Henrywood and Agarwal, Equation (12)

Average Accuracy: 58.1% → 76.6%

Time: 1.0min

Precision: binary64

Cost: 26700

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

Math

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

↓

\[\begin{array}{l} t_0 := {\left(\frac{d}{\ell}\right)}^{0.5}\\ t_1 := 1 + h \cdot \frac{-0.5}{\left(2 \cdot \frac{\frac{d}{M}}{D}\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}}\\ t_2 := \frac{1}{\sqrt{\frac{h}{d}}}\\ t_3 := \sqrt{-d}\\ t_4 := \frac{t_3}{\sqrt{-h}}\\ t_5 := \frac{t_3}{\sqrt{-\ell}}\\ \mathbf{if}\;h \leq -3.7 \cdot 10^{+167}:\\ \;\;\;\;\left(t_4 \cdot t_0\right) \cdot t_1\\ \mathbf{elif}\;h \leq -6.6 \cdot 10^{-98}:\\ \;\;\;\;t_1 \cdot \left(t_2 \cdot t_5\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;t_4 \cdot t_5\\ \mathbf{elif}\;h \leq 2.2 \cdot 10^{-123}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(t_2 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (/ d l) 0.5))
        (t_1
         (+
          1.0
          (*
           h
           (/ -0.5 (* (* 2.0 (/ (/ d M) D)) (/ l (* 0.5 (/ D (/ d M)))))))))
        (t_2 (/ 1.0 (sqrt (/ h d))))
        (t_3 (sqrt (- d)))
        (t_4 (/ t_3 (sqrt (- h))))
        (t_5 (/ t_3 (sqrt (- l)))))
   (if (<= h -3.7e+167)
     (* (* t_4 t_0) t_1)
     (if (<= h -6.6e-98)
       (* t_1 (* t_2 t_5))
       (if (<= h -5e-311)
         (* t_4 t_5)
         (if (<= h 2.2e-123)
           (* t_1 (* t_0 (/ (sqrt d) (sqrt h))))
           (* t_1 (* t_2 (/ (sqrt d) (sqrt l))))))))))

C

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

↓

double code(double d, double h, double l, double M, double D) {
	double t_0 = pow((d / l), 0.5);
	double t_1 = 1.0 + (h * (-0.5 / ((2.0 * ((d / M) / D)) * (l / (0.5 * (D / (d / M)))))));
	double t_2 = 1.0 / sqrt((h / d));
	double t_3 = sqrt(-d);
	double t_4 = t_3 / sqrt(-h);
	double t_5 = t_3 / sqrt(-l);
	double tmp;
	if (h <= -3.7e+167) {
		tmp = (t_4 * t_0) * t_1;
	} else if (h <= -6.6e-98) {
		tmp = t_1 * (t_2 * t_5);
	} else if (h <= -5e-311) {
		tmp = t_4 * t_5;
	} else if (h <= 2.2e-123) {
		tmp = t_1 * (t_0 * (sqrt(d) / sqrt(h)));
	} else {
		tmp = t_1 * (t_2 * (sqrt(d) / sqrt(l)));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
end function

↓

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = (d / l) ** 0.5d0
    t_1 = 1.0d0 + (h * ((-0.5d0) / ((2.0d0 * ((d / m) / d_1)) * (l / (0.5d0 * (d_1 / (d / m)))))))
    t_2 = 1.0d0 / sqrt((h / d))
    t_3 = sqrt(-d)
    t_4 = t_3 / sqrt(-h)
    t_5 = t_3 / sqrt(-l)
    if (h <= (-3.7d+167)) then
        tmp = (t_4 * t_0) * t_1
    else if (h <= (-6.6d-98)) then
        tmp = t_1 * (t_2 * t_5)
    else if (h <= (-5d-311)) then
        tmp = t_4 * t_5
    else if (h <= 2.2d-123) then
        tmp = t_1 * (t_0 * (sqrt(d) / sqrt(h)))
    else
        tmp = t_1 * (t_2 * (sqrt(d) / sqrt(l)))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.pow((d / l), 0.5);
	double t_1 = 1.0 + (h * (-0.5 / ((2.0 * ((d / M) / D)) * (l / (0.5 * (D / (d / M)))))));
	double t_2 = 1.0 / Math.sqrt((h / d));
	double t_3 = Math.sqrt(-d);
	double t_4 = t_3 / Math.sqrt(-h);
	double t_5 = t_3 / Math.sqrt(-l);
	double tmp;
	if (h <= -3.7e+167) {
		tmp = (t_4 * t_0) * t_1;
	} else if (h <= -6.6e-98) {
		tmp = t_1 * (t_2 * t_5);
	} else if (h <= -5e-311) {
		tmp = t_4 * t_5;
	} else if (h <= 2.2e-123) {
		tmp = t_1 * (t_0 * (Math.sqrt(d) / Math.sqrt(h)));
	} else {
		tmp = t_1 * (t_2 * (Math.sqrt(d) / Math.sqrt(l)));
	}
	return tmp;
}

