Henrywood and Agarwal, Equation (12)

Average Error: 27.2 → 16.7

Time: 48.7s

Precision: binary64

Cost: 27920

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := 1 + \left(h \cdot \frac{{\left(\frac{D}{\frac{d + d}{M}}\right)}^{2}}{\ell}\right) \cdot -0.5\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := \sqrt{-d}\\ \mathbf{if}\;h \leq -2.35 \cdot 10^{-70}:\\ \;\;\;\;\frac{t_3}{\sqrt{-\ell}} \cdot \left(t_0 \cdot t_1\right)\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t_2 \cdot \left(\frac{t_3}{\sqrt{-h}} \cdot \left(1 + \frac{{\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{h}} \cdot -0.5\right)\right)\\ \mathbf{elif}\;h \leq 4.8 \cdot 10^{-60}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;h \leq 2.4 \cdot 10^{+168}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(t_0 \cdot \left(1 + \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(t_1 \cdot \frac{1}{\sqrt{h} \cdot \sqrt{\frac{1}{d}}}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (+ 1.0 (* (* h (/ (pow (/ D (/ (+ d d) M)) 2.0) l)) -0.5)))
        (t_2 (sqrt (/ d l)))
        (t_3 (sqrt (- d))))
   (if (<= h -2.35e-70)
     (* (/ t_3 (sqrt (- l))) (* t_0 t_1))
     (if (<= h -4e-310)
       (*
        t_2
        (*
         (/ t_3 (sqrt (- h)))
         (+ 1.0 (* (/ (pow (* (* 0.5 M) (/ D d)) 2.0) (/ l h)) -0.5))))
       (if (<= h 4.8e-60)
         (* d (/ (sqrt (/ 1.0 l)) (sqrt h)))
         (if (<= h 2.4e+168)
           (*
            (/ (sqrt d) (sqrt l))
            (*
             t_0
             (+ 1.0 (* (* (pow (* (/ D d) (/ M 2.0)) 2.0) (/ h l)) -0.5))))
           (* t_2 (* t_1 (/ 1.0 (* (sqrt h) (sqrt (/ 1.0 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 = sqrt((d / h));
	double t_1 = 1.0 + ((h * (pow((D / ((d + d) / M)), 2.0) / l)) * -0.5);
	double t_2 = sqrt((d / l));
	double t_3 = sqrt(-d);
	double tmp;
	if (h <= -2.35e-70) {
		tmp = (t_3 / sqrt(-l)) * (t_0 * t_1);
	} else if (h <= -4e-310) {
		tmp = t_2 * ((t_3 / sqrt(-h)) * (1.0 + ((pow(((0.5 * M) * (D / d)), 2.0) / (l / h)) * -0.5)));
	} else if (h <= 4.8e-60) {
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	} else if (h <= 2.4e+168) {
		tmp = (sqrt(d) / sqrt(l)) * (t_0 * (1.0 + ((pow(((D / d) * (M / 2.0)), 2.0) * (h / l)) * -0.5)));
	} else {
		tmp = t_2 * (t_1 * (1.0 / (sqrt(h) * sqrt((1.0 / 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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sqrt((d / h))
    t_1 = 1.0d0 + ((h * (((d_1 / ((d + d) / m)) ** 2.0d0) / l)) * (-0.5d0))
    t_2 = sqrt((d / l))
    t_3 = sqrt(-d)
    if (h <= (-2.35d-70)) then
        tmp = (t_3 / sqrt(-l)) * (t_0 * t_1)
    else if (h <= (-4d-310)) then
        tmp = t_2 * ((t_3 / sqrt(-h)) * (1.0d0 + (((((0.5d0 * m) * (d_1 / d)) ** 2.0d0) / (l / h)) * (-0.5d0))))
    else if (h <= 4.8d-60) then
        tmp = d * (sqrt((1.0d0 / l)) / sqrt(h))
    else if (h <= 2.4d+168) then
        tmp = (sqrt(d) / sqrt(l)) * (t_0 * (1.0d0 + (((((d_1 / d) * (m / 2.0d0)) ** 2.0d0) * (h / l)) * (-0.5d0))))
    else
        tmp = t_2 * (t_1 * (1.0d0 / (sqrt(h) * sqrt((1.0d0 / 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.sqrt((d / h));
	double t_1 = 1.0 + ((h * (Math.pow((D / ((d + d) / M)), 2.0) / l)) * -0.5);
	double t_2 = Math.sqrt((d / l));
	double t_3 = Math.sqrt(-d);
	double tmp;
	if (h <= -2.35e-70) {
		tmp = (t_3 / Math.sqrt(-l)) * (t_0 * t_1);
	} else if (h <= -4e-310) {
