Henrywood and Agarwal, Equation (12)

Average Error: 26.3 → 14.1

Time: 45.9s

Precision: binary64

Cost: 104784

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{0.5}{d} \cdot \left(M \cdot D\right)\\ t_1 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {t_0}^{2}\right)\right)\\ t_2 := {\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\\ t_3 := t_2 \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ t_4 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;t_3 \leq -2 \cdot 10^{-128}:\\ \;\;\;\;t_2 \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(t_0 \cdot \sqrt{0.5}\right)\right)}^{2}\right)\\ \mathbf{elif}\;t_3 \leq 10^{-306}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{+273}:\\ \;\;\;\;t_4 \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{D \cdot \left(M \cdot \frac{0.5}{d}\right)}{d \cdot \frac{\frac{\ell}{h}}{M \cdot \left(D \cdot -0.25\right)}}\right)\right)\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(t_4 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \frac{\frac{h}{d} \cdot \left(D \cdot \left(M \cdot D\right)\right)}{d \cdot \frac{\ell}{M}}\right)\right)\\ \end{array} \]

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 (* (/ 0.5 d) (* M D)))
        (t_1
         (*
          (fabs (/ d (sqrt (* h l))))
          (+ 1.0 (* -0.5 (* (/ h l) (pow t_0 2.0))))))
        (t_2 (* (pow (/ d h) 0.5) (pow (/ d l) 0.5)))
        (t_3
         (* t_2 (+ 1.0 (* (/ h l) (* (pow (/ (* M D) (* d 2.0)) 2.0) -0.5)))))
        (t_4 (sqrt (/ d h))))
   (if (<= t_3 -2e-128)
     (* t_2 (- 1.0 (pow (* (sqrt (/ h l)) (* t_0 (sqrt 0.5))) 2.0)))
     (if (<= t_3 1e-306)
       t_1
       (if (<= t_3 5e+273)
         (*
          t_4
          (*
           (sqrt (/ d l))
           (+
            1.0
            (/ (* D (* M (/ 0.5 d))) (* d (/ (/ l h) (* M (* D -0.25))))))))
         (if (<= t_3 INFINITY)
           t_1
           (*
            (* t_4 (/ (sqrt d) (sqrt l)))
            (+
             1.0
             (*
              -0.5
              (* 0.25 (/ (* (/ h d) (* D (* M D))) (* d (/ l M)))))))))))))