Python

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

↓

def code(d, h, l, M, D):
	t_0 = math.pow((d / l), 0.5)
	t_1 = 1.0 + (h * (-0.5 / ((2.0 * ((d / M) / D)) * (l / (0.5 * (D / (d / M)))))))
	t_2 = 1.0 / math.sqrt((h / d))
	t_3 = math.sqrt(-d)
	t_4 = t_3 / math.sqrt(-h)
	t_5 = t_3 / math.sqrt(-l)
	tmp = 0
	if h <= -3.7e+167:
		tmp = (t_4 * t_0) * t_1
	elif h <= -6.6e-98:
		tmp = t_1 * (t_2 * t_5)
	elif h <= -5e-311:
		tmp = t_4 * t_5
	elif h <= 2.2e-123:
		tmp = t_1 * (t_0 * (math.sqrt(d) / math.sqrt(h)))
	else:
		tmp = t_1 * (t_2 * (math.sqrt(d) / math.sqrt(l)))
	return tmp

Julia

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

↓

function code(d, h, l, M, D)
	t_0 = Float64(d / l) ^ 0.5
	t_1 = Float64(1.0 + Float64(h * Float64(-0.5 / Float64(Float64(2.0 * Float64(Float64(d / M) / D)) * Float64(l / Float64(0.5 * Float64(D / Float64(d / M))))))))
	t_2 = Float64(1.0 / sqrt(Float64(h / d)))
	t_3 = sqrt(Float64(-d))
	t_4 = Float64(t_3 / sqrt(Float64(-h)))
	t_5 = Float64(t_3 / sqrt(Float64(-l)))
	tmp = 0.0
	if (h <= -3.7e+167)
		tmp = Float64(Float64(t_4 * t_0) * t_1);
	elseif (h <= -6.6e-98)
		tmp = Float64(t_1 * Float64(t_2 * t_5));
	elseif (h <= -5e-311)
		tmp = Float64(t_4 * t_5);
	elseif (h <= 2.2e-123)
		tmp = Float64(t_1 * Float64(t_0 * Float64(sqrt(d) / sqrt(h))));
	else
		tmp = Float64(t_1 * Float64(t_2 * Float64(sqrt(d) / sqrt(l))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(d, h, l, M, D)
	t_0 = (d / l) ^ 0.5;
	t_1 = 1.0 + (h * (-0.5 / ((2.0 * ((d / M) / D)) * (l / (0.5 * (D / (d / M)))))));
	t_2 = 1.0 / sqrt((h / d));
	t_3 = sqrt(-d);
	t_4 = t_3 / sqrt(-h);
	t_5 = t_3 / sqrt(-l);
	tmp = 0.0;
	if (h <= -3.7e+167)
		tmp = (t_4 * t_0) * t_1;
	elseif (h <= -6.6e-98)
		tmp = t_1 * (t_2 * t_5);
	elseif (h <= -5e-311)
		tmp = t_4 * t_5;
	elseif (h <= 2.2e-123)
		tmp = t_1 * (t_0 * (sqrt(d) / sqrt(h)));
	else
		tmp = t_1 * (t_2 * (sqrt(d) / sqrt(l)));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(h * N[(-0.5 / N[(N[(2.0 * N[(N[(d / M), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision] * N[(l / N[(0.5 * N[(D / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -3.7e+167], N[(N[(t$95$4 * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[h, -6.6e-98], N[(t$95$1 * N[(t$95$2 * t$95$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-311], N[(t$95$4 * t$95$5), $MachinePrecision], If[LessEqual[h, 2.2e-123], N[(t$95$1 * N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(t$95$2 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if h < -3.7000000000000001e167

   1. Initial program 50.6%

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

   2. Applied egg-rr 50.6%

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

      [Start] 50.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      expm1-log1p-u [=>] 49.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]
      expm1-udef [=>] 49.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} - 1\right)}\right) \]
      log1p-udef [=>] 49.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(e^{\color{blue}{\log \left(1 + \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}} - 1\right)\right) \]
      add-exp-log [<=] 50.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(1 + \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} - 1\right)\right) \]
      associate-*l* [=>] 50.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) - 1\right)\right) \]
      metadata-eval [=>] 50.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]
      *-un-lft-identity [=>] 50.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + 0.5 \cdot \left({\left(\frac{\color{blue}{1 \cdot \left(M \cdot D\right)}}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]
      times-frac [=>] 50.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + 0.5 \cdot \left({\color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]
      metadata-eval [=>] 50.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + 0.5 \cdot \left({\left(\color{blue}{0.5} \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]