		tmp = t_2 * ((t_3 / Math.sqrt(-h)) * (1.0 + ((Math.pow(((0.5 * M) * (D / d)), 2.0) / (l / h)) * -0.5)));
	} else if (h <= 4.8e-60) {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	} else if (h <= 2.4e+168) {
		tmp = (Math.sqrt(d) / Math.sqrt(l)) * (t_0 * (1.0 + ((Math.pow(((D / d) * (M / 2.0)), 2.0) * (h / l)) * -0.5)));
	} else {
		tmp = t_2 * (t_1 * (1.0 / (Math.sqrt(h) * Math.sqrt((1.0 / 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.sqrt((d / h))
	t_1 = 1.0 + ((h * (math.pow((D / ((d + d) / M)), 2.0) / l)) * -0.5)
	t_2 = math.sqrt((d / l))
	t_3 = math.sqrt(-d)
	tmp = 0
	if h <= -2.35e-70:
		tmp = (t_3 / math.sqrt(-l)) * (t_0 * t_1)
	elif h <= -4e-310:
		tmp = t_2 * ((t_3 / math.sqrt(-h)) * (1.0 + ((math.pow(((0.5 * M) * (D / d)), 2.0) / (l / h)) * -0.5)))
	elif h <= 4.8e-60:
		tmp = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	elif h <= 2.4e+168:
		tmp = (math.sqrt(d) / math.sqrt(l)) * (t_0 * (1.0 + ((math.pow(((D / d) * (M / 2.0)), 2.0) * (h / l)) * -0.5)))
	else:
		tmp = t_2 * (t_1 * (1.0 / (math.sqrt(h) * math.sqrt((1.0 / 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 = sqrt(Float64(d / h))
	t_1 = Float64(1.0 + Float64(Float64(h * Float64((Float64(D / Float64(Float64(d + d) / M)) ^ 2.0) / l)) * -0.5))
	t_2 = sqrt(Float64(d / l))
	t_3 = sqrt(Float64(-d))
	tmp = 0.0
	if (h <= -2.35e-70)
		tmp = Float64(Float64(t_3 / sqrt(Float64(-l))) * Float64(t_0 * t_1));
	elseif (h <= -4e-310)
		tmp = Float64(t_2 * Float64(Float64(t_3 / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64((Float64(Float64(0.5 * M) * Float64(D / d)) ^ 2.0) / Float64(l / h)) * -0.5))));
	elseif (h <= 4.8e-60)
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	elseif (h <= 2.4e+168)
		tmp = Float64(Float64(sqrt(d) / sqrt(l)) * Float64(t_0 * Float64(1.0 + Float64(Float64((Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0) * Float64(h / l)) * -0.5))));
	else
		tmp = Float64(t_2 * Float64(t_1 * Float64(1.0 / Float64(sqrt(h) * sqrt(Float64(1.0 / 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 = sqrt((d / h));
	t_1 = 1.0 + ((h * (((D / ((d + d) / M)) ^ 2.0) / l)) * -0.5);
	t_2 = sqrt((d / l));
	t_3 = sqrt(-d);
	tmp = 0.0;
	if (h <= -2.35e-70)
		tmp = (t_3 / sqrt(-l)) * (t_0 * t_1);
	elseif (h <= -4e-310)
		tmp = t_2 * ((t_3 / sqrt(-h)) * (1.0 + (((((0.5 * M) * (D / d)) ^ 2.0) / (l / h)) * -0.5)));
	elseif (h <= 4.8e-60)
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	elseif (h <= 2.4e+168)
		tmp = (sqrt(d) / sqrt(l)) * (t_0 * (1.0 + (((((D / d) * (M / 2.0)) ^ 2.0) * (h / l)) * -0.5)));
	else
		tmp = t_2 * (t_1 * (1.0 / (sqrt(h) * sqrt((1.0 / 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[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[(h * N[(N[Power[N[(D / N[(N[(d + d), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[h, -2.35e-70], N[(N[(t$95$3 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -4e-310], N[(t$95$2 * N[(N[(t$95$3 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[Power[N[(N[(0.5 * M), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 4.8e-60], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 2.4e+168], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(1.0 + N[(N[(N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(t$95$1 * N[(1.0 / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[N[(1.0 / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if h < -2.3499999999999999e-70

