Henrywood and Agarwal, Equation (12)

Average Error: 27.2 → 16.2

Time: 1.9min

Precision: binary64

Cost: 34064

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := 1 + \frac{h}{\frac{\ell}{\frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{4}}} \cdot -0.5\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := \sqrt{-d}\\ \mathbf{if}\;h \leq -2.35 \cdot 10^{-70}:\\ \;\;\;\;\frac{t_3}{\sqrt{-\ell}} \cdot \left(t_0 \cdot t_1\right)\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t_2 \cdot \left(\frac{t_3}{\sqrt{-h}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{elif}\;h \leq 2.4 \cdot 10^{-83}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq 2.4 \cdot 10^{+167}:\\ \;\;\;\;\left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 + -0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(t_1 \cdot \frac{1}{\sqrt{h} \cdot \sqrt{\frac{1}{d}}}\right)\\ \end{array} \]

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 (/ l (/ (pow (* D (/ M d)) 2.0) 4.0))) -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 (* 0.5 (* (pow (* (/ M 2.0) (/ D d)) 2.0) (/ h l))))))
       (if (<= h 2.4e-83)
         (*
          (/ d (* (sqrt l) (sqrt h)))
          (+ 1.0 (* -0.5 (* (/ h l) (pow (* D (/ M (* d 2.0))) 2.0)))))
         (if (<= h 2.4e+167)
           (*
            (* t_0 (/ (sqrt d) (sqrt l)))
            (+
             1.0
             (* -0.5 (pow (* (* (/ D d) (* 0.5 M)) (sqrt (/ h l))) 2.0))))
           (* 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 / (l / (pow((D * (M / d)), 2.0) / 4.0))) * -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 - (0.5 * (pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))));
	} else if (h <= 2.4e-83) {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + (-0.5 * ((h / l) * pow((D * (M / (d * 2.0))), 2.0))));
	} else if (h <= 2.4e+167) {
		tmp = (t_0 * (sqrt(d) / sqrt(l))) * (1.0 + (-0.5 * pow((((D / d) * (0.5 * M)) * sqrt((h / l))), 2.0)));
	} 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 / (l / (((d_1 * (m / d)) ** 2.0d0) / 4.0d0))) * (-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 / 2.0d0) * (d_1 / d)) ** 2.0d0) * (h / l)))))
    else if (h <= 2.4d-83) then
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 + ((-0.5d0) * ((h / l) * ((d_1 * (m / (d * 2.0d0))) ** 2.0d0))))
    else if (h <= 2.4d+167) then
        tmp = (t_0 * (sqrt(d) / sqrt(l))) * (1.0d0 + ((-0.5d0) * ((((d_1 / d) * (0.5d0 * m)) * sqrt((h / l))) ** 2.0d0)))
    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 / (l / (Math.pow((D * (M / d)), 2.0) / 4.0))) * -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 - (0.5 * (Math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))));
	} else if (h <= 2.4e-83) {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 + (-0.5 * ((h / l) * Math.pow((D * (M / (d * 2.0))), 2.0))));
	} else if (h <= 2.4e+167) {
		tmp = (t_0 * (Math.sqrt(d) / Math.sqrt(l))) * (1.0 + (-0.5 * Math.pow((((D / d) * (0.5 * M)) * Math.sqrt((h / l))), 2.0)));
	} 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 / (l / (math.pow((D * (M / d)), 2.0) / 4.0))) * -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 - (0.5 * (math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))))
	elif h <= 2.4e-83:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 + (-0.5 * ((h / l) * math.pow((D * (M / (d * 2.0))), 2.0))))
	elif h <= 2.4e+167:
		tmp = (t_0 * (math.sqrt(d) / math.sqrt(l))) * (1.0 + (-0.5 * math.pow((((D / d) * (0.5 * M)) * math.sqrt((h / l))), 2.0)))
	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(l / Float64((Float64(D * Float64(M / d)) ^ 2.0) / 4.0))) * -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(0.5 * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * Float64(h / l))))));
	elseif (h <= 2.4e-83)
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(D * Float64(M / Float64(d * 2.0))) ^ 2.0)))));
	elseif (h <= 2.4e+167)
		tmp = Float64(Float64(t_0 * Float64(sqrt(d) / sqrt(l))) * Float64(1.0 + Float64(-0.5 * (Float64(Float64(Float64(D / d) * Float64(0.5 * M)) * sqrt(Float64(h / l))) ^ 2.0))));
	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 / (l / (((D * (M / d)) ^ 2.0) / 4.0))) * -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 / 2.0) * (D / d)) ^ 2.0) * (h / l)))));
	elseif (h <= 2.4e-83)
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + (-0.5 * ((h / l) * ((D * (M / (d * 2.0))) ^ 2.0))));
	elseif (h <= 2.4e+167)
		tmp = (t_0 * (sqrt(d) / sqrt(l))) * (1.0 + (-0.5 * ((((D / d) * (0.5 * M)) * sqrt((h / l))) ^ 2.0)));
	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[(l / N[(N[Power[N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / 4.0), $MachinePrecision]), $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[(0.5 * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 2.4e-83], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 2.4e+167], N[(N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[Power[N[(N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $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