   3. Simplified 59.0%

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

      [Start] 50.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + 0.5 \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]
      +-commutative [=>] 50.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(0.5 \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) + 1\right)} - 1\right)\right) \]
      associate--l+ [=>] 50.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) + \left(1 - 1\right)\right)}\right) \]
      metadata-eval [=>] 50.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(0.5 \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) + \color{blue}{0}\right)\right) \]
      associate-*r* [=>] 50.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(0.5 \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}\right) \cdot \frac{h}{\ell}} + 0\right)\right) \]
      associate-*r/ [=>] 55.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{\left(0.5 \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}\right) \cdot h}{\ell}} + 0\right)\right) \]
      associate-*l/ [<=] 59.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{0.5 \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}}{\ell} \cdot h} + 0\right)\right) \]
      *-commutative [=>] 59.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{h \cdot \frac{0.5 \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}}{\ell}} + 0\right)\right) \]
      associate-/l* [=>] 59.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \color{blue}{\frac{0.5}{\frac{\ell}{{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}}}} + 0\right)\right) \]
      associate-*r/ [=>] 59.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\frac{\ell}{{\color{blue}{\left(\frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)}}^{2}}} + 0\right)\right) \]
      associate-/l* [=>] 59.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\frac{\ell}{{\color{blue}{\left(\frac{0.5}{\frac{d}{M \cdot D}}\right)}}^{2}}} + 0\right)\right) \]
      *-commutative [=>] 59.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\frac{\ell}{{\left(\frac{0.5}{\frac{d}{\color{blue}{D \cdot M}}}\right)}^{2}}} + 0\right)\right) \]

   4. Applied egg-rr 59.9%

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

      [Start] 59.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\frac{\ell}{{\left(\frac{0.5}{\frac{d}{D \cdot M}}\right)}^{2}}} + 0\right)\right) \]
      *-un-lft-identity [=>] 59.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\frac{\color{blue}{1 \cdot \ell}}{{\left(\frac{0.5}{\frac{d}{D \cdot M}}\right)}^{2}}} + 0\right)\right) \]
      unpow2 [=>] 59.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\frac{1 \cdot \ell}{\color{blue}{\frac{0.5}{\frac{d}{D \cdot M}} \cdot \frac{0.5}{\frac{d}{D \cdot M}}}}} + 0\right)\right) \]
      times-frac [=>] 60.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\color{blue}{\frac{1}{\frac{0.5}{\frac{d}{D \cdot M}}} \cdot \frac{\ell}{\frac{0.5}{\frac{d}{D \cdot M}}}}} + 0\right)\right) \]
      clear-num [<=] 60.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\color{blue}{\frac{\frac{d}{D \cdot M}}{0.5}} \cdot \frac{\ell}{\frac{0.5}{\frac{d}{D \cdot M}}}} + 0\right)\right) \]
      div-inv [=>] 60.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\color{blue}{\left(\frac{d}{D \cdot M} \cdot \frac{1}{0.5}\right)} \cdot \frac{\ell}{\frac{0.5}{\frac{d}{D \cdot M}}}} + 0\right)\right) \]
      *-commutative [=>] 60.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{d}{\color{blue}{M \cdot D}} \cdot \frac{1}{0.5}\right) \cdot \frac{\ell}{\frac{0.5}{\frac{d}{D \cdot M}}}} + 0\right)\right) \]
      associate-/r* [=>] 59.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\color{blue}{\frac{\frac{d}{M}}{D}} \cdot \frac{1}{0.5}\right) \cdot \frac{\ell}{\frac{0.5}{\frac{d}{D \cdot M}}}} + 0\right)\right) \]
      metadata-eval [=>] 59.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot \color{blue}{2}\right) \cdot \frac{\ell}{\frac{0.5}{\frac{d}{D \cdot M}}}} + 0\right)\right) \]
      div-inv [=>] 59.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{\color{blue}{0.5 \cdot \frac{1}{\frac{d}{D \cdot M}}}}} + 0\right)\right) \]
      clear-num [<=] 59.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \color{blue}{\frac{D \cdot M}{d}}}} + 0\right)\right) \]
      associate-/l* [=>] 59.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \color{blue}{\frac{D}{\frac{d}{M}}}}} + 0\right)\right) \]

   5. Applied egg-rr 67.7%

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

      [Start] 59.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}} + 0\right)\right) \]
      metadata-eval [=>] 59.9	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}} + 0\right)\right) \]
      unpow1/2 [=>] 59.9	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}} + 0\right)\right) \]
      frac-2neg [=>] 59.9	\[ \left(\sqrt{\color{blue}{\frac{-d}{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}} + 0\right)\right) \]
      sqrt-div [=>] 67.7	\[ \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}} + 0\right)\right) \]

   if -3.7000000000000001e167 < h < -6.6000000000000002e-98

   1. Initial program 68.4%

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

   2. Applied egg-rr 68.4%

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

      [Start] 68.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      expm1-log1p-u [=>] 67.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]
      expm1-udef [=>] 67.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} - 1\right)}\right) \]
      log1p-udef [=>] 67.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(e^{\color{blue}{\log \left(1 + \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}} - 1\right)\right) \]
      add-exp-log [<=] 68.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(1 + \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} - 1\right)\right) \]
      associate-*l* [=>] 68.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) - 1\right)\right) \]
      metadata-eval [=>] 68.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]
      *-un-lft-identity [=>] 68.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + 0.5 \cdot \left({\left(\frac{\color{blue}{1 \cdot \left(M \cdot D\right)}}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]
      times-frac [=>] 68.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + 0.5 \cdot \left({\color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]
      metadata-eval [=>] 68.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + 0.5 \cdot \left({\left(\color{blue}{0.5} \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]

