Henrywood and Agarwal, Equation (12)

Percentage Accurate: 67.2% → 75.5%

Time: 30.2s

Alternatives: 26

Speedup: 1.4×

Specification

Math

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

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)))))

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)));
}

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

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)));
}

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)))

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

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

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]

Initial Program: 67.2% accurate, 1.0× speedup

Math

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

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)))))

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)));
}

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

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)));
}

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)))

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

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

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]

Alternative 1: 75.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\\ t_1 := \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot t_0\right)\right)\\ \mathbf{if}\;d \leq -6.7 \cdot 10^{+109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -4.4 \cdot 10^{-56}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{1}{\frac{2 \cdot d}{D \cdot M}}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.5\right)\right)\\ \mathbf{elif}\;d \leq -4.8 \cdot 10^{-238}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot t_0\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (* D (/ M (* 2.0 d))) 2.0))
        (t_1
         (* (* d (sqrt (/ 1.0 (* h l)))) (- -1.0 (* -0.5 (* (/ h l) t_0))))))
   (if (<= d -6.7e+109)
     t_1
     (if (<= d -4.4e-56)
       (*
        (* (sqrt (/ d h)) (sqrt (/ d l)))
        (- 1.0 (* (pow (/ 1.0 (/ (* 2.0 d) (* D M))) 2.0) (* (/ h l) 0.5))))
       (if (<= d -4.8e-238)
         t_1
         (if (<= d 4.6e-303)
           (* (sqrt (/ h (pow l 3.0))) (* 0.125 (* M (/ (* D M) (/ d D)))))
           (*
            (* (/ (sqrt d) (sqrt h)) (pow (/ d l) 0.5))
            (- 1.0 (/ (* h (* 0.5 t_0)) l)))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = pow((D * (M / (2.0 * d))), 2.0);
	double t_1 = (d * sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * t_0)));
	double tmp;
	if (d <= -6.7e+109) {
		tmp = t_1;
	} else if (d <= -4.4e-56) {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - (pow((1.0 / ((2.0 * d) / (D * M))), 2.0) * ((h / l) * 0.5)));
	} else if (d <= -4.8e-238) {
		tmp = t_1;
	} else if (d <= 4.6e-303) {
		tmp = sqrt((h / pow(l, 3.0))) * (0.125 * (M * ((D * M) / (d / D))));
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * pow((d / l), 0.5)) * (1.0 - ((h * (0.5 * t_0)) / l));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this 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) :: tmp
    t_0 = (d_1 * (m / (2.0d0 * d))) ** 2.0d0
    t_1 = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) - ((-0.5d0) * ((h / l) * t_0)))
    if (d <= (-6.7d+109)) then
        tmp = t_1
    else if (d <= (-4.4d-56)) then
        tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0d0 - (((1.0d0 / ((2.0d0 * d) / (d_1 * m))) ** 2.0d0) * ((h / l) * 0.5d0)))
    else if (d <= (-4.8d-238)) then
        tmp = t_1
    else if (d <= 4.6d-303) then
        tmp = sqrt((h / (l ** 3.0d0))) * (0.125d0 * (m * ((d_1 * m) / (d / d_1))))
    else
        tmp = ((sqrt(d) / sqrt(h)) * ((d / l) ** 0.5d0)) * (1.0d0 - ((h * (0.5d0 * t_0)) / l))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.pow((D * (M / (2.0 * d))), 2.0);
	double t_1 = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * t_0)));
	double tmp;
	if (d <= -6.7e+109) {
		tmp = t_1;
	} else if (d <= -4.4e-56) {
		tmp = (Math.sqrt((d / h)) * Math.sqrt((d / l))) * (1.0 - (Math.pow((1.0 / ((2.0 * d) / (D * M))), 2.0) * ((h / l) * 0.5)));
	} else if (d <= -4.8e-238) {
		tmp = t_1;
	} else if (d <= 4.6e-303) {
		tmp = Math.sqrt((h / Math.pow(l, 3.0))) * (0.125 * (M * ((D * M) / (d / D))));
	} else {
		tmp = ((Math.sqrt(d) / Math.sqrt(h)) * Math.pow((d / l), 0.5)) * (1.0 - ((h * (0.5 * t_0)) / l));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.pow((D * (M / (2.0 * d))), 2.0)
	t_1 = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * t_0)))
	tmp = 0
	if d <= -6.7e+109:
		tmp = t_1
	elif d <= -4.4e-56:
		tmp = (math.sqrt((d / h)) * math.sqrt((d / l))) * (1.0 - (math.pow((1.0 / ((2.0 * d) / (D * M))), 2.0) * ((h / l) * 0.5)))
	elif d <= -4.8e-238:
		tmp = t_1
	elif d <= 4.6e-303:
		tmp = math.sqrt((h / math.pow(l, 3.0))) * (0.125 * (M * ((D * M) / (d / D))))
	else:
		tmp = ((math.sqrt(d) / math.sqrt(h)) * math.pow((d / l), 0.5)) * (1.0 - ((h * (0.5 * t_0)) / l))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(D * Float64(M / Float64(2.0 * d))) ^ 2.0
	t_1 = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 - Float64(-0.5 * Float64(Float64(h / l) * t_0))))
	tmp = 0.0
	if (d <= -6.7e+109)
		tmp = t_1;
	elseif (d <= -4.4e-56)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * Float64(1.0 - Float64((Float64(1.0 / Float64(Float64(2.0 * d) / Float64(D * M))) ^ 2.0) * Float64(Float64(h / l) * 0.5))));
	elseif (d <= -4.8e-238)
		tmp = t_1;
	elseif (d <= 4.6e-303)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(0.125 * Float64(M * Float64(Float64(D * M) / Float64(d / D)))));
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(h * Float64(0.5 * t_0)) / l)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (D * (M / (2.0 * d))) ^ 2.0;
	t_1 = (d * sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * t_0)));
	tmp = 0.0;
	if (d <= -6.7e+109)
		tmp = t_1;
	elseif (d <= -4.4e-56)
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - (((1.0 / ((2.0 * d) / (D * M))) ^ 2.0) * ((h / l) * 0.5)));
	elseif (d <= -4.8e-238)
		tmp = t_1;
	elseif (d <= 4.6e-303)
		tmp = sqrt((h / (l ^ 3.0))) * (0.125 * (M * ((D * M) / (d / D))));
	else
		tmp = ((sqrt(d) / sqrt(h)) * ((d / l) ^ 0.5)) * (1.0 - ((h * (0.5 * t_0)) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(D * N[(M / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[(h / l), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -6.7e+109], t$95$1, If[LessEqual[d, -4.4e-56], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Power[N[(1.0 / N[(N[(2.0 * d), $MachinePrecision] / N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4.8e-238], t$95$1, If[LessEqual[d, 4.6e-303], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.125 * N[(M * N[(N[(D * M), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h * N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -6.70000000000000036e109 or -4.40000000000000008e-56 < d < -4.7999999999999997e-238

   1. Initial program 67.6%

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

   3. Applied egg-rr 22.4%

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

      1. expm1-def 27.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 56.4%

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

   5. Simplified 56.4%

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

   6. Taylor expanded in d around -inf 81.8%

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

      1. mul-1-neg 81.8%

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

      2. distribute-rgt-neg-in 81.8%

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

      3. *-commutative 81.8%

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

   8. Simplified 81.8%

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

   if -6.70000000000000036e109 < d < -4.40000000000000008e-56

   1. Initial program 91.6%

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

      1. metadata-eval 91.6%

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

      2. unpow1/2 91.6%

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

      3. metadata-eval 91.6%

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

      4. unpow1/2 91.6%

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

      5. *-commutative 91.6%

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

      6. associate-*l* 91.6%

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

      7. times-frac 91.6%

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

      8. metadata-eval 91.6%

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

   3. Simplified 91.6%

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

      1. frac-times 91.6%

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

      2. clear-num 91.6%

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

      3. *-commutative 91.6%

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

   5. Applied egg-rr 91.6%

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

   if -4.7999999999999997e-238 < d < 4.59999999999999991e-303

   1. Initial program 28.2%

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

   3. Applied egg-rr 5.6%

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

      1. expm1-def 5.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 22.7%

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

   5. Simplified 22.7%

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

   6. Taylor expanded in d around -inf 33.8%

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

      1. mul-1-neg 33.8%

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

      2. distribute-rgt-neg-in 33.8%

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

      3. *-commutative 33.8%

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

   8. Simplified 33.8%

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

   9. Taylor expanded in D around inf 61.9%

      \[\leadsto \color{blue}{0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 61.9%

          \[\leadsto 0.125 \cdot \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

       2. unpow2 61.9%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{d}\right) \]

       3. unpow2 61.9%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{d}\right) \]

       4. associate-*l/ 56.4%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)}\right) \]

       5. *-commutative 56.4%

          \[\leadsto \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)\right) \cdot 0.125} \]

       6. associate-*l* 56.4%

          \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right) \cdot 0.125\right)} \]

       7. *-commutative 56.4%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(0.125 \cdot \left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)\right)} \]

       8. associate-*r* 57.2%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \color{blue}{\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot M\right)}\right) \]

       9. *-commutative 57.2%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \color{blue}{\left(M \cdot \left(\frac{D \cdot D}{d} \cdot M\right)\right)}\right) \]

       10. associate-/l* 57.1%

           \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \left(\color{blue}{\frac{D}{\frac{d}{D}}} \cdot M\right)\right)\right) \]

       11. associate-*l/ 62.5%

           \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \color{blue}{\frac{D \cdot M}{\frac{d}{D}}}\right)\right) \]

   11. Simplified 62.5%

       \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)} \]

   if 4.59999999999999991e-303 < d

   1. Initial program 66.0%

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

      1. metadata-eval 66.0%

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

      2. pow1/2 66.0%

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

      3. sqrt-div 78.6%

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

   3. Applied egg-rr 78.6%

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

      1. associate-*r/ 80.3%

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

      2. metadata-eval 80.3%

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

      3. *-commutative 80.3%

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

      4. *-commutative 80.3%

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

      5. associate-*r/ 79.5%

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

   5. Applied egg-rr 79.5%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.7 \cdot 10^{+109}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -4.4 \cdot 10^{-56}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{1}{\frac{2 \cdot d}{D \cdot M}}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.5\right)\right)\\ \mathbf{elif}\;d \leq -4.8 \cdot 10^{-238}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)}{\ell}\right)\\ \end{array} \]

Alternative 2: 75.3% accurate, 0.6× speedup

Math

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

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= h -2e-310)
   (*
    (* d (sqrt (/ 1.0 (* h l))))
    (- -1.0 (* -0.5 (* (/ h l) (pow (* D (/ M (* 2.0 d))) 2.0)))))
   (*
    (* (/ (sqrt d) (sqrt h)) (/ (sqrt d) (sqrt l)))
    (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* D M) (* 2.0 d)) 2.0)))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -2e-310) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * pow((D * (M / (2.0 * d))), 2.0))));
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * (sqrt(d) / sqrt(l))) * (1.0 - ((h / l) * (0.5 * pow(((D * M) / (2.0 * d)), 2.0))));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this 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) :: tmp
    if (h <= (-2d-310)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) - ((-0.5d0) * ((h / l) * ((d_1 * (m / (2.0d0 * d))) ** 2.0d0))))
    else
        tmp = ((sqrt(d) / sqrt(h)) * (sqrt(d) / sqrt(l))) * (1.0d0 - ((h / l) * (0.5d0 * (((d_1 * m) / (2.0d0 * d)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -2e-310) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * Math.pow((D * (M / (2.0 * d))), 2.0))));
	} else {
		tmp = ((Math.sqrt(d) / Math.sqrt(h)) * (Math.sqrt(d) / Math.sqrt(l))) * (1.0 - ((h / l) * (0.5 * Math.pow(((D * M) / (2.0 * d)), 2.0))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if h <= -2e-310:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * math.pow((D * (M / (2.0 * d))), 2.0))))
	else:
		tmp = ((math.sqrt(d) / math.sqrt(h)) * (math.sqrt(d) / math.sqrt(l))) * (1.0 - ((h / l) * (0.5 * math.pow(((D * M) / (2.0 * d)), 2.0))))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -2e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 - Float64(-0.5 * Float64(Float64(h / l) * (Float64(D * Float64(M / Float64(2.0 * d))) ^ 2.0)))));
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * Float64(sqrt(d) / sqrt(l))) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= -2e-310)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * ((D * (M / (2.0 * d))) ^ 2.0))));
	else
		tmp = ((sqrt(d) / sqrt(h)) * (sqrt(d) / sqrt(l))) * (1.0 - ((h / l) * (0.5 * (((D * M) / (2.0 * d)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[h, -2e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -1.999999999999994e-310