   1. Initial program 24.2

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

   2. Simplified 24.2

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} \]
      Proof
      (*.f64 (sqrt.f64 (/.f64 d l)) (*.f64 (sqrt.f64 (/.f64 d h)) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d l) 1/2)) (*.f64 (sqrt.f64 (/.f64 d h)) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d l) (Rewrite<= metadata-eval (/.f64 1 2))) (*.f64 (sqrt.f64 (/.f64 d h)) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (*.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d h) 1/2)) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d h) (Rewrite<= metadata-eval (/.f64 1 2))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (-.f64 1 (*.f64 (Rewrite<= metadata-eval (/.f64 1 2)) (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (-.f64 1 (*.f64 (/.f64 1 2) (*.f64 (pow.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 M D) (*.f64 2 d))) 2) (/.f64 h l)))))): 6 points increase in error, 7 points decrease in error
      (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (-.f64 1 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (pow.f64 (/.f64 d h) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))): 5 points increase in error, 7 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.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 points increase in error, 0 points decrease in error

   3. Applied egg-rr 24.7

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

   4. Simplified 21.7

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right)}^{2}}{\ell}\right)}\right)\right) \]
      Proof
      (*.f64 h (/.f64 (pow.f64 (*.f64 D (/.f64 M (/.f64 d 1/2))) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (*.f64 D (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 M 1/2) d))) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (*.f64 M 1/2) d) D)) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (*.f64 M 1/2) D) d)) 2) l)): 14 points increase in error, 17 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (*.f64 M 1/2) (/.f64 D d))) 2) l)): 23 points increase in error, 14 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (pow.f64 (*.f64 (*.f64 M 1/2) (/.f64 D d)) 2) l) h)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (pow.f64 (*.f64 (*.f64 M 1/2) (/.f64 D d)) 2) h) l)): 29 points increase in error, 14 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 (pow.f64 (*.f64 (*.f64 M 1/2) (/.f64 D d)) 2) (/.f64 h l))): 36 points increase in error, 22 points decrease in error
      (Rewrite<= expm1-log1p_binary64 (expm1.f64 (log1p.f64 (*.f64 (pow.f64 (*.f64 (*.f64 M 1/2) (/.f64 D d)) 2) (/.f64 h l))))): 17 points increase in error, 13 points decrease in error
      (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (log1p.f64 (*.f64 (pow.f64 (*.f64 (*.f64 M 1/2) (/.f64 D d)) 2) (/.f64 h l)))) 1)): 35 points increase in error, 9 points decrease in error

   5. Applied egg-rr 21.6

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

   6. Applied egg-rr 15.3

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

   if -2.3499999999999999e-70 < h < -3.999999999999988e-310

   1. Initial program 31.1

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

   2. Simplified 31.7

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} \]
      Proof
      (*.f64 (sqrt.f64 (/.f64 d l)) (*.f64 (sqrt.f64 (/.f64 d h)) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d l) 1/2)) (*.f64 (sqrt.f64 (/.f64 d h)) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d l) (Rewrite<= metadata-eval (/.f64 1 2))) (*.f64 (sqrt.f64 (/.f64 d h)) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (*.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d h) 1/2)) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d h) (Rewrite<= metadata-eval (/.f64 1 2))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (-.f64 1 (*.f64 (Rewrite<= metadata-eval (/.f64 1 2)) (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (-.f64 1 (*.f64 (/.f64 1 2) (*.f64 (pow.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 M D) (*.f64 2 d))) 2) (/.f64 h l)))))): 6 points increase in error, 7 points decrease in error
      (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (-.f64 1 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (pow.f64 (/.f64 d h) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))): 5 points increase in error, 7 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.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 points increase in error, 0 points decrease in error

