Henrywood and Agarwal, Equation (12)

Average Error: 26.6 → 13.6

Time: 42.7s

Precision: binary64

Cost: 83661

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

Math

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

↓

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}{\ell \cdot \frac{\frac{d}{D}}{0.5 \cdot M}}\right)\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{-300} \lor \neg \left(t_0 \leq 2 \cdot 10^{+286}\right):\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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))))))
   (if (<= t_0 -1e-134)
     (*
      (* (sqrt (/ d h)) (sqrt (/ d l)))
      (-
       1.0
       (* 0.5 (/ (* h (* M (/ 0.5 (/ d D)))) (* l (/ (/ d D) (* 0.5 M)))))))
     (if (or (<= t_0 4e-300) (not (<= t_0 2e+286)))
       (fabs (* d (pow (* h l) -0.5)))
       t_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 = (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 tmp;
	if (t_0 <= -1e-134) {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - (0.5 * ((h * (M * (0.5 / (d / D)))) / (l * ((d / D) / (0.5 * M))))));
	} else if ((t_0 <= 4e-300) || !(t_0 <= 2e+286)) {
		tmp = fabs((d * pow((h * l), -0.5)));
	} else {
		tmp = t_0;
	}
	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)) * (1.0d0 + ((h / l) * ((((m * d_1) / (d * 2.0d0)) ** 2.0d0) * (-0.5d0))))
    if (t_0 <= (-1d-134)) then
        tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0d0 - (0.5d0 * ((h * (m * (0.5d0 / (d / d_1)))) / (l * ((d / d_1) / (0.5d0 * m))))))
    else if ((t_0 <= 4d-300) .or. (.not. (t_0 <= 2d+286))) then
        tmp = abs((d * ((h * l) ** (-0.5d0))))
    else
        tmp = t_0
    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 tmp;
	if (t_0 <= -1e-134) {
		tmp = (Math.sqrt((d / h)) * Math.sqrt((d / l))) * (1.0 - (0.5 * ((h * (M * (0.5 / (d / D)))) / (l * ((d / D) / (0.5 * M))))));
	} else if ((t_0 <= 4e-300) || !(t_0 <= 2e+286)) {
		tmp = Math.abs((d * Math.pow((h * l), -0.5)));
	} else {
		tmp = t_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.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)))
	tmp = 0
	if t_0 <= -1e-134:
		tmp = (math.sqrt((d / h)) * math.sqrt((d / l))) * (1.0 - (0.5 * ((h * (M * (0.5 / (d / D)))) / (l * ((d / D) / (0.5 * M))))))
	elif (t_0 <= 4e-300) or not (t_0 <= 2e+286):
		tmp = math.fabs((d * math.pow((h * l), -0.5)))
	else:
		tmp = t_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 = 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))))
	tmp = 0.0
	if (t_0 <= -1e-134)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * Float64(M * Float64(0.5 / Float64(d / D)))) / Float64(l * Float64(Float64(d / D) / Float64(0.5 * M)))))));
	elseif ((t_0 <= 4e-300) || !(t_0 <= 2e+286))
		tmp = abs(Float64(d * (Float64(h * l) ^ -0.5)));
	else
		tmp = t_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 = (((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 + ((h / l) * ((((M * D) / (d * 2.0)) ^ 2.0) * -0.5)));
	tmp = 0.0;
	if (t_0 <= -1e-134)
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - (0.5 * ((h * (M * (0.5 / (d / D)))) / (l * ((d / D) / (0.5 * M))))));
	elseif ((t_0 <= 4e-300) || ~((t_0 <= 2e+286)))
		tmp = abs((d * ((h * l) ^ -0.5)));
	else
		tmp = t_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[(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]}, If[LessEqual[t$95$0, -1e-134], 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[(h * N[(M * N[(0.5 / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(N[(d / D), $MachinePrecision] / N[(0.5 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, 4e-300], N[Not[LessEqual[t$95$0, 2e+286]], $MachinePrecision]], N[Abs[N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 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)))) < -1.00000000000000004e-134

   1. Initial program 29.3

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

   2. Simplified 30.5

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

      [Start] 29.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 29.3	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 29.3	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 29.3	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 29.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 29.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 29.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 30.5	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 32.9

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)} - 1\right)}\right) \]

   4. Simplified 30.4

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

      [Start] 32.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(e^{\mathsf{log1p}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)} - 1\right)\right) \]
      expm1-def [=>] 32.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]
      expm1-log1p [=>] 30.5	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      associate-*r/ [=>] 31.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}}\right) \]
      associate-*l/ [<=] 30.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} \cdot h\right)}\right) \]
      *-commutative [=>] 30.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)}\right) \]
      *-commutative [=>] 30.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(M \cdot \color{blue}{\left(\frac{D}{d} \cdot 0.5\right)}\right)}^{2}}{\ell}\right)\right) \]
      associate-*r* [=>] 30.4	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot 0.5\right)}}^{2}}{\ell}\right)\right) \]