   1. Initial program 69.2%

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

   3. Applied egg-rr 21.9%

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

      1. expm1-def 30.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 57.9%

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

   5. Simplified 57.9%

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

   6. Taylor expanded in d around -inf 73.7%

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

      1. mul-1-neg 73.7%

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

      2. distribute-rgt-neg-in 73.7%

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

      3. *-commutative 73.7%

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

   8. Simplified 73.7%

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

   if -1.999999999999994e-310 < h

   1. Initial program 65.5%

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

      1. metadata-eval 65.5%

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

      2. pow1/2 65.5%

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

      3. sqrt-div 78.0%

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

   3. Applied egg-rr 78.0%

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

      1. metadata-eval 78.0%

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

      2. pow1/2 78.0%

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

      3. sqrt-div 81.8%

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

   5. Applied egg-rr 81.8%

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

4. Final simplification 77.8%

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

Alternative 3: 70.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\\ t_3 := \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot t_2\right)\right)\\ t_4 := \sqrt{\frac{h}{{\ell}^{3}}}\\ t_5 := M \cdot \frac{D}{d}\\ t_6 := M \cdot \frac{D \cdot M}{\frac{d}{D}}\\ \mathbf{if}\;d \leq -2.45 \cdot 10^{+113}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq -3.7 \cdot 10^{-56}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(t_5 \cdot t_5\right)\right) \cdot 0.25\right)\right)\right)\\ \mathbf{elif}\;d \leq -5.4 \cdot 10^{-238}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_4 \cdot \left(0.125 \cdot t_6\right)\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-112}:\\ \;\;\;\;t_4 \cdot \left(t_6 \cdot -0.125\right)\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{+133}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \frac{h \cdot t_2}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (sqrt (/ d l)))
        (t_2 (pow (* D (/ M (* 2.0 d))) 2.0))
        (t_3
         (* (* d (sqrt (/ 1.0 (* h l)))) (- -1.0 (* -0.5 (* (/ h l) t_2)))))
        (t_4 (sqrt (/ h (pow l 3.0))))
        (t_5 (* M (/ D d)))
        (t_6 (* M (/ (* D M) (/ d D)))))
   (if (<= d -2.45e+113)
     t_3
     (if (<= d -3.7e-56)
       (* t_0 (* t_1 (- 1.0 (* 0.5 (* (* (/ h l) (* t_5 t_5)) 0.25)))))
       (if (<= d -5.4e-238)
         t_3
         (if (<= d -2e-310)
           (* t_4 (* 0.125 t_6))
           (if (<= d 1.05e-112)
             (* t_4 (* t_6 -0.125))
             (if (<= d 9.5e+133)
               (* t_0 (* t_1 (- 1.0 (* 0.5 (/ (* h t_2) l)))))
               (* d (/ (pow h -0.5) (sqrt l)))))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = sqrt((d / l));
	double t_2 = pow((D * (M / (2.0 * d))), 2.0);
	double t_3 = (d * sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * t_2)));
	double t_4 = sqrt((h / pow(l, 3.0)));
	double t_5 = M * (D / d);
	double t_6 = M * ((D * M) / (d / D));
	double tmp;
	if (d <= -2.45e+113) {
		tmp = t_3;
	} else if (d <= -3.7e-56) {
		tmp = t_0 * (t_1 * (1.0 - (0.5 * (((h / l) * (t_5 * t_5)) * 0.25))));
	} else if (d <= -5.4e-238) {
		tmp = t_3;
	} else if (d <= -2e-310) {
		tmp = t_4 * (0.125 * t_6);
	} else if (d <= 1.05e-112) {
		tmp = t_4 * (t_6 * -0.125);
	} else if (d <= 9.5e+133) {
		tmp = t_0 * (t_1 * (1.0 - (0.5 * ((h * t_2) / l))));
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this 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) :: t_6
    real(8) :: tmp
    t_0 = sqrt((d / h))
    t_1 = sqrt((d / l))
    t_2 = (d_1 * (m / (2.0d0 * d))) ** 2.0d0
    t_3 = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) - ((-0.5d0) * ((h / l) * t_2)))
    t_4 = sqrt((h / (l ** 3.0d0)))
    t_5 = m * (d_1 / d)
    t_6 = m * ((d_1 * m) / (d / d_1))
    if (d <= (-2.45d+113)) then
        tmp = t_3
    else if (d <= (-3.7d-56)) then
        tmp = t_0 * (t_1 * (1.0d0 - (0.5d0 * (((h / l) * (t_5 * t_5)) * 0.25d0))))
    else if (d <= (-5.4d-238)) then
        tmp = t_3
    else if (d <= (-2d-310)) then
        tmp = t_4 * (0.125d0 * t_6)
    else if (d <= 1.05d-112) then
        tmp = t_4 * (t_6 * (-0.125d0))
    else if (d <= 9.5d+133) then
        tmp = t_0 * (t_1 * (1.0d0 - (0.5d0 * ((h * t_2) / l))))
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / h));
	double t_1 = Math.sqrt((d / l));
	double t_2 = Math.pow((D * (M / (2.0 * d))), 2.0);
	double t_3 = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * t_2)));
	double t_4 = Math.sqrt((h / Math.pow(l, 3.0)));
	double t_5 = M * (D / d);
	double t_6 = M * ((D * M) / (d / D));
	double tmp;
	if (d <= -2.45e+113) {
		tmp = t_3;
	} else if (d <= -3.7e-56) {
		tmp = t_0 * (t_1 * (1.0 - (0.5 * (((h / l) * (t_5 * t_5)) * 0.25))));
	} else if (d <= -5.4e-238) {
		tmp = t_3;
	} else if (d <= -2e-310) {
		tmp = t_4 * (0.125 * t_6);
	} else if (d <= 1.05e-112) {
		tmp = t_4 * (t_6 * -0.125);
	} else if (d <= 9.5e+133) {
		tmp = t_0 * (t_1 * (1.0 - (0.5 * ((h * t_2) / l))));
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt((d / h))
	t_1 = math.sqrt((d / l))
	t_2 = math.pow((D * (M / (2.0 * d))), 2.0)
	t_3 = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * t_2)))
	t_4 = math.sqrt((h / math.pow(l, 3.0)))
	t_5 = M * (D / d)
	t_6 = M * ((D * M) / (d / D))
	tmp = 0
	if d <= -2.45e+113:
		tmp = t_3
	elif d <= -3.7e-56:
		tmp = t_0 * (t_1 * (1.0 - (0.5 * (((h / l) * (t_5 * t_5)) * 0.25))))
	elif d <= -5.4e-238:
		tmp = t_3
	elif d <= -2e-310:
		tmp = t_4 * (0.125 * t_6)
	elif d <= 1.05e-112:
		tmp = t_4 * (t_6 * -0.125)
	elif d <= 9.5e+133:
		tmp = t_0 * (t_1 * (1.0 - (0.5 * ((h * t_2) / l))))
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(d / l))
	t_2 = Float64(D * Float64(M / Float64(2.0 * d))) ^ 2.0
	t_3 = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 - Float64(-0.5 * Float64(Float64(h / l) * t_2))))
	t_4 = sqrt(Float64(h / (l ^ 3.0)))
	t_5 = Float64(M * Float64(D / d))
	t_6 = Float64(M * Float64(Float64(D * M) / Float64(d / D)))
	tmp = 0.0
	if (d <= -2.45e+113)
		tmp = t_3;
	elseif (d <= -3.7e-56)
		tmp = Float64(t_0 * Float64(t_1 * Float64(1.0 - Float64(0.5 * Float64(Float64(Float64(h / l) * Float64(t_5 * t_5)) * 0.25)))));
	elseif (d <= -5.4e-238)
		tmp = t_3;
	elseif (d <= -2e-310)
		tmp = Float64(t_4 * Float64(0.125 * t_6));
	elseif (d <= 1.05e-112)
		tmp = Float64(t_4 * Float64(t_6 * -0.125));
	elseif (d <= 9.5e+133)
		tmp = Float64(t_0 * Float64(t_1 * Float64(1.0 - Float64(0.5 * Float64(Float64(h * t_2) / l)))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / h));
	t_1 = sqrt((d / l));
	t_2 = (D * (M / (2.0 * d))) ^ 2.0;
	t_3 = (d * sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * t_2)));
	t_4 = sqrt((h / (l ^ 3.0)));
	t_5 = M * (D / d);
	t_6 = M * ((D * M) / (d / D));
	tmp = 0.0;
	if (d <= -2.45e+113)
		tmp = t_3;
	elseif (d <= -3.7e-56)
		tmp = t_0 * (t_1 * (1.0 - (0.5 * (((h / l) * (t_5 * t_5)) * 0.25))));
	elseif (d <= -5.4e-238)
		tmp = t_3;
	elseif (d <= -2e-310)
		tmp = t_4 * (0.125 * t_6);
	elseif (d <= 1.05e-112)
		tmp = t_4 * (t_6 * -0.125);
	elseif (d <= 9.5e+133)
		tmp = t_0 * (t_1 * (1.0 - (0.5 * ((h * t_2) / l))));
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(D * N[(M / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[(h / l), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(M * N[(N[(D * M), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.45e+113], t$95$3, If[LessEqual[d, -3.7e-56], N[(t$95$0 * N[(t$95$1 * N[(1.0 - N[(0.5 * N[(N[(N[(h / l), $MachinePrecision] * N[(t$95$5 * t$95$5), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5.4e-238], t$95$3, If[LessEqual[d, -2e-310], N[(t$95$4 * N[(0.125 * t$95$6), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.05e-112], N[(t$95$4 * N[(t$95$6 * -0.125), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 9.5e+133], N[(t$95$0 * N[(t$95$1 * N[(1.0 - N[(0.5 * N[(N[(h * t$95$2), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if d < -2.45000000000000011e113 or -3.7000000000000002e-56 < d < -5.39999999999999981e-238

   1. Initial program 67.6%

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

   3. Applied egg-rr 22.4%

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

      1. expm1-def 27.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 56.4%

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

   5. Simplified 56.4%

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

   6. Taylor expanded in d around -inf 81.8%

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

      1. mul-1-neg 81.8%

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

      2. distribute-rgt-neg-in 81.8%

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

      3. *-commutative 81.8%

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

   8. Simplified 81.8%

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

   if -2.45000000000000011e113 < d < -3.7000000000000002e-56

   1. Initial program 91.6%

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

      1. associate-*l* 91.6%

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

      2. metadata-eval 91.6%

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

      3. unpow1/2 91.6%

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

      4. metadata-eval 91.6%

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

      5. unpow1/2 91.6%

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

      6. associate-*l* 91.6%

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

      7. metadata-eval 91.6%

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

      8. times-frac 91.6%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in M around 0 80.1%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)}\right)\right) \]
   5. Step-by-step derivation

      1. associate-*r/ 80.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right)\right) \]

      2. *-commutative 80.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right) \]

      3. associate-*r/ 80.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      4. *-commutative 80.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot 0.25\right)}\right)\right) \]

      5. associate-*r* 77.3%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell} \cdot 0.25\right)\right)\right) \]

      6. *-commutative 77.3%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{\color{blue}{h \cdot \left({D}^{2} \cdot {M}^{2}\right)}}{{d}^{2} \cdot \ell} \cdot 0.25\right)\right)\right) \]

      7. *-commutative 77.3%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h \cdot \left({D}^{2} \cdot {M}^{2}\right)}{\color{blue}{\ell \cdot {d}^{2}}} \cdot 0.25\right)\right)\right) \]

      8. times-frac 80.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\color{blue}{\left(\frac{h}{\ell} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)} \cdot 0.25\right)\right)\right) \]

      9. *-commutative 80.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}}\right) \cdot 0.25\right)\right)\right) \]

      10. unpow2 80.2%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot {D}^{2}}{{d}^{2}}\right) \cdot 0.25\right)\right)\right) \]

      11. unpow2 80.2%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot \color{blue}{\left(D \cdot D\right)}}{{d}^{2}}\right) \cdot 0.25\right)\right)\right) \]

      12. unswap-sqr 88.7%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{\color{blue}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}}{{d}^{2}}\right) \cdot 0.25\right)\right)\right) \]

      13. unpow2 88.7%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{d \cdot d}}\right) \cdot 0.25\right)\right)\right) \]