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}}{\ell}\right)}\right)\right) \]
      Proof
      (*.f64 h (/.f64 (pow.f64 (*.f64 D (/.f64 (/.f64 M d) 2)) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (*.f64 D (Rewrite<= associate-/r*_binary64 (/.f64 M (*.f64 d 2)))) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 M (*.f64 d 2)) D)) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 M D) (*.f64 d 2))) 2) l)): 14 points increase in error, 17 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (/.f64 (*.f64 M D) (Rewrite=> *-commutative_binary64 (*.f64 2 d))) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 (*.f64 M D) 2) d)) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (/.f64 (Rewrite=> associate-/l*_binary64 (/.f64 M (/.f64 2 D))) d) 2) l)): 6 points increase in error, 4 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (/.f64 (/.f64 M (/.f64 (Rewrite<= metadata-eval (/.f64 1 1/2)) D)) d) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (/.f64 (/.f64 M (Rewrite<= associate-/r*_binary64 (/.f64 1 (*.f64 1/2 D)))) d) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 M (*.f64 1/2 D)) 1)) d) 2) l)): 4 points increase in error, 6 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (/.f64 (Rewrite=> /-rgt-identity_binary64 (*.f64 M (*.f64 1/2 D))) d) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (/.f64 (Rewrite=> associate-*r*_binary64 (*.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-*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 21.7

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{h}{\frac{\ell}{\frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{4}}}}\right)\right) \]

   6. Applied egg-rr 15.4

      \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - 0.5 \cdot \frac{h}{\frac{\ell}{\frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{4}}}\right)\right) \]

   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

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

   if -3.999999999999988e-310 < h < 2.4000000000000001e-83

   1. Initial program 31.7

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

   2. Simplified 32.6

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
      Proof
      (*.f64 (*.f64 (sqrt.f64 (/.f64 d h)) (sqrt.f64 (/.f64 d l))) (-.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 (*.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d h) 1/2)) (sqrt.f64 (/.f64 d l))) (-.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 (*.f64 (pow.f64 (/.f64 d h) (Rewrite<= metadata-eval (/.f64 1 2))) (sqrt.f64 (/.f64 d l))) (-.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 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d l) 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 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (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 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.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 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.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))))): 4 points increase in error, 8 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.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

   3. Applied egg-rr 27.9

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

   4. Simplified 27.8

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

   5. Applied egg-rr 13.3

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

   6. Simplified 14.2

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left({\left(\frac{M}{d \cdot 2} \cdot D\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
      Proof
      (*.f64 (/.f64 d (*.f64 (sqrt.f64 l) (sqrt.f64 h))) (+.f64 1 (*.f64 -1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M (*.f64 d 2)) D) 2) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 d (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 h) (sqrt.f64 l)))) (+.f64 1 (*.f64 -1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M (*.f64 d 2)) D) 2) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 d (*.f64 (sqrt.f64 h) (sqrt.f64 l))) (+.f64 1 (*.f64 -1/2 (*.f64 (pow.f64 (Rewrite<= associate-/r/_binary64 (/.f64 M (/.f64 (*.f64 d 2) D))) 2) (/.f64 h l))))): 5 points increase in error, 4 points decrease in error
      (*.f64 (/.f64 d (*.f64 (sqrt.f64 h) (sqrt.f64 l))) (+.f64 1 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 -1/2 (pow.f64 (/.f64 M (/.f64 (*.f64 d 2) D)) 2)) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> distribute-rgt-in_binary64 (+.f64 (*.f64 1 (/.f64 d (*.f64 (sqrt.f64 h) (sqrt.f64 l)))) (*.f64 (*.f64 (*.f64 -1/2 (pow.f64 (/.f64 M (/.f64 (*.f64 d 2) D)) 2)) (/.f64 h l)) (/.f64 d (*.f64 (sqrt.f64 h) (sqrt.f64 l)))))): 0 points increase in error, 1 points decrease in error
      (+.f64 (Rewrite=> *-lft-identity_binary64 (/.f64 d (*.f64 (sqrt.f64 h) (sqrt.f64 l)))) (*.f64 (*.f64 (*.f64 -1/2 (pow.f64 (/.f64 M (/.f64 (*.f64 d 2) D)) 2)) (/.f64 h l)) (/.f64 d (*.f64 (sqrt.f64 h) (sqrt.f64 l))))): 0 points increase in error, 0 points decrease in error