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 = (0.5 / d) * (M * D);
	double t_1 = fabs((d / sqrt((h * l)))) * (1.0 + (-0.5 * ((h / l) * pow(t_0, 2.0))));
	double t_2 = pow((d / h), 0.5) * pow((d / l), 0.5);
	double t_3 = t_2 * (1.0 + ((h / l) * (pow(((M * D) / (d * 2.0)), 2.0) * -0.5)));
	double t_4 = sqrt((d / h));
	double tmp;
	if (t_3 <= -2e-128) {
		tmp = t_2 * (1.0 - pow((sqrt((h / l)) * (t_0 * sqrt(0.5))), 2.0));
	} else if (t_3 <= 1e-306) {
		tmp = t_1;
	} else if (t_3 <= 5e+273) {
		tmp = t_4 * (sqrt((d / l)) * (1.0 + ((D * (M * (0.5 / d))) / (d * ((l / h) / (M * (D * -0.25)))))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = (t_4 * (sqrt(d) / sqrt(l))) * (1.0 + (-0.5 * (0.25 * (((h / d) * (D * (M * D))) / (d * (l / M))))));
	}
	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 = (0.5 / d) * (M * D);
	double t_1 = Math.abs((d / Math.sqrt((h * l)))) * (1.0 + (-0.5 * ((h / l) * Math.pow(t_0, 2.0))));
	double t_2 = Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5);
	double t_3 = t_2 * (1.0 + ((h / l) * (Math.pow(((M * D) / (d * 2.0)), 2.0) * -0.5)));
	double t_4 = Math.sqrt((d / h));
	double tmp;
	if (t_3 <= -2e-128) {
		tmp = t_2 * (1.0 - Math.pow((Math.sqrt((h / l)) * (t_0 * Math.sqrt(0.5))), 2.0));
	} else if (t_3 <= 1e-306) {
		tmp = t_1;
	} else if (t_3 <= 5e+273) {
		tmp = t_4 * (Math.sqrt((d / l)) * (1.0 + ((D * (M * (0.5 / d))) / (d * ((l / h) / (M * (D * -0.25)))))));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = (t_4 * (Math.sqrt(d) / Math.sqrt(l))) * (1.0 + (-0.5 * (0.25 * (((h / d) * (D * (M * D))) / (d * (l / M))))));
	}
	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 = (0.5 / d) * (M * D)
	t_1 = math.fabs((d / math.sqrt((h * l)))) * (1.0 + (-0.5 * ((h / l) * math.pow(t_0, 2.0))))
	t_2 = math.pow((d / h), 0.5) * math.pow((d / l), 0.5)
	t_3 = t_2 * (1.0 + ((h / l) * (math.pow(((M * D) / (d * 2.0)), 2.0) * -0.5)))
	t_4 = math.sqrt((d / h))
	tmp = 0
	if t_3 <= -2e-128:
		tmp = t_2 * (1.0 - math.pow((math.sqrt((h / l)) * (t_0 * math.sqrt(0.5))), 2.0))
	elif t_3 <= 1e-306:
		tmp = t_1
	elif t_3 <= 5e+273:
		tmp = t_4 * (math.sqrt((d / l)) * (1.0 + ((D * (M * (0.5 / d))) / (d * ((l / h) / (M * (D * -0.25)))))))
	elif t_3 <= math.inf:
		tmp = t_1
	else:
		tmp = (t_4 * (math.sqrt(d) / math.sqrt(l))) * (1.0 + (-0.5 * (0.25 * (((h / d) * (D * (M * D))) / (d * (l / M))))))
	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 = Float64(Float64(0.5 / d) * Float64(M * D))
	t_1 = Float64(abs(Float64(d / sqrt(Float64(h * l)))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (t_0 ^ 2.0)))))
	t_2 = Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5))
	t_3 = Float64(t_2 * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * -0.5))))
	t_4 = sqrt(Float64(d / h))
	tmp = 0.0
	if (t_3 <= -2e-128)
		tmp = Float64(t_2 * Float64(1.0 - (Float64(sqrt(Float64(h / l)) * Float64(t_0 * sqrt(0.5))) ^ 2.0)));
	elseif (t_3 <= 1e-306)
		tmp = t_1;
	elseif (t_3 <= 5e+273)
		tmp = Float64(t_4 * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64(D * Float64(M * Float64(0.5 / d))) / Float64(d * Float64(Float64(l / h) / Float64(M * Float64(D * -0.25))))))));
	elseif (t_3 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(t_4 * Float64(sqrt(d) / sqrt(l))) * Float64(1.0 + Float64(-0.5 * Float64(0.25 * Float64(Float64(Float64(h / d) * Float64(D * Float64(M * D))) / Float64(d * Float64(l / M)))))));
	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 = (0.5 / d) * (M * D);
	t_1 = abs((d / sqrt((h * l)))) * (1.0 + (-0.5 * ((h / l) * (t_0 ^ 2.0))));
	t_2 = ((d / h) ^ 0.5) * ((d / l) ^ 0.5);
	t_3 = t_2 * (1.0 + ((h / l) * ((((M * D) / (d * 2.0)) ^ 2.0) * -0.5)));
	t_4 = sqrt((d / h));
	tmp = 0.0;
	if (t_3 <= -2e-128)
		tmp = t_2 * (1.0 - ((sqrt((h / l)) * (t_0 * sqrt(0.5))) ^ 2.0));
	elseif (t_3 <= 1e-306)
		tmp = t_1;
	elseif (t_3 <= 5e+273)
		tmp = t_4 * (sqrt((d / l)) * (1.0 + ((D * (M * (0.5 / d))) / (d * ((l / h) / (M * (D * -0.25)))))));
	elseif (t_3 <= Inf)
		tmp = t_1;
	else
		tmp = (t_4 * (sqrt(d) / sqrt(l))) * (1.0 + (-0.5 * (0.25 * (((h / d) * (D * (M * D))) / (d * (l / M))))));
	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[(N[(0.5 / d), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * 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$4 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, -2e-128], N[(t$95$2 * N[(1.0 - N[Power[N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e-306], t$95$1, If[LessEqual[t$95$3, 5e+273], N[(t$95$4 * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(N[(l / h), $MachinePrecision] / N[(M * N[(D * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$1, N[(N[(t$95$4 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(0.25 * N[(N[(N[(h / d), $MachinePrecision] * N[(D * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 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)))) < -2.00000000000000011e-128