   3. Applied egg-rr 18.0

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

   4. Applied egg-rr 18.0

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

   if -3.999999999999988e-310 < h < 4.80000000000000019e-60

   1. Initial program 30.5

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

   2. Taylor expanded in d around inf 22.7

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

   3. Applied egg-rr 15.4

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

   if 4.80000000000000019e-60 < h < 2.40000000000000009e168

   1. Initial program 21.5

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

   2. Simplified 21.5

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} \]
      Proof
      (*.f64 (sqrt.f64 (/.f64 d l)) (*.f64 (sqrt.f64 (/.f64 d h)) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d l) 1/2)) (*.f64 (sqrt.f64 (/.f64 d h)) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d l) (Rewrite<= metadata-eval (/.f64 1 2))) (*.f64 (sqrt.f64 (/.f64 d h)) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (*.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d h) 1/2)) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d h) (Rewrite<= metadata-eval (/.f64 1 2))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (-.f64 1 (*.f64 (Rewrite<= metadata-eval (/.f64 1 2)) (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (-.f64 1 (*.f64 (/.f64 1 2) (*.f64 (pow.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 M D) (*.f64 2 d))) 2) (/.f64 h l)))))): 6 points increase in error, 7 points decrease in error
      (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (-.f64 1 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (pow.f64 (/.f64 d h) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))): 5 points increase in error, 7 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.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 points increase in error, 0 points decrease in error

   3. Applied egg-rr 16.1

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

   4. Simplified 16.1

      \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      Proof
      (/.f64 (sqrt.f64 d) (sqrt.f64 l)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (sqrt.f64 d) 1)) (sqrt.f64 l)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 (sqrt.f64 d) (/.f64 1 (sqrt.f64 l)))): 20 points increase in error, 19 points decrease in error

   if 2.40000000000000009e168 < h

   1. Initial program 32.7

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

   2. Simplified 32.8

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} \]
      Proof
      (*.f64 (sqrt.f64 (/.f64 d l)) (*.f64 (sqrt.f64 (/.f64 d h)) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d l) 1/2)) (*.f64 (sqrt.f64 (/.f64 d h)) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d l) (Rewrite<= metadata-eval (/.f64 1 2))) (*.f64 (sqrt.f64 (/.f64 d h)) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (*.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d h) 1/2)) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d h) (Rewrite<= metadata-eval (/.f64 1 2))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (-.f64 1 (*.f64 (Rewrite<= metadata-eval (/.f64 1 2)) (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (-.f64 1 (*.f64 (/.f64 1 2) (*.f64 (pow.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 M D) (*.f64 2 d))) 2) (/.f64 h l)))))): 6 points increase in error, 7 points decrease in error
      (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (-.f64 1 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (pow.f64 (/.f64 d h) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))): 5 points increase in error, 7 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.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 points increase in error, 0 points decrease in error

   3. Applied egg-rr 33.2

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

   4. Simplified 27.8

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right)}^{2}}{\ell}\right)}\right)\right) \]
      Proof
      (*.f64 h (/.f64 (pow.f64 (*.f64 D (/.f64 M (/.f64 d 1/2))) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (*.f64 D (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 M 1/2) d))) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (*.f64 M 1/2) d) D)) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (*.f64 M 1/2) D) d)) 2) l)): 14 points increase in error, 17 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (*.f64 M 1/2) (/.f64 D d))) 2) l)): 23 points increase in error, 14 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (pow.f64 (*.f64 (*.f64 M 1/2) (/.f64 D d)) 2) l) h)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (pow.f64 (*.f64 (*.f64 M 1/2) (/.f64 D d)) 2) h) l)): 29 points increase in error, 14 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 (pow.f64 (*.f64 (*.f64 M 1/2) (/.f64 D d)) 2) (/.f64 h l))): 36 points increase in error, 22 points decrease in error
      (Rewrite<= expm1-log1p_binary64 (expm1.f64 (log1p.f64 (*.f64 (pow.f64 (*.f64 (*.f64 M 1/2) (/.f64 D d)) 2) (/.f64 h l))))): 17 points increase in error, 13 points decrease in error
      (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (log1p.f64 (*.f64 (pow.f64 (*.f64 (*.f64 M 1/2) (/.f64 D d)) 2) (/.f64 h l)))) 1)): 35 points increase in error, 9 points decrease in error