      14. times-frac 91.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \color{blue}{\left(\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d}\right)}\right) \cdot 0.25\right)\right)\right) \]

      15. associate-*r/ 91.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(\color{blue}{\left(M \cdot \frac{D}{d}\right)} \cdot \frac{M \cdot D}{d}\right)\right) \cdot 0.25\right)\right)\right) \]

      16. associate-*r/ 91.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(M \cdot \frac{D}{d}\right)}\right)\right) \cdot 0.25\right)\right)\right) \]

   6. Simplified 91.6%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\left(\frac{h}{\ell} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)\right) \cdot 0.25\right)}\right)\right) \]

   if -5.39999999999999981e-238 < d < -1.999999999999994e-310

   1. Initial program 29.9%

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

   3. Applied egg-rr 5.9%

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

      1. expm1-def 5.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 24.0%

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

   5. Simplified 24.0%

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

   6. Taylor expanded in d around -inf 35.8%

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

      1. mul-1-neg 35.8%

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

      2. distribute-rgt-neg-in 35.8%

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

      3. *-commutative 35.8%

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

   8. Simplified 35.8%

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

   9. Taylor expanded in D around inf 65.3%

      \[\leadsto \color{blue}{0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 65.3%

          \[\leadsto 0.125 \cdot \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

       2. unpow2 65.3%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{d}\right) \]

       3. unpow2 65.3%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{d}\right) \]

       4. associate-*l/ 59.5%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)}\right) \]

       5. *-commutative 59.5%

          \[\leadsto \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)\right) \cdot 0.125} \]

       6. associate-*l* 59.5%

          \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right) \cdot 0.125\right)} \]

       7. *-commutative 59.5%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(0.125 \cdot \left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)\right)} \]

       8. associate-*r* 60.3%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \color{blue}{\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot M\right)}\right) \]

       9. *-commutative 60.3%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \color{blue}{\left(M \cdot \left(\frac{D \cdot D}{d} \cdot M\right)\right)}\right) \]

       10. associate-/l* 60.2%

           \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \left(\color{blue}{\frac{D}{\frac{d}{D}}} \cdot M\right)\right)\right) \]

       11. associate-*l/ 65.9%

           \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \color{blue}{\frac{D \cdot M}{\frac{d}{D}}}\right)\right) \]

   11. Simplified 65.9%

       \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)} \]

   if -1.999999999999994e-310 < d < 1.05e-112

   1. Initial program 35.1%

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

      1. metadata-eval 35.1%

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

      2. unpow1/2 35.1%

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

      3. metadata-eval 35.1%

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

      4. unpow1/2 35.1%

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

      5. *-commutative 35.1%

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

      6. associate-*l* 35.1%

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

      7. times-frac 35.2%

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

      8. metadata-eval 35.2%

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

   3. Simplified 35.2%

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

      1. frac-times 35.1%

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

      2. clear-num 35.2%

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

      3. *-commutative 35.2%

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

   5. Applied egg-rr 35.2%

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

      1. associate-*r* 35.2%

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

      2. *-commutative 35.2%

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

      3. clear-num 35.1%

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

      4. *-commutative 35.1%

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

      5. metadata-eval 35.1%

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

      6. associate-*r/ 37.4%

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

      7. metadata-eval 37.4%

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

      8. times-frac 37.5%

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

   7. Applied egg-rr 37.5%

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

   8. Taylor expanded in d around 0 40.1%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 40.1%

         \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative 40.1%

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      3. unpow2 40.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{d}\right) \]

      4. unpow2 40.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{d}\right) \]

      5. associate-*l/ 39.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)}\right) \]

      6. associate-*r* 42.7%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot M\right)}\right) \]

      7. *-commutative 42.7%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(M \cdot \left(\frac{D \cdot D}{d} \cdot M\right)\right)}\right) \]

      8. associate-/l* 48.0%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(M \cdot \left(\color{blue}{\frac{D}{\frac{d}{D}}} \cdot M\right)\right)\right) \]

      9. associate-*l/ 50.5%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(M \cdot \color{blue}{\frac{D \cdot M}{\frac{d}{D}}}\right)\right) \]

   10. Simplified 50.5%

       \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)} \]

   if 1.05e-112 < d < 9.49999999999999996e133

   1. Initial program 85.2%

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

      1. associate-*l* 85.1%

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

      2. metadata-eval 85.1%

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

      3. unpow1/2 85.1%

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

      4. metadata-eval 85.1%

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

      5. unpow1/2 85.1%

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

      6. associate-*l* 85.1%

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

      7. metadata-eval 85.1%

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

      8. times-frac 85.2%

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

   3. Simplified 85.2%

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

      1. associate-*r/ 85.3%

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

      2. frac-times 85.2%

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

      3. *-commutative 85.2%

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

      4. *-commutative 85.2%

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

      5. associate-*r/ 81.8%

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

   5. Applied egg-rr 81.8%

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

   if 9.49999999999999996e133 < d

   1. Initial program 66.3%

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

   2. Taylor expanded in d around inf 79.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-commutative 79.7%

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

      2. *-commutative 79.7%

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

      3. associate-/r* 79.7%

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

   4. Simplified 79.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   5. Step-by-step derivation

      1. sqrt-div 90.7%

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

      2. inv-pow 90.7%

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

   6. Applied egg-rr 90.7%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}}}{\sqrt{\ell}}} \]
   7. Step-by-step derivation

      1. sqr-pow 90.7%

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

      2. rem-sqrt-square 90.7%

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

      3. metadata-eval 90.7%

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

      4. sqr-pow 90.8%

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

      5. fabs-sqr 90.8%

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

      6. sqr-pow 90.7%

         \[\leadsto d \cdot \frac{\color{blue}{{h}^{-0.5}}}{\sqrt{\ell}} \]

   8. Simplified 90.7%

      \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.45 \cdot 10^{+113}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -3.7 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)\right) \cdot 0.25\right)\right)\right)\\ \mathbf{elif}\;d \leq -5.4 \cdot 10^{-238}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right) \cdot -0.125\right)\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{+133}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 4: 70.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{h}{{\ell}^{3}}}\\ t_3 := \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)\\ t_4 := M \cdot \frac{D}{d}\\ t_5 := M \cdot \frac{D \cdot M}{\frac{d}{D}}\\ \mathbf{if}\;d \leq -1.1 \cdot 10^{+109}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq -4.4 \cdot 10^{-56}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(t_4 \cdot t_4\right)\right) \cdot 0.25\right)\right)\right)\\ \mathbf{elif}\;d \leq -4.4 \cdot 10^{-238}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_2 \cdot \left(0.125 \cdot t_5\right)\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{-113}:\\ \;\;\;\;t_2 \cdot \left(t_5 \cdot -0.125\right)\\ \mathbf{elif}\;d \leq 1.08 \cdot 10^{+134}:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot 0.5\right) \cdot {\left(\frac{\frac{D \cdot M}{2}}{d}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (sqrt (/ d l)))
        (t_2 (sqrt (/ h (pow l 3.0))))
        (t_3
         (*
          (* d (sqrt (/ 1.0 (* h l))))
          (- -1.0 (* -0.5 (* (/ h l) (pow (* D (/ M (* 2.0 d))) 2.0))))))
        (t_4 (* M (/ D d)))
        (t_5 (* M (/ (* D M) (/ d D)))))
   (if (<= d -1.1e+109)
     t_3
     (if (<= d -4.4e-56)
       (* t_0 (* t_1 (- 1.0 (* 0.5 (* (* (/ h l) (* t_4 t_4)) 0.25)))))
       (if (<= d -4.4e-238)
         t_3
         (if (<= d -2e-310)
           (* t_2 (* 0.125 t_5))
           (if (<= d 7.2e-113)
             (* t_2 (* t_5 -0.125))
             (if (<= d 1.08e+134)
               (*
                (* t_0 t_1)
                (- 1.0 (* (* (/ h l) 0.5) (pow (/ (/ (* D M) 2.0) d) 2.0))))
               (* d (/ (pow h -0.5) (sqrt l)))))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = sqrt((d / l));
	double t_2 = sqrt((h / pow(l, 3.0)));
	double t_3 = (d * sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * pow((D * (M / (2.0 * d))), 2.0))));
	double t_4 = M * (D / d);
	double t_5 = M * ((D * M) / (d / D));
	double tmp;
	if (d <= -1.1e+109) {
		tmp = t_3;
	} else if (d <= -4.4e-56) {
		tmp = t_0 * (t_1 * (1.0 - (0.5 * (((h / l) * (t_4 * t_4)) * 0.25))));
	} else if (d <= -4.4e-238) {
		tmp = t_3;
	} else if (d <= -2e-310) {
		tmp = t_2 * (0.125 * t_5);
	} else if (d <= 7.2e-113) {
		tmp = t_2 * (t_5 * -0.125);
	} else if (d <= 1.08e+134) {
		tmp = (t_0 * t_1) * (1.0 - (((h / l) * 0.5) * pow((((D * M) / 2.0) / d), 2.0)));
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this 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 = sqrt((d / h))
    t_1 = sqrt((d / l))
    t_2 = sqrt((h / (l ** 3.0d0)))
    t_3 = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) - ((-0.5d0) * ((h / l) * ((d_1 * (m / (2.0d0 * d))) ** 2.0d0))))
    t_4 = m * (d_1 / d)
    t_5 = m * ((d_1 * m) / (d / d_1))
    if (d <= (-1.1d+109)) then
        tmp = t_3
    else if (d <= (-4.4d-56)) then
        tmp = t_0 * (t_1 * (1.0d0 - (0.5d0 * (((h / l) * (t_4 * t_4)) * 0.25d0))))
    else if (d <= (-4.4d-238)) then
        tmp = t_3
    else if (d <= (-2d-310)) then
        tmp = t_2 * (0.125d0 * t_5)
    else if (d <= 7.2d-113) then
        tmp = t_2 * (t_5 * (-0.125d0))
    else if (d <= 1.08d+134) then
        tmp = (t_0 * t_1) * (1.0d0 - (((h / l) * 0.5d0) * ((((d_1 * m) / 2.0d0) / d) ** 2.0d0)))
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / h));
	double t_1 = Math.sqrt((d / l));
	double t_2 = Math.sqrt((h / Math.pow(l, 3.0)));
	double t_3 = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * Math.pow((D * (M / (2.0 * d))), 2.0))));
	double t_4 = M * (D / d);
	double t_5 = M * ((D * M) / (d / D));
	double tmp;
	if (d <= -1.1e+109) {
		tmp = t_3;
	} else if (d <= -4.4e-56) {
		tmp = t_0 * (t_1 * (1.0 - (0.5 * (((h / l) * (t_4 * t_4)) * 0.25))));
	} else if (d <= -4.4e-238) {
		tmp = t_3;
	} else if (d <= -2e-310) {
		tmp = t_2 * (0.125 * t_5);
	} else if (d <= 7.2e-113) {
		tmp = t_2 * (t_5 * -0.125);
	} else if (d <= 1.08e+134) {
		tmp = (t_0 * t_1) * (1.0 - (((h / l) * 0.5) * Math.pow((((D * M) / 2.0) / d), 2.0)));
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt((d / h))
	t_1 = math.sqrt((d / l))
	t_2 = math.sqrt((h / math.pow(l, 3.0)))
	t_3 = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * math.pow((D * (M / (2.0 * d))), 2.0))))
	t_4 = M * (D / d)
	t_5 = M * ((D * M) / (d / D))
	tmp = 0
	if d <= -1.1e+109:
		tmp = t_3
	elif d <= -4.4e-56:
		tmp = t_0 * (t_1 * (1.0 - (0.5 * (((h / l) * (t_4 * t_4)) * 0.25))))
	elif d <= -4.4e-238:
		tmp = t_3
	elif d <= -2e-310:
		tmp = t_2 * (0.125 * t_5)
	elif d <= 7.2e-113:
		tmp = t_2 * (t_5 * -0.125)
	elif d <= 1.08e+134:
		tmp = (t_0 * t_1) * (1.0 - (((h / l) * 0.5) * math.pow((((D * M) / 2.0) / d), 2.0)))
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(d / l))
	t_2 = sqrt(Float64(h / (l ^ 3.0)))
	t_3 = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 - Float64(-0.5 * Float64(Float64(h / l) * (Float64(D * Float64(M / Float64(2.0 * d))) ^ 2.0)))))
	t_4 = Float64(M * Float64(D / d))
	t_5 = Float64(M * Float64(Float64(D * M) / Float64(d / D)))
	tmp = 0.0
	if (d <= -1.1e+109)
		tmp = t_3;
	elseif (d <= -4.4e-56)
		tmp = Float64(t_0 * Float64(t_1 * Float64(1.0 - Float64(0.5 * Float64(Float64(Float64(h / l) * Float64(t_4 * t_4)) * 0.25)))));
	elseif (d <= -4.4e-238)
		tmp = t_3;
	elseif (d <= -2e-310)
		tmp = Float64(t_2 * Float64(0.125 * t_5));
	elseif (d <= 7.2e-113)
		tmp = Float64(t_2 * Float64(t_5 * -0.125));
	elseif (d <= 1.08e+134)
		tmp = Float64(Float64(t_0 * t_1) * Float64(1.0 - Float64(Float64(Float64(h / l) * 0.5) * (Float64(Float64(Float64(D * M) / 2.0) / d) ^ 2.0))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / h));
	t_1 = sqrt((d / l));
	t_2 = sqrt((h / (l ^ 3.0)));
	t_3 = (d * sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * ((D * (M / (2.0 * d))) ^ 2.0))));
	t_4 = M * (D / d);
	t_5 = M * ((D * M) / (d / D));
	tmp = 0.0;
	if (d <= -1.1e+109)
		tmp = t_3;
	elseif (d <= -4.4e-56)
		tmp = t_0 * (t_1 * (1.0 - (0.5 * (((h / l) * (t_4 * t_4)) * 0.25))));
	elseif (d <= -4.4e-238)
		tmp = t_3;
	elseif (d <= -2e-310)
		tmp = t_2 * (0.125 * t_5);
	elseif (d <= 7.2e-113)
		tmp = t_2 * (t_5 * -0.125);
	elseif (d <= 1.08e+134)
		tmp = (t_0 * t_1) * (1.0 - (((h / l) * 0.5) * ((((D * M) / 2.0) / d) ^ 2.0)));
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(M * N[(N[(D * M), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.1e+109], t$95$3, If[LessEqual[d, -4.4e-56], N[(t$95$0 * N[(t$95$1 * N[(1.0 - N[(0.5 * N[(N[(N[(h / l), $MachinePrecision] * N[(t$95$4 * t$95$4), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4.4e-238], t$95$3, If[LessEqual[d, -2e-310], N[(t$95$2 * N[(0.125 * t$95$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.2e-113], N[(t$95$2 * N[(t$95$5 * -0.125), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.08e+134], N[(N[(t$95$0 * t$95$1), $MachinePrecision] * N[(1.0 - N[(N[(N[(h / l), $MachinePrecision] * 0.5), $MachinePrecision] * N[Power[N[(N[(N[(D * M), $MachinePrecision] / 2.0), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if d < -1.1e109 or -4.40000000000000008e-56 < d < -4.39999999999999982e-238

