Henrywood and Agarwal, Equation (12)

Average Accuracy: 58.9% → 78.9%

Time: 48.6s

Precision: binary64

Cost: 104593

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \frac{\left(t_0 \cdot -0.125\right) \cdot \left(M \cdot \frac{M}{\frac{\ell}{h}}\right)}{{\left(\frac{d}{D}\right)}^{2}}\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-293}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0 \lor \neg \left(t_1 \leq 2 \cdot 10^{+221}\right):\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(t_0 \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)\\ \end{array} \]

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))))))
   (if (<= t_1 (- INFINITY))
     (*
      (sqrt (/ d h))
      (/ (* (* t_0 -0.125) (* M (/ M (/ l h)))) (pow (/ d D) 2.0)))
     (if (<= t_1 -2e-293)
       t_1
       (if (or (<= t_1 0.0) (not (<= t_1 2e+221)))
         (fabs (/ d (sqrt (* h l))))
         (*
          (/ 1.0 (sqrt (/ h d)))
          (*
           t_0
           (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M 2.0) (/ D d)) 2.0)))))))))))

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 tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = sqrt((d / h)) * (((t_0 * -0.125) * (M * (M / (l / h)))) / pow((d / D), 2.0));
	} else if (t_1 <= -2e-293) {
		tmp = t_1;
	} else if ((t_1 <= 0.0) || !(t_1 <= 2e+221)) {
		tmp = fabs((d / sqrt((h * l))));
	} else {
		tmp = (1.0 / sqrt((h / d))) * (t_0 * (1.0 - (0.5 * ((h / l) * pow(((M / 2.0) * (D / d)), 2.0)))));
	}
	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 / l));
	double t_1 = (Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((0.5 * Math.pow(((M * D) / (d * 2.0)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((d / h)) * (((t_0 * -0.125) * (M * (M / (l / h)))) / Math.pow((d / D), 2.0));
	} else if (t_1 <= -2e-293) {
		tmp = t_1;
	} else if ((t_1 <= 0.0) || !(t_1 <= 2e+221)) {
		tmp = Math.abs((d / Math.sqrt((h * l))));
	} else {
		tmp = (1.0 / Math.sqrt((h / d))) * (t_0 * (1.0 - (0.5 * ((h / l) * Math.pow(((M / 2.0) * (D / d)), 2.0)))));
	}
	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 / l))
	t_1 = (math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 - ((0.5 * math.pow(((M * D) / (d * 2.0)), 2.0)) * (h / l)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = math.sqrt((d / h)) * (((t_0 * -0.125) * (M * (M / (l / h)))) / math.pow((d / D), 2.0))
	elif t_1 <= -2e-293:
		tmp = t_1
	elif (t_1 <= 0.0) or not (t_1 <= 2e+221):
		tmp = math.fabs((d / math.sqrt((h * l))))
	else:
		tmp = (1.0 / math.sqrt((h / d))) * (t_0 * (1.0 - (0.5 * ((h / l) * math.pow(((M / 2.0) * (D / d)), 2.0)))))
	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))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(Float64(t_0 * -0.125) * Float64(M * Float64(M / Float64(l / h)))) / (Float64(d / D) ^ 2.0)));
	elseif (t_1 <= -2e-293)
		tmp = t_1;
	elseif ((t_1 <= 0.0) || !(t_1 <= 2e+221))
		tmp = abs(Float64(d / sqrt(Float64(h * l))));
	else
		tmp = Float64(Float64(1.0 / sqrt(Float64(h / d))) * Float64(t_0 * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0))))));
	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 / l));
	t_1 = (((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((0.5 * (((M * D) / (d * 2.0)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = sqrt((d / h)) * (((t_0 * -0.125) * (M * (M / (l / h)))) / ((d / D) ^ 2.0));
	elseif (t_1 <= -2e-293)
		tmp = t_1;
	elseif ((t_1 <= 0.0) || ~((t_1 <= 2e+221)))
		tmp = abs((d / sqrt((h * l))));
	else
		tmp = (1.0 / sqrt((h / d))) * (t_0 * (1.0 - (0.5 * ((h / l) * (((M / 2.0) * (D / d)) ^ 2.0)))));
	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 / 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]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(t$95$0 * -0.125), $MachinePrecision] * N[(M * N[(M / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e-293], t$95$1, If[Or[LessEqual[t$95$1, 0.0], N[Not[LessEqual[t$95$1, 2e+221]], $MachinePrecision]], N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $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)))) < -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 4.5%