   1. Initial program 27.7

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

   2. Applied egg-rr 19.8

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

   if -2.00000000000000011e-128 < (*.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)))) < 1.00000000000000003e-306 or 4.99999999999999961e273 < (*.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 50.7

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

   2. Applied egg-rr 53.7

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

   3. Applied egg-rr 8.9

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

   if 1.00000000000000003e-306 < (*.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)))) < 4.99999999999999961e273

   1. Initial program 1.1

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

   2. Simplified 1.1

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

      [Start] 1.1	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 1.1	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 1.1	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 1.1	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 1.1	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 1.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      cancel-sign-sub-inv [=>] 1.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]
      +-commutative [=>] 1.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell} + 1\right)}\right) \]
      *-commutative [=>] 1.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}\right) \cdot \frac{h}{\ell} + 1\right)\right) \]
      distribute-rgt-neg-in [=>] 1.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(-\frac{1}{2}\right)\right)} \cdot \frac{h}{\ell} + 1\right)\right) \]
      associate-*l* [=>] 1.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\left(-\frac{1}{2}\right) \cdot \frac{h}{\ell}\right)} + 1\right)\right) \]
      fma-def [=>] 1.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}, \left(-\frac{1}{2}\right) \cdot \frac{h}{\ell}, 1\right)}\right) \]

   3. Applied egg-rr 1.1

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

   4. Applied egg-rr 1.2

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

   5. Applied egg-rr 1.4

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

   6. Simplified 1.4

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\left(M \cdot \frac{0.5}{d}\right) \cdot D}{\color{blue}{d \cdot \frac{\frac{\ell}{h}}{M \cdot \left(-0.25 \cdot D\right)}}} + 1\right)\right) \]
      Proof

