Henrywood and Agarwal, Equation (12)

Average Accuracy: 58.2% → 73.3%

Time: 54.9s

Precision: binary64

Cost: 40652

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

Math

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

↓

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{M \cdot D}{d}\\ t_1 := \sqrt{-h}\\ t_2 := t_0 \cdot \sqrt{0.5}\\ \mathbf{if}\;d \leq -4.8 \cdot 10^{+166}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{{t_0}^{2} \cdot \left(h \cdot 0.5\right)}{\ell} + -1\right)\\ \mathbf{elif}\;d \leq -5.5 \cdot 10^{+70}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{t_1} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{elif}\;d \leq -6.2 \cdot 10^{-285}:\\ \;\;\;\;\left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot \frac{-1}{\sqrt{\frac{h}{d}}}\right) \cdot \left({\left(t_2 \cdot \frac{t_1}{\sqrt{-\ell}}\right)}^{2} + -1\right)\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{+31}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - {\left(t_2 \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

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 (/ (* M D) d)))
        (t_1 (sqrt (- h)))
        (t_2 (* t_0 (sqrt 0.5))))
   (if (<= d -4.8e+166)
     (*
      (* d (sqrt (/ 1.0 (* l h))))
      (+ (/ (* (pow t_0 2.0) (* h 0.5)) l) -1.0))
     (if (<= d -5.5e+70)
       (*
        (* (/ (sqrt (- d)) t_1) (sqrt (/ d l)))
        (+ 1.0 (* (pow (* (/ M 2.0) (/ D d)) 2.0) (* (/ h l) -0.5))))
       (if (<= d -6.2e-285)
         (*
          (* (pow (/ d l) 0.5) (/ -1.0 (sqrt (/ h d))))
          (+ (pow (* t_2 (/ t_1 (sqrt (- l)))) 2.0) -1.0))
         (if (<= d 2.05e+31)
           (*
            (* (pow (/ d h) 0.5) (/ (sqrt d) (sqrt l)))
            (- 1.0 (pow (* t_2 (sqrt (/ h l))) 2.0)))
           (* d (* (pow h -0.5) (pow l -0.5)))))))))

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 * ((M * D) / d);
	double t_1 = sqrt(-h);
	double t_2 = t_0 * sqrt(0.5);
	double tmp;
	if (d <= -4.8e+166) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (((pow(t_0, 2.0) * (h * 0.5)) / l) + -1.0);
	} else if (d <= -5.5e+70) {
		tmp = ((sqrt(-d) / t_1) * sqrt((d / l))) * (1.0 + (pow(((M / 2.0) * (D / d)), 2.0) * ((h / l) * -0.5)));
	} else if (d <= -6.2e-285) {
		tmp = (pow((d / l), 0.5) * (-1.0 / sqrt((h / d)))) * (pow((t_2 * (t_1 / sqrt(-l))), 2.0) + -1.0);
	} else if (d <= 2.05e+31) {
		tmp = (pow((d / h), 0.5) * (sqrt(d) / sqrt(l))) * (1.0 - pow((t_2 * sqrt((h / l))), 2.0));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
end function

↓

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 0.5d0 * ((m * d_1) / d)
    t_1 = sqrt(-h)
    t_2 = t_0 * sqrt(0.5d0)
    if (d <= (-4.8d+166)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((((t_0 ** 2.0d0) * (h * 0.5d0)) / l) + (-1.0d0))
    else if (d <= (-5.5d+70)) then
        tmp = ((sqrt(-d) / t_1) * sqrt((d / l))) * (1.0d0 + ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * ((h / l) * (-0.5d0))))
    else if (d <= (-6.2d-285)) then
        tmp = (((d / l) ** 0.5d0) * ((-1.0d0) / sqrt((h / d)))) * (((t_2 * (t_1 / sqrt(-l))) ** 2.0d0) + (-1.0d0))
    else if (d <= 2.05d+31) then
        tmp = (((d / h) ** 0.5d0) * (sqrt(d) / sqrt(l))) * (1.0d0 - ((t_2 * sqrt((h / l))) ** 2.0d0))
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