   3. Simplified 70.3%

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

      [Start] 68.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + 0.5 \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]
      +-commutative [=>] 68.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(0.5 \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) + 1\right)} - 1\right)\right) \]
      associate--l+ [=>] 68.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) + \left(1 - 1\right)\right)}\right) \]
      metadata-eval [=>] 68.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(0.5 \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) + \color{blue}{0}\right)\right) \]
      associate-*r* [=>] 68.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(0.5 \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}\right) \cdot \frac{h}{\ell}} + 0\right)\right) \]
      associate-*r/ [=>] 69.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{\left(0.5 \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}\right) \cdot h}{\ell}} + 0\right)\right) \]
      associate-*l/ [<=] 70.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{0.5 \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}}{\ell} \cdot h} + 0\right)\right) \]
      *-commutative [=>] 70.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{h \cdot \frac{0.5 \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}}{\ell}} + 0\right)\right) \]
      associate-/l* [=>] 70.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \color{blue}{\frac{0.5}{\frac{\ell}{{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}}}} + 0\right)\right) \]
      associate-*r/ [=>] 70.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\frac{\ell}{{\color{blue}{\left(\frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)}}^{2}}} + 0\right)\right) \]
      associate-/l* [=>] 70.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\frac{\ell}{{\color{blue}{\left(\frac{0.5}{\frac{d}{M \cdot D}}\right)}}^{2}}} + 0\right)\right) \]
      *-commutative [=>] 70.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\frac{\ell}{{\left(\frac{0.5}{\frac{d}{\color{blue}{D \cdot M}}}\right)}^{2}}} + 0\right)\right) \]

   4. Applied egg-rr 71.8%

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

      [Start] 70.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\frac{\ell}{{\left(\frac{0.5}{\frac{d}{D \cdot M}}\right)}^{2}}} + 0\right)\right) \]
      *-un-lft-identity [=>] 70.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\frac{\color{blue}{1 \cdot \ell}}{{\left(\frac{0.5}{\frac{d}{D \cdot M}}\right)}^{2}}} + 0\right)\right) \]
      unpow2 [=>] 70.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\frac{1 \cdot \ell}{\color{blue}{\frac{0.5}{\frac{d}{D \cdot M}} \cdot \frac{0.5}{\frac{d}{D \cdot M}}}}} + 0\right)\right) \]
      times-frac [=>] 72.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\color{blue}{\frac{1}{\frac{0.5}{\frac{d}{D \cdot M}}} \cdot \frac{\ell}{\frac{0.5}{\frac{d}{D \cdot M}}}}} + 0\right)\right) \]
      clear-num [<=] 72.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\color{blue}{\frac{\frac{d}{D \cdot M}}{0.5}} \cdot \frac{\ell}{\frac{0.5}{\frac{d}{D \cdot M}}}} + 0\right)\right) \]
      div-inv [=>] 72.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\color{blue}{\left(\frac{d}{D \cdot M} \cdot \frac{1}{0.5}\right)} \cdot \frac{\ell}{\frac{0.5}{\frac{d}{D \cdot M}}}} + 0\right)\right) \]
      *-commutative [=>] 72.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{d}{\color{blue}{M \cdot D}} \cdot \frac{1}{0.5}\right) \cdot \frac{\ell}{\frac{0.5}{\frac{d}{D \cdot M}}}} + 0\right)\right) \]
      associate-/r* [=>] 70.7	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\color{blue}{\frac{\frac{d}{M}}{D}} \cdot \frac{1}{0.5}\right) \cdot \frac{\ell}{\frac{0.5}{\frac{d}{D \cdot M}}}} + 0\right)\right) \]
      metadata-eval [=>] 70.7	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot \color{blue}{2}\right) \cdot \frac{\ell}{\frac{0.5}{\frac{d}{D \cdot M}}}} + 0\right)\right) \]
      div-inv [=>] 70.7	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{\color{blue}{0.5 \cdot \frac{1}{\frac{d}{D \cdot M}}}}} + 0\right)\right) \]
      clear-num [<=] 70.7	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \color{blue}{\frac{D \cdot M}{d}}}} + 0\right)\right) \]
      associate-/l* [=>] 71.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \color{blue}{\frac{D}{\frac{d}{M}}}}} + 0\right)\right) \]

   5. Applied egg-rr 72.0%

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

      [Start] 71.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}} + 0\right)\right) \]
      metadata-eval [=>] 71.8	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}} + 0\right)\right) \]
      unpow1/2 [=>] 71.8	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}} + 0\right)\right) \]
      clear-num [=>] 71.6	\[ \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}} + 0\right)\right) \]
      sqrt-div [=>] 72.0	\[ \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}} + 0\right)\right) \]
      metadata-eval [=>] 72.0	\[ \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}} + 0\right)\right) \]