      [Start] 1.4	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\left(M \cdot \frac{0.5}{d}\right) \cdot D}{\sqrt{\frac{\ell}{h}} \cdot \left(\frac{\sqrt{\frac{\ell}{h}}}{\left(M \cdot 0.5\right) \cdot \left(D \cdot -0.5\right)} \cdot d\right)} + 1\right)\right) \]
      associate-*r* [=>] 1.4	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\left(M \cdot \frac{0.5}{d}\right) \cdot D}{\color{blue}{\left(\sqrt{\frac{\ell}{h}} \cdot \frac{\sqrt{\frac{\ell}{h}}}{\left(M \cdot 0.5\right) \cdot \left(D \cdot -0.5\right)}\right) \cdot d}} + 1\right)\right) \]
      *-commutative [=>] 1.4	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\left(M \cdot \frac{0.5}{d}\right) \cdot D}{\color{blue}{d \cdot \left(\sqrt{\frac{\ell}{h}} \cdot \frac{\sqrt{\frac{\ell}{h}}}{\left(M \cdot 0.5\right) \cdot \left(D \cdot -0.5\right)}\right)}} + 1\right)\right) \]
      associate-*r/ [=>] 1.4	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\left(M \cdot \frac{0.5}{d}\right) \cdot D}{d \cdot \color{blue}{\frac{\sqrt{\frac{\ell}{h}} \cdot \sqrt{\frac{\ell}{h}}}{\left(M \cdot 0.5\right) \cdot \left(D \cdot -0.5\right)}}} + 1\right)\right) \]
      rem-square-sqrt [=>] 1.4	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\left(M \cdot \frac{0.5}{d}\right) \cdot D}{d \cdot \frac{\color{blue}{\frac{\ell}{h}}}{\left(M \cdot 0.5\right) \cdot \left(D \cdot -0.5\right)}} + 1\right)\right) \]
      associate-*l* [=>] 1.4	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\left(M \cdot \frac{0.5}{d}\right) \cdot D}{d \cdot \frac{\frac{\ell}{h}}{\color{blue}{M \cdot \left(0.5 \cdot \left(D \cdot -0.5\right)\right)}}} + 1\right)\right) \]
      *-commutative [=>] 1.4	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\left(M \cdot \frac{0.5}{d}\right) \cdot D}{d \cdot \frac{\frac{\ell}{h}}{M \cdot \left(0.5 \cdot \color{blue}{\left(-0.5 \cdot D\right)}\right)}} + 1\right)\right) \]
      associate-*r* [=>] 1.4	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\left(M \cdot \frac{0.5}{d}\right) \cdot D}{d \cdot \frac{\frac{\ell}{h}}{M \cdot \color{blue}{\left(\left(0.5 \cdot -0.5\right) \cdot D\right)}}} + 1\right)\right) \]
      metadata-eval [=>] 1.4	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\left(M \cdot \frac{0.5}{d}\right) \cdot D}{d \cdot \frac{\frac{\ell}{h}}{M \cdot \left(\color{blue}{-0.25} \cdot D\right)}} + 1\right)\right) \]

   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))))

   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 63.9

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
      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 [=>] 63.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Taylor expanded in M around 0 58.1

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)}\right) \]

   4. Simplified 57.6

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \left(\frac{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}{\ell} \cdot \frac{\frac{h}{d}}{d}\right)\right)}\right) \]
      Proof

      [Start] 58.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)\right) \]
      associate-/r* [=>] 57.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{\frac{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}}\right)\right) \]
      associate-/l/ [=>] 58.1	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right)\right) \]
      associate-*r* [=>] 58.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}\right)\right) \]
      *-commutative [<=] 58.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}{\color{blue}{\ell \cdot {d}^{2}}}\right)\right) \]
      times-frac [=>] 58.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{\ell} \cdot \frac{h}{{d}^{2}}\right)}\right)\right) \]
      unpow2 [=>] 58.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{{D}^{2} \cdot \color{blue}{\left(M \cdot M\right)}}{\ell} \cdot \frac{h}{{d}^{2}}\right)\right)\right) \]
      associate-*r* [=>] 57.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\color{blue}{\left({D}^{2} \cdot M\right) \cdot M}}{\ell} \cdot \frac{h}{{d}^{2}}\right)\right)\right) \]
      unpow2 [=>] 57.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot M\right) \cdot M}{\ell} \cdot \frac{h}{{d}^{2}}\right)\right)\right) \]
      unpow2 [=>] 57.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}{\ell} \cdot \frac{h}{\color{blue}{d \cdot d}}\right)\right)\right) \]
      associate-/r* [=>] 57.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}{\ell} \cdot \color{blue}{\frac{\frac{h}{d}}{d}}\right)\right)\right) \]

   5. Applied egg-rr 53.7

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left(\sqrt{d} \cdot \frac{1}{\sqrt{\ell}}\right)}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}{\ell} \cdot \frac{\frac{h}{d}}{d}\right)\right)\right) \]

   6. Simplified 53.7

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}{\ell} \cdot \frac{\frac{h}{d}}{d}\right)\right)\right) \]
      Proof

      [Start] 53.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot \frac{1}{\sqrt{\ell}}\right)\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}{\ell} \cdot \frac{\frac{h}{d}}{d}\right)\right)\right) \]
      associate-*r/ [=>] 53.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{d} \cdot 1}{\sqrt{\ell}}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}{\ell} \cdot \frac{\frac{h}{d}}{d}\right)\right)\right) \]
      *-rgt-identity [=>] 53.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}{\ell} \cdot \frac{\frac{h}{d}}{d}\right)\right)\right) \]

