Henrywood and Agarwal, Equation (12)

Average Error: 27.4 → 22.3

Time: 27.7s

Precision: binary64

Cost: 41796

Math

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

↓

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

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))))
   (if (<=
        (* t_0 (- 1.0 (* (* 0.5 (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
        1e+264)
     (* (- 1.0 (* 0.5 (/ h (/ l (pow (* (/ 0.5 d) (* D M)) 2.0))))) t_0)
     (* (sqrt (/ 1.0 (* l h))) d))))

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);
	double tmp;
	if ((t_0 * (1.0 - ((0.5 * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= 1e+264) {
		tmp = (1.0 - (0.5 * (h / (l / pow(((0.5 / d) * (D * M)), 2.0))))) * t_0;
	} else {
		tmp = sqrt((1.0 / (l * h))) * d;
	}
	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) :: tmp
    t_0 = ((d / h) ** 0.5d0) * ((d / l) ** 0.5d0)
    if ((t_0 * (1.0d0 - ((0.5d0 * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= 1d+264) then
        tmp = (1.0d0 - (0.5d0 * (h / (l / (((0.5d0 / d) * (d_1 * m)) ** 2.0d0))))) * t_0
    else
        tmp = sqrt((1.0d0 / (l * h))) * d
    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);
	double tmp;
	if ((t_0 * (1.0 - ((0.5 * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= 1e+264) {
		tmp = (1.0 - (0.5 * (h / (l / Math.pow(((0.5 / d) * (D * M)), 2.0))))) * t_0;
	} else {
		tmp = Math.sqrt((1.0 / (l * h))) * d;
	}
	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)
	tmp = 0
	if (t_0 * (1.0 - ((0.5 * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= 1e+264:
		tmp = (1.0 - (0.5 * (h / (l / math.pow(((0.5 / d) * (D * M)), 2.0))))) * t_0
	else:
		tmp = math.sqrt((1.0 / (l * h))) * d
	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(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5))
	tmp = 0.0
	if (Float64(t_0 * Float64(1.0 - Float64(Float64(0.5 * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= 1e+264)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(h / Float64(l / (Float64(Float64(0.5 / d) * Float64(D * M)) ^ 2.0))))) * t_0);
	else
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * d);
	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);
	tmp = 0.0;
	if ((t_0 * (1.0 - ((0.5 * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= 1e+264)
		tmp = (1.0 - (0.5 * (h / (l / (((0.5 / d) * (D * M)) ^ 2.0))))) * t_0;
	else
		tmp = sqrt((1.0 / (l * h))) * d;
	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[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(1.0 - N[(N[(0.5 * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+264], N[(N[(1.0 - N[(0.5 * N[(h / N[(l / N[Power[N[(N[(0.5 / d), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]]]

Derivation

1. Split input into 2 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)))) < 1.00000000000000004e264

   1. Initial program 12.9

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

   2. Simplified 13.6

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

      [Start] 12.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      rational.json-simplify-2 [=>] 12.9	\[ \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]
      rational.json-simplify-43 [=>] 13.0	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 13.0	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 13.0	\[ {\left(\frac{d}{h}\right)}^{0.5} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      rational.json-simplify-2 [=>] 13.0	\[ {\left(\frac{d}{h}\right)}^{0.5} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right)\right) \]
      rational.json-simplify-43 [=>] 13.0	\[ {\left(\frac{d}{h}\right)}^{0.5} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)\right) \]
      metadata-eval [=>] 13.0	\[ {\left(\frac{d}{h}\right)}^{0.5} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      rational.json-simplify-2 [=>] 13.0	\[ {\left(\frac{d}{h}\right)}^{0.5} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      rational.json-simplify-49 [=>] 13.6	\[ {\left(\frac{d}{h}\right)}^{0.5} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      rational.json-simplify-2 [=>] 13.6	\[ {\left(\frac{d}{h}\right)}^{0.5} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left({\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      rational.json-simplify-46 [=>] 13.6	\[ {\left(\frac{d}{h}\right)}^{0.5} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left({\left(M \cdot \color{blue}{\frac{\frac{D}{d}}{2}}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   3. Applied egg-rr 13.6

      \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{0.5} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.5\right)\right)\right) + 0} \]

   4. Simplified 12.9

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

      [Start] 13.6	\[ {\left(\frac{d}{h}\right)}^{0.5} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.5\right)\right)\right) + 0 \]
      rational.json-simplify-4 [=>] 13.6	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{0.5} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(1 - {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.5\right)\right)\right)} \]
      rational.json-simplify-43 [<=] 13.4	\[ \color{blue}{\left(1 - {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.5\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right)} \]
      rational.json-simplify-43 [<=] 13.4	\[ \left(1 - \color{blue}{0.5 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \]
      rational.json-simplify-2 [=>] 13.4	\[ \left(1 - \color{blue}{\left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.5}\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \]
      metadata-eval [<=] 13.4	\[ \left(1 - \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left(1 \cdot 0.5\right)}\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \]
      rational.json-simplify-43 [<=] 13.4	\[ \left(1 - \color{blue}{0.5 \cdot \left(\left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot 1\right)}\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \]
      rational.json-simplify-2 [=>] 13.4	\[ \left(1 - 0.5 \cdot \color{blue}{\left(1 \cdot \left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \]
      rational.json-simplify-6 [=>] 13.4	\[ \left(1 - 0.5 \cdot \color{blue}{\left({\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \]
      rational.json-simplify-2 [=>] 13.4	\[ \left(1 - 0.5 \cdot \left({\left(M \cdot \color{blue}{\left(\frac{0.5}{d} \cdot D\right)}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \]
      rational.json-simplify-43 [=>] 12.9	\[ \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{0.5}{d} \cdot \left(D \cdot M\right)\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \]

   5. Applied egg-rr 13.1

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

   if 1.00000000000000004e264 < (*.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 62.6

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

   2. Taylor expanded in d around inf 44.8

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+264}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h}{\frac{\ell}{{\left(\frac{0.5}{d} \cdot \left(D \cdot M\right)\right)}^{2}}}\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot d\\ \end{array} \]

Alternatives

Alternative 1
Error	24.9
Cost	21128

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - h \cdot \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right)\\ \mathbf{if}\;h \leq 9 \cdot 10^{-277}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq 4.4 \cdot 10^{-104}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	24.9
Cost	21128

\[\begin{array}{l} t_0 := {\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\\ \mathbf{if}\;h \leq 9 \cdot 10^{-277}:\\ \;\;\;\;t_0 \cdot \left(1 - h \cdot \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq 1.65 \cdot 10^{-103}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(1 - h \cdot \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell + \ell}\right)\\ \end{array} \]

Alternative 3
Error	43.9
Cost	6976

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

Alternative 4
Error	43.8
Cost	6848

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

Alternative 5
Error	43.9
Cost	6848

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

Reproduce

herbie shell --seed 2023064 
(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)))))