   5. Applied egg-rr 22.3

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

   6. Applied egg-rr 22.2

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

   if -1.00000000000000004e-134 < (*.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)))) < 4.0000000000000001e-300 or 2.00000000000000007e286 < (*.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 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. Simplified 57.0

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

      [Start] 56.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 56.5	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 56.5	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 56.5	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 56.5	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 56.5	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 56.5	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 57.0	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Taylor expanded in d around inf 41.0

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

   4. Simplified 41.0

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

      [Start] 41.0	\[ \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]
      *-commutative [=>] 41.0	\[ \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
      associate-/r* [=>] 41.0	\[ d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   5. Applied egg-rr 41.0

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

   6. Taylor expanded in l around 0 44.6

      \[\leadsto d \cdot {\color{blue}{\left(e^{0.25 \cdot \left(-1 \cdot \log \ell + \log \left(\frac{1}{h}\right)\right)}\right)}}^{2} \]

   7. Simplified 41.0

      \[\leadsto d \cdot {\color{blue}{\left({\left(\ell \cdot h\right)}^{-0.25}\right)}}^{2} \]
      Proof

      [Start] 44.6	\[ d \cdot {\left(e^{0.25 \cdot \left(-1 \cdot \log \ell + \log \left(\frac{1}{h}\right)\right)}\right)}^{2} \]
      distribute-rgt-in [=>] 44.6	\[ d \cdot {\left(e^{\color{blue}{\left(-1 \cdot \log \ell\right) \cdot 0.25 + \log \left(\frac{1}{h}\right) \cdot 0.25}}\right)}^{2} \]
      mul-1-neg [=>] 44.6	\[ d \cdot {\left(e^{\color{blue}{\left(-\log \ell\right)} \cdot 0.25 + \log \left(\frac{1}{h}\right) \cdot 0.25}\right)}^{2} \]
      log-rec [<=] 44.6	\[ d \cdot {\left(e^{\color{blue}{\log \left(\frac{1}{\ell}\right)} \cdot 0.25 + \log \left(\frac{1}{h}\right) \cdot 0.25}\right)}^{2} \]
      log-rec [=>] 44.6	\[ d \cdot {\left(e^{\log \left(\frac{1}{\ell}\right) \cdot 0.25 + \color{blue}{\left(-\log h\right)} \cdot 0.25}\right)}^{2} \]
      mul-1-neg [<=] 44.6	\[ d \cdot {\left(e^{\log \left(\frac{1}{\ell}\right) \cdot 0.25 + \color{blue}{\left(-1 \cdot \log h\right)} \cdot 0.25}\right)}^{2} \]
      distribute-rgt-in [<=] 44.6	\[ d \cdot {\left(e^{\color{blue}{0.25 \cdot \left(\log \left(\frac{1}{\ell}\right) + -1 \cdot \log h\right)}}\right)}^{2} \]
      distribute-lft-in [=>] 44.6	\[ d \cdot {\left(e^{\color{blue}{0.25 \cdot \log \left(\frac{1}{\ell}\right) + 0.25 \cdot \left(-1 \cdot \log h\right)}}\right)}^{2} \]
      log-rec [=>] 44.6	\[ d \cdot {\left(e^{0.25 \cdot \color{blue}{\left(-\log \ell\right)} + 0.25 \cdot \left(-1 \cdot \log h\right)}\right)}^{2} \]
      mul-1-neg [<=] 44.6	\[ d \cdot {\left(e^{0.25 \cdot \color{blue}{\left(-1 \cdot \log \ell\right)} + 0.25 \cdot \left(-1 \cdot \log h\right)}\right)}^{2} \]
      associate-*r* [=>] 44.6	\[ d \cdot {\left(e^{\color{blue}{\left(0.25 \cdot -1\right) \cdot \log \ell} + 0.25 \cdot \left(-1 \cdot \log h\right)}\right)}^{2} \]
      metadata-eval [=>] 44.6	\[ d \cdot {\left(e^{\color{blue}{-0.25} \cdot \log \ell + 0.25 \cdot \left(-1 \cdot \log h\right)}\right)}^{2} \]
      associate-*r* [=>] 44.6	\[ d \cdot {\left(e^{-0.25 \cdot \log \ell + \color{blue}{\left(0.25 \cdot -1\right) \cdot \log h}}\right)}^{2} \]
      metadata-eval [=>] 44.6	\[ d \cdot {\left(e^{-0.25 \cdot \log \ell + \color{blue}{-0.25} \cdot \log h}\right)}^{2} \]
      distribute-lft-in [<=] 44.6	\[ d \cdot {\left(e^{\color{blue}{-0.25 \cdot \left(\log \ell + \log h\right)}}\right)}^{2} \]
      log-prod [<=] 41.9	\[ d \cdot {\left(e^{-0.25 \cdot \color{blue}{\log \left(\ell \cdot h\right)}}\right)}^{2} \]
      log-pow [<=] 41.9	\[ d \cdot {\left(e^{\color{blue}{\log \left({\left(\ell \cdot h\right)}^{-0.25}\right)}}\right)}^{2} \]
      rem-exp-log [=>] 41.0	\[ d \cdot {\color{blue}{\left({\left(\ell \cdot h\right)}^{-0.25}\right)}}^{2} \]