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 * ((M * D) / d);
	double t_1 = Math.sqrt(-h);
	double t_2 = t_0 * Math.sqrt(0.5);
	double tmp;
	if (d <= -4.8e+166) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (((Math.pow(t_0, 2.0) * (h * 0.5)) / l) + -1.0);
	} else if (d <= -5.5e+70) {
		tmp = ((Math.sqrt(-d) / t_1) * Math.sqrt((d / l))) * (1.0 + (Math.pow(((M / 2.0) * (D / d)), 2.0) * ((h / l) * -0.5)));
	} else if (d <= -6.2e-285) {
		tmp = (Math.pow((d / l), 0.5) * (-1.0 / Math.sqrt((h / d)))) * (Math.pow((t_2 * (t_1 / Math.sqrt(-l))), 2.0) + -1.0);
	} else if (d <= 2.05e+31) {
		tmp = (Math.pow((d / h), 0.5) * (Math.sqrt(d) / Math.sqrt(l))) * (1.0 - Math.pow((t_2 * Math.sqrt((h / l))), 2.0));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	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 * ((M * D) / d)
	t_1 = math.sqrt(-h)
	t_2 = t_0 * math.sqrt(0.5)
	tmp = 0
	if d <= -4.8e+166:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (((math.pow(t_0, 2.0) * (h * 0.5)) / l) + -1.0)
	elif d <= -5.5e+70:
		tmp = ((math.sqrt(-d) / t_1) * math.sqrt((d / l))) * (1.0 + (math.pow(((M / 2.0) * (D / d)), 2.0) * ((h / l) * -0.5)))
	elif d <= -6.2e-285:
		tmp = (math.pow((d / l), 0.5) * (-1.0 / math.sqrt((h / d)))) * (math.pow((t_2 * (t_1 / math.sqrt(-l))), 2.0) + -1.0)
	elif d <= 2.05e+31:
		tmp = (math.pow((d / h), 0.5) * (math.sqrt(d) / math.sqrt(l))) * (1.0 - math.pow((t_2 * math.sqrt((h / l))), 2.0))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	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(0.5 * Float64(Float64(M * D) / d))
	t_1 = sqrt(Float64(-h))
	t_2 = Float64(t_0 * sqrt(0.5))
	tmp = 0.0
	if (d <= -4.8e+166)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(Float64(Float64((t_0 ^ 2.0) * Float64(h * 0.5)) / l) + -1.0));
	elseif (d <= -5.5e+70)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / t_1) * sqrt(Float64(d / l))) * Float64(1.0 + Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * Float64(Float64(h / l) * -0.5))));
	elseif (d <= -6.2e-285)
		tmp = Float64(Float64((Float64(d / l) ^ 0.5) * Float64(-1.0 / sqrt(Float64(h / d)))) * Float64((Float64(t_2 * Float64(t_1 / sqrt(Float64(-l)))) ^ 2.0) + -1.0));
	elseif (d <= 2.05e+31)
		tmp = Float64(Float64((Float64(d / h) ^ 0.5) * Float64(sqrt(d) / sqrt(l))) * Float64(1.0 - (Float64(t_2 * sqrt(Float64(h / l))) ^ 2.0)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	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 * ((M * D) / d);
	t_1 = sqrt(-h);
	t_2 = t_0 * sqrt(0.5);
	tmp = 0.0;
	if (d <= -4.8e+166)
		tmp = (d * sqrt((1.0 / (l * h)))) * ((((t_0 ^ 2.0) * (h * 0.5)) / l) + -1.0);
	elseif (d <= -5.5e+70)
		tmp = ((sqrt(-d) / t_1) * sqrt((d / l))) * (1.0 + ((((M / 2.0) * (D / d)) ^ 2.0) * ((h / l) * -0.5)));
	elseif (d <= -6.2e-285)
		tmp = (((d / l) ^ 0.5) * (-1.0 / sqrt((h / d)))) * (((t_2 * (t_1 / sqrt(-l))) ^ 2.0) + -1.0);
	elseif (d <= 2.05e+31)
		tmp = (((d / h) ^ 0.5) * (sqrt(d) / sqrt(l))) * (1.0 - ((t_2 * sqrt((h / l))) ^ 2.0));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	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[(0.5 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-h)], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -4.8e+166], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(h * 0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5.5e+70], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / t$95$1), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -6.2e-285], N[(N[(N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision] * N[(-1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(t$95$2 * N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.05e+31], N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Power[N[(t$95$2 * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -4.79999999999999984e166

   1. Initial program 56.5%

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

   2. Applied egg-rr 38.6%

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

   3. Taylor expanded in d around -inf 75.4%

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

   4. Simplified 75.4%

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

      [Start] 75.4	\[ {\left(\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right) \cdot \left(1 - \frac{{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1} \]
      mul-1-neg [=>] 75.4	\[ {\left(\color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1} \]
      *-commutative [<=] 75.4	\[ {\left(\left(-\color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d}\right) \cdot \left(1 - \frac{{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1} \]
      distribute-rgt-neg-in [=>] 75.4	\[ {\left(\color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right)} \cdot \left(1 - \frac{{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\right)}^{1} \]

   if -4.79999999999999984e166 < d < -5.49999999999999986e70

   1. Initial program 66.4%

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

   2. Simplified 66.7%

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

      [Start] 66.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 66.4	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 66.4	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 66.4	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 66.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      *-commutative [=>] 66.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 66.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]
      times-frac [=>] 66.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 66.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 82.5%