   if 2.4000000000000001e-83 < h < 2.39999999999999999e167

   1. Initial program 21.3

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

   2. Simplified 21.3

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
      Proof
      (*.f64 (*.f64 (sqrt.f64 (/.f64 d h)) (sqrt.f64 (/.f64 d l))) (-.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 (*.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d h) 1/2)) (sqrt.f64 (/.f64 d l))) (-.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 (*.f64 (pow.f64 (/.f64 d h) (Rewrite<= metadata-eval (/.f64 1 2))) (sqrt.f64 (/.f64 d l))) (-.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 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d l) 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 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (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 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.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 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.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))))): 4 points increase in error, 8 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.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

   3. Applied egg-rr 16.1

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

   4. Simplified 16.1

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

   5. Applied egg-rr 14.4

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

   if 2.39999999999999999e167 < h

   1. Initial program 32.5

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

   2. Simplified 32.6

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} \]
      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 33.1

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

   4. Simplified 27.7

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}}{\ell}\right)}\right)\right) \]
      Proof
      (*.f64 h (/.f64 (pow.f64 (*.f64 D (/.f64 (/.f64 M d) 2)) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (*.f64 D (Rewrite<= associate-/r*_binary64 (/.f64 M (*.f64 d 2)))) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 M (*.f64 d 2)) D)) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 M D) (*.f64 d 2))) 2) l)): 14 points increase in error, 17 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (/.f64 (*.f64 M D) (Rewrite=> *-commutative_binary64 (*.f64 2 d))) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 (*.f64 M D) 2) d)) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (/.f64 (Rewrite=> associate-/l*_binary64 (/.f64 M (/.f64 2 D))) d) 2) l)): 6 points increase in error, 4 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (/.f64 (/.f64 M (/.f64 (Rewrite<= metadata-eval (/.f64 1 1/2)) D)) d) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (/.f64 (/.f64 M (Rewrite<= associate-/r*_binary64 (/.f64 1 (*.f64 1/2 D)))) d) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 M (*.f64 1/2 D)) 1)) d) 2) l)): 4 points increase in error, 6 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (/.f64 (Rewrite=> /-rgt-identity_binary64 (*.f64 M (*.f64 1/2 D))) d) 2) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 h (/.f64 (pow.f64 (/.f64 (Rewrite=> associate-*r*_binary64 (*.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-*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 27.7

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{h}{\frac{\ell}{\frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{4}}}}\right)\right) \]

   6. Applied egg-rr 28.2

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot \left(1 - 0.5 \cdot \frac{h}{\frac{\ell}{\frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{4}}}\right)\right) \]

   7. Applied egg-rr 22.1

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\frac{1}{\color{blue}{\sqrt{h} \cdot \sqrt{\frac{1}{d}}}} \cdot \left(1 - 0.5 \cdot \frac{h}{\frac{\ell}{\frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{4}}}\right)\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 16.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.35 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\frac{\ell}{\frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{4}}} \cdot -0.5\right)\right)\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{elif}\;h \leq 2.4 \cdot 10^{-83}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq 2.4 \cdot 10^{+167}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 + -0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{h}{\frac{\ell}{\frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{4}}} \cdot -0.5\right) \cdot \frac{1}{\sqrt{h} \cdot \sqrt{\frac{1}{d}}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	18.5
Cost	27528