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

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

   3. Taylor expanded in M around inf 5.7%

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

   4. Simplified 3.9%

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

      [Start] 5.7	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)\right) \]
      times-frac [=>] 3.9	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right)\right) \]
      unpow2 [=>] 3.9	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right)\right) \]
      unpow2 [=>] 3.9	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right)\right) \]
      unpow2 [=>] 3.9	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell}\right)\right)\right) \]

   5. Applied egg-rr 13.8%

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

      [Start] 3.9	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right)\right)\right) \]
      clear-num [=>] 3.9	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\color{blue}{\frac{1}{\frac{d \cdot d}{D \cdot D}}} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right)\right)\right) \]
      frac-2neg [=>] 3.9	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\frac{1}{\frac{d \cdot d}{D \cdot D}} \cdot \color{blue}{\frac{-\left(M \cdot M\right) \cdot h}{-\ell}}\right)\right)\right) \]
      frac-times [=>] 5.2	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \color{blue}{\frac{1 \cdot \left(-\left(M \cdot M\right) \cdot h\right)}{\frac{d \cdot d}{D \cdot D} \cdot \left(-\ell\right)}}\right)\right) \]
      *-un-lft-identity [<=] 5.2	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \frac{\color{blue}{-\left(M \cdot M\right) \cdot h}}{\frac{d \cdot d}{D \cdot D} \cdot \left(-\ell\right)}\right)\right) \]
      associate-*l* [=>] 6.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \frac{-\color{blue}{M \cdot \left(M \cdot h\right)}}{\frac{d \cdot d}{D \cdot D} \cdot \left(-\ell\right)}\right)\right) \]
      distribute-rgt-neg-in [=>] 6.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \frac{\color{blue}{M \cdot \left(-M \cdot h\right)}}{\frac{d \cdot d}{D \cdot D} \cdot \left(-\ell\right)}\right)\right) \]
      times-frac [=>] 13.8	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \frac{M \cdot \left(-M \cdot h\right)}{\color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)} \cdot \left(-\ell\right)}\right)\right) \]

   6. Simplified 14.6%

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

      [Start] 13.8	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \frac{M \cdot \left(-M \cdot h\right)}{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \left(-\ell\right)}\right)\right) \]
      times-frac [=>] 14.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \color{blue}{\left(\frac{M}{\frac{d}{D} \cdot \frac{d}{D}} \cdot \frac{-M \cdot h}{-\ell}\right)}\right)\right) \]
      distribute-rgt-neg-in [=>] 14.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\frac{M}{\frac{d}{D} \cdot \frac{d}{D}} \cdot \frac{\color{blue}{M \cdot \left(-h\right)}}{-\ell}\right)\right)\right) \]

   7. Applied egg-rr 20.2%

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

      [Start] 14.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\frac{M}{\frac{d}{D} \cdot \frac{d}{D}} \cdot \frac{M \cdot \left(-h\right)}{-\ell}\right)\right)\right) \]
      associate-*r* [=>] 14.6	\[ \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(\left(\sqrt{\frac{d}{\ell}} \cdot -0.125\right) \cdot \left(\frac{M}{\frac{d}{D} \cdot \frac{d}{D}} \cdot \frac{M \cdot \left(-h\right)}{-\ell}\right)\right)} \]
      associate-*l/ [=>] 16.8	\[ \sqrt{\frac{d}{h}} \cdot \left(\left(\sqrt{\frac{d}{\ell}} \cdot -0.125\right) \cdot \color{blue}{\frac{M \cdot \frac{M \cdot \left(-h\right)}{-\ell}}{\frac{d}{D} \cdot \frac{d}{D}}}\right) \]
      associate-*r/ [=>] 21.0	\[ \sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\left(\sqrt{\frac{d}{\ell}} \cdot -0.125\right) \cdot \left(M \cdot \frac{M \cdot \left(-h\right)}{-\ell}\right)}{\frac{d}{D} \cdot \frac{d}{D}}} \]