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

   if -5.49999999999999986e70 < d < -6.2000000000000002e-285

   1. Initial program 58.3%

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

   2. Applied egg-rr 62.1%

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

   3. Applied egg-rr 61.9%

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

   4. Applied egg-rr 66.9%

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

   if -6.2000000000000002e-285 < d < 2.0500000000000001e31

   1. Initial program 53.4%

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

   2. Applied egg-rr 57.7%

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

   3. Applied egg-rr 65.3%

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

   if 2.0500000000000001e31 < d

   1. Initial program 61.9%

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

   2. Taylor expanded in d around inf 69.9%

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

   3. Simplified 70.6%

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

      [Start] 69.9	\[ \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]
      *-commutative [=>] 69.9	\[ \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
      *-commutative [=>] 69.9	\[ d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]
      associate-/r* [=>] 70.6	\[ d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   4. Applied egg-rr 86.8%

      \[\leadsto d \cdot \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.8 \cdot 10^{+166}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell} + -1\right)\\ \mathbf{elif}\;d \leq -5.5 \cdot 10^{+70}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{elif}\;d \leq -6.2 \cdot 10^{-285}:\\ \;\;\;\;\left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot \frac{-1}{\sqrt{\frac{h}{d}}}\right) \cdot \left({\left(\left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \frac{\sqrt{-h}}{\sqrt{-\ell}}\right)}^{2} + -1\right)\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{+31}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - {\left(\left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	75.0%
Cost	131540

\[\begin{array}{l} t_0 := 1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \mathsf{fma}\left({\left(\frac{0.5 \cdot M}{\frac{d}{D}}\right)}^{2}, \frac{h \cdot -0.5}{\ell}, 1\right)\\ t_3 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_4 := \left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot t_3\right) \cdot t_0\\ t_5 := t_0 \cdot \left(t_1 \cdot t_3\right)\\ \mathbf{if}\;t_4 \leq -\infty:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t_1 \cdot \left(-0.125 \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\frac{d \cdot \ell}{\frac{h}{d}}}\right)\right)\\ \mathbf{elif}\;t_4 \leq -2 \cdot 10^{-158}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;t_4 \leq 0:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_4 \leq 4 \cdot 10^{+262}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{d \cdot \frac{\frac{d}{\ell}}{h}} \cdot \left(1 + \frac{\left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(h \cdot M\right)\right)\right) \cdot -0.125}{\ell}\right)\\ \end{array} \]

Alternative 2
Accuracy	75.5%
Cost	131540

\[\begin{array}{l} t_0 := 1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \mathsf{fma}\left({\left(\frac{0.5 \cdot M}{\frac{d}{D}}\right)}^{2}, \frac{h \cdot -0.5}{\ell}, 1\right)\\ t_3 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_4 := \left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot t_3\right) \cdot t_0\\ t_5 := t_0 \cdot \left(t_1 \cdot t_3\right)\\ \mathbf{if}\;t_4 \leq -\infty:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t_1 \cdot \left(-0.125 \cdot {\left(\sqrt{\frac{h}{\ell}} \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;t_4 \leq -2 \cdot 10^{-158}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;t_4 \leq 0:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_4 \leq 4 \cdot 10^{+262}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{d \cdot \frac{\frac{d}{\ell}}{h}} \cdot \left(1 + \frac{\left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(h \cdot M\right)\right)\right) \cdot -0.125}{\ell}\right)\\ \end{array} \]

Alternative 3
Accuracy	75.9%
Cost	131540

\[\begin{array}{l} t_0 := 1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \mathsf{fma}\left({\left(\frac{0.5 \cdot M}{\frac{d}{D}}\right)}^{2}, \frac{h \cdot -0.5}{\ell}, 1\right)\\ t_3 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_4 := \left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot t_3\right) \cdot t_0\\ t_5 := t_0 \cdot \left(t_1 \cdot t_3\right)\\ \mathbf{if}\;t_4 \leq -\infty:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t_1 \cdot \left(1 + -0.5 \cdot {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;t_4 \leq -2 \cdot 10^{-158}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;t_4 \leq 0:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_4 \leq 4 \cdot 10^{+262}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{d \cdot \frac{\frac{d}{\ell}}{h}} \cdot \left(1 + \frac{\left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(h \cdot M\right)\right)\right) \cdot -0.125}{\ell}\right)\\ \end{array} \]

