) * (t_3 / l))));
	elseif ((t_2 <= -5e-213) || (~((t_2 <= 0.0)) && (t_2 <= 5e+281)))
		tmp = t_0 * (t_1 * (1.0 - (0.5 * ((h / l) * (((M * (0.5 * D)) / d) ^ 2.0)))));
	else
		tmp = abs((d / sqrt((h * l))));
	end
	tmp_2 = 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[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = 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[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(t$95$0 * t$95$1), $MachinePrecision] * N[(1.0 - N[(0.125 * N[(N[(t$95$3 / N[(1.0 / h), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$2, -5e-213], And[N[Not[LessEqual[t$95$2, 0.0]], $MachinePrecision], LessEqual[t$95$2, 5e+281]]], N[(t$95$0 * N[(t$95$1 * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M * N[(0.5 * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 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 0.0%

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

   2. Simplified 3.4%

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

      [Start] 0.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 0.0	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 0.0	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 0.0	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 0.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      *-commutative [=>] 0.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 0.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]
      times-frac [=>] 3.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 3.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Taylor expanded in M around 0 7.0%

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

   4. Simplified 8.2%

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

      [Start] 7.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right) \]
      associate-*r/ [=>] 7.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right) \]
      *-commutative [=>] 7.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right) \]
      associate-*r/ [<=] 7.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
      times-frac [=>] 7.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \]
      unpow2 [=>] 7.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \]
      unpow2 [=>] 7.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \]
      times-frac [=>] 11.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \]
      associate-/l* [=>] 8.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\frac{{M}^{2}}{\frac{\ell}{h}}}\right)\right) \]
      unpow2 [=>] 8.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{\color{blue}{M \cdot M}}{\frac{\ell}{h}}\right)\right) \]

   5. Applied egg-rr 16.1%

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

      [Start] 8.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right) \]
      associate-*r/ [=>] 2.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot M\right)}{\frac{\ell}{h}}}\right) \]
      div-inv [=>] 2.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot M\right)}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]
      associate-/r* [=>] 6.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{\frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot M\right)}{\ell}}{\frac{1}{h}}}\right) \]
      pow2 [=>] 6.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{\frac{\color{blue}{{\left(\frac{D}{d}\right)}^{2}} \cdot \left(M \cdot M\right)}{\ell}}{\frac{1}{h}}\right) \]
      pow2 [=>] 6.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{\frac{{\left(\frac{D}{d}\right)}^{2} \cdot \color{blue}{{M}^{2}}}{\ell}}{\frac{1}{h}}\right) \]
      pow-prod-down [=>] 16.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{\frac{\color{blue}{{\left(\frac{D}{d} \cdot M\right)}^{2}}}{\ell}}{\frac{1}{h}}\right) \]

   6. Applied egg-rr 38.1%

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

      [Start] 16.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{\frac{{\left(\frac{D}{d} \cdot M\right)}^{2}}{\ell}}{\frac{1}{h}}\right) \]
      associate-/l/ [=>] 5.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{{\left(\frac{D}{d} \cdot M\right)}^{2}}{\frac{1}{h} \cdot \ell}}\right) \]
      unpow2 [=>] 5.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{\left(\frac{D}{d} \cdot M\right) \cdot \left(\frac{D}{d} \cdot M\right)}}{\frac{1}{h} \cdot \ell}\right) \]
      times-frac [=>] 38.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\left(\frac{\frac{D}{d} \cdot M}{\frac{1}{h}} \cdot \frac{\frac{D}{d} \cdot M}{\ell}\right)}\right) \]

   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)))) < -4.99999999999999977e-213 or 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)))) < 5.00000000000000016e281

   1. Initial program 98.5%

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

   2. Simplified 97.0%

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

      [Start] 98.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 98.3	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 98.3	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 98.3	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 98.3	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 98.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      associate-*l* [=>] 98.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)\right) \]
      metadata-eval [=>] 98.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      times-frac [=>] 97.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   3. Applied egg-rr 98.3%