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := D \cdot \frac{M}{d}\\ t_3 := \sqrt{-d}\\ t_4 := \frac{t_3}{\sqrt{-\ell}}\\ t_5 := t_4 \cdot t_0\\ \mathbf{if}\;d \leq -2.8 \cdot 10^{+205}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;d \leq -2 \cdot 10^{+144}:\\ \;\;\;\;t_1 \cdot \left(\frac{t_3}{\sqrt{-h}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{elif}\;d \leq -3.9 \cdot 10^{+84}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;d \leq -9.5 \cdot 10^{-20}:\\ \;\;\;\;t_1 \cdot \left(\left(1 + \frac{h}{\frac{\ell}{\frac{{t_2}^{2}}{4}}} \cdot -0.5\right) \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-311}:\\ \;\;\;\;t_4 \cdot \left(t_0 \cdot \left(1 + 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(0.5 \cdot t_2\right)\right) \cdot \left(t_2 \cdot -0.5\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 2
Error	16.8
Cost	27528

\[\begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;h \leq -1 \cdot 10^{-68}:\\ \;\;\;\;\frac{t_0}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{t_0}{\sqrt{-h}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 3
Error	18.4
Cost	27396

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(t_0 \cdot \left(1 + \frac{h}{\frac{\ell}{\frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{4}}} \cdot -0.5\right)\right)\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{-129}:\\ \;\;\;\;\left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \frac{D \cdot \frac{M \cdot M}{\frac{d}{h}}}{d \cdot \frac{\ell}{D}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 4
Error	20.2
Cost	21452

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ t_2 := D \cdot \frac{M}{d}\\ \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+119}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \left(1 + 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(0.5 \cdot t_2\right)\right) \cdot \left(t_2 \cdot -0.5\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_1 \cdot t_0\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-129}:\\ \;\;\;\;\left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \frac{D \cdot \frac{M \cdot M}{\frac{d}{h}}}{d \cdot \frac{\ell}{D}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 5
Error	20.2
Cost	21392

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{-d}\\ t_2 := \frac{t_1}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -1.22 \cdot 10^{+218}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -4.8 \cdot 10^{+144}:\\ \;\;\;\;t_0 \cdot \frac{t_1}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq -1.35 \cdot 10^{+102}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -1.2 \cdot 10^{-19}:\\ \;\;\;\;t_0 \cdot \left(\left(1 + \frac{h}{\frac{\ell}{\frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{4}}} \cdot -0.5\right) \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-311}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 6
Error	20.0
Cost	20872

\[\begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;h \leq -7 \cdot 10^{-69}:\\ \;\;\;\;\frac{t_0}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \frac{t_0}{\sqrt{-h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 7
Error	21.9
Cost	20308

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := D \cdot \frac{M}{d}\\ t_2 := \sqrt{-d}\\ t_3 := \sqrt{\frac{d}{\ell}}\\ t_4 := t_1 \cdot -0.5\\ \mathbf{if}\;h \leq -2.1 \cdot 10^{-67}:\\ \;\;\;\;\frac{t_2}{\sqrt{-\ell}} \cdot t_0\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t_3 \cdot \frac{t_2}{\sqrt{-h}}\\ \mathbf{elif}\;h \leq 6 \cdot 10^{+47}:\\ \;\;\;\;\frac{d}{{\left({\ell}^{0.25} \cdot {h}^{0.25}\right)}^{2}}\\ \mathbf{elif}\;h \leq 1.35 \cdot 10^{+85}:\\ \;\;\;\;t_3 \cdot \left(t_0 \cdot \left(1 + 0.5 \cdot \left(\frac{h \cdot \left(0.5 \cdot \frac{D \cdot M}{d}\right)}{\ell} \cdot t_4\right)\right)\right)\\ \mathbf{elif}\;h \leq 2.3 \cdot 10^{+153}:\\ \;\;\;\;\frac{t_0 \cdot \sqrt{d}}{\sqrt{\ell}}\\ \mathbf{elif}\;h \leq 2.5 \cdot 10^{+284}:\\ \;\;\;\;t_3 \cdot \left(t_0 \cdot \left(1 + 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(0.5 \cdot t_1\right)\right) \cdot t_4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 8
Error	21.8
Cost	20108