   1. Initial program 67.6%

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

   3. Applied egg-rr 22.4%

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

      1. expm1-def 27.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 56.4%

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

   5. Simplified 56.4%

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

   6. Taylor expanded in d around -inf 81.8%

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

      1. mul-1-neg 81.8%

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

      2. distribute-rgt-neg-in 81.8%

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

      3. *-commutative 81.8%

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

   8. Simplified 81.8%

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

   if -1.1e109 < d < -4.40000000000000008e-56

   1. Initial program 91.6%

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

      1. associate-*l* 91.6%

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

      2. metadata-eval 91.6%

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

      3. unpow1/2 91.6%

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

      4. metadata-eval 91.6%

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

      5. unpow1/2 91.6%

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

      6. associate-*l* 91.6%

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

      7. metadata-eval 91.6%

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

      8. times-frac 91.6%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in M around 0 80.1%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)}\right)\right) \]
   5. Step-by-step derivation

      1. associate-*r/ 80.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right)\right) \]

      2. *-commutative 80.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right) \]

      3. associate-*r/ 80.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      4. *-commutative 80.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot 0.25\right)}\right)\right) \]

      5. associate-*r* 77.3%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell} \cdot 0.25\right)\right)\right) \]

      6. *-commutative 77.3%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{\color{blue}{h \cdot \left({D}^{2} \cdot {M}^{2}\right)}}{{d}^{2} \cdot \ell} \cdot 0.25\right)\right)\right) \]

      7. *-commutative 77.3%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h \cdot \left({D}^{2} \cdot {M}^{2}\right)}{\color{blue}{\ell \cdot {d}^{2}}} \cdot 0.25\right)\right)\right) \]

      8. times-frac 80.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\color{blue}{\left(\frac{h}{\ell} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)} \cdot 0.25\right)\right)\right) \]

      9. *-commutative 80.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}}\right) \cdot 0.25\right)\right)\right) \]

      10. unpow2 80.2%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot {D}^{2}}{{d}^{2}}\right) \cdot 0.25\right)\right)\right) \]

      11. unpow2 80.2%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot \color{blue}{\left(D \cdot D\right)}}{{d}^{2}}\right) \cdot 0.25\right)\right)\right) \]

      12. unswap-sqr 88.7%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{\color{blue}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}}{{d}^{2}}\right) \cdot 0.25\right)\right)\right) \]

      13. unpow2 88.7%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{d \cdot d}}\right) \cdot 0.25\right)\right)\right) \]

      14. times-frac 91.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \color{blue}{\left(\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d}\right)}\right) \cdot 0.25\right)\right)\right) \]

      15. associate-*r/ 91.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(\color{blue}{\left(M \cdot \frac{D}{d}\right)} \cdot \frac{M \cdot D}{d}\right)\right) \cdot 0.25\right)\right)\right) \]

      16. associate-*r/ 91.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(M \cdot \frac{D}{d}\right)}\right)\right) \cdot 0.25\right)\right)\right) \]

   6. Simplified 91.6%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\left(\frac{h}{\ell} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)\right) \cdot 0.25\right)}\right)\right) \]

   if -4.39999999999999982e-238 < d < -1.999999999999994e-310

   1. Initial program 29.9%

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

   3. Applied egg-rr 5.9%

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

      1. expm1-def 5.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 24.0%

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

   5. Simplified 24.0%

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

   6. Taylor expanded in d around -inf 35.8%

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

      1. mul-1-neg 35.8%

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

      2. distribute-rgt-neg-in 35.8%

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

      3. *-commutative 35.8%

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

   8. Simplified 35.8%

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

   9. Taylor expanded in D around inf 65.3%

      \[\leadsto \color{blue}{0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 65.3%

          \[\leadsto 0.125 \cdot \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

       2. unpow2 65.3%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{d}\right) \]

       3. unpow2 65.3%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{d}\right) \]

       4. associate-*l/ 59.5%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)}\right) \]

       5. *-commutative 59.5%

          \[\leadsto \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)\right) \cdot 0.125} \]

       6. associate-*l* 59.5%

          \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right) \cdot 0.125\right)} \]

       7. *-commutative 59.5%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(0.125 \cdot \left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)\right)} \]

       8. associate-*r* 60.3%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \color{blue}{\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot M\right)}\right) \]

       9. *-commutative 60.3%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \color{blue}{\left(M \cdot \left(\frac{D \cdot D}{d} \cdot M\right)\right)}\right) \]

       10. associate-/l* 60.2%

           \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \left(\color{blue}{\frac{D}{\frac{d}{D}}} \cdot M\right)\right)\right) \]

       11. associate-*l/ 65.9%

           \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \color{blue}{\frac{D \cdot M}{\frac{d}{D}}}\right)\right) \]

   11. Simplified 65.9%

       \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)} \]

   if -1.999999999999994e-310 < d < 7.1999999999999995e-113

   1. Initial program 35.1%

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

      1. metadata-eval 35.1%

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

      2. unpow1/2 35.1%

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

      3. metadata-eval 35.1%

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

      4. unpow1/2 35.1%

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

      5. *-commutative 35.1%

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

      6. associate-*l* 35.1%

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

      7. times-frac 35.2%

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

      8. metadata-eval 35.2%

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

   3. Simplified 35.2%

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

      1. frac-times 35.1%

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

      2. clear-num 35.2%

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

      3. *-commutative 35.2%

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

   5. Applied egg-rr 35.2%

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

      1. associate-*r* 35.2%

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

      2. *-commutative 35.2%

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

      3. clear-num 35.1%

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

      4. *-commutative 35.1%

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

      5. metadata-eval 35.1%

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

      6. associate-*r/ 37.4%

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

      7. metadata-eval 37.4%

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

      8. times-frac 37.5%

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

   7. Applied egg-rr 37.5%

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

   8. Taylor expanded in d around 0 40.1%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 40.1%

         \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative 40.1%

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      3. unpow2 40.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{d}\right) \]

      4. unpow2 40.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{d}\right) \]

      5. associate-*l/ 39.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)}\right) \]

      6. associate-*r* 42.7%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot M\right)}\right) \]

      7. *-commutative 42.7%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(M \cdot \left(\frac{D \cdot D}{d} \cdot M\right)\right)}\right) \]

      8. associate-/l* 48.0%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(M \cdot \left(\color{blue}{\frac{D}{\frac{d}{D}}} \cdot M\right)\right)\right) \]

      9. associate-*l/ 50.5%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(M \cdot \color{blue}{\frac{D \cdot M}{\frac{d}{D}}}\right)\right) \]

   10. Simplified 50.5%

       \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)} \]

   if 7.1999999999999995e-113 < d < 1.0800000000000001e134

   1. Initial program 85.2%

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

      1. metadata-eval 85.2%

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

      2. unpow1/2 85.2%

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

      3. metadata-eval 85.2%

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

      4. unpow1/2 85.2%

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

      5. *-commutative 85.2%

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

      6. associate-*l* 85.2%

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

      7. times-frac 85.2%

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

      8. metadata-eval 85.2%

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

   3. Simplified 85.2%

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

      1. frac-times 87.0%

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

      2. associate-/r* 87.0%

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

   5. Applied egg-rr 85.2%

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

   if 1.0800000000000001e134 < d

   1. Initial program 66.3%

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

   2. Taylor expanded in d around inf 79.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-commutative 79.7%

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

      2. *-commutative 79.7%

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

      3. associate-/r* 79.7%

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

   4. Simplified 79.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   5. Step-by-step derivation

      1. sqrt-div 90.7%

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

      2. inv-pow 90.7%

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

   6. Applied egg-rr 90.7%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}}}{\sqrt{\ell}}} \]
   7. Step-by-step derivation

      1. sqr-pow 90.7%

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

      2. rem-sqrt-square 90.7%

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

      3. metadata-eval 90.7%

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

      4. sqr-pow 90.8%

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

      5. fabs-sqr 90.8%

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

      6. sqr-pow 90.7%

         \[\leadsto d \cdot \frac{\color{blue}{{h}^{-0.5}}}{\sqrt{\ell}} \]