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

      [Start] 97.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      associate-*r/ [=>] 98.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{\frac{M}{2} \cdot D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      div-inv [=>] 98.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      metadata-eval [=>] 98.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\left(M \cdot \color{blue}{0.5}\right) \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      associate-*l* [=>] 98.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\color{blue}{M \cdot \left(0.5 \cdot D\right)}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   if -4.99999999999999977e-213 < (*.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 or 5.00000000000000016e281 < (*.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 10.1%

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

   2. Simplified 9.6%

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

      [Start] 10.1	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 10.1	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 10.1	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 10.1	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 10.1	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 10.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      sub-neg [=>] 10.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]
      +-commutative [=>] 10.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]
      *-commutative [=>] 10.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) + 1\right)\right) \]
      distribute-rgt-neg-in [=>] 10.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{h}{\ell} \cdot \left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} + 1\right)\right) \]
      fma-def [=>] 10.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, -\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}, 1\right)}\right) \]

   3. Taylor expanded in h around 0 15.9%

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

   4. Applied egg-rr 36.4%

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

      [Start] 15.9	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right) \]
      add-sqr-sqrt [=>] 15.8	\[ \color{blue}{\sqrt{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right)} \cdot \sqrt{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right)}} \]
      pow2 [=>] 15.8	\[ \color{blue}{{\left(\sqrt{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right)}\right)}^{2}} \]
      sqrt-div [=>] 22.2	\[ {\left(\sqrt{\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right)}\right)}^{2} \]
      *-rgt-identity [=>] 22.2	\[ {\left(\sqrt{\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}}\right)}^{2} \]
      sqrt-div [=>] 33.7	\[ {\left(\sqrt{\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}}\right)}^{2} \]
      frac-times [=>] 33.7	\[ {\left(\sqrt{\color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}}}\right)}^{2} \]
      add-sqr-sqrt [<=] 33.8	\[ {\left(\sqrt{\frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}}}\right)}^{2} \]
      pow1/2 [=>] 33.8	\[ {\left(\sqrt{\frac{d}{\color{blue}{{h}^{0.5}} \cdot \sqrt{\ell}}}\right)}^{2} \]
      pow1/2 [=>] 33.8	\[ {\left(\sqrt{\frac{d}{{h}^{0.5} \cdot \color{blue}{{\ell}^{0.5}}}}\right)}^{2} \]
      pow-prod-down [=>] 36.4	\[ {\left(\sqrt{\frac{d}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}}}\right)}^{2} \]

   5. Applied egg-rr 21.7%

      \[\leadsto {\color{blue}{\left({\left(\frac{d \cdot d}{h \cdot \ell}\right)}^{0.25}\right)}}^{2} \]
      Proof