Alternative 4
Accuracy	75.5%
Cost	110608

\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \mathsf{fma}\left({\left(\frac{0.5 \cdot M}{\frac{d}{D}}\right)}^{2}, \frac{h \cdot -0.5}{\ell}, 1\right)\\ t_1 := 1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\\ t_2 := {\left(\frac{d}{\ell}\right)}^{0.5}\\ t_3 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_4 := \left(t_2 \cdot t_3\right) \cdot t_1\\ \mathbf{if}\;t_4 \leq -2 \cdot 10^{-158}:\\ \;\;\;\;\left({\left(\left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2} + -1\right) \cdot \left(t_2 \cdot \frac{-1}{\sqrt{\frac{h}{d}}}\right)\\ \mathbf{elif}\;t_4 \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_4 \leq 4 \cdot 10^{+262}:\\ \;\;\;\;t_1 \cdot \left(\sqrt{\frac{d}{\ell}} \cdot t_3\right)\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{d \cdot \frac{\frac{d}{\ell}}{h}} \cdot \left(1 + \frac{\left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(h \cdot M\right)\right)\right) \cdot -0.125}{\ell}\right)\\ \end{array} \]

Alternative 5
Accuracy	75.5%
Cost	110608

\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \mathsf{fma}\left({\left(\frac{0.5 \cdot M}{\frac{d}{D}}\right)}^{2}, \frac{h \cdot -0.5}{\ell}, 1\right)\\ t_1 := {\left(\left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\\ t_2 := {\left(\frac{d}{\ell}\right)}^{0.5}\\ t_3 := t_2 \cdot {\left(\frac{d}{h}\right)}^{0.5}\\ t_4 := t_3 \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{if}\;t_4 \leq -2 \cdot 10^{-158}:\\ \;\;\;\;\left(t_1 + -1\right) \cdot \left(t_2 \cdot \frac{-1}{\sqrt{\frac{h}{d}}}\right)\\ \mathbf{elif}\;t_4 \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_4 \leq 4 \cdot 10^{+262}:\\ \;\;\;\;\left(1 - t_1\right) \cdot t_3\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{d \cdot \frac{\frac{d}{\ell}}{h}} \cdot \left(1 + \frac{\left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(h \cdot M\right)\right)\right) \cdot -0.125}{\ell}\right)\\ \end{array} \]

Alternative 6
Accuracy	66.9%
Cost	21320

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := t_0 \cdot \sqrt{\frac{d}{h}}\\ t_2 := \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell} + -1\right)\\ t_3 := {\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2}\\ \mathbf{if}\;d \leq -3.5 \cdot 10^{+157}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -3.1 \cdot 10^{+38}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t_0\right) \cdot \left(1 + \frac{M \cdot M}{\frac{\ell}{h}} \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot -0.125\right)\right)\\ \mathbf{elif}\;d \leq -780000:\\ \;\;\;\;t_1 \cdot \left(1 - h \cdot \frac{t_3}{\frac{\ell}{0.5}}\right)\\ \mathbf{elif}\;d \leq -1.04 \cdot 10^{-45}:\\ \;\;\;\;t_1 \cdot \left(1 + \frac{M \cdot \left(-0.125 \cdot {\left(\frac{D}{d}\right)}^{2}\right)}{\frac{\ell}{h \cdot M}}\right)\\ \mathbf{elif}\;d \leq -9.5 \cdot 10^{-267}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t_3 \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 7
Accuracy	70.6%
Cost	21264

\[\begin{array}{l} t_0 := {\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2}\\ t_1 := \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - h \cdot \frac{t_0}{\frac{\ell}{0.5}}\right)\\ \mathbf{if}\;h \leq -4.8 \cdot 10^{+83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{0.5 \cdot M}{\frac{d}{D}}\right)}^{2}, \frac{h \cdot -0.5}{\ell}, 1\right) \cdot \left(d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\right)\\ \mathbf{elif}\;h \leq 1.06 \cdot 10^{+150}:\\ \;\;\;\;\left(1 + t_0 \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{elif}\;h \leq 2.25 \cdot 10^{+257}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Accuracy	62.9%
Cost	21009