   8. Simplified 90.7%

      \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{+109}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -4.4 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)\right) \cdot 0.25\right)\right)\right)\\ \mathbf{elif}\;d \leq -4.4 \cdot 10^{-238}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{-113}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right) \cdot -0.125\right)\\ \mathbf{elif}\;d \leq 1.08 \cdot 10^{+134}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot 0.5\right) \cdot {\left(\frac{\frac{D \cdot M}{2}}{d}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 5: 71.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)\\ t_3 := M \cdot \frac{D}{d}\\ \mathbf{if}\;d \leq -3.15 \cdot 10^{+111}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-56}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(t_3 \cdot t_3\right)\right) \cdot 0.25\right)\right)\right)\\ \mathbf{elif}\;d \leq -4.4 \cdot 10^{-238}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-236}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right)\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{+134}:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (sqrt (/ d l)))
        (t_2
         (*
          (* d (sqrt (/ 1.0 (* h l))))
          (- -1.0 (* -0.5 (* (/ h l) (pow (* D (/ M (* 2.0 d))) 2.0))))))
        (t_3 (* M (/ D d))))
   (if (<= d -3.15e+111)
     t_2
     (if (<= d -5e-56)
       (* t_0 (* t_1 (- 1.0 (* 0.5 (* (* (/ h l) (* t_3 t_3)) 0.25)))))
       (if (<= d -4.4e-238)
         t_2
         (if (<= d -2e-310)
           (* (sqrt (/ h (pow l 3.0))) (* 0.125 (* M (/ (* D M) (/ d D)))))
           (if (<= d 4e-236)
             (* -0.125 (* (/ (* D D) (/ d (* M M))) (/ (sqrt h) (pow l 1.5))))
             (if (<= d 3.5e+134)
               (*
                (* t_0 t_1)
                (- 1.0 (/ (* h (* 0.5 (pow (* (/ D d) (/ M 2.0)) 2.0))) l)))
               (* d (/ (pow h -0.5) (sqrt l)))))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = sqrt((d / l));
	double t_2 = (d * sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * pow((D * (M / (2.0 * d))), 2.0))));
	double t_3 = M * (D / d);
	double tmp;
	if (d <= -3.15e+111) {
		tmp = t_2;
	} else if (d <= -5e-56) {
		tmp = t_0 * (t_1 * (1.0 - (0.5 * (((h / l) * (t_3 * t_3)) * 0.25))));
	} else if (d <= -4.4e-238) {
		tmp = t_2;
	} else if (d <= -2e-310) {
		tmp = sqrt((h / pow(l, 3.0))) * (0.125 * (M * ((D * M) / (d / D))));
	} else if (d <= 4e-236) {
		tmp = -0.125 * (((D * D) / (d / (M * M))) * (sqrt(h) / pow(l, 1.5)));
	} else if (d <= 3.5e+134) {
		tmp = (t_0 * t_1) * (1.0 - ((h * (0.5 * pow(((D / d) * (M / 2.0)), 2.0))) / l));
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this 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 = sqrt((d / l))
    t_2 = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) - ((-0.5d0) * ((h / l) * ((d_1 * (m / (2.0d0 * d))) ** 2.0d0))))
    t_3 = m * (d_1 / d)
    if (d <= (-3.15d+111)) then
        tmp = t_2
    else if (d <= (-5d-56)) then
        tmp = t_0 * (t_1 * (1.0d0 - (0.5d0 * (((h / l) * (t_3 * t_3)) * 0.25d0))))
    else if (d <= (-4.4d-238)) then
        tmp = t_2
    else if (d <= (-2d-310)) then
        tmp = sqrt((h / (l ** 3.0d0))) * (0.125d0 * (m * ((d_1 * m) / (d / d_1))))
    else if (d <= 4d-236) then
        tmp = (-0.125d0) * (((d_1 * d_1) / (d / (m * m))) * (sqrt(h) / (l ** 1.5d0)))
    else if (d <= 3.5d+134) then
        tmp = (t_0 * t_1) * (1.0d0 - ((h * (0.5d0 * (((d_1 / d) * (m / 2.0d0)) ** 2.0d0))) / l))
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / h));
	double t_1 = Math.sqrt((d / l));
	double t_2 = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * Math.pow((D * (M / (2.0 * d))), 2.0))));
	double t_3 = M * (D / d);
	double tmp;
	if (d <= -3.15e+111) {
		tmp = t_2;
	} else if (d <= -5e-56) {
		tmp = t_0 * (t_1 * (1.0 - (0.5 * (((h / l) * (t_3 * t_3)) * 0.25))));
	} else if (d <= -4.4e-238) {
		tmp = t_2;
	} else if (d <= -2e-310) {
		tmp = Math.sqrt((h / Math.pow(l, 3.0))) * (0.125 * (M * ((D * M) / (d / D))));
	} else if (d <= 4e-236) {
		tmp = -0.125 * (((D * D) / (d / (M * M))) * (Math.sqrt(h) / Math.pow(l, 1.5)));
	} else if (d <= 3.5e+134) {
		tmp = (t_0 * t_1) * (1.0 - ((h * (0.5 * Math.pow(((D / d) * (M / 2.0)), 2.0))) / l));
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt((d / h))
	t_1 = math.sqrt((d / l))
	t_2 = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * math.pow((D * (M / (2.0 * d))), 2.0))))
	t_3 = M * (D / d)
	tmp = 0
	if d <= -3.15e+111:
		tmp = t_2
	elif d <= -5e-56:
		tmp = t_0 * (t_1 * (1.0 - (0.5 * (((h / l) * (t_3 * t_3)) * 0.25))))
	elif d <= -4.4e-238:
		tmp = t_2
	elif d <= -2e-310:
		tmp = math.sqrt((h / math.pow(l, 3.0))) * (0.125 * (M * ((D * M) / (d / D))))
	elif d <= 4e-236:
		tmp = -0.125 * (((D * D) / (d / (M * M))) * (math.sqrt(h) / math.pow(l, 1.5)))
	elif d <= 3.5e+134:
		tmp = (t_0 * t_1) * (1.0 - ((h * (0.5 * math.pow(((D / d) * (M / 2.0)), 2.0))) / l))
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(d / l))
	t_2 = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 - Float64(-0.5 * Float64(Float64(h / l) * (Float64(D * Float64(M / Float64(2.0 * d))) ^ 2.0)))))
	t_3 = Float64(M * Float64(D / d))
	tmp = 0.0
	if (d <= -3.15e+111)
		tmp = t_2;
	elseif (d <= -5e-56)
		tmp = Float64(t_0 * Float64(t_1 * Float64(1.0 - Float64(0.5 * Float64(Float64(Float64(h / l) * Float64(t_3 * t_3)) * 0.25)))));
	elseif (d <= -4.4e-238)
		tmp = t_2;
	elseif (d <= -2e-310)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(0.125 * Float64(M * Float64(Float64(D * M) / Float64(d / D)))));
	elseif (d <= 4e-236)
		tmp = Float64(-0.125 * Float64(Float64(Float64(D * D) / Float64(d / Float64(M * M))) * Float64(sqrt(h) / (l ^ 1.5))));
	elseif (d <= 3.5e+134)
		tmp = Float64(Float64(t_0 * t_1) * Float64(1.0 - Float64(Float64(h * Float64(0.5 * (Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0))) / l)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / h));
	t_1 = sqrt((d / l));
	t_2 = (d * sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * ((D * (M / (2.0 * d))) ^ 2.0))));
	t_3 = M * (D / d);
	tmp = 0.0;
	if (d <= -3.15e+111)
		tmp = t_2;
	elseif (d <= -5e-56)
		tmp = t_0 * (t_1 * (1.0 - (0.5 * (((h / l) * (t_3 * t_3)) * 0.25))));
	elseif (d <= -4.4e-238)
		tmp = t_2;
	elseif (d <= -2e-310)
		tmp = sqrt((h / (l ^ 3.0))) * (0.125 * (M * ((D * M) / (d / D))));
	elseif (d <= 4e-236)
		tmp = -0.125 * (((D * D) / (d / (M * M))) * (sqrt(h) / (l ^ 1.5)));
	elseif (d <= 3.5e+134)
		tmp = (t_0 * t_1) * (1.0 - ((h * (0.5 * (((D / d) * (M / 2.0)) ^ 2.0))) / l));
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.15e+111], t$95$2, If[LessEqual[d, -5e-56], N[(t$95$0 * N[(t$95$1 * N[(1.0 - N[(0.5 * N[(N[(N[(h / l), $MachinePrecision] * N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4.4e-238], t$95$2, If[LessEqual[d, -2e-310], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.125 * N[(M * N[(N[(D * M), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4e-236], N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] / N[(d / N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.5e+134], N[(N[(t$95$0 * t$95$1), $MachinePrecision] * N[(1.0 - N[(N[(h * N[(0.5 * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if d < -3.1500000000000001e111 or -4.99999999999999997e-56 < d < -4.39999999999999982e-238

   1. Initial program 67.6%

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

   3. Applied egg-rr 22.4%

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

      1. expm1-def 27.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 56.4%

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

   5. Simplified 56.4%

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

   6. Taylor expanded in d around -inf 81.8%

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

      1. mul-1-neg 81.8%

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

      2. distribute-rgt-neg-in 81.8%

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

      3. *-commutative 81.8%

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

   8. Simplified 81.8%

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

   if -3.1500000000000001e111 < d < -4.99999999999999997e-56

   1. Initial program 91.6%

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

      1. associate-*l* 91.6%

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

      2. metadata-eval 91.6%

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

      3. unpow1/2 91.6%

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

      4. metadata-eval 91.6%

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

      5. unpow1/2 91.6%

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

      6. associate-*l* 91.6%

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

      7. metadata-eval 91.6%

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

      8. times-frac 91.6%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in M around 0 80.1%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)}\right)\right) \]
   5. Step-by-step derivation

      1. associate-*r/ 80.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right)\right) \]

      2. *-commutative 80.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right) \]

      3. associate-*r/ 80.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      4. *-commutative 80.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot 0.25\right)}\right)\right) \]

      5. associate-*r* 77.3%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell} \cdot 0.25\right)\right)\right) \]

      6. *-commutative 77.3%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{\color{blue}{h \cdot \left({D}^{2} \cdot {M}^{2}\right)}}{{d}^{2} \cdot \ell} \cdot 0.25\right)\right)\right) \]

      7. *-commutative 77.3%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h \cdot \left({D}^{2} \cdot {M}^{2}\right)}{\color{blue}{\ell \cdot {d}^{2}}} \cdot 0.25\right)\right)\right) \]

      8. times-frac 80.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\color{blue}{\left(\frac{h}{\ell} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)} \cdot 0.25\right)\right)\right) \]

      9. *-commutative 80.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}}\right) \cdot 0.25\right)\right)\right) \]

      10. unpow2 80.2%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot {D}^{2}}{{d}^{2}}\right) \cdot 0.25\right)\right)\right) \]

      11. unpow2 80.2%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot \color{blue}{\left(D \cdot D\right)}}{{d}^{2}}\right) \cdot 0.25\right)\right)\right) \]

      12. unswap-sqr 88.7%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{\color{blue}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}}{{d}^{2}}\right) \cdot 0.25\right)\right)\right) \]

      13. unpow2 88.7%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{d \cdot d}}\right) \cdot 0.25\right)\right)\right) \]

      14. times-frac 91.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \color{blue}{\left(\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d}\right)}\right) \cdot 0.25\right)\right)\right) \]

      15. associate-*r/ 91.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(\color{blue}{\left(M \cdot \frac{D}{d}\right)} \cdot \frac{M \cdot D}{d}\right)\right) \cdot 0.25\right)\right)\right) \]

      16. associate-*r/ 91.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(M \cdot \frac{D}{d}\right)}\right)\right) \cdot 0.25\right)\right)\right) \]

   6. Simplified 91.6%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\left(\frac{h}{\ell} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)\right) \cdot 0.25\right)}\right)\right) \]

   if -4.39999999999999982e-238 < d < -1.999999999999994e-310

   1. Initial program 29.9%

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

   3. Applied egg-rr 5.9%

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

      1. expm1-def 5.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 24.0%

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

   5. Simplified 24.0%

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

   6. Taylor expanded in d around -inf 35.8%

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

      1. mul-1-neg 35.8%

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

      2. distribute-rgt-neg-in 35.8%

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

      3. *-commutative 35.8%

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

   8. Simplified 35.8%

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

   9. Taylor expanded in D around inf 65.3%

      \[\leadsto \color{blue}{0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 65.3%

          \[\leadsto 0.125 \cdot \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

       2. unpow2 65.3%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{d}\right) \]

       3. unpow2 65.3%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{d}\right) \]

       4. associate-*l/ 59.5%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)}\right) \]

       5. *-commutative 59.5%

          \[\leadsto \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)\right) \cdot 0.125} \]

       6. associate-*l* 59.5%

          \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right) \cdot 0.125\right)} \]

       7. *-commutative 59.5%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(0.125 \cdot \left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)\right)} \]

       8. associate-*r* 60.3%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \color{blue}{\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot M\right)}\right) \]

       9. *-commutative 60.3%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \color{blue}{\left(M \cdot \left(\frac{D \cdot D}{d} \cdot M\right)\right)}\right) \]

       10. associate-/l* 60.2%

           \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \left(\color{blue}{\frac{D}{\frac{d}{D}}} \cdot M\right)\right)\right) \]

       11. associate-*l/ 65.9%

           \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \color{blue}{\frac{D \cdot M}{\frac{d}{D}}}\right)\right) \]

   11. Simplified 65.9%

       \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)} \]

   if -1.999999999999994e-310 < d < 4.0000000000000002e-236

   1. Initial program 33.4%

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

   2. Taylor expanded in d around 0 49.2%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   3. Step-by-step derivation

      1. associate-/l* 44.3%

         \[\leadsto -0.125 \cdot \left(\color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      2. unpow2 44.3%

         \[\leadsto -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\frac{d}{{M}^{2}}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      3. unpow2 44.3%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{\color{blue}{M \cdot M}}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