      [Start] 36.4	\[ {\left(\sqrt{\frac{d}{{\left(h \cdot \ell\right)}^{0.5}}}\right)}^{2} \]
      pow1/2 [=>] 36.4	\[ {\color{blue}{\left({\left(\frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\right)}^{0.5}\right)}}^{2} \]
      metadata-eval [<=] 36.4	\[ {\left({\left(\frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\right)}^{\color{blue}{\left(0.25 + 0.25\right)}}\right)}^{2} \]
      metadata-eval [<=] 36.4	\[ {\left({\left(\frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\right)}^{\left(\color{blue}{0.5 \cdot 0.5} + 0.25\right)}\right)}^{2} \]
      metadata-eval [<=] 36.4	\[ {\left({\left(\frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\right)}^{\left(0.5 \cdot 0.5 + \color{blue}{0.5 \cdot 0.5}\right)}\right)}^{2} \]
      pow-prod-up [<=] 36.2	\[ {\color{blue}{\left({\left(\frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\right)}^{\left(0.5 \cdot 0.5\right)} \cdot {\left(\frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\right)}^{\left(0.5 \cdot 0.5\right)}\right)}}^{2} \]
      pow-prod-down [=>] 24.3	\[ {\color{blue}{\left({\left(\frac{d}{{\left(h \cdot \ell\right)}^{0.5}} \cdot \frac{d}{{\left(h \cdot \ell\right)}^{0.5}}\right)}^{\left(0.5 \cdot 0.5\right)}\right)}}^{2} \]
      frac-times [=>] 21.7	\[ {\left({\color{blue}{\left(\frac{d \cdot d}{{\left(h \cdot \ell\right)}^{0.5} \cdot {\left(h \cdot \ell\right)}^{0.5}}\right)}}^{\left(0.5 \cdot 0.5\right)}\right)}^{2} \]
      unpow1/2 [=>] 21.7	\[ {\left({\left(\frac{d \cdot d}{\color{blue}{\sqrt{h \cdot \ell}} \cdot {\left(h \cdot \ell\right)}^{0.5}}\right)}^{\left(0.5 \cdot 0.5\right)}\right)}^{2} \]
      unpow1/2 [=>] 21.7	\[ {\left({\left(\frac{d \cdot d}{\sqrt{h \cdot \ell} \cdot \color{blue}{\sqrt{h \cdot \ell}}}\right)}^{\left(0.5 \cdot 0.5\right)}\right)}^{2} \]
      add-sqr-sqrt [<=] 21.7	\[ {\left({\left(\frac{d \cdot d}{\color{blue}{h \cdot \ell}}\right)}^{\left(0.5 \cdot 0.5\right)}\right)}^{2} \]
      metadata-eval [=>] 21.7	\[ {\left({\left(\frac{d \cdot d}{h \cdot \ell}\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   6. Applied egg-rr 63.6%

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

      [Start] 21.7	\[ {\left({\left(\frac{d \cdot d}{h \cdot \ell}\right)}^{0.25}\right)}^{2} \]
      add-sqr-sqrt [=>] 21.7	\[ \color{blue}{\sqrt{{\left({\left(\frac{d \cdot d}{h \cdot \ell}\right)}^{0.25}\right)}^{2}} \cdot \sqrt{{\left({\left(\frac{d \cdot d}{h \cdot \ell}\right)}^{0.25}\right)}^{2}}} \]
      sqrt-prod [<=] 21.7	\[ \color{blue}{\sqrt{{\left({\left(\frac{d \cdot d}{h \cdot \ell}\right)}^{0.25}\right)}^{2} \cdot {\left({\left(\frac{d \cdot d}{h \cdot \ell}\right)}^{0.25}\right)}^{2}}} \]
      rem-sqrt-square [=>] 21.7	\[ \color{blue}{\left|{\left({\left(\frac{d \cdot d}{h \cdot \ell}\right)}^{0.25}\right)}^{2}\right|} \]
      pow-pow [=>] 21.8	\[ \left|\color{blue}{{\left(\frac{d \cdot d}{h \cdot \ell}\right)}^{\left(0.25 \cdot 2\right)}}\right| \]
      metadata-eval [=>] 21.8	\[ \left|{\left(\frac{d \cdot d}{h \cdot \ell}\right)}^{\color{blue}{0.5}}\right| \]
      pow1/2 [<=] 21.8	\[ \left|\color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}}\right| \]
      sqrt-div [=>] 30.5	\[ \left|\color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}}\right| \]
      sqrt-prod [=>] 32.1	\[ \left|\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}}\right| \]
      add-sqr-sqrt [<=] 63.6	\[ \left|\frac{\color{blue}{d}}{\sqrt{h \cdot \ell}}\right| \]

3. Recombined 3 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq -\infty:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{M \cdot \frac{D}{d}}{\frac{1}{h}} \cdot \frac{M \cdot \frac{D}{d}}{\ell}\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq -5 \cdot 10^{-213} \lor \neg \left(\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq 0\right) \land \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq 5 \cdot 10^{+281}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \left(0.5 \cdot D\right)}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	79.8%
Cost	83533