   7. Applied egg-rr 50.7

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{\frac{\frac{h}{d} \cdot \left(D \cdot \left(D \cdot M\right)\right)}{d \cdot \frac{\ell}{M}}}\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 14.1

   \[\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 -2 \cdot 10^{-128}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right) \cdot \sqrt{0.5}\right)\right)}^{2}\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 10^{-306}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}\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^{+273}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{D \cdot \left(M \cdot \frac{0.5}{d}\right)}{d \cdot \frac{\frac{\ell}{h}}{M \cdot \left(D \cdot -0.25\right)}}\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 \infty:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \frac{\frac{h}{d} \cdot \left(D \cdot \left(M \cdot D\right)\right)}{d \cdot \frac{\ell}{M}}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	18.6
Cost	104784

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := D \cdot \left(M \cdot \frac{0.5}{d}\right)\\ t_3 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ \mathbf{if}\;t_3 \leq -2 \cdot 10^{-128}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 + \frac{t_2}{\frac{\frac{\ell}{h}}{-0.5 \cdot \left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}}\right)\right)\\ \mathbf{elif}\;t_3 \leq 10^{-306}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{+273}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 + \frac{t_2}{d \cdot \frac{\frac{\ell}{h}}{M \cdot \left(D \cdot -0.25\right)}}\right)\right)\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \frac{\frac{h}{d} \cdot \left(D \cdot \left(M \cdot D\right)\right)}{d \cdot \frac{\ell}{M}}\right)\right)\\ \end{array} \]

Alternative 2
Error	14.4
Cost	104784

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}\right)\right)\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := D \cdot \left(M \cdot \frac{0.5}{d}\right)\\ t_4 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ \mathbf{if}\;t_4 \leq -2 \cdot 10^{-128}:\\ \;\;\;\;t_0 \cdot \left(t_2 \cdot \left(1 + \frac{t_3}{\frac{\frac{\ell}{h}}{-0.5 \cdot \left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}}\right)\right)\\ \mathbf{elif}\;t_4 \leq 10^{-306}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_4 \leq 5 \cdot 10^{+273}:\\ \;\;\;\;t_0 \cdot \left(t_2 \cdot \left(1 + \frac{t_3}{d \cdot \frac{\frac{\ell}{h}}{M \cdot \left(D \cdot -0.25\right)}}\right)\right)\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \frac{\frac{h}{d} \cdot \left(D \cdot \left(M \cdot D\right)\right)}{d \cdot \frac{\ell}{M}}\right)\right)\\ \end{array} \]

Alternative 3
Error	21.5
Cost	21716

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := d \cdot \frac{d}{h}\\ \mathbf{if}\;d \leq -6.7 \cdot 10^{+15}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;d \leq -1.85 \cdot 10^{-131}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 + \frac{D \cdot \left(M \cdot \frac{0.5}{d}\right)}{\frac{\frac{\ell}{h}}{-0.5 \cdot \left(0.5 \cdot \frac{M}{\frac{d}{D}}\right)}}\right)\right)\\ \mathbf{elif}\;d \leq -3.05 \cdot 10^{-192}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{-211}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(t_1 \cdot \left(1 + {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{-129}:\\ \;\;\;\;\left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(-0.25 \cdot \frac{D \cdot \left(M \cdot D\right)}{\frac{\ell}{M} \cdot t_2}\right)\right)\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+32}:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(1 + 0.5 \cdot \left(-0.25 \cdot \frac{M \cdot D}{t_2 \cdot \frac{\ell}{M \cdot D}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 4
Error	17.9
Cost	21716