   4. Simplified 44.3%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   5. Step-by-step derivation

      1. sqrt-div 44.3%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}}\right) \]

   6. Applied egg-rr 44.3%

      \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}}\right) \]
   7. Step-by-step derivation

      1. sqr-pow 44.3%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{\sqrt{h}}{\sqrt{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)} \cdot {\ell}^{\left(\frac{3}{2}\right)}}}}\right) \]

      2. rem-sqrt-square 54.3%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{\sqrt{h}}{\color{blue}{\left|{\ell}^{\left(\frac{3}{2}\right)}\right|}}\right) \]

      3. sqr-pow 54.3%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{\sqrt{h}}{\left|\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}\right|}\right) \]

      4. fabs-sqr 54.3%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}}\right) \]

      5. sqr-pow 54.3%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)}}}\right) \]

      6. metadata-eval 54.3%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}}\right) \]

   8. Simplified 54.3%

      \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \color{blue}{\frac{\sqrt{h}}{{\ell}^{1.5}}}\right) \]

   if 4.0000000000000002e-236 < d < 3.50000000000000003e134

   1. Initial program 73.6%

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

      1. metadata-eval 73.6%

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

      2. unpow1/2 73.6%

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

      3. metadata-eval 73.6%

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

      4. unpow1/2 73.6%

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

      5. *-commutative 73.6%

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

      6. associate-*l* 73.6%

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

      7. times-frac 73.7%

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

      8. metadata-eval 73.7%

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

   3. Simplified 73.7%

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

      1. frac-times 73.6%

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

      2. clear-num 73.7%

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

      3. *-commutative 73.7%

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

   5. Applied egg-rr 73.7%

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

      1. associate-*r* 73.7%

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

      2. *-commutative 73.7%

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

      3. clear-num 73.6%

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

      4. *-commutative 73.6%

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

      5. metadata-eval 73.6%

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

      6. associate-*r/ 76.2%

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

      7. metadata-eval 76.2%

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

      8. times-frac 76.3%

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

   7. Applied egg-rr 76.3%

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

   if 3.50000000000000003e134 < d

   1. Initial program 66.3%

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

   2. Taylor expanded in d around inf 79.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-commutative 79.7%

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

      2. *-commutative 79.7%

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

      3. associate-/r* 79.7%

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

   4. Simplified 79.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   5. Step-by-step derivation

      1. sqrt-div 90.7%

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

      2. inv-pow 90.7%

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

   6. Applied egg-rr 90.7%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}}}{\sqrt{\ell}}} \]
   7. Step-by-step derivation

      1. sqr-pow 90.7%

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

      2. rem-sqrt-square 90.7%

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

      3. metadata-eval 90.7%

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

      4. sqr-pow 90.8%

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

      5. fabs-sqr 90.8%

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

      6. sqr-pow 90.7%

         \[\leadsto d \cdot \frac{\color{blue}{{h}^{-0.5}}}{\sqrt{\ell}} \]

   8. Simplified 90.7%

      \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.15 \cdot 10^{+111}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)\right) \cdot 0.25\right)\right)\right)\\ \mathbf{elif}\;d \leq -4.4 \cdot 10^{-238}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-236}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right)\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{+134}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 6: 71.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)\\ t_3 := M \cdot \frac{D}{d}\\ \mathbf{if}\;d \leq -8 \cdot 10^{+113}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -5.2 \cdot 10^{-56}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(t_3 \cdot t_3\right)\right) \cdot 0.25\right)\right)\right)\\ \mathbf{elif}\;d \leq -5.2 \cdot 10^{-238}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-236}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right)\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{+134}:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{\frac{D \cdot M}{2}}{d}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (sqrt (/ d l)))
        (t_2
         (*
          (* d (sqrt (/ 1.0 (* h l))))
          (- -1.0 (* -0.5 (* (/ h l) (pow (* D (/ M (* 2.0 d))) 2.0))))))
        (t_3 (* M (/ D d))))
   (if (<= d -8e+113)
     t_2
     (if (<= d -5.2e-56)
       (* t_0 (* t_1 (- 1.0 (* 0.5 (* (* (/ h l) (* t_3 t_3)) 0.25)))))
       (if (<= d -5.2e-238)
         t_2
         (if (<= d -2e-310)
           (* (sqrt (/ h (pow l 3.0))) (* 0.125 (* M (/ (* D M) (/ d D)))))
           (if (<= d 4e-236)
             (* -0.125 (* (/ (* D D) (/ d (* M M))) (/ (sqrt h) (pow l 1.5))))
             (if (<= d 1.9e+134)
               (*
                (* t_0 t_1)
                (- 1.0 (/ (* h (* 0.5 (pow (/ (/ (* D M) 2.0) d) 2.0))) l)))
               (* d (/ (pow h -0.5) (sqrt l)))))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = sqrt((d / l));
	double t_2 = (d * sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * pow((D * (M / (2.0 * d))), 2.0))));
	double t_3 = M * (D / d);
	double tmp;
	if (d <= -8e+113) {
		tmp = t_2;
	} else if (d <= -5.2e-56) {
		tmp = t_0 * (t_1 * (1.0 - (0.5 * (((h / l) * (t_3 * t_3)) * 0.25))));
	} else if (d <= -5.2e-238) {
		tmp = t_2;
	} else if (d <= -2e-310) {
		tmp = sqrt((h / pow(l, 3.0))) * (0.125 * (M * ((D * M) / (d / D))));
	} else if (d <= 4e-236) {
		tmp = -0.125 * (((D * D) / (d / (M * M))) * (sqrt(h) / pow(l, 1.5)));
	} else if (d <= 1.9e+134) {
		tmp = (t_0 * t_1) * (1.0 - ((h * (0.5 * pow((((D * M) / 2.0) / d), 2.0))) / l));
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this 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 = sqrt((d / l))
    t_2 = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) - ((-0.5d0) * ((h / l) * ((d_1 * (m / (2.0d0 * d))) ** 2.0d0))))
    t_3 = m * (d_1 / d)
    if (d <= (-8d+113)) then
        tmp = t_2
    else if (d <= (-5.2d-56)) then
        tmp = t_0 * (t_1 * (1.0d0 - (0.5d0 * (((h / l) * (t_3 * t_3)) * 0.25d0))))
    else if (d <= (-5.2d-238)) then
        tmp = t_2
    else if (d <= (-2d-310)) then
        tmp = sqrt((h / (l ** 3.0d0))) * (0.125d0 * (m * ((d_1 * m) / (d / d_1))))
    else if (d <= 4d-236) then
        tmp = (-0.125d0) * (((d_1 * d_1) / (d / (m * m))) * (sqrt(h) / (l ** 1.5d0)))
    else if (d <= 1.9d+134) then
        tmp = (t_0 * t_1) * (1.0d0 - ((h * (0.5d0 * ((((d_1 * m) / 2.0d0) / d) ** 2.0d0))) / l))
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / h));
	double t_1 = Math.sqrt((d / l));
	double t_2 = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * Math.pow((D * (M / (2.0 * d))), 2.0))));
	double t_3 = M * (D / d);
	double tmp;
	if (d <= -8e+113) {
		tmp = t_2;
	} else if (d <= -5.2e-56) {
		tmp = t_0 * (t_1 * (1.0 - (0.5 * (((h / l) * (t_3 * t_3)) * 0.25))));
	} else if (d <= -5.2e-238) {
		tmp = t_2;
	} else if (d <= -2e-310) {
		tmp = Math.sqrt((h / Math.pow(l, 3.0))) * (0.125 * (M * ((D * M) / (d / D))));
	} else if (d <= 4e-236) {
		tmp = -0.125 * (((D * D) / (d / (M * M))) * (Math.sqrt(h) / Math.pow(l, 1.5)));
	} else if (d <= 1.9e+134) {
		tmp = (t_0 * t_1) * (1.0 - ((h * (0.5 * Math.pow((((D * M) / 2.0) / d), 2.0))) / l));
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt((d / h))
	t_1 = math.sqrt((d / l))
	t_2 = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * math.pow((D * (M / (2.0 * d))), 2.0))))
	t_3 = M * (D / d)
	tmp = 0
	if d <= -8e+113:
		tmp = t_2
	elif d <= -5.2e-56:
		tmp = t_0 * (t_1 * (1.0 - (0.5 * (((h / l) * (t_3 * t_3)) * 0.25))))
	elif d <= -5.2e-238:
		tmp = t_2
	elif d <= -2e-310:
		tmp = math.sqrt((h / math.pow(l, 3.0))) * (0.125 * (M * ((D * M) / (d / D))))
	elif d <= 4e-236:
		tmp = -0.125 * (((D * D) / (d / (M * M))) * (math.sqrt(h) / math.pow(l, 1.5)))
	elif d <= 1.9e+134:
		tmp = (t_0 * t_1) * (1.0 - ((h * (0.5 * math.pow((((D * M) / 2.0) / d), 2.0))) / l))
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(d / l))
	t_2 = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 - Float64(-0.5 * Float64(Float64(h / l) * (Float64(D * Float64(M / Float64(2.0 * d))) ^ 2.0)))))
	t_3 = Float64(M * Float64(D / d))
	tmp = 0.0
	if (d <= -8e+113)
		tmp = t_2;
	elseif (d <= -5.2e-56)
		tmp = Float64(t_0 * Float64(t_1 * Float64(1.0 - Float64(0.5 * Float64(Float64(Float64(h / l) * Float64(t_3 * t_3)) * 0.25)))));
	elseif (d <= -5.2e-238)
		tmp = t_2;
	elseif (d <= -2e-310)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(0.125 * Float64(M * Float64(Float64(D * M) / Float64(d / D)))));
	elseif (d <= 4e-236)
		tmp = Float64(-0.125 * Float64(Float64(Float64(D * D) / Float64(d / Float64(M * M))) * Float64(sqrt(h) / (l ^ 1.5))));
	elseif (d <= 1.9e+134)
		tmp = Float64(Float64(t_0 * t_1) * Float64(1.0 - Float64(Float64(h * Float64(0.5 * (Float64(Float64(Float64(D * M) / 2.0) / d) ^ 2.0))) / l)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / h));
	t_1 = sqrt((d / l));
	t_2 = (d * sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * ((D * (M / (2.0 * d))) ^ 2.0))));
	t_3 = M * (D / d);
	tmp = 0.0;
	if (d <= -8e+113)
		tmp = t_2;
	elseif (d <= -5.2e-56)
		tmp = t_0 * (t_1 * (1.0 - (0.5 * (((h / l) * (t_3 * t_3)) * 0.25))));
	elseif (d <= -5.2e-238)
		tmp = t_2;
	elseif (d <= -2e-310)
		tmp = sqrt((h / (l ^ 3.0))) * (0.125 * (M * ((D * M) / (d / D))));
	elseif (d <= 4e-236)
		tmp = -0.125 * (((D * D) / (d / (M * M))) * (sqrt(h) / (l ^ 1.5)));
	elseif (d <= 1.9e+134)
		tmp = (t_0 * t_1) * (1.0 - ((h * (0.5 * ((((D * M) / 2.0) / d) ^ 2.0))) / l));
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -8e+113], t$95$2, If[LessEqual[d, -5.2e-56], N[(t$95$0 * N[(t$95$1 * N[(1.0 - N[(0.5 * N[(N[(N[(h / l), $MachinePrecision] * N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5.2e-238], t$95$2, If[LessEqual[d, -2e-310], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.125 * N[(M * N[(N[(D * M), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4e-236], N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] / N[(d / N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.9e+134], N[(N[(t$95$0 * t$95$1), $MachinePrecision] * N[(1.0 - N[(N[(h * N[(0.5 * N[Power[N[(N[(N[(D * M), $MachinePrecision] / 2.0), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if d < -8e113 or -5.19999999999999994e-56 < d < -5.2000000000000002e-238

   1. Initial program 67.6%

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

   3. Applied egg-rr 22.4%

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

      1. expm1-def 27.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 56.4%

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

   5. Simplified 56.4%

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

   6. Taylor expanded in d around -inf 81.8%

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

      1. mul-1-neg 81.8%

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

      2. distribute-rgt-neg-in 81.8%

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

      3. *-commutative 81.8%

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

   8. Simplified 81.8%

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

   if -8e113 < d < -5.19999999999999994e-56

   1. Initial program 91.6%

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

      1. associate-*l* 91.6%

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

      2. metadata-eval 91.6%

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

      3. unpow1/2 91.6%

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

      4. metadata-eval 91.6%

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

      5. unpow1/2 91.6%

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

      6. associate-*l* 91.6%

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

      7. metadata-eval 91.6%

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

      8. times-frac 91.6%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in M around 0 80.1%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)}\right)\right) \]
   5. Step-by-step derivation