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{-213}:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(1 - {\left(\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right) \cdot \sqrt{\frac{h \cdot 0.5}{\ell}}\right)}^{2}\right)\\ \mathbf{elif}\;t_2 \leq 0 \lor \neg \left(t_2 \leq 5 \cdot 10^{+281}\right):\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \left(0.5 \cdot D\right)}{d}\right)}^{2}\right)\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	67.2%
Cost	21264

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := M \cdot \frac{D}{d}\\ t_3 := \left(t_0 \cdot t_1\right) \cdot \left(1 - 0.125 \cdot \left(\frac{t_2}{\frac{1}{h}} \cdot \frac{t_2}{\ell}\right)\right)\\ \mathbf{if}\;d \leq -3.6 \cdot 10^{+79}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;d \leq -1.55 \cdot 10^{-66}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq -8 \cdot 10^{-289}:\\ \;\;\;\;t_0 \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq 8.6 \cdot 10^{-179}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;d \leq 8.6 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{+52}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 3
Accuracy	67.1%
Cost	21136

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := t_0 \cdot \sqrt{\frac{d}{\ell}}\\ t_2 := M \cdot \frac{D}{d}\\ t_3 := t_1 \cdot \left(1 - 0.125 \cdot \left(\frac{t_2}{\frac{1}{h}} \cdot \frac{t_2}{\ell}\right)\right)\\ \mathbf{if}\;d \leq -3.9 \cdot 10^{+79}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;d \leq -1.65 \cdot 10^{-67}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq -8.6 \cdot 10^{-288}:\\ \;\;\;\;t_0 \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq 6.7 \cdot 10^{-189}:\\ \;\;\;\;t_1 \cdot \left(1 + \left(h \cdot \frac{{t_2}^{2}}{\ell}\right) \cdot -0.125\right)\\ \mathbf{elif}\;d \leq 8.2 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;d \leq 2.35 \cdot 10^{+52}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 4
Accuracy	67.2%
Cost	20172

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := M \cdot \frac{D}{d}\\ t_2 := t_0 \cdot \sqrt{\frac{d}{\ell}}\\ t_3 := \frac{t_1}{\ell}\\ t_4 := t_2 \cdot \left(1 - 0.125 \cdot \left(\frac{t_1}{\frac{1}{h}} \cdot t_3\right)\right)\\ \mathbf{if}\;d \leq -4.9 \cdot 10^{+79}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;d \leq -1.1 \cdot 10^{-67}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;d \leq -6.9 \cdot 10^{-286}:\\ \;\;\;\;t_0 \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{-192}:\\ \;\;\;\;t_2 \cdot \left(1 + \left(h \cdot \left(t_1 \cdot t_3\right)\right) \cdot -0.125\right)\\ \mathbf{elif}\;d \leq 8.5 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;d \leq 9.4 \cdot 10^{+51}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 5
Accuracy	63.0%
Cost	15452

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ t_1 := t_0 \cdot \left(1 + \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \left(M \cdot \frac{M}{\ell}\right)\right)\right) \cdot -0.125\right)\\ t_2 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_3 := \frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \mathbf{if}\;d \leq -1.8 \cdot 10^{+80}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -8.1 \cdot 10^{-66}:\\ \;\;\;\;t_0 \cdot \left(1 - 0.125 \cdot \frac{\frac{D}{d} \cdot \frac{M}{\frac{\ell}{h \cdot M}}}{\frac{d}{D}}\right)\\ \mathbf{elif}\;d \leq 5.6 \cdot 10^{-291}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{-212}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq 6 \cdot 10^{-194}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;d \leq 2.75 \cdot 10^{+47}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 6
Accuracy	62.9%
Cost	15452