   6. Applied egg-rr 81.4%

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

      [Start] 72.0	\[ \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}} + 0\right)\right) \]
      metadata-eval [=>] 72.0	\[ \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}} + 0\right)\right) \]
      unpow1/2 [=>] 72.0	\[ \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}} + 0\right)\right) \]
      frac-2neg [=>] 72.0	\[ \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \sqrt{\color{blue}{\frac{-d}{-\ell}}}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}} + 0\right)\right) \]
      sqrt-div [=>] 81.4	\[ \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}} + 0\right)\right) \]

   if -6.6000000000000002e-98 < h < -5.00000000000023e-311

   1. Initial program 48.0%

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

   2. Simplified 46.4%

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

      [Start] 48.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 48.0	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 48.0	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 48.0	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 48.0	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 48.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      cancel-sign-sub-inv [=>] 48.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]
      +-commutative [=>] 48.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell} + 1\right)}\right) \]
      *-commutative [=>] 48.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}\right) \cdot \frac{h}{\ell} + 1\right)\right) \]
      distribute-rgt-neg-in [=>] 48.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(-\frac{1}{2}\right)\right)} \cdot \frac{h}{\ell} + 1\right)\right) \]
      associate-*l* [=>] 48.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\left(-\frac{1}{2}\right) \cdot \frac{h}{\ell}\right)} + 1\right)\right) \]
      fma-def [=>] 48.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}, \left(-\frac{1}{2}\right) \cdot \frac{h}{\ell}, 1\right)}\right) \]

   3. Taylor expanded in M around 0 43.0%

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

   4. Applied egg-rr 52.4%

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

      [Start] 43.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right) \]
      frac-2neg [=>] 43.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\color{blue}{\frac{-d}{-\ell}}} \cdot 1\right) \]
      sqrt-div [=>] 52.4	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}} \cdot 1\right) \]

   5. Applied egg-rr 76.7%

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

      [Start] 52.4	\[ \sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot 1\right) \]
      frac-2neg [=>] 52.4	\[ \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot 1\right) \]
      sqrt-div [=>] 76.7	\[ \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot 1\right) \]

   if -5.00000000000023e-311 < h < 2.20000000000000006e-123

   1. Initial program 48.4%

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

   2. Applied egg-rr 48.4%

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

      [Start] 48.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      expm1-log1p-u [=>] 48.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]
      expm1-udef [=>] 48.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} - 1\right)}\right) \]
      log1p-udef [=>] 48.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(e^{\color{blue}{\log \left(1 + \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}} - 1\right)\right) \]
      add-exp-log [<=] 48.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(1 + \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} - 1\right)\right) \]
      associate-*l* [=>] 48.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) - 1\right)\right) \]
      metadata-eval [=>] 48.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]
      *-un-lft-identity [=>] 48.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + 0.5 \cdot \left({\left(\frac{\color{blue}{1 \cdot \left(M \cdot D\right)}}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]
      times-frac [=>] 48.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + 0.5 \cdot \left({\color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]
      metadata-eval [=>] 48.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + 0.5 \cdot \left({\left(\color{blue}{0.5} \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]

   3. Simplified 46.4%

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

      [Start] 48.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + 0.5 \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]
      +-commutative [=>] 48.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(0.5 \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) + 1\right)} - 1\right)\right) \]
      associate--l+ [=>] 48.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) + \left(1 - 1\right)\right)}\right) \]
      metadata-eval [=>] 48.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(0.5 \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) + \color{blue}{0}\right)\right) \]
      associate-*r* [=>] 48.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(0.5 \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}\right) \cdot \frac{h}{\ell}} + 0\right)\right) \]
      associate-*r/ [=>] 48.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{\left(0.5 \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}\right) \cdot h}{\ell}} + 0\right)\right) \]
      associate-*l/ [<=] 46.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{0.5 \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}}{\ell} \cdot h} + 0\right)\right) \]
      *-commutative [=>] 46.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{h \cdot \frac{0.5 \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}}{\ell}} + 0\right)\right) \]
      associate-/l* [=>] 46.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \color{blue}{\frac{0.5}{\frac{\ell}{{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}}}} + 0\right)\right) \]
      associate-*r/ [=>] 46.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\frac{\ell}{{\color{blue}{\left(\frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)}}^{2}}} + 0\right)\right) \]
      associate-/l* [=>] 46.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\frac{\ell}{{\color{blue}{\left(\frac{0.5}{\frac{d}{M \cdot D}}\right)}}^{2}}} + 0\right)\right) \]
      *-commutative [=>] 46.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\frac{\ell}{{\left(\frac{0.5}{\frac{d}{\color{blue}{D \cdot M}}}\right)}^{2}}} + 0\right)\right) \]