      1. associate-*r/ 80.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right)\right) \]

      2. *-commutative 80.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right) \]

      3. associate-*r/ 80.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      4. *-commutative 80.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot 0.25\right)}\right)\right) \]

      5. associate-*r* 77.3%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell} \cdot 0.25\right)\right)\right) \]

      6. *-commutative 77.3%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{\color{blue}{h \cdot \left({D}^{2} \cdot {M}^{2}\right)}}{{d}^{2} \cdot \ell} \cdot 0.25\right)\right)\right) \]

      7. *-commutative 77.3%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h \cdot \left({D}^{2} \cdot {M}^{2}\right)}{\color{blue}{\ell \cdot {d}^{2}}} \cdot 0.25\right)\right)\right) \]

      8. times-frac 80.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\color{blue}{\left(\frac{h}{\ell} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)} \cdot 0.25\right)\right)\right) \]

      9. *-commutative 80.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}}\right) \cdot 0.25\right)\right)\right) \]

      10. unpow2 80.2%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot {D}^{2}}{{d}^{2}}\right) \cdot 0.25\right)\right)\right) \]

      11. unpow2 80.2%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot \color{blue}{\left(D \cdot D\right)}}{{d}^{2}}\right) \cdot 0.25\right)\right)\right) \]

      12. unswap-sqr 88.7%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{\color{blue}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}}{{d}^{2}}\right) \cdot 0.25\right)\right)\right) \]

      13. unpow2 88.7%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{d \cdot d}}\right) \cdot 0.25\right)\right)\right) \]

      14. times-frac 91.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \color{blue}{\left(\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d}\right)}\right) \cdot 0.25\right)\right)\right) \]

      15. associate-*r/ 91.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(\color{blue}{\left(M \cdot \frac{D}{d}\right)} \cdot \frac{M \cdot D}{d}\right)\right) \cdot 0.25\right)\right)\right) \]

      16. associate-*r/ 91.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(M \cdot \frac{D}{d}\right)}\right)\right) \cdot 0.25\right)\right)\right) \]

   6. Simplified 91.6%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\left(\frac{h}{\ell} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)\right) \cdot 0.25\right)}\right)\right) \]

   if -5.2000000000000002e-238 < d < -1.999999999999994e-310

   1. Initial program 29.9%

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

   3. Applied egg-rr 5.9%

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

      1. expm1-def 5.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 24.0%

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

   5. Simplified 24.0%

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

   6. Taylor expanded in d around -inf 35.8%

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

      1. mul-1-neg 35.8%

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

      2. distribute-rgt-neg-in 35.8%

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

      3. *-commutative 35.8%

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

   8. Simplified 35.8%

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

   9. Taylor expanded in D around inf 65.3%

      \[\leadsto \color{blue}{0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 65.3%

          \[\leadsto 0.125 \cdot \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

       2. unpow2 65.3%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{d}\right) \]

       3. unpow2 65.3%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{d}\right) \]

       4. associate-*l/ 59.5%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)}\right) \]

       5. *-commutative 59.5%

          \[\leadsto \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)\right) \cdot 0.125} \]

       6. associate-*l* 59.5%

          \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right) \cdot 0.125\right)} \]

       7. *-commutative 59.5%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(0.125 \cdot \left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)\right)} \]

       8. associate-*r* 60.3%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \color{blue}{\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot M\right)}\right) \]

       9. *-commutative 60.3%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \color{blue}{\left(M \cdot \left(\frac{D \cdot D}{d} \cdot M\right)\right)}\right) \]

       10. associate-/l* 60.2%

           \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \left(\color{blue}{\frac{D}{\frac{d}{D}}} \cdot M\right)\right)\right) \]

       11. associate-*l/ 65.9%

           \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \color{blue}{\frac{D \cdot M}{\frac{d}{D}}}\right)\right) \]

   11. Simplified 65.9%

       \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)} \]

   if -1.999999999999994e-310 < d < 4.0000000000000002e-236

   1. Initial program 33.4%

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

   2. Taylor expanded in d around 0 49.2%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   3. Step-by-step derivation

      1. associate-/l* 44.3%

         \[\leadsto -0.125 \cdot \left(\color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      2. unpow2 44.3%

         \[\leadsto -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\frac{d}{{M}^{2}}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      3. unpow2 44.3%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{\color{blue}{M \cdot M}}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

   4. Simplified 44.3%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   5. Step-by-step derivation

      1. sqrt-div 44.3%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}}\right) \]

   6. Applied egg-rr 44.3%

      \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}}\right) \]
   7. Step-by-step derivation

      1. sqr-pow 44.3%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{\sqrt{h}}{\sqrt{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)} \cdot {\ell}^{\left(\frac{3}{2}\right)}}}}\right) \]

      2. rem-sqrt-square 54.3%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{\sqrt{h}}{\color{blue}{\left|{\ell}^{\left(\frac{3}{2}\right)}\right|}}\right) \]

      3. sqr-pow 54.3%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{\sqrt{h}}{\left|\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}\right|}\right) \]

      4. fabs-sqr 54.3%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}}\right) \]

      5. sqr-pow 54.3%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)}}}\right) \]

      6. metadata-eval 54.3%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}}\right) \]

   8. Simplified 54.3%

      \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \color{blue}{\frac{\sqrt{h}}{{\ell}^{1.5}}}\right) \]

   if 4.0000000000000002e-236 < d < 1.89999999999999999e134

   1. Initial program 73.6%

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

      1. metadata-eval 73.6%

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

      2. unpow1/2 73.6%

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

      3. metadata-eval 73.6%

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

      4. unpow1/2 73.6%

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

      5. *-commutative 73.6%

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

      6. associate-*l* 73.6%

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

      7. times-frac 73.7%

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

      8. metadata-eval 73.7%

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

   3. Simplified 73.7%

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

      1. frac-times 73.6%

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

      2. clear-num 73.7%

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

      3. *-commutative 73.7%

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

   5. Applied egg-rr 73.7%

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

      1. associate-*r* 73.7%

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

      2. *-commutative 73.7%

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

      3. clear-num 73.6%

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

      4. *-commutative 73.6%

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

      5. metadata-eval 73.6%

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

      6. associate-*r/ 76.2%

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

      7. metadata-eval 76.2%

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

      8. times-frac 76.3%

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

   7. Applied egg-rr 76.3%

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

      1. frac-times 76.2%

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

      2. associate-/r* 76.2%

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

   9. Applied egg-rr 76.2%

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

   if 1.89999999999999999e134 < d

   1. Initial program 66.3%

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

   2. Taylor expanded in d around inf 79.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-commutative 79.7%

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

      2. *-commutative 79.7%

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

      3. associate-/r* 79.7%

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

   4. Simplified 79.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   5. Step-by-step derivation

      1. sqrt-div 90.7%

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

      2. inv-pow 90.7%

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

   6. Applied egg-rr 90.7%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}}}{\sqrt{\ell}}} \]
   7. Step-by-step derivation

      1. sqr-pow 90.7%

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

      2. rem-sqrt-square 90.7%

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

      3. metadata-eval 90.7%

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

      4. sqr-pow 90.8%

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

      5. fabs-sqr 90.8%

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

      6. sqr-pow 90.7%

         \[\leadsto d \cdot \frac{\color{blue}{{h}^{-0.5}}}{\sqrt{\ell}} \]

   8. Simplified 90.7%

      \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8 \cdot 10^{+113}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -5.2 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)\right) \cdot 0.25\right)\right)\right)\\ \mathbf{elif}\;d \leq -5.2 \cdot 10^{-238}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-236}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right)\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{+134}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{\frac{D \cdot M}{2}}{d}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 7: 71.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ t_1 := \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)\\ \mathbf{if}\;d \leq -8 \cdot 10^{+113}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-56}:\\ \;\;\;\;t_0 \cdot \left(1 - {\left(\frac{1}{\frac{2 \cdot d}{D \cdot M}}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.5\right)\right)\\ \mathbf{elif}\;d \leq -5.1 \cdot 10^{-238}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{-239}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right)\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+134}:\\ \;\;\;\;t_0 \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{\frac{D \cdot M}{2}}{d}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (sqrt (/ d h)) (sqrt (/ d l))))
        (t_1
         (*
          (* d (sqrt (/ 1.0 (* h l))))
          (- -1.0 (* -0.5 (* (/ h l) (pow (* D (/ M (* 2.0 d))) 2.0)))))))
   (if (<= d -8e+113)
     t_1
     (if (<= d -3.5e-56)
       (*
        t_0
        (- 1.0 (* (pow (/ 1.0 (/ (* 2.0 d) (* D M))) 2.0) (* (/ h l) 0.5))))
       (if (<= d -5.1e-238)
         t_1
         (if (<= d -2e-310)
           (* (sqrt (/ h (pow l 3.0))) (* 0.125 (* M (/ (* D M) (/ d D)))))
           (if (<= d 6.8e-239)
             (* -0.125 (* (/ (* D D) (/ d (* M M))) (/ (sqrt h) (pow l 1.5))))
             (if (<= d 2e+134)
               (*
                t_0
                (- 1.0 (/ (* h (* 0.5 (pow (/ (/ (* D M) 2.0) d) 2.0))) l)))
               (* d (/ (pow h -0.5) (sqrt l)))))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h)) * sqrt((d / l));
	double t_1 = (d * sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * pow((D * (M / (2.0 * d))), 2.0))));
	double tmp;
	if (d <= -8e+113) {
		tmp = t_1;
	} else if (d <= -3.5e-56) {
		tmp = t_0 * (1.0 - (pow((1.0 / ((2.0 * d) / (D * M))), 2.0) * ((h / l) * 0.5)));
	} else if (d <= -5.1e-238) {
		tmp = t_1;
	} else if (d <= -2e-310) {
		tmp = sqrt((h / pow(l, 3.0))) * (0.125 * (M * ((D * M) / (d / D))));
	} else if (d <= 6.8e-239) {
		tmp = -0.125 * (((D * D) / (d / (M * M))) * (sqrt(h) / pow(l, 1.5)));
	} else if (d <= 2e+134) {
		tmp = t_0 * (1.0 - ((h * (0.5 * pow((((D * M) / 2.0) / d), 2.0))) / l));
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this 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) :: tmp
    t_0 = sqrt((d / h)) * sqrt((d / l))
    t_1 = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) - ((-0.5d0) * ((h / l) * ((d_1 * (m / (2.0d0 * d))) ** 2.0d0))))
    if (d <= (-8d+113)) then
        tmp = t_1
    else if (d <= (-3.5d-56)) then
        tmp = t_0 * (1.0d0 - (((1.0d0 / ((2.0d0 * d) / (d_1 * m))) ** 2.0d0) * ((h / l) * 0.5d0)))
    else if (d <= (-5.1d-238)) then
        tmp = t_1
    else if (d <= (-2d-310)) then
        tmp = sqrt((h / (l ** 3.0d0))) * (0.125d0 * (m * ((d_1 * m) / (d / d_1))))
    else if (d <= 6.8d-239) then
        tmp = (-0.125d0) * (((d_1 * d_1) / (d / (m * m))) * (sqrt(h) / (l ** 1.5d0)))
    else if (d <= 2d+134) then
        tmp = t_0 * (1.0d0 - ((h * (0.5d0 * ((((d_1 * m) / 2.0d0) / d) ** 2.0d0))) / l))
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / h)) * Math.sqrt((d / l));
	double t_1 = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * Math.pow((D * (M / (2.0 * d))), 2.0))));
	double tmp;
	if (d <= -8e+113) {
		tmp = t_1;
	} else if (d <= -3.5e-56) {
		tmp = t_0 * (1.0 - (Math.pow((1.0 / ((2.0 * d) / (D * M))), 2.0) * ((h / l) * 0.5)));
	} else if (d <= -5.1e-238) {
		tmp = t_1;
	} else if (d <= -2e-310) {
		tmp = Math.sqrt((h / Math.pow(l, 3.0))) * (0.125 * (M * ((D * M) / (d / D))));
	} else if (d <= 6.8e-239) {
		tmp = -0.125 * (((D * D) / (d / (M * M))) * (Math.sqrt(h) / Math.pow(l, 1.5)));
	} else if (d <= 2e+134) {
		tmp = t_0 * (1.0 - ((h * (0.5 * Math.pow((((D * M) / 2.0) / d), 2.0))) / l));
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt((d / h)) * math.sqrt((d / l))
	t_1 = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * math.pow((D * (M / (2.0 * d))), 2.0))))
	tmp = 0
	if d <= -8e+113:
		tmp = t_1
	elif d <= -3.5e-56:
		tmp = t_0 * (1.0 - (math.pow((1.0 / ((2.0 * d) / (D * M))), 2.0) * ((h / l) * 0.5)))
	elif d <= -5.1e-238:
		tmp = t_1
	elif d <= -2e-310:
		tmp = math.sqrt((h / math.pow(l, 3.0))) * (0.125 * (M * ((D * M) / (d / D))))
	elif d <= 6.8e-239:
		tmp = -0.125 * (((D * D) / (d / (M * M))) * (math.sqrt(h) / math.pow(l, 1.5)))
	elif d <= 2e+134:
		tmp = t_0 * (1.0 - ((h * (0.5 * math.pow((((D * M) / 2.0) / d), 2.0))) / l))
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)))
	t_1 = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 - Float64(-0.5 * Float64(Float64(h / l) * (Float64(D * Float64(M / Float64(2.0 * d))) ^ 2.0)))))
	tmp = 0.0
	if (d <= -8e+113)
		tmp = t_1;
	elseif (d <= -3.5e-56)
		tmp = Float64(t_0 * Float64(1.0 - Float64((Float64(1.0 / Float64(Float64(2.0 * d) / Float64(D * M))) ^ 2.0) * Float64(Float64(h / l) * 0.5))));
	elseif (d <= -5.1e-238)
		tmp = t_1;
	elseif (d <= -2e-310)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(0.125 * Float64(M * Float64(Float64(D * M) / Float64(d / D)))));
	elseif (d <= 6.8e-239)
		tmp = Float64(-0.125 * Float64(Float64(Float64(D * D) / Float64(d / Float64(M * M))) * Float64(sqrt(h) / (l ^ 1.5))));
	elseif (d <= 2e+134)
		tmp = Float64(t_0 * Float64(1.0 - Float64(Float64(h * Float64(0.5 * (Float64(Float64(Float64(D * M) / 2.0) / d) ^ 2.0))) / l)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / h)) * sqrt((d / l));
	t_1 = (d * sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * ((D * (M / (2.0 * d))) ^ 2.0))));
	tmp = 0.0;
	if (d <= -8e+113)
		tmp = t_1;
	elseif (d <= -3.5e-56)
		tmp = t_0 * (1.0 - (((1.0 / ((2.0 * d) / (D * M))) ^ 2.0) * ((h / l) * 0.5)));
	elseif (d <= -5.1e-238)
		tmp = t_1;
	elseif (d <= -2e-310)
		tmp = sqrt((h / (l ^ 3.0))) * (0.125 * (M * ((D * M) / (d / D))));
	elseif (d <= 6.8e-239)
		tmp = -0.125 * (((D * D) / (d / (M * M))) * (sqrt(h) / (l ^ 1.5)));
	elseif (d <= 2e+134)
		tmp = t_0 * (1.0 - ((h * (0.5 * ((((D * M) / 2.0) / d) ^ 2.0))) / l));
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -8e+113], t$95$1, If[LessEqual[d, -3.5e-56], N[(t$95$0 * N[(1.0 - N[(N[Power[N[(1.0 / N[(N[(2.0 * d), $MachinePrecision] / N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5.1e-238], t$95$1, If[LessEqual[d, -2e-310], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.125 * N[(M * N[(N[(D * M), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6.8e-239], N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] / N[(d / N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2e+134], N[(t$95$0 * N[(1.0 - N[(N[(h * N[(0.5 * N[Power[N[(N[(N[(D * M), $MachinePrecision] / 2.0), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if d < -8e113 or -3.4999999999999998e-56 < d < -5.1000000000000001e-238

