Henrywood and Agarwal, Equation (12)

Average Error: 26.9 → 13.9

Time: 52.3s

Precision: binary64

Cost: 83660

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

Math

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

↓

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ t_1 := M \cdot \frac{D}{d}\\ t_2 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-71}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(0.5 \cdot t_1\right)\right) \cdot \left(-0.5 \cdot t_1\right)\right)\right)\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;t_2 \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;t_0 \leq 10^{+248}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(1 + -0.5 \cdot \frac{h \cdot {\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}}{\ell}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (+ 1.0 (* (/ h l) (* (pow (/ (* M D) (* d 2.0)) 2.0) -0.5)))))
        (t_1 (* M (/ D d)))
        (t_2 (fabs (/ d (sqrt (* h l))))))
   (if (<= t_0 -5e-71)
     (*
      (* (sqrt (/ d h)) (sqrt (/ d l)))
      (+ 1.0 (* 0.5 (* (* (/ h l) (* 0.5 t_1)) (* -0.5 t_1)))))
     (if (<= t_0 0.0)
       (* t_2 (+ 1.0 (* -0.5 (* (/ h l) (pow (* (/ 0.5 d) (* M D)) 2.0)))))
       (if (<= t_0 1e+248)
         t_0
         (*
          t_2
          (+ 1.0 (* -0.5 (/ (* h (pow (* M (/ (* 0.5 D) d)) 2.0)) l)))))))))

C

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

↓

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 + ((h / l) * (pow(((M * D) / (d * 2.0)), 2.0) * -0.5)));
	double t_1 = M * (D / d);
	double t_2 = fabs((d / sqrt((h * l))));
	double tmp;
	if (t_0 <= -5e-71) {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 + (0.5 * (((h / l) * (0.5 * t_1)) * (-0.5 * t_1))));
	} else if (t_0 <= 0.0) {
		tmp = t_2 * (1.0 + (-0.5 * ((h / l) * pow(((0.5 / d) * (M * D)), 2.0))));
	} else if (t_0 <= 1e+248) {
		tmp = t_0;
	} else {
		tmp = t_2 * (1.0 + (-0.5 * ((h * pow((M * ((0.5 * D) / d)), 2.0)) / l)));
	}
	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 = (((d / h) ** 0.5d0) * ((d / l) ** 0.5d0)) * (1.0d0 + ((h / l) * ((((m * d_1) / (d * 2.0d0)) ** 2.0d0) * (-0.5d0))))
    t_1 = m * (d_1 / d)
    t_2 = abs((d / sqrt((h * l))))
    if (t_0 <= (-5d-71)) then
        tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0d0 + (0.5d0 * (((h / l) * (0.5d0 * t_1)) * ((-0.5d0) * t_1))))
    else if (t_0 <= 0.0d0) then
        tmp = t_2 * (1.0d0 + ((-0.5d0) * ((h / l) * (((0.5d0 / d) * (m * d_1)) ** 2.0d0))))
    else if (t_0 <= 1d+248) then
        tmp = t_0
    else
        tmp = t_2 * (1.0d0 + ((-0.5d0) * ((h * ((m * ((0.5d0 * d_1) / d)) ** 2.0d0)) / l)))
    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 = (Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 + ((h / l) * (Math.pow(((M * D) / (d * 2.0)), 2.0) * -0.5)));
	double t_1 = M * (D / d);
	double t_2 = Math.abs((d / Math.sqrt((h * l))));
	double tmp;
	if (t_0 <= -5e-71) {
		tmp = (Math.sqrt((d / h)) * Math.sqrt((d / l))) * (1.0 + (0.5 * (((h / l) * (0.5 * t_1)) * (-0.5 * t_1))));
	} else if (t_0 <= 0.0) {
		tmp = t_2 * (1.0 + (-0.5 * ((h / l) * Math.pow(((0.5 / d) * (M * D)), 2.0))));
	} else if (t_0 <= 1e+248) {
		tmp = t_0;
	} else {
		tmp = t_2 * (1.0 + (-0.5 * ((h * Math.pow((M * ((0.5 * D) / d)), 2.0)) / l)));
	}
	return tmp;
}

Python

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

↓

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 + ((h / l) * (math.pow(((M * D) / (d * 2.0)), 2.0) * -0.5)))
	t_1 = M * (D / d)
	t_2 = math.fabs((d / math.sqrt((h * l))))
	tmp = 0
	if t_0 <= -5e-71:
		tmp = (math.sqrt((d / h)) * math.sqrt((d / l))) * (1.0 + (0.5 * (((h / l) * (0.5 * t_1)) * (-0.5 * t_1))))
	elif t_0 <= 0.0:
		tmp = t_2 * (1.0 + (-0.5 * ((h / l) * math.pow(((0.5 / d) * (M * D)), 2.0))))
	elif t_0 <= 1e+248:
		tmp = t_0
	else:
		tmp = t_2 * (1.0 + (-0.5 * ((h * math.pow((M * ((0.5 * D) / d)), 2.0)) / l)))
	return tmp