   5. Applied egg-rr 27.8

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

   6. Applied egg-rr 28.4

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

   7. Applied egg-rr 22.1

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\frac{1}{\color{blue}{\sqrt{h} \cdot \sqrt{\frac{1}{d}}}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{D}{\frac{d + d}{M}}\right)}^{2}}{\ell}\right)\right)\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 16.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.35 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot \frac{{\left(\frac{D}{\frac{d + d}{M}}\right)}^{2}}{\ell}\right) \cdot -0.5\right)\right)\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{{\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{h}} \cdot -0.5\right)\right)\\ \mathbf{elif}\;h \leq 4.8 \cdot 10^{-60}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;h \leq 2.4 \cdot 10^{+168}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \left(h \cdot \frac{{\left(\frac{D}{\frac{d + d}{M}}\right)}^{2}}{\ell}\right) \cdot -0.5\right) \cdot \frac{1}{\sqrt{h} \cdot \sqrt{\frac{1}{d}}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	20.3
Cost	28060

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;d \leq -2.2 \cdot 10^{+225}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -1.5 \cdot 10^{+143}:\\ \;\;\;\;t_1 \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{{\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{h}} \cdot -0.5\right)\right)\\ \mathbf{elif}\;d \leq -5.3 \cdot 10^{+81}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -3.3 \cdot 10^{-20}:\\ \;\;\;\;t_1 \cdot \left(\left(1 + \left(h \cdot \frac{{\left(\frac{D}{\frac{d + d}{M}}\right)}^{2}}{\ell}\right) \cdot -0.5\right) \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right)\\ \mathbf{elif}\;d \leq -5.5 \cdot 10^{-89}:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(1 + 0.5 \cdot \left(\frac{0.25}{\ell} \cdot \frac{D \cdot \left(M \cdot \left(h \cdot M\right)\right)}{d \cdot \frac{-d}{D}}\right)\right)\\ \mathbf{elif}\;d \leq 1.86 \cdot 10^{-298}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq 0.0035:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(t_0 \cdot \left(1 + \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 2
Error	22.1
Cost	27928

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := t_1 \cdot \left(t_0 \cdot \left(1 + \left(h \cdot \frac{{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right)}^{2}}{\ell}\right) \cdot -0.5\right)\right)\\ t_3 := \left(t_0 \cdot t_1\right) \cdot \left(1 + -0.5 \cdot {\left(\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{if}\;h \leq -1.5 \cdot 10^{+193}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;h \leq -1.15 \cdot 10^{-233}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;h \leq -1.4 \cdot 10^{-263}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq 1.12 \cdot 10^{-66}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;h \leq 9.6 \cdot 10^{+84}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;h \leq 3.8 \cdot 10^{+152}:\\ \;\;\;\;\frac{1}{\sqrt{\ell}} \cdot \left(d \cdot {h}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	21.6
Cost	27928

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := t_0 \cdot \left(t_1 \cdot \left(1 + \left(h \cdot \frac{{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right)}^{2}}{\ell}\right) \cdot -0.5\right)\right)\\ \mathbf{if}\;h \leq -1.2 \cdot 10^{+194}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;h \leq -5.9 \cdot 10^{-235}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;h \leq -1.35 \cdot 10^{-261}:\\ \;\;\;\;\left(t_1 \cdot t_0\right) \cdot \left(1 + -0.5 \cdot {\left(\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq 2.3 \cdot 10^{-58}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;h \leq 1.75 \cdot 10^{+163}:\\ \;\;\;\;\left(1 + \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5\right) \cdot \left(t_1 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	21.5
Cost	27928