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}} \cdot \left(t_0 \cdot \left(1 + 0.5 \cdot \left(\frac{h \cdot \left(0.5 \cdot \frac{D \cdot M}{d}\right)}{\ell} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot -0.5\right)\right)\right)\right)\\ \mathbf{if}\;\ell \leq 1.6 \cdot 10^{-293}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-124}:\\ \;\;\;\;\frac{t_0 \cdot \sqrt{d}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \leq 9.6 \cdot 10^{+76}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{{\left({\ell}^{0.25} \cdot {h}^{0.25}\right)}^{2}}\\ \end{array} \]

Alternative 9
Error	21.8
Cost	20108

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}} \cdot \left(t_0 \cdot \left(1 + 0.5 \cdot \left(\frac{h \cdot \left(0.5 \cdot \frac{D \cdot M}{d}\right)}{\ell} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot -0.5\right)\right)\right)\right)\\ \mathbf{if}\;\ell \leq 1.95 \cdot 10^{-293}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.06 \cdot 10^{-124}:\\ \;\;\;\;t_0 \cdot \frac{1}{\frac{\sqrt{\ell}}{\sqrt{d}}}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{{\left({\ell}^{0.25} \cdot {h}^{0.25}\right)}^{2}}\\ \end{array} \]

Alternative 10
Error	22.4
Cost	20108

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot t_0\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{-123}:\\ \;\;\;\;t_0 \cdot \frac{1}{\frac{\sqrt{\ell}}{\sqrt{d}}}\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{+77}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(t_0 \cdot \left(1 + 0.5 \cdot \left(\frac{h \cdot \left(0.5 \cdot \frac{D \cdot M}{d}\right)}{\ell} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot -0.5\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{{\left({\ell}^{0.25} \cdot {h}^{0.25}\right)}^{2}}\\ \end{array} \]

Alternative 11
Error	22.1
Cost	19912

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}} \cdot \left(t_0 \cdot \left(1 + 0.5 \cdot \left(\frac{h \cdot \left(0.5 \cdot \frac{D \cdot M}{d}\right)}{\ell} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot -0.5\right)\right)\right)\right)\\ \mathbf{if}\;\ell \leq 1.2 \cdot 10^{-291}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{-124}:\\ \;\;\;\;\frac{t_0 \cdot \sqrt{d}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \leq 8.2 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 12
Error	23.0
Cost	15180

\[\begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ t_1 := \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(0.5 \cdot t_0\right)\right) \cdot \left(t_0 \cdot -0.5\right)\right)\right)\right)\\ \mathbf{if}\;\ell \leq -5.5 \cdot 10^{-298}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.85 \cdot 10^{-123}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 13
Error	21.9
Cost	15180

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + 0.5 \cdot \left(\frac{h \cdot \left(0.5 \cdot \frac{D \cdot M}{d}\right)}{\ell} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot -0.5\right)\right)\right)\right)\\ \mathbf{if}\;\ell \leq 1.2 \cdot 10^{-291}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{-124}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+78}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 14
Error	26.7
Cost	15052

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -5.5 \cdot 10^{-298}:\\ \;\;\;\;t_0 \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{-89}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{+49}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + 0.5 \cdot \left(\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\ell} \cdot \frac{h}{d \cdot d}\right) \cdot -0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 15
Error	26.4
Cost	13508

\[\begin{array}{l} \mathbf{if}\;d \leq 7.6 \cdot 10^{-212}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 16
Error	30.2
Cost	13384

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell \cdot \frac{h}{d}}}\\ \mathbf{if}\;d \leq 1.95 \cdot 10^{-298}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{+227}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{+307}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 17
Error	26.6
Cost	13380

\[\begin{array}{l} \mathbf{if}\;d \leq 1.45 \cdot 10^{-211}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 18
Error	26.6
Cost	13380

\[\begin{array}{l} \mathbf{if}\;d \leq 4 \cdot 10^{-212}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 19
Error	30.9
Cost	13252

\[\begin{array}{l} \mathbf{if}\;d \leq 5.5 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\frac{d}{\ell \cdot \frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 20
Error	34.3
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq 1.86 \cdot 10^{-298}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 21
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 22
Error	33.8
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq 7 \cdot 10^{-155}:\\ \;\;\;\;\sqrt{\frac{d}{\ell \cdot \frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 23
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)))))