   8. Applied egg-rr 24.8

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

   if 4.0000000000000001e-300 < (*.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)))) < 2.00000000000000007e286

   1. Initial program 0.8

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 13.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot \left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}{\ell \cdot \frac{\frac{d}{D}}{0.5 \cdot M}}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq 4 \cdot 10^{-300} \lor \neg \left(\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq 2 \cdot 10^{+286}\right):\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	19.0
Cost	21384

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{if}\;\ell \leq -2.35 \cdot 10^{-113}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 + 0.5 \cdot \frac{h \cdot \left(M \cdot \frac{-0.5}{\frac{d}{D}}\right)}{\frac{\ell}{M \cdot \frac{0.5}{\frac{d}{D}}}}\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - \frac{0.125}{\ell} \cdot \left(\frac{h \cdot M}{\frac{d}{D}} \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-149}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 2
Error	19.8
Cost	21320

\[\begin{array}{l} t_0 := M \cdot \frac{0.5}{\frac{d}{D}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -2 \cdot 10^{-57}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot t_1\right) \cdot \left(1 + 0.5 \cdot \left(h \cdot \left(t_0 \cdot \left(t_0 \cdot \frac{-1}{\ell}\right)\right)\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(1 - \frac{0.125}{\ell} \cdot \left(\frac{h \cdot M}{\frac{d}{D}} \cdot \left(D \cdot \frac{M}{d}\right)\right)\right) \cdot \left(t_1 \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 3
Error	22.3
Cost	15308

\[\begin{array}{l} t_0 := M \cdot \frac{0.5}{\frac{d}{D}}\\ t_1 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;h \leq -2.25 \cdot 10^{-60}:\\ \;\;\;\;\left(t_1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(h \cdot \left(t_0 \cdot \left(t_0 \cdot \frac{-1}{\ell}\right)\right)\right)\right)\\ \mathbf{elif}\;h \leq -1.6 \cdot 10^{-209}:\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{elif}\;h \leq 9.5 \cdot 10^{-293}:\\ \;\;\;\;\left(t_1 \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 + 0.5 \cdot \frac{h \cdot \left(M \cdot \frac{-0.5}{\frac{d}{D}}\right)}{\frac{\ell}{t_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 4
Error	22.4
Cost	15180

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ t_1 := \frac{0.5}{\frac{d}{D}}\\ \mathbf{if}\;h \leq -1.3 \cdot 10^{-59}:\\ \;\;\;\;t_0 \cdot \left(1 + 0.5 \cdot \left(h \cdot \left(\frac{M}{\frac{\ell}{t_1}} \cdot \left(M \cdot \frac{-0.5}{\frac{d}{D}}\right)\right)\right)\right)\\ \mathbf{elif}\;h \leq -9 \cdot 10^{-209}:\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;t_0 \cdot \left(1 + -0.5 \cdot \frac{D \cdot \frac{h \cdot M}{\frac{d}{0.5}}}{\frac{\ell}{M \cdot t_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 5
Error	22.1
Cost	15180

\[\begin{array}{l} t_0 := M \cdot \frac{0.5}{\frac{d}{D}}\\ t_1 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -2.55 \cdot 10^{-67}:\\ \;\;\;\;t_1 \cdot \left(1 - 0.5 \cdot \frac{h \cdot t_0}{\ell \cdot \frac{\frac{d}{D}}{0.5 \cdot M}}\right)\\ \mathbf{elif}\;h \leq -9 \cdot 10^{-210}:\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;t_1 \cdot \left(1 + -0.5 \cdot \frac{D \cdot \frac{h \cdot M}{\frac{d}{0.5}}}{\frac{\ell}{t_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 6
Error	22.1
Cost	15180