   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)))) < -2.0000000000000001e-293

   1. Initial program 98.1%

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

   if -2.0000000000000001e-293 < (*.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 2.0000000000000001e221 < (*.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 14.6%

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

   2. Taylor expanded in d around inf 35.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   3. Simplified 35.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
      Proof

      [Start] 35.8	\[ \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]
      *-commutative [=>] 35.8	\[ \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
      associate-/r* [=>] 35.9	\[ d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   4. Applied egg-rr 19.2%

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

      [Start] 35.9	\[ d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}} \]
      add-sqr-sqrt [=>] 34.6	\[ \color{blue}{\sqrt{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot \sqrt{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}} \]
      sqrt-unprod [=>] 21.8	\[ \color{blue}{\sqrt{\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}} \]
      pow1/2 [=>] 21.8	\[ \color{blue}{{\left(\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)}^{0.5}} \]
      *-commutative [=>] 21.8	\[ {\left(\color{blue}{\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d\right)} \cdot \left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)}^{0.5} \]
      *-commutative [=>] 21.8	\[ {\left(\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d\right) \cdot \color{blue}{\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d\right)}\right)}^{0.5} \]
      swap-sqr [=>] 19.1	\[ {\color{blue}{\left(\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(d \cdot d\right)\right)}}^{0.5} \]
      add-sqr-sqrt [<=] 19.2	\[ {\left(\color{blue}{\frac{\frac{1}{\ell}}{h}} \cdot \left(d \cdot d\right)\right)}^{0.5} \]

   5. Simplified 21.9%

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

      [Start] 19.2	\[ {\left(\frac{\frac{1}{\ell}}{h} \cdot \left(d \cdot d\right)\right)}^{0.5} \]
      unpow1/2 [=>] 19.2	\[ \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h} \cdot \left(d \cdot d\right)}} \]
      associate-/r* [<=] 19.2	\[ \sqrt{\color{blue}{\frac{1}{\ell \cdot h}} \cdot \left(d \cdot d\right)} \]
      associate-*l/ [=>] 19.2	\[ \sqrt{\color{blue}{\frac{1 \cdot \left(d \cdot d\right)}{\ell \cdot h}}} \]
      *-lft-identity [=>] 19.2	\[ \sqrt{\frac{\color{blue}{d \cdot d}}{\ell \cdot h}} \]
      associate-*r/ [<=] 21.9	\[ \sqrt{\color{blue}{d \cdot \frac{d}{\ell \cdot h}}} \]

   6. Applied egg-rr 62.7%

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

      [Start] 21.9	\[ \sqrt{d \cdot \frac{d}{\ell \cdot h}} \]
      add-sqr-sqrt [=>] 21.9	\[ \sqrt{\color{blue}{\sqrt{d \cdot \frac{d}{\ell \cdot h}} \cdot \sqrt{d \cdot \frac{d}{\ell \cdot h}}}} \]
      rem-sqrt-square [=>] 21.9	\[ \color{blue}{\left|\sqrt{d \cdot \frac{d}{\ell \cdot h}}\right|} \]
      associate-*r/ [=>] 19.2	\[ \left|\sqrt{\color{blue}{\frac{d \cdot d}{\ell \cdot h}}}\right| \]
      sqrt-div [=>] 28.2	\[ \left|\color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}}}\right| \]
      sqrt-unprod [<=] 30.5	\[ \left|\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{\ell \cdot h}}\right| \]
      add-sqr-sqrt [<=] 62.7	\[ \left|\frac{\color{blue}{d}}{\sqrt{\ell \cdot h}}\right| \]

   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)))) < 2.0000000000000001e221

   1. Initial program 98.8%

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

   2. Simplified 98.5%

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

      [Start] 98.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 98.8	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 98.8	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 98.8	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 98.8	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 98.8	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      associate-*l* [=>] 98.8	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)\right) \]
      metadata-eval [=>] 98.8	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      times-frac [=>] 98.5	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   3. Applied egg-rr 98.2%