   4. Applied egg-rr 48.5%

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

      [Start] 46.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\frac{\ell}{{\left(\frac{0.5}{\frac{d}{D \cdot M}}\right)}^{2}}} + 0\right)\right) \]
      *-un-lft-identity [=>] 46.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\frac{\color{blue}{1 \cdot \ell}}{{\left(\frac{0.5}{\frac{d}{D \cdot M}}\right)}^{2}}} + 0\right)\right) \]
      unpow2 [=>] 46.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\frac{1 \cdot \ell}{\color{blue}{\frac{0.5}{\frac{d}{D \cdot M}} \cdot \frac{0.5}{\frac{d}{D \cdot M}}}}} + 0\right)\right) \]
      times-frac [=>] 49.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\color{blue}{\frac{1}{\frac{0.5}{\frac{d}{D \cdot M}}} \cdot \frac{\ell}{\frac{0.5}{\frac{d}{D \cdot M}}}}} + 0\right)\right) \]
      clear-num [<=] 49.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\color{blue}{\frac{\frac{d}{D \cdot M}}{0.5}} \cdot \frac{\ell}{\frac{0.5}{\frac{d}{D \cdot M}}}} + 0\right)\right) \]
      div-inv [=>] 49.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\color{blue}{\left(\frac{d}{D \cdot M} \cdot \frac{1}{0.5}\right)} \cdot \frac{\ell}{\frac{0.5}{\frac{d}{D \cdot M}}}} + 0\right)\right) \]
      *-commutative [=>] 49.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{d}{\color{blue}{M \cdot D}} \cdot \frac{1}{0.5}\right) \cdot \frac{\ell}{\frac{0.5}{\frac{d}{D \cdot M}}}} + 0\right)\right) \]
      associate-/r* [=>] 48.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\color{blue}{\frac{\frac{d}{M}}{D}} \cdot \frac{1}{0.5}\right) \cdot \frac{\ell}{\frac{0.5}{\frac{d}{D \cdot M}}}} + 0\right)\right) \]
      metadata-eval [=>] 48.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot \color{blue}{2}\right) \cdot \frac{\ell}{\frac{0.5}{\frac{d}{D \cdot M}}}} + 0\right)\right) \]
      div-inv [=>] 48.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{\color{blue}{0.5 \cdot \frac{1}{\frac{d}{D \cdot M}}}}} + 0\right)\right) \]
      clear-num [<=] 48.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \color{blue}{\frac{D \cdot M}{d}}}} + 0\right)\right) \]
      associate-/l* [=>] 48.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \color{blue}{\frac{D}{\frac{d}{M}}}}} + 0\right)\right) \]

   5. Applied egg-rr 73.7%

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

      [Start] 48.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}} + 0\right)\right) \]
      metadata-eval [=>] 48.5	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}} + 0\right)\right) \]
      unpow1/2 [=>] 48.5	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}} + 0\right)\right) \]
      sqrt-div [=>] 73.7	\[ \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}} + 0\right)\right) \]

   if 2.20000000000000006e-123 < h

   1. Initial program 62.8%

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

   2. Applied egg-rr 62.8%

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

      [Start] 62.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      expm1-log1p-u [=>] 61.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]
      expm1-udef [=>] 61.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} - 1\right)}\right) \]
      log1p-udef [=>] 61.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(e^{\color{blue}{\log \left(1 + \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}} - 1\right)\right) \]
      add-exp-log [<=] 62.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(1 + \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} - 1\right)\right) \]
      associate-*l* [=>] 62.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) - 1\right)\right) \]
      metadata-eval [=>] 62.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]
      *-un-lft-identity [=>] 62.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + 0.5 \cdot \left({\left(\frac{\color{blue}{1 \cdot \left(M \cdot D\right)}}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]
      times-frac [=>] 62.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + 0.5 \cdot \left({\color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]
      metadata-eval [=>] 62.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + 0.5 \cdot \left({\left(\color{blue}{0.5} \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]

   3. Simplified 66.2%

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

      [Start] 62.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(1 + 0.5 \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) - 1\right)\right) \]
      +-commutative [=>] 62.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(0.5 \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) + 1\right)} - 1\right)\right) \]
      associate--l+ [=>] 62.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) + \left(1 - 1\right)\right)}\right) \]
      metadata-eval [=>] 62.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(0.5 \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) + \color{blue}{0}\right)\right) \]
      associate-*r* [=>] 62.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(0.5 \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}\right) \cdot \frac{h}{\ell}} + 0\right)\right) \]
      associate-*r/ [=>] 64.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{\left(0.5 \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}\right) \cdot h}{\ell}} + 0\right)\right) \]
      associate-*l/ [<=] 66.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{0.5 \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}}{\ell} \cdot h} + 0\right)\right) \]
      *-commutative [=>] 66.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{h \cdot \frac{0.5 \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}}{\ell}} + 0\right)\right) \]
      associate-/l* [=>] 66.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \color{blue}{\frac{0.5}{\frac{\ell}{{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}}}} + 0\right)\right) \]
      associate-*r/ [=>] 66.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\frac{\ell}{{\color{blue}{\left(\frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)}}^{2}}} + 0\right)\right) \]
      associate-/l* [=>] 66.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\frac{\ell}{{\color{blue}{\left(\frac{0.5}{\frac{d}{M \cdot D}}\right)}}^{2}}} + 0\right)\right) \]
      *-commutative [=>] 66.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\frac{\ell}{{\left(\frac{0.5}{\frac{d}{\color{blue}{D \cdot M}}}\right)}^{2}}} + 0\right)\right) \]