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{+195}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + \left(M \cdot \left(h \cdot \frac{M}{\ell}\right)\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot -0.125\right)\right)\\ \mathbf{elif}\;\ell \leq -1.9 \cdot 10^{-308}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{-130} \lor \neg \left(\ell \leq 1.8 \cdot 10^{+55}\right):\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{0.5 \cdot M}{\frac{d}{D}}\right)}^{2}, \frac{h \cdot -0.5}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]

Alternative 9
Accuracy	63.9%
Cost	21004

\[\begin{array}{l} \mathbf{if}\;h \leq -3.6 \cdot 10^{+86}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{\ell}}{\frac{h}{d}}} \cdot \left(1 + \frac{-0.125 \cdot \left(D \cdot \left(\frac{h}{d} \cdot \left(D \cdot \frac{M}{\frac{d}{M}}\right)\right)\right)}{\ell}\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;h \leq 2.95 \cdot 10^{+149}:\\ \;\;\;\;\left(1 + {\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{elif}\;h \leq 3.2 \cdot 10^{+257}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 10
Accuracy	65.7%
Cost	21004

\[\begin{array}{l} \mathbf{if}\;h \leq -1.55 \cdot 10^{+100}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{\ell}}{\frac{h}{d}}} \cdot \left(1 + \frac{-0.125 \cdot \left(D \cdot \left(\frac{h}{d} \cdot \left(D \cdot \frac{M}{\frac{d}{M}}\right)\right)\right)}{\ell}\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{0.5 \cdot M}{\frac{d}{D}}\right)}^{2}, \frac{h \cdot -0.5}{\ell}, 1\right) \cdot \left(d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\right)\\ \mathbf{elif}\;h \leq 2.9 \cdot 10^{+149}:\\ \;\;\;\;\left(1 + {\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{elif}\;h \leq 2.4 \cdot 10^{+258}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 11
Accuracy	63.6%
Cost	13644

\[\begin{array}{l} \mathbf{if}\;d \leq -1.55 \cdot 10^{+54}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;d \leq -1.6 \cdot 10^{-62}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 12
Accuracy	62.4%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;h \leq -6.2 \cdot 10^{+85}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{\ell}}{\frac{h}{d}}} \cdot \left(1 + \frac{-0.125 \cdot \left(D \cdot \left(\frac{h}{d} \cdot \left(D \cdot \frac{M}{\frac{d}{M}}\right)\right)\right)}{\ell}\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 13
Accuracy	62.4%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 14
Accuracy	62.4%
Cost	13316

\[\begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 15
Accuracy	54.5%
Cost	7376

\[\begin{array}{l} \mathbf{if}\;d \leq -5.5 \cdot 10^{-266}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;d \leq 8.2 \cdot 10^{-227}:\\ \;\;\;\;\sqrt{d \cdot \frac{\frac{d}{h}}{\ell}}\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{+77}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+121}:\\ \;\;\;\;\sqrt{\frac{d \cdot \frac{d}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]

Alternative 16
Accuracy	46.5%
Cost	7245

\[\begin{array}{l} \mathbf{if}\;d \leq 3.8 \cdot 10^{-227}:\\ \;\;\;\;\sqrt{\frac{d}{\ell \cdot \frac{h}{d}}}\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{+77} \lor \neg \left(d \leq 3.3 \cdot 10^{+128}\right):\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d \cdot \frac{d}{h}}{\ell}}\\ \end{array} \]

Alternative 17
Accuracy	46.5%
Cost	7244

\[\begin{array}{l} \mathbf{if}\;d \leq 4.6 \cdot 10^{-225}:\\ \;\;\;\;\sqrt{\frac{d}{\ell \cdot \frac{h}{d}}}\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{+77}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq 4.1 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{\frac{d \cdot \frac{d}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]

Alternative 18
Accuracy	46.8%
Cost	7244

\[\begin{array}{l} \mathbf{if}\;d \leq 3.15 \cdot 10^{-226}:\\ \;\;\;\;\sqrt{\frac{d}{\ell \cdot \frac{h}{d}}}\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{+77}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+121}:\\ \;\;\;\;\sqrt{\frac{d \cdot \frac{d}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]

Alternative 19
Accuracy	45.1%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq -4.8 \cdot 10^{-178}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]

Alternative 20
Accuracy	47.9%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq 1.1 \cdot 10^{-226}:\\ \;\;\;\;\sqrt{d \cdot \frac{\frac{d}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]

Alternative 21
Accuracy	47.9%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq 4.2 \cdot 10^{-227}:\\ \;\;\;\;\sqrt{\frac{d}{\ell \cdot \frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]

Alternative 22
Accuracy	31.5%
Cost	6784

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

Reproduce

herbie shell --seed 2023125 
(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)))))