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

      [Start] 98.5	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      clear-num [=>] 98.2	\[ \sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      sqrt-div [=>] 98.2	\[ \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      metadata-eval [=>] 98.2	\[ \frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -\infty:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \frac{\left(\sqrt{\frac{d}{\ell}} \cdot -0.125\right) \cdot \left(M \cdot \frac{M}{\frac{\ell}{h}}\right)}{{\left(\frac{d}{D}\right)}^{2}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -2 \cdot 10^{-293}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0 \lor \neg \left(\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{+221}\right):\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	69.0%
Cost	21136

\[\begin{array}{l} \mathbf{if}\;h \leq -2.6 \cdot 10^{+225}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq -1.7 \cdot 10^{+164}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2} \cdot -0.5\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-1 + h \cdot \left(\frac{0.5}{\ell} \cdot {\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}\right)}{\frac{\sqrt{h \cdot \ell}}{d}}\\ \mathbf{elif}\;h \leq 3.9 \cdot 10^{+197}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + {\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{-0.5}}{\frac{\sqrt{h}}{d}}\\ \end{array} \]

Alternative 2
Accuracy	69.4%
Cost	21136

\[\begin{array}{l} t_0 := -1 + h \cdot \left(\frac{0.5}{\ell} \cdot {\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}\right)\\ \mathbf{if}\;h \leq -2.6 \cdot 10^{+225}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq -3.2 \cdot 10^{+158}:\\ \;\;\;\;\frac{t_0}{-\sqrt{\frac{h}{d} \cdot \frac{\ell}{d}}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{t_0}{\frac{\sqrt{h \cdot \ell}}{d}}\\ \mathbf{elif}\;h \leq 8.5 \cdot 10^{+197}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \frac{{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right)}^{2}}{\frac{\ell}{h \cdot -0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{-0.5}}{\frac{\sqrt{h}}{d}}\\ \end{array} \]

Alternative 3
Accuracy	70.4%
Cost	21136

\[\begin{array}{l} t_0 := \frac{-1 + h \cdot \left(\frac{0.5}{\ell} \cdot {\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}\right)}{\frac{\sqrt{h \cdot \ell}}{d}}\\ \mathbf{if}\;d \leq -4.4 \cdot 10^{+63}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.3 \cdot 10^{-181}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq -1.3 \cdot 10^{-298}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{+175}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \frac{{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right)}^{2}}{\frac{\ell}{h \cdot -0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]

Alternative 4
Accuracy	71.6%
Cost	21004

\[\begin{array}{l} \mathbf{if}\;h \leq -3.7 \cdot 10^{+158}:\\ \;\;\;\;\frac{1 - h \cdot \left(\frac{0.5}{\ell} \cdot {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}\right)}{\frac{\sqrt{\frac{\ell}{d}}}{\sqrt{\frac{d}{h}}}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-1 + h \cdot \left(\frac{0.5}{\ell} \cdot {\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}\right)}{\frac{\sqrt{h \cdot \ell}}{d}}\\ \mathbf{elif}\;h \leq 7 \cdot 10^{+197}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \frac{{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right)}^{2}}{\frac{\ell}{h \cdot -0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{-0.5}}{\frac{\sqrt{h}}{d}}\\ \end{array} \]

Alternative 5
Accuracy	64.4%
Cost	14732

\[\begin{array}{l} \mathbf{if}\;h \leq -2.3 \cdot 10^{+225}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq -1.15 \cdot 10^{+164}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2} \cdot -0.5\right)\right)\\ \mathbf{elif}\;h \leq -1.35 \cdot 10^{+87}:\\ \;\;\;\;\sqrt{d \cdot \frac{\frac{d}{h}}{\ell}} \cdot \left(1 - h \cdot \left(\frac{0.5}{\ell} \cdot {\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 6
Accuracy	65.6%
Cost	14668

\[\begin{array}{l} t_0 := \sqrt{h \cdot \ell}\\ \mathbf{if}\;\ell \leq -3.05 \cdot 10^{-276}:\\ \;\;\;\;\left|\frac{d}{t_0}\right|\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{-133}:\\ \;\;\;\;\frac{{\ell}^{-0.5}}{\frac{\sqrt{h}}{d}}\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+103}:\\ \;\;\;\;\frac{-1 + h \cdot \left(\frac{0.5}{\ell} \cdot {\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}\right)}{\frac{-t_0}{d}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \end{array} \]