   4. Applied egg-rr 67.3%

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

      [Start] 66.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\frac{\ell}{{\left(\frac{0.5}{\frac{d}{D \cdot M}}\right)}^{2}}} + 0\right)\right) \]
      *-un-lft-identity [=>] 66.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\frac{\color{blue}{1 \cdot \ell}}{{\left(\frac{0.5}{\frac{d}{D \cdot M}}\right)}^{2}}} + 0\right)\right) \]
      unpow2 [=>] 66.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\frac{1 \cdot \ell}{\color{blue}{\frac{0.5}{\frac{d}{D \cdot M}} \cdot \frac{0.5}{\frac{d}{D \cdot M}}}}} + 0\right)\right) \]
      times-frac [=>] 67.7	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\color{blue}{\frac{1}{\frac{0.5}{\frac{d}{D \cdot M}}} \cdot \frac{\ell}{\frac{0.5}{\frac{d}{D \cdot M}}}}} + 0\right)\right) \]
      clear-num [<=] 67.7	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\color{blue}{\frac{\frac{d}{D \cdot M}}{0.5}} \cdot \frac{\ell}{\frac{0.5}{\frac{d}{D \cdot M}}}} + 0\right)\right) \]
      div-inv [=>] 67.7	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\color{blue}{\left(\frac{d}{D \cdot M} \cdot \frac{1}{0.5}\right)} \cdot \frac{\ell}{\frac{0.5}{\frac{d}{D \cdot M}}}} + 0\right)\right) \]
      *-commutative [=>] 67.7	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{d}{\color{blue}{M \cdot D}} \cdot \frac{1}{0.5}\right) \cdot \frac{\ell}{\frac{0.5}{\frac{d}{D \cdot M}}}} + 0\right)\right) \]
      associate-/r* [=>] 66.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\color{blue}{\frac{\frac{d}{M}}{D}} \cdot \frac{1}{0.5}\right) \cdot \frac{\ell}{\frac{0.5}{\frac{d}{D \cdot M}}}} + 0\right)\right) \]
      metadata-eval [=>] 66.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot \color{blue}{2}\right) \cdot \frac{\ell}{\frac{0.5}{\frac{d}{D \cdot M}}}} + 0\right)\right) \]
      div-inv [=>] 66.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{\color{blue}{0.5 \cdot \frac{1}{\frac{d}{D \cdot M}}}}} + 0\right)\right) \]
      clear-num [<=] 66.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \color{blue}{\frac{D \cdot M}{d}}}} + 0\right)\right) \]
      associate-/l* [=>] 67.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \color{blue}{\frac{D}{\frac{d}{M}}}}} + 0\right)\right) \]

   5. Applied egg-rr 67.2%

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

      [Start] 67.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}} + 0\right)\right) \]
      metadata-eval [=>] 67.3	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}} + 0\right)\right) \]
      unpow1/2 [=>] 67.3	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}} + 0\right)\right) \]
      clear-num [=>] 66.9	\[ \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}} + 0\right)\right) \]
      sqrt-div [=>] 67.2	\[ \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}} + 0\right)\right) \]
      metadata-eval [=>] 67.2	\[ \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}} + 0\right)\right) \]

   6. Applied egg-rr 77.6%

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

      [Start] 67.2	\[ \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}} + 0\right)\right) \]
      metadata-eval [=>] 67.2	\[ \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}} + 0\right)\right) \]
      unpow1/2 [=>] 67.2	\[ \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}} + 0\right)\right) \]
      sqrt-div [=>] 77.6	\[ \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(h \cdot \frac{0.5}{\left(\frac{\frac{d}{M}}{D} \cdot 2\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}} + 0\right)\right) \]

3. Recombined 5 regimes into one program.

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -3.7 \cdot 10^{+167}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + h \cdot \frac{-0.5}{\left(2 \cdot \frac{\frac{d}{M}}{D}\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}}\right)\\ \mathbf{elif}\;h \leq -6.6 \cdot 10^{-98}:\\ \;\;\;\;\left(1 + h \cdot \frac{-0.5}{\left(2 \cdot \frac{\frac{d}{M}}{D}\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}}\right) \cdot \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;h \leq 2.2 \cdot 10^{-123}:\\ \;\;\;\;\left(1 + h \cdot \frac{-0.5}{\left(2 \cdot \frac{\frac{d}{M}}{D}\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + h \cdot \frac{-0.5}{\left(2 \cdot \frac{\frac{d}{M}}{D}\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}}\right) \cdot \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	72.7%
Cost	21840

\[\begin{array}{l} t_0 := 1 + h \cdot \frac{-0.5}{\left(2 \cdot \frac{\frac{d}{M}}{D}\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -7.6 \cdot 10^{-59}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot {\left(\frac{d}{h}\right)}^{0.5}\right)\\ \mathbf{elif}\;h \leq -1.05 \cdot 10^{-123}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\sqrt{-d} \cdot t_1}{\sqrt{-h}}\\ \mathbf{elif}\;h \leq 2.05 \cdot 10^{-113}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \]

Alternative 2
Accuracy	72.2%
Cost	21840

\[\begin{array}{l} t_0 := 1 + h \cdot \frac{-0.5}{\left(2 \cdot \frac{\frac{d}{M}}{D}\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -5.1 \cdot 10^{-59}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot {\left(\frac{d}{h}\right)}^{0.5}\right)\\ \mathbf{elif}\;h \leq -1.48 \cdot 10^{-123}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\sqrt{-d} \cdot t_1}{\sqrt{-h}}\\ \mathbf{elif}\;h \leq 1.1 \cdot 10^{-122}:\\ \;\;\;\;t_0 \cdot \left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \]