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := t_0 \cdot \left(t_1 \cdot \left(1 + \left(h \cdot \frac{{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right)}^{2}}{\ell}\right) \cdot -0.5\right)\right)\\ \mathbf{if}\;h \leq -1.5 \cdot 10^{+193}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;h \leq -9 \cdot 10^{-238}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;h \leq -1.15 \cdot 10^{-261}:\\ \;\;\;\;\left(t_1 \cdot t_0\right) \cdot \left(1 + -0.5 \cdot {\left(\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq 9.5 \cdot 10^{-59}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;h \leq 7.4 \cdot 10^{+182}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(t_1 \cdot \left(1 + \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	20.0
Cost	27928

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -5.8 \cdot 10^{+89}:\\ \;\;\;\;t_0 \cdot \left(\frac{t_2}{\sqrt{-h}} \cdot \left(1 + \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5\right)\right)\\ \mathbf{elif}\;\ell \leq -2.1 \cdot 10^{-133}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq -3.1 \cdot 10^{-204}:\\ \;\;\;\;\left(t_1 \cdot t_0\right) \cdot \left(1 + 0.5 \cdot \left(\frac{h \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}}{d \cdot \ell} \cdot -0.25\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-308}:\\ \;\;\;\;\left(\frac{t_2}{\sqrt{-\ell}} \cdot t_1\right) \cdot \left(1 + \left(\frac{0.25}{\ell} \cdot \left(\frac{D \cdot D}{d} \cdot \frac{M \cdot \left(h \cdot M\right)}{d}\right)\right) \cdot -0.5\right)\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{-123}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{1}{h}}\right)\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+187}:\\ \;\;\;\;\left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d \cdot \frac{2}{M}}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\frac{\sqrt{\ell}}{{h}^{-0.5}}}\\ \end{array} \]

Alternative 6
Error	17.4
Cost	27664

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := t_0 \cdot \left(1 + \left(h \cdot \frac{{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right)}^{2}}{\ell}\right) \cdot -0.5\right)\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := \sqrt{-d}\\ \mathbf{if}\;h \leq -9 \cdot 10^{-70}:\\ \;\;\;\;\frac{t_3}{\sqrt{-\ell}} \cdot t_1\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t_2 \cdot \left(\frac{t_3}{\sqrt{-h}} \cdot \left(1 + \frac{{\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{h}} \cdot -0.5\right)\right)\\ \mathbf{elif}\;h \leq 2.2 \cdot 10^{-58}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;h \leq 2.85 \cdot 10^{+183}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(t_0 \cdot \left(1 + \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot t_1\\ \end{array} \]

Alternative 7
Error	17.4
Cost	27664

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;h \leq -2.1 \cdot 10^{-71}:\\ \;\;\;\;\frac{t_2}{\sqrt{-\ell}} \cdot \left(t_0 \cdot \left(1 + \left(h \cdot \frac{{\left(\frac{D}{\frac{d + d}{M}}\right)}^{2}}{\ell}\right) \cdot -0.5\right)\right)\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t_1 \cdot \left(\frac{t_2}{\sqrt{-h}} \cdot \left(1 + \frac{{\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{h}} \cdot -0.5\right)\right)\\ \mathbf{elif}\;h \leq 5 \cdot 10^{-61}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;h \leq 5.6 \cdot 10^{+182}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(t_0 \cdot \left(1 + \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \left(1 + \left(h \cdot \frac{{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right)}^{2}}{\ell}\right) \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 8
Error	22.3
Cost	21660

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := t_0 \cdot \left(t_1 \cdot \left(1 + \left(h \cdot \frac{{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right)}^{2}}{\ell}\right) \cdot -0.5\right)\right)\\ \mathbf{if}\;h \leq -1.5 \cdot 10^{+193}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;h \leq -6.5 \cdot 10^{-243}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;h \leq -1.35 \cdot 10^{-261}:\\ \;\;\;\;\left(t_1 \cdot t_0\right) \cdot \left(1 + 0.5 \cdot \left(\frac{0.25}{\ell} \cdot \frac{D \cdot \left(M \cdot \left(h \cdot M\right)\right)}{d \cdot \frac{-d}{D}}\right)\right)\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq 1.85 \cdot 10^{-65}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;h \leq 4.1 \cdot 10^{+83}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \left(1 + {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;h \leq 8.2 \cdot 10^{+152}:\\ \;\;\;\;\frac{1}{\sqrt{\ell}} \cdot \left(d \cdot {h}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	22.3
Cost	21660