Alternative 7
Accuracy	68.9%
Cost	14668

\[\begin{array}{l} t_0 := -1 + h \cdot \left(\frac{0.5}{\ell} \cdot {\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}\right)\\ t_1 := \sqrt{h \cdot \ell}\\ \mathbf{if}\;\ell \leq -3.05 \cdot 10^{-276}:\\ \;\;\;\;\frac{t_0}{\frac{t_1}{d}}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{-133}:\\ \;\;\;\;\frac{{\ell}^{-0.5}}{\frac{\sqrt{h}}{d}}\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+100}:\\ \;\;\;\;\frac{t_0}{\frac{-t_1}{d}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \end{array} \]

Alternative 8
Accuracy	64.0%
Cost	13580

\[\begin{array}{l} \mathbf{if}\;h \leq -2.1 \cdot 10^{+225}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq -9 \cdot 10^{+157}:\\ \;\;\;\;\frac{-1 + h \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\left(\frac{M}{d} \cdot \frac{M}{d}\right) \cdot 0.125\right)\right)}{-\sqrt{\frac{h}{d} \cdot \frac{\ell}{d}}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]

Alternative 9
Accuracy	57.6%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;h \leq -2.6 \cdot 10^{+225}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq -2.6 \cdot 10^{+158}:\\ \;\;\;\;\frac{-1 + h \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\left(\frac{M}{d} \cdot \frac{M}{d}\right) \cdot 0.125\right)\right)}{-\sqrt{\frac{h}{d} \cdot \frac{\ell}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]

Alternative 10
Accuracy	64.1%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;d \leq 4 \cdot 10^{-302}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 11
Accuracy	57.5%
Cost	8456

\[\begin{array}{l} \mathbf{if}\;h \leq -2.6 \cdot 10^{+225}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq -1.15 \cdot 10^{+158}:\\ \;\;\;\;\frac{-1 + h \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\left(\frac{M}{d} \cdot \frac{M}{d}\right) \cdot 0.125\right)\right)}{-\sqrt{\frac{h}{d} \cdot \frac{\ell}{d}}}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-306}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h} \cdot \frac{1}{\ell}}\\ \end{array} \]

Alternative 12
Accuracy	57.4%
Cost	7372

\[\begin{array}{l} \mathbf{if}\;h \leq -7.8 \cdot 10^{+245}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq -7 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-306}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h} \cdot \frac{1}{\ell}}\\ \end{array} \]

Alternative 13
Accuracy	57.6%
Cost	7308

\[\begin{array}{l} t_0 := d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{if}\;h \leq -1.05 \cdot 10^{+246}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq -4 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;h \leq 6 \cdot 10^{-309}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]

Alternative 14
Accuracy	57.4%
Cost	7308

\[\begin{array}{l} t_0 := \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;h \leq -9 \cdot 10^{+241}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq -4.8 \cdot 10^{+158}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;h \leq 6 \cdot 10^{-309}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]

Alternative 15
Accuracy	57.5%
Cost	7308

\[\begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{if}\;h \leq -1.02 \cdot 10^{+245}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq -2.9 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-306}:\\ \;\;\;\;d \cdot \left(-t_0\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot t_0\\ \end{array} \]

Alternative 16
Accuracy	47.9%
Cost	7244

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq 2.7 \cdot 10^{-259}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{+220}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+292}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 17
Accuracy	47.8%
Cost	7244

\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.1 \cdot 10^{-260}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 6.8 \cdot 10^{+218}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{+292}:\\ \;\;\;\;\sqrt{\frac{d \cdot \frac{d}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 18
Accuracy	47.9%
Cost	7244

\[\begin{array}{l} \mathbf{if}\;\ell \leq 2.55 \cdot 10^{-260}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+219}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{+292}:\\ \;\;\;\;\sqrt{\frac{d \cdot \frac{d}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]

Alternative 19
Accuracy	46.0%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.3 \cdot 10^{-268}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 20
Accuracy	31.4%
Cost	6784

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

Alternative 21
Accuracy	31.4%
Cost	6720

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

Reproduce

herbie shell --seed 2023146 
(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)))))