Julia

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

↓

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * -0.5))))
	t_1 = Float64(M * Float64(D / d))
	t_2 = abs(Float64(d / sqrt(Float64(h * l))))
	tmp = 0.0
	if (t_0 <= -5e-71)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * Float64(1.0 + Float64(0.5 * Float64(Float64(Float64(h / l) * Float64(0.5 * t_1)) * Float64(-0.5 * t_1)))));
	elseif (t_0 <= 0.0)
		tmp = Float64(t_2 * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(Float64(0.5 / d) * Float64(M * D)) ^ 2.0)))));
	elseif (t_0 <= 1e+248)
		tmp = t_0;
	else
		tmp = Float64(t_2 * Float64(1.0 + Float64(-0.5 * Float64(Float64(h * (Float64(M * Float64(Float64(0.5 * D) / d)) ^ 2.0)) / l))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 + ((h / l) * ((((M * D) / (d * 2.0)) ^ 2.0) * -0.5)));
	t_1 = M * (D / d);
	t_2 = abs((d / sqrt((h * l))));
	tmp = 0.0;
	if (t_0 <= -5e-71)
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 + (0.5 * (((h / l) * (0.5 * t_1)) * (-0.5 * t_1))));
	elseif (t_0 <= 0.0)
		tmp = t_2 * (1.0 + (-0.5 * ((h / l) * (((0.5 / d) * (M * D)) ^ 2.0))));
	elseif (t_0 <= 1e+248)
		tmp = t_0;
	else
		tmp = t_2 * (1.0 + (-0.5 * ((h * ((M * ((0.5 * D) / d)) ^ 2.0)) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -5e-71], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(0.5 * N[(N[(N[(h / l), $MachinePrecision] * N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(-0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(t$95$2 * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(0.5 / d), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+248], t$95$0, N[(t$95$2 * N[(1.0 + N[(-0.5 * N[(N[(h * N[Power[N[(M * N[(N[(0.5 * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $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)))) < -4.99999999999999998e-71

   1. Initial program 30.9

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

   2. Simplified 32.2

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

      [Start] 30.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 30.9	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 30.9	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 30.9	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 30.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 30.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 30.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 32.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 31.7

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

   4. Applied egg-rr 24.0

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

   if -4.99999999999999998e-71 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 0.0

   1. Initial program 35.2

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

   2. Applied egg-rr 41.6

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

   3. Applied egg-rr 13.0

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 1.00000000000000005e248

   1. Initial program 0.9

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

   if 1.00000000000000005e248 < (*.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 61.3

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

   2. Applied egg-rr 64.0

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

   3. Applied egg-rr 40.5

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

   4. Applied egg-rr 28.2

      \[\leadsto {\left(\left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(1 + -0.5 \cdot \color{blue}{\frac{h \cdot {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\ell}}\right)\right)}^{1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 13.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq -5 \cdot 10^{-71}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)\right) \cdot \left(-0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq 10^{+248}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(1 + -0.5 \cdot \frac{h \cdot {\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}}{\ell}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	13.9
Cost	83660

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ t_1 := M \cdot \frac{D}{d}\\ t_2 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-71}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(0.5 \cdot t_1\right)\right) \cdot \left(-0.5 \cdot t_1\right)\right)\right)\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;t_2 \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;t_0 \leq 10^{+285}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(h \cdot M\right)}{\ell}\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	19.6
Cost	27608

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ t_1 := 0.5 \cdot \left(M \cdot \frac{D}{d}\right)\\ \mathbf{if}\;d \leq -1.2 \cdot 10^{+112}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-99}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{t_1}{\ell} \cdot \frac{t_1}{\frac{1}{h}}\right)\right)\\ \mathbf{elif}\;d \leq -6.6 \cdot 10^{-268}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{-258}:\\ \;\;\;\;\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\left(\frac{D}{\frac{\sqrt{d}}{M}}\right)}^{2}\right)}{{\ell}^{1.5}}\\ \mathbf{elif}\;d \leq 4.9 \cdot 10^{-241}:\\ \;\;\;\;t_0 \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{-102}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} + \frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left({\left(M \cdot D\right)}^{2} \cdot \frac{-0.125}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	19.6
Cost	27476