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := D \cdot \left(M \cdot \frac{0.5}{d}\right)\\ t_3 := 1 + \frac{t_2}{\frac{\frac{\ell}{h}}{-0.5 \cdot t_2}}\\ t_4 := \sqrt{-d}\\ t_5 := \frac{t_4}{\sqrt{-h}} \cdot \left(t_0 \cdot t_3\right)\\ \mathbf{if}\;d \leq -1.8 \cdot 10^{-71}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;d \leq -2.4 \cdot 10^{-213}:\\ \;\;\;\;t_1 \cdot \left(t_3 \cdot \frac{t_4}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;d \leq -2.4 \cdot 10^{-308}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;d \leq 3.25 \cdot 10^{-187}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-0.125}{d} \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)\right)\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{-115}:\\ \;\;\;\;\left(t_1 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \frac{\frac{h}{d} \cdot \left(D \cdot \left(M \cdot D\right)\right)}{d \cdot \frac{\ell}{M}}\right)\right)\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{+33}:\\ \;\;\;\;\left(t_1 \cdot t_0\right) \cdot \left(1 + 0.5 \cdot \left(-0.25 \cdot \frac{M \cdot D}{\left(d \cdot \frac{d}{h}\right) \cdot \frac{\ell}{M \cdot D}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 5
Error	19.3
Cost	21452

\[\begin{array}{l} t_0 := D \cdot \left(M \cdot \frac{0.5}{d}\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -3 \cdot 10^{-309}:\\ \;\;\;\;t_1 \cdot \left(\left(1 + \frac{t_0}{\frac{\frac{\ell}{h}}{-0.5 \cdot t_0}}\right) \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;d \leq 3.25 \cdot 10^{-187}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-0.125}{d} \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)\right)\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{-115}:\\ \;\;\;\;\left(t_1 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \frac{\frac{h}{d} \cdot \left(D \cdot \left(M \cdot D\right)\right)}{d \cdot \frac{\ell}{M}}\right)\right)\\ \mathbf{elif}\;d \leq 2.35 \cdot 10^{+33}:\\ \;\;\;\;\left(t_1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(-0.25 \cdot \frac{M \cdot D}{\left(d \cdot \frac{d}{h}\right) \cdot \frac{\ell}{M \cdot D}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 6
Error	20.4
Cost	21136

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \frac{h}{\ell} \cdot -0.5\\ t_3 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -4.8 \cdot 10^{-5}:\\ \;\;\;\;t_1 \cdot \left(t_3 \cdot \left(1 + {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot t_2\right)\right)\\ \mathbf{elif}\;h \leq -9.5 \cdot 10^{-276}:\\ \;\;\;\;\frac{-d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;h \leq 3.2 \cdot 10^{-294}:\\ \;\;\;\;\left(t_1 \cdot t_3\right) \cdot \left(1 + 0.5 \cdot \left(-0.25 \cdot \frac{M \cdot D}{\left(d \cdot \frac{d}{h}\right) \cdot \frac{\ell}{M \cdot D}}\right)\right)\\ \mathbf{elif}\;h \leq 1.15 \cdot 10^{+52}:\\ \;\;\;\;t_0 \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot t_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	20.3
Cost	21136

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ t_1 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;h \leq -6.8 \cdot 10^{+16}:\\ \;\;\;\;\left(t_1 \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq -1.08 \cdot 10^{-275}:\\ \;\;\;\;\frac{-d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;h \leq 4.5 \cdot 10^{-294}:\\ \;\;\;\;\left(t_1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(-0.25 \cdot \frac{M \cdot D}{\left(d \cdot \frac{d}{h}\right) \cdot \frac{\ell}{M \cdot D}}\right)\right)\\ \mathbf{elif}\;h \leq 1.8 \cdot 10^{+47}:\\ \;\;\;\;t_0 \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	19.5
Cost	21136

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ t_2 := \frac{h}{\ell} \cdot -0.5\\ \mathbf{if}\;h \leq -2.2 \cdot 10^{+101}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(t_0 \cdot \left(1 + {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot t_2\right)\right)\\ \mathbf{elif}\;h \leq -9.5 \cdot 10^{-276}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq 3.2 \cdot 10^{-294}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot t_0\right) \cdot \left(1 + 0.5 \cdot \left(-0.25 \cdot \frac{M \cdot D}{\left(d \cdot \frac{d}{h}\right) \cdot \frac{\ell}{M \cdot D}}\right)\right)\\ \mathbf{elif}\;h \leq 5.5 \cdot 10^{+51}:\\ \;\;\;\;t_1 \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot t_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	21.8
Cost	15184