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := t_0 \cdot \left(t_1 \cdot \left(1 + \left(h \cdot \frac{{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right)}^{2}}{\ell}\right) \cdot -0.5\right)\right)\\ \mathbf{if}\;h \leq -2.3 \cdot 10^{+193}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;h \leq -9 \cdot 10^{-238}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-261}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 + \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.5\right)\right)\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq 6 \cdot 10^{-66}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;h \leq 6.2 \cdot 10^{+83}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \left(1 + {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;h \leq 4.3 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\sqrt{\ell}} \cdot \left(d \cdot {h}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Error	22.5
Cost	21264

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;d \leq -3.45 \cdot 10^{+46}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -1.8 \cdot 10^{-88}:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(1 + 0.5 \cdot \left(\frac{0.25}{\ell} \cdot \frac{D \cdot \left(M \cdot \left(h \cdot M\right)\right)}{d \cdot \frac{-d}{D}}\right)\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-293}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{-55}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 + {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 11
Error	21.9
Cost	15316

\[\begin{array}{l} t_0 := \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(\left(D \cdot M\right) \cdot \frac{D \cdot M}{d \cdot d}\right)\right) \cdot -0.25\right)\right)\\ t_1 := \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;d \leq -7.8 \cdot 10^{+22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -1.06 \cdot 10^{-88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-298}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-156}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 12
Error	21.9
Cost	15316

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ t_1 := \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;d \leq -7.8 \cdot 10^{+22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -2.3 \cdot 10^{-89}:\\ \;\;\;\;t_0 \cdot \left(1 + 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(\left(D \cdot M\right) \cdot \frac{D \cdot M}{d \cdot d}\right)\right) \cdot -0.25\right)\right)\\ \mathbf{elif}\;d \leq 1.86 \cdot 10^{-298}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{-155}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{-57}:\\ \;\;\;\;t_0 \cdot \left(1 + 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot d}\right) \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 13
Error	22.0
Cost	15316

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ t_1 := \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;d \leq -8 \cdot 10^{+44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -1.5 \cdot 10^{-86}:\\ \;\;\;\;t_0 \cdot \left(1 + 0.5 \cdot \left(\frac{0.25}{\ell} \cdot \frac{D \cdot \left(M \cdot \left(h \cdot M\right)\right)}{d \cdot \frac{-d}{D}}\right)\right)\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-298}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{-156}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-57}:\\ \;\;\;\;t_0 \cdot \left(1 + 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot d}\right) \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 14
Error	22.4
Cost	13580

\[\begin{array}{l} t_0 := \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;d \leq -8.8 \cdot 10^{+42}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -5.5 \cdot 10^{-88}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.86 \cdot 10^{-298}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\frac{\sqrt{\ell}}{{h}^{-0.5}}}\\ \end{array} \]

Alternative 15
Error	23.3
Cost	13316

\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{-293}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]

Alternative 16
Error	23.3
Cost	13316

\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{-293}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\frac{\sqrt{\ell}}{{h}^{-0.5}}}\\ \end{array} \]

Alternative 17
Error	23.3
Cost	13252

\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{-293}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 18
Error	27.0
Cost	7044

\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.9 \cdot 10^{-288}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]

Alternative 19
Error	34.3
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq 2.1 \cdot 10^{-298}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 20
Error	33.5
Cost	6980

\[\begin{array}{l} \mathbf{if}\;h \leq -1.4 \cdot 10^{-261}:\\ \;\;\;\;\sqrt{d \cdot \frac{\frac{d}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 21
Error	33.4
Cost	6980

\[\begin{array}{l} \mathbf{if}\;h \leq -1.4 \cdot 10^{-261}:\\ \;\;\;\;\sqrt{d \cdot \frac{\frac{d}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]

Alternative 22
Error	43.5
Cost	6720

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

Reproduce

herbie shell --seed 2022325 
(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)))))