\[\begin{array}{l} t_0 := M \cdot \frac{D}{d}\\ t_1 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ t_2 := 0.5 \cdot t_0\\ \mathbf{if}\;d \leq -1.4 \cdot 10^{+112}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-97}:\\ \;\;\;\;t_1 \cdot \left(1 + -0.5 \cdot \left(\frac{t_2}{\ell} \cdot \frac{t_2}{\frac{1}{h}}\right)\right)\\ \mathbf{elif}\;d \leq -6.6 \cdot 10^{-268}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{-221}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left({t_2}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-104}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} + \frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left({\left(M \cdot D\right)}^{2} \cdot \frac{-0.125}{d}\right)\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-11}:\\ \;\;\;\;t_1 \cdot \left(1 + 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot t_2\right) \cdot \left(-0.5 \cdot t_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 4
Error	20.6
Cost	26960

\[\begin{array}{l} t_0 := 0.5 \cdot \left(M \cdot \frac{D}{d}\right)\\ \mathbf{if}\;d \leq -1.25 \cdot 10^{+112}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-95}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{t_0}{\ell} \cdot \frac{t_0}{\frac{1}{h}}\right)\right)\\ \mathbf{elif}\;d \leq -6.6 \cdot 10^{-268}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{-187}:\\ \;\;\;\;\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\left(\frac{D}{\frac{\sqrt{d}}{M}}\right)}^{2}\right)}{{\ell}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 5
Error	20.4
Cost	21456

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ t_2 := 0.5 \cdot \left(M \cdot \frac{D}{d}\right)\\ t_3 := \left(\sqrt{\frac{d}{h}} \cdot t_0\right) \cdot \left(1 + -0.5 \cdot \left(\frac{t_2}{\ell} \cdot \frac{t_2}{\frac{1}{h}}\right)\right)\\ \mathbf{if}\;d \leq -2.8 \cdot 10^{+111}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -2.5 \cdot 10^{-98}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq -6.6 \cdot 10^{-268}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 10^{-274}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t_0 \cdot \left(1 + -0.125 \cdot \frac{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(h \cdot \left(M \cdot \frac{M}{d}\right)\right)}{d}\right)\right)\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-232}:\\ \;\;\;\;t_1 \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{-196}:\\ \;\;\;\;\frac{D \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot -0.125\right)}{\frac{d}{M \cdot \left(M \cdot D\right)}}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-11}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	19.6
Cost	21328

\[\begin{array}{l} t_0 := 0.5 \cdot \left(M \cdot \frac{D}{d}\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \left(\sqrt{\frac{d}{h}} \cdot t_1\right) \cdot \left(1 + -0.5 \cdot \left(\frac{t_0}{\ell} \cdot \frac{t_0}{\frac{1}{h}}\right)\right)\\ t_3 := \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{if}\;d \leq -7.5 \cdot 10^{+110}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -5.8 \cdot 10^{-93}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -1.7 \cdot 10^{-264}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{-290}:\\ \;\;\;\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot t_1\right)\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-230}:\\ \;\;\;\;t_3 \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{-197}:\\ \;\;\;\;\frac{D \cdot \left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot -0.125\right)}{\frac{d}{M \cdot \left(M \cdot D\right)}}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-11}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 7
Error	19.7
Cost	15440

\[\begin{array}{l} t_0 := 0.5 \cdot \left(M \cdot \frac{D}{d}\right)\\ t_1 := \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{t_0}{\ell} \cdot \frac{t_0}{\frac{1}{h}}\right)\right)\\ \mathbf{if}\;d \leq -3.8 \cdot 10^{+111}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -2.3 \cdot 10^{-97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-266}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(-1 + {\left(D \cdot \left(M \cdot \frac{-0.5}{d}\right)\right)}^{2} \cdot \frac{h \cdot 0.5}{\ell}\right)\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 8
Error	19.7
Cost	15440

\[\begin{array}{l} t_0 := 0.5 \cdot \left(M \cdot \frac{D}{d}\right)\\ t_1 := \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{t_0}{\ell} \cdot \frac{t_0}{\frac{1}{h}}\right)\right)\\ \mathbf{if}\;d \leq -5 \cdot 10^{+111}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -2.5 \cdot 10^{-94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -8 \cdot 10^{-264}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 9
Error	19.8
Cost	15176

\[\begin{array}{l} t_0 := 0.5 \cdot \left(M \cdot \frac{D}{d}\right)\\ \mathbf{if}\;d \leq -1.4 \cdot 10^{+111}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-11}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{t_0}{\ell} \cdot \frac{t_0}{\frac{1}{h}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 10
Error	20.9
Cost	15048