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{D \cdot \left(M \cdot \frac{0.5}{d}\right)}{d \cdot \frac{\frac{\ell}{h}}{M \cdot \left(D \cdot -0.25\right)}}\right)\right)\\ \mathbf{if}\;d \leq -6.7 \cdot 10^{+15}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;d \leq 8.5 \cdot 10^{-293}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{-186}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-0.125}{d} \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)\right)\\ \mathbf{elif}\;d \leq 2.35 \cdot 10^{+33}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 10
Error	22.0
Cost	15184

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -6.7 \cdot 10^{+15}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-293}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 + \frac{D \cdot \left(M \cdot \frac{0.5}{d}\right)}{d \cdot \frac{\frac{\ell}{h}}{M \cdot \left(D \cdot -0.25\right)}}\right)\right)\\ \mathbf{elif}\;d \leq 5.2 \cdot 10^{-185}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-0.125}{d} \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)\right)\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{+32}:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(1 + 0.5 \cdot \left(-0.25 \cdot \frac{M \cdot D}{\left(d \cdot \frac{d}{h}\right) \cdot \frac{\ell}{M \cdot D}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 11
Error	21.9
Cost	15184

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -6.7 \cdot 10^{+15}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-293}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 + \frac{D \cdot \left(M \cdot \frac{0.5}{d}\right)}{\frac{\frac{\ell}{h}}{-0.5 \cdot \left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}}\right)\right)\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{-185}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-0.125}{d} \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)\right)\\ \mathbf{elif}\;d \leq 2.55 \cdot 10^{+33}:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(1 + 0.5 \cdot \left(-0.25 \cdot \frac{M \cdot D}{\left(d \cdot \frac{d}{h}\right) \cdot \frac{\ell}{M \cdot D}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 12
Error	22.3
Cost	15184

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -6.7 \cdot 10^{+15}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{-294}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 + \frac{D \cdot \left(M \cdot \frac{0.5}{d}\right)}{\frac{\frac{\ell}{h}}{-0.5 \cdot \left(0.5 \cdot \frac{M}{\frac{d}{D}}\right)}}\right)\right)\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{-185}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-0.125}{d} \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)\right)\\ \mathbf{elif}\;d \leq 9 \cdot 10^{+32}:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(1 + 0.5 \cdot \left(-0.25 \cdot \frac{M \cdot D}{\left(d \cdot \frac{d}{h}\right) \cdot \frac{\ell}{M \cdot D}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 13
Error	23.1
Cost	13384

\[\begin{array}{l} \mathbf{if}\;h \leq -3.45 \cdot 10^{+106}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 14
Error	22.3
Cost	13384

\[\begin{array}{l} \mathbf{if}\;h \leq -1.5 \cdot 10^{+103}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 15
Error	22.3
Cost	13384

\[\begin{array}{l} \mathbf{if}\;h \leq -2.8 \cdot 10^{+105}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 16
Error	26.8
Cost	7244

\[\begin{array}{l} t_0 := \sqrt{h \cdot \ell}\\ t_1 := \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{if}\;h \leq -1.12 \cdot 10^{+106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-d}{t_0}\\ \mathbf{elif}\;h \leq 1.42 \cdot 10^{+150}:\\ \;\;\;\;\frac{d}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 17
Error	28.1
Cost	6916

\[\begin{array}{l} t_0 := \sqrt{h \cdot \ell}\\ \mathbf{if}\;d \leq 9.8 \cdot 10^{-203}:\\ \;\;\;\;\frac{-d}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t_0}\\ \end{array} \]

Alternative 18
Error	43.2
Cost	6720

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

Reproduce

herbie shell --seed 2023073 
(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)))))