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ t_1 := \frac{\ell}{M \cdot \frac{0.5}{\frac{d}{D}}}\\ \mathbf{if}\;h \leq -7.6 \cdot 10^{-69}:\\ \;\;\;\;t_0 \cdot \left(1 + 0.5 \cdot \frac{h \cdot \left(M \cdot \frac{-0.5}{\frac{d}{D}}\right)}{t_1}\right)\\ \mathbf{elif}\;h \leq -9.6 \cdot 10^{-210}:\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;t_0 \cdot \left(1 + -0.5 \cdot \frac{D \cdot \frac{h \cdot M}{\frac{d}{0.5}}}{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 7
Error	21.9
Cost	15180

\[\begin{array}{l} t_0 := M \cdot \frac{0.5}{\frac{d}{D}}\\ t_1 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -8 \cdot 10^{-68}:\\ \;\;\;\;t_1 \cdot \left(1 + 0.5 \cdot \left(h \cdot \left(t_0 \cdot \left(t_0 \cdot \frac{-1}{\ell}\right)\right)\right)\right)\\ \mathbf{elif}\;h \leq -5.6 \cdot 10^{-210}:\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;t_1 \cdot \left(1 + -0.5 \cdot \frac{D \cdot \frac{h \cdot M}{\frac{d}{0.5}}}{\frac{\ell}{t_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 8
Error	23.6
Cost	14988

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125}{\ell} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \left(h \cdot \frac{M}{d}\right)\right)\right)\right)\\ \mathbf{if}\;h \leq -5.4 \cdot 10^{-92}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq -7.5 \cdot 10^{-210}:\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{elif}\;h \leq 9.5 \cdot 10^{-293}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 9
Error	22.4
Cost	14988

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -5.8 \cdot 10^{-58}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot t_0\right) \cdot \left(1 + 0.5 \cdot \left(h \cdot \left(\frac{M}{\frac{\ell}{\frac{0.5}{\frac{d}{D}}}} \cdot \left(M \cdot \frac{-0.5}{\frac{d}{D}}\right)\right)\right)\right)\\ \mathbf{elif}\;h \leq -4.4 \cdot 10^{-209}:\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot t_0\right) \cdot \left(1 - \frac{M \cdot \frac{D}{d}}{\left(\ell \cdot 8\right) \cdot \frac{\frac{d}{D}}{h \cdot M}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 10
Error	21.5
Cost	14856

\[\begin{array}{l} \mathbf{if}\;d \leq -3.2 \cdot 10^{+147}:\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{-305}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{M \cdot \frac{D}{d}}{\left(\ell \cdot 8\right) \cdot \frac{\frac{d}{D}}{h \cdot M}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 11
Error	22.2
Cost	14792

\[\begin{array}{l} \mathbf{if}\;d \leq -2.9 \cdot 10^{+140}:\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{-308}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125}{\ell} \cdot \left(\frac{h \cdot M}{\frac{d}{D}} \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 12
Error	25.6
Cost	14468

\[\begin{array}{l} t_0 := \left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{if}\;\ell \leq -1.06 \cdot 10^{+206}:\\ \;\;\;\;\left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{\ell \cdot \frac{h}{d}}}\\ \mathbf{elif}\;\ell \leq -8.2 \cdot 10^{-94}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -2.15 \cdot 10^{-271}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq -1.05 \cdot 10^{-303}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 13
Error	24.9
Cost	13644

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -5 \cdot 10^{+84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq -7.5 \cdot 10^{-210}:\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{elif}\;h \leq 9.5 \cdot 10^{-293}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 14
Error	24.6
Cost	13380

\[\begin{array}{l} \mathbf{if}\;h \leq 3.9 \cdot 10^{-288}:\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\frac{1}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 15
Error	24.5
Cost	13380

\[\begin{array}{l} \mathbf{if}\;h \leq 9.5 \cdot 10^{-293}:\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 16
Error	24.6
Cost	13316

\[\begin{array}{l} \mathbf{if}\;h \leq 2.2 \cdot 10^{-287}:\\ \;\;\;\;\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 17
Error	29.9
Cost	13252

\[\begin{array}{l} \mathbf{if}\;h \leq 2.7 \cdot 10^{-306}:\\ \;\;\;\;\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 18
Error	33.9
Cost	7377

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{if}\;h \leq -7.5 \cdot 10^{+84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-203}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq 8 \cdot 10^{-285} \lor \neg \left(h \leq 2.5 \cdot 10^{+193}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 19
Error	33.8
Cost	7245

\[\begin{array}{l} \mathbf{if}\;\ell \leq -6.9 \cdot 10^{-270}:\\ \;\;\;\;\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+202} \lor \neg \left(\ell \leq 2.25 \cdot 10^{+224}\right):\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \end{array} \]

Alternative 20
Error	35.1
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\ell \leq -6.9 \cdot 10^{-270}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 21
Error	44.4
Cost	6784

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

Alternative 22
Error	44.4
Cost	6720

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

Reproduce

herbie shell --seed 2023054 
(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)))))