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := t_0 \cdot t_1\\ t_3 := \frac{D}{d} \cdot \frac{D}{d}\\ t_4 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_5 := \frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \mathbf{if}\;d \leq -4.6 \cdot 10^{+79}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;d \leq -1.4 \cdot 10^{-63}:\\ \;\;\;\;t_2 \cdot \left(1 - 0.125 \cdot \frac{\frac{D}{d} \cdot \frac{M}{\frac{\ell}{h \cdot M}}}{\frac{d}{D}}\right)\\ \mathbf{elif}\;d \leq 5 \cdot 10^{-288}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{-212}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-194}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 + \left(t_3 \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot 0.25\right)\right) \cdot -0.5\right)\right)\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{+49}:\\ \;\;\;\;t_2 \cdot \left(1 + \left(t_3 \cdot \left(h \cdot \left(M \cdot \frac{M}{\ell}\right)\right)\right) \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;t_5\\ \end{array} \]

Alternative 7
Accuracy	62.9%
Cost	15188

\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_1 := \frac{D}{d} \cdot \frac{D}{d}\\ t_2 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -1.75 \cdot 10^{+119}:\\ \;\;\;\;t_2 \cdot \left(1 + \left(t_1 \cdot \left(h \cdot \left(M \cdot \frac{M}{\ell}\right)\right)\right) \cdot -0.125\right)\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq 7.5 \cdot 10^{-93}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;h \leq 6.5 \cdot 10^{+29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq 3.8 \cdot 10^{+123}:\\ \;\;\;\;t_2 \cdot \left(1 + \left(t_1 \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right) \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 8
Accuracy	62.7%
Cost	15188

\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_1 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -2.15 \cdot 10^{+119}:\\ \;\;\;\;t_1 \cdot \left(1 + \left(\frac{D}{\frac{d}{D}} \cdot \frac{h \cdot \frac{M}{\frac{\ell}{M}}}{d}\right) \cdot -0.125\right)\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq 8 \cdot 10^{-91}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;h \leq 1.95 \cdot 10^{+30}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq 3.8 \cdot 10^{+123}:\\ \;\;\;\;t_1 \cdot \left(1 + \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right) \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 9
Accuracy	67.1%
Cost	15056

\[\begin{array}{l} t_0 := M \cdot \frac{D}{d}\\ t_1 := \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + \left(h \cdot \left(t_0 \cdot \frac{t_0}{\ell}\right)\right) \cdot -0.125\right)\\ t_2 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;h \leq -2.2 \cdot 10^{+118}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;h \leq 3.15 \cdot 10^{-90}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;h \leq 1.02 \cdot 10^{+30}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Accuracy	62.5%
Cost	14660

\[\begin{array}{l} \mathbf{if}\;h \leq -2.7 \cdot 10^{+121}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \left(M \cdot \frac{M}{\ell}\right)\right)\right) \cdot -0.125\right)\\ \mathbf{elif}\;h \leq 3.3 \cdot 10^{-296}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 11
Accuracy	62.3%
Cost	13508

\[\begin{array}{l} \mathbf{if}\;h \leq -6 \cdot 10^{+252}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;h \leq 3.4 \cdot 10^{-303}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 12
Accuracy	57.6%
Cost	13385

\[\begin{array}{l} \mathbf{if}\;h \leq -4.4 \cdot 10^{+252} \lor \neg \left(h \leq 7.2 \cdot 10^{+54}\right):\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]

Alternative 13
Accuracy	62.1%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;h \leq -1.8 \cdot 10^{+254}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;h \leq 1.75 \cdot 10^{-301}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 14
Accuracy	62.3%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;h \leq -1.15 \cdot 10^{+253}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq 2.9 \cdot 10^{-303}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 15
Accuracy	48.1%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;h \leq -7.2 \cdot 10^{-290} \lor \neg \left(h \leq 1.25 \cdot 10^{+56}\right):\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 16
Accuracy	46.8%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.05 \cdot 10^{-294}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 17
Accuracy	32.1%
Cost	6720

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

Reproduce

herbie shell --seed 2023136 
(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)))))