\[\begin{array}{l} t_0 := M \cdot \frac{D}{d}\\ \mathbf{if}\;d \leq -2 \cdot 10^{+123}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-11}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(0.5 \cdot t_0\right)\right) \cdot \left(-0.5 \cdot t_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 11
Error	21.7
Cost	14856

\[\begin{array}{l} t_0 := d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{if}\;d \leq -1.6 \cdot 10^{+123}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -2.3 \cdot 10^{-132}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{M \cdot \left(D \cdot \frac{M}{\frac{\ell}{D}}\right)}{\frac{d \cdot \left(-d\right)}{h}} \cdot 0.125\right)\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-198}:\\ \;\;\;\;\frac{\frac{\sqrt{h} \cdot -0.125}{{\ell}^{1.5}}}{\frac{d}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 12
Error	23.5
Cost	14792

\[\begin{array}{l} t_0 := d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{if}\;d \leq -1.95 \cdot 10^{+108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.4 \cdot 10^{-27}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{-196}:\\ \;\;\;\;\frac{\frac{\sqrt{h} \cdot -0.125}{{\ell}^{1.5}}}{\frac{d}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 13
Error	23.1
Cost	14792

\[\begin{array}{l} t_0 := d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{if}\;d \leq -3.5 \cdot 10^{+111}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -2.2 \cdot 10^{-132}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \frac{D \cdot \left(M \cdot M\right)}{\frac{\ell}{D} \cdot \left(d \cdot \frac{d}{h}\right)}\right)\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{-192}:\\ \;\;\;\;\frac{\frac{\sqrt{h} \cdot -0.125}{{\ell}^{1.5}}}{\frac{d}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 14
Error	23.0
Cost	14792

\[\begin{array}{l} t_0 := d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{if}\;d \leq -9.2 \cdot 10^{+107}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.75 \cdot 10^{-31}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \frac{\left(h \cdot M\right) \cdot \left(M \cdot \left(D \cdot \frac{D}{\ell}\right)\right)}{d \cdot d}\right)\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-182}:\\ \;\;\;\;\frac{\frac{\sqrt{h} \cdot -0.125}{{\ell}^{1.5}}}{\frac{d}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 15
Error	22.6
Cost	14792

\[\begin{array}{l} \mathbf{if}\;d \leq -1.12 \cdot 10^{+111}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-199}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \frac{\frac{D \cdot \left(M \cdot \frac{M}{d}\right)}{\frac{\frac{\ell}{D}}{h}}}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 16
Error	21.6
Cost	14792

\[\begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{+111}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 4.7 \cdot 10^{-85}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + -0.125 \cdot \frac{\frac{D \cdot \frac{M}{d}}{\frac{\frac{\ell}{D}}{h \cdot M}}}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 17
Error	23.2
Cost	14352

\[\begin{array}{l} t_0 := d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{if}\;d \leq -3.5 \cdot 10^{+111}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.4 \cdot 10^{-95}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-307}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-199}:\\ \;\;\;\;\left(\frac{\sqrt{h}}{d} \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)\right) \cdot \frac{-0.125}{{\ell}^{1.5}}\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 18
Error	22.8
Cost	14352

\[\begin{array}{l} t_0 := d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{if}\;d \leq -4.8 \cdot 10^{+112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -6.5 \cdot 10^{-91}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 6 \cdot 10^{-198}:\\ \;\;\;\;\frac{\frac{\sqrt{h} \cdot -0.125}{{\ell}^{1.5}}}{\frac{d}{D \cdot \left(M \cdot \left(M \cdot D\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 19
Error	23.0
Cost	13516

\[\begin{array}{l} t_0 := d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{if}\;d \leq -1.5 \cdot 10^{+112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -6.8 \cdot 10^{-96}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 2.45 \cdot 10^{-255}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 20
Error	29.4
Cost	13252

\[\begin{array}{l} \mathbf{if}\;d \leq 2.45 \cdot 10^{-255}:\\ \;\;\;\;\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 21
Error	23.9
Cost	13252

\[\begin{array}{l} \mathbf{if}\;d \leq 2.45 \cdot 10^{-255}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 22
Error	32.9
Cost	7113

\[\begin{array}{l} \mathbf{if}\;h \leq -3.3 \cdot 10^{-307} \lor \neg \left(h \leq 1.9 \cdot 10^{+158}\right):\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 23
Error	33.1
Cost	7113

\[\begin{array}{l} \mathbf{if}\;h \leq 1.34 \cdot 10^{-306} \lor \neg \left(h \leq 1.2 \cdot 10^{+159}\right):\\ \;\;\;\;\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 24
Error	43.7
Cost	6784

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

Alternative 25
Error	43.7
Cost	6720

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

Reproduce

herbie shell --seed 2023059 
(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)))))