Alternative 3
Accuracy	75.9%
Cost	21840

\[\begin{array}{l} t_0 := {\left(\frac{d}{\ell}\right)}^{0.5}\\ t_1 := 1 + h \cdot \frac{-0.5}{\left(2 \cdot \frac{\frac{d}{M}}{D}\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}}\\ t_2 := \sqrt{-d}\\ t_3 := \left(\frac{t_2}{\sqrt{-h}} \cdot t_0\right) \cdot t_1\\ t_4 := \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{if}\;h \leq -1 \cdot 10^{+172}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;h \leq -1.22 \cdot 10^{-122}:\\ \;\;\;\;t_1 \cdot \left(t_4 \cdot \frac{t_2}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;h \leq 5 \cdot 10^{-122}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(t_4 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \]

Alternative 4
Accuracy	75.2%
Cost	21708

\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := 1 + h \cdot \frac{-0.5}{\left(2 \cdot \frac{\frac{d}{M}}{D}\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}}\\ t_2 := \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{if}\;h \leq -1.7 \cdot 10^{-126}:\\ \;\;\;\;t_1 \cdot \left(t_2 \cdot \frac{t_0}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{t_0 \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{-h}}\\ \mathbf{elif}\;h \leq 3.7 \cdot 10^{-124}:\\ \;\;\;\;t_1 \cdot \left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(t_2 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \]

Alternative 5
Accuracy	70.8%
Cost	21004

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -2.65 \cdot 10^{-58}:\\ \;\;\;\;\left(1 + h \cdot \frac{-0.5}{\left(2 \cdot \frac{\frac{d}{M}}{D}\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}}\right) \cdot \left(t_0 \cdot {\left(\frac{d}{h}\right)}^{0.5}\right)\\ \mathbf{elif}\;h \leq -2.05 \cdot 10^{-126}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\sqrt{-d} \cdot t_0}{\sqrt{-h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	69.6%
Cost	20172

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \left(1 + h \cdot \frac{-0.5}{\left(2 \cdot \frac{\frac{d}{M}}{D}\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}}\right) \cdot \left(t_0 \cdot {\left(\frac{d}{h}\right)}^{0.5}\right)\\ \mathbf{if}\;h \leq -5.4 \cdot 10^{-59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;h \leq -7.6 \cdot 10^{-126}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\sqrt{-d} \cdot t_0}{\sqrt{-h}}\\ \mathbf{elif}\;h \leq 3.8 \cdot 10^{-25}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Accuracy	65.8%
Cost	15577

\[\begin{array}{l} t_0 := D \cdot \left(M \cdot \frac{0.5}{d}\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{d}{h}} \cdot \left(t_1 \cdot \left(1 + \frac{t_0}{\frac{\frac{\ell}{h}}{-0.5 \cdot t_0}}\right)\right)\\ \mathbf{if}\;h \leq -1.32 \cdot 10^{-58}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;h \leq -1.5 \cdot 10^{-208}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq -4.2 \cdot 10^{-264}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{t_1}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;h \leq 2 \cdot 10^{-27} \lor \neg \left(h \leq 8.5 \cdot 10^{+89}\right):\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Accuracy	63.5%
Cost	15317

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -3 \cdot 10^{+99}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -3.1 \cdot 10^{-78}:\\ \;\;\;\;\frac{t_0}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq 4.05 \cdot 10^{-308}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{-74} \lor \neg \left(d \leq 2.25 \cdot 10^{+47}\right):\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}{\ell} \cdot \frac{\frac{h}{d}}{d}\right)\right)\right)\\ \end{array} \]

Alternative 9
Accuracy	68.0%
Cost	15176

\[\begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{+162}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq 3.45 \cdot 10^{+128}:\\ \;\;\;\;\left(1 + h \cdot \frac{-0.5}{\left(2 \cdot \frac{\frac{d}{M}}{D}\right) \cdot \frac{\ell}{0.5 \cdot \frac{D}{\frac{d}{M}}}}\right) \cdot \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]

Alternative 10
Accuracy	63.7%
Cost	13580

\[\begin{array}{l} \mathbf{if}\;d \leq -2.1 \cdot 10^{+99}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -1.1 \cdot 10^{-77}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]

Alternative 11
Accuracy	63.8%
Cost	13580

\[\begin{array}{l} \mathbf{if}\;d \leq -6 \cdot 10^{+99}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -5.2 \cdot 10^{-77}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-308}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]

Alternative 12
Accuracy	62.4%
Cost	13316

\[\begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]

Alternative 13
Accuracy	62.4%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 14
Accuracy	56.4%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{-294}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 15
Accuracy	56.2%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{-294}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 16
Accuracy	45.5%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq 5.5 \cdot 10^{-283}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 17
Accuracy	47.8%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq 4.5 \cdot 10^{-283}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 18
Accuracy	47.8%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq 4.8 \cdot 10^{-282}:\\ \;\;\;\;\sqrt{\frac{d}{h \cdot \frac{\ell}{d}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 19
Accuracy	47.9%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq 10^{-282}:\\ \;\;\;\;\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 20
Accuracy	31.4%
Cost	6784

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

Alternative 21
Accuracy	31.4%
Cost	6720

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

Reproduce

herbie shell --seed 2023129 
(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)))))