   1. Initial program 67.6%

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

   3. Applied egg-rr 22.4%

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

      1. expm1-def 27.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 56.4%

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

   5. Simplified 56.4%

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

   6. Taylor expanded in d around -inf 81.8%

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

      1. mul-1-neg 81.8%

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

      2. distribute-rgt-neg-in 81.8%

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

      3. *-commutative 81.8%

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

   8. Simplified 81.8%

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

   if -8e113 < d < -3.4999999999999998e-56

   1. Initial program 91.6%

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

      1. metadata-eval 91.6%

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

      2. unpow1/2 91.6%

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

      3. metadata-eval 91.6%

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

      4. unpow1/2 91.6%

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

      5. *-commutative 91.6%

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

      6. associate-*l* 91.6%

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

      7. times-frac 91.6%

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

      8. metadata-eval 91.6%

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

   3. Simplified 91.6%

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

      1. frac-times 91.6%

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

      2. clear-num 91.6%

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

      3. *-commutative 91.6%

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

   5. Applied egg-rr 91.6%

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

   if -5.1000000000000001e-238 < d < -1.999999999999994e-310

   1. Initial program 29.9%

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

   3. Applied egg-rr 5.9%

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

      1. expm1-def 5.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 24.0%

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

   5. Simplified 24.0%

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

   6. Taylor expanded in d around -inf 35.8%

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

      1. mul-1-neg 35.8%

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

      2. distribute-rgt-neg-in 35.8%

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

      3. *-commutative 35.8%

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

   8. Simplified 35.8%

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

   9. Taylor expanded in D around inf 65.3%

      \[\leadsto \color{blue}{0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 65.3%

          \[\leadsto 0.125 \cdot \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

       2. unpow2 65.3%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{d}\right) \]

       3. unpow2 65.3%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{d}\right) \]

       4. associate-*l/ 59.5%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)}\right) \]

       5. *-commutative 59.5%

          \[\leadsto \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)\right) \cdot 0.125} \]

       6. associate-*l* 59.5%

          \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right) \cdot 0.125\right)} \]

       7. *-commutative 59.5%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(0.125 \cdot \left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)\right)} \]

       8. associate-*r* 60.3%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \color{blue}{\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot M\right)}\right) \]

       9. *-commutative 60.3%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \color{blue}{\left(M \cdot \left(\frac{D \cdot D}{d} \cdot M\right)\right)}\right) \]

       10. associate-/l* 60.2%

           \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \left(\color{blue}{\frac{D}{\frac{d}{D}}} \cdot M\right)\right)\right) \]

       11. associate-*l/ 65.9%

           \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \color{blue}{\frac{D \cdot M}{\frac{d}{D}}}\right)\right) \]

   11. Simplified 65.9%

       \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)} \]

   if -1.999999999999994e-310 < d < 6.8e-239

   1. Initial program 33.4%

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

   2. Taylor expanded in d around 0 49.2%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   3. Step-by-step derivation

      1. associate-/l* 44.3%

         \[\leadsto -0.125 \cdot \left(\color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      2. unpow2 44.3%

         \[\leadsto -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\frac{d}{{M}^{2}}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      3. unpow2 44.3%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{\color{blue}{M \cdot M}}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

   4. Simplified 44.3%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   5. Step-by-step derivation

      1. sqrt-div 44.3%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}}\right) \]

   6. Applied egg-rr 44.3%

      \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}}\right) \]
   7. Step-by-step derivation

      1. sqr-pow 44.3%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{\sqrt{h}}{\sqrt{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)} \cdot {\ell}^{\left(\frac{3}{2}\right)}}}}\right) \]

      2. rem-sqrt-square 54.3%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{\sqrt{h}}{\color{blue}{\left|{\ell}^{\left(\frac{3}{2}\right)}\right|}}\right) \]

      3. sqr-pow 54.3%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{\sqrt{h}}{\left|\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}\right|}\right) \]

      4. fabs-sqr 54.3%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}}\right) \]

      5. sqr-pow 54.3%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)}}}\right) \]

      6. metadata-eval 54.3%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}}\right) \]

   8. Simplified 54.3%

      \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \color{blue}{\frac{\sqrt{h}}{{\ell}^{1.5}}}\right) \]

   if 6.8e-239 < d < 1.99999999999999984e134

   1. Initial program 73.6%

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

      1. metadata-eval 73.6%

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

      2. unpow1/2 73.6%

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

      3. metadata-eval 73.6%

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

      4. unpow1/2 73.6%

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

      5. *-commutative 73.6%

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

      6. associate-*l* 73.6%

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

      7. times-frac 73.7%

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

      8. metadata-eval 73.7%

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

   3. Simplified 73.7%

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

      1. frac-times 73.6%

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

      2. clear-num 73.7%

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

      3. *-commutative 73.7%

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

   5. Applied egg-rr 73.7%

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

      1. associate-*r* 73.7%

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

      2. *-commutative 73.7%

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

      3. clear-num 73.6%

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

      4. *-commutative 73.6%

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

      5. metadata-eval 73.6%

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

      6. associate-*r/ 76.2%

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

      7. metadata-eval 76.2%

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

      8. times-frac 76.3%

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

   7. Applied egg-rr 76.3%

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

      1. frac-times 76.2%

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

      2. associate-/r* 76.2%

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

   9. Applied egg-rr 76.2%

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

   if 1.99999999999999984e134 < d

   1. Initial program 66.3%

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

   2. Taylor expanded in d around inf 79.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-commutative 79.7%

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

      2. *-commutative 79.7%

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

      3. associate-/r* 79.7%

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

   4. Simplified 79.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   5. Step-by-step derivation

      1. sqrt-div 90.7%

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

      2. inv-pow 90.7%

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

   6. Applied egg-rr 90.7%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}}}{\sqrt{\ell}}} \]
   7. Step-by-step derivation

      1. sqr-pow 90.7%

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

      2. rem-sqrt-square 90.7%

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

      3. metadata-eval 90.7%

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

      4. sqr-pow 90.8%

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

      5. fabs-sqr 90.8%

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

      6. sqr-pow 90.7%

         \[\leadsto d \cdot \frac{\color{blue}{{h}^{-0.5}}}{\sqrt{\ell}} \]

   8. Simplified 90.7%

      \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8 \cdot 10^{+113}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-56}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{1}{\frac{2 \cdot d}{D \cdot M}}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.5\right)\right)\\ \mathbf{elif}\;d \leq -5.1 \cdot 10^{-238}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{-239}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right)\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+134}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{\frac{D \cdot M}{2}}{d}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 8: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)\\ t_1 := M \cdot \frac{D}{d}\\ \mathbf{if}\;d \leq -1.15 \cdot 10^{+115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-56}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{1}{\frac{2 \cdot d}{D \cdot M}}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.5\right)\right)\\ \mathbf{elif}\;d \leq -5.9 \cdot 10^{-238}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot \left(t_1 \cdot t_1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* d (sqrt (/ 1.0 (* h l))))
          (- -1.0 (* -0.5 (* (/ h l) (pow (* D (/ M (* 2.0 d))) 2.0))))))
        (t_1 (* M (/ D d))))
   (if (<= d -1.15e+115)
     t_0
     (if (<= d -3.5e-56)
       (*
        (* (sqrt (/ d h)) (sqrt (/ d l)))
        (- 1.0 (* (pow (/ 1.0 (/ (* 2.0 d) (* D M))) 2.0) (* (/ h l) 0.5))))
       (if (<= d -5.9e-238)
         t_0
         (if (<= d 4.6e-303)
           (* (sqrt (/ h (pow l 3.0))) (* 0.125 (* M (/ (* D M) (/ d D)))))
           (*
            (* (/ (sqrt d) (sqrt h)) (pow (/ d l) 0.5))
            (- 1.0 (* 0.125 (* (/ h l) (* t_1 t_1)))))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (d * sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * pow((D * (M / (2.0 * d))), 2.0))));
	double t_1 = M * (D / d);
	double tmp;
	if (d <= -1.15e+115) {
		tmp = t_0;
	} else if (d <= -3.5e-56) {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - (pow((1.0 / ((2.0 * d) / (D * M))), 2.0) * ((h / l) * 0.5)));
	} else if (d <= -5.9e-238) {
		tmp = t_0;
	} else if (d <= 4.6e-303) {
		tmp = sqrt((h / pow(l, 3.0))) * (0.125 * (M * ((D * M) / (d / D))));
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * pow((d / l), 0.5)) * (1.0 - (0.125 * ((h / l) * (t_1 * t_1))));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this 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) :: tmp
    t_0 = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) - ((-0.5d0) * ((h / l) * ((d_1 * (m / (2.0d0 * d))) ** 2.0d0))))
    t_1 = m * (d_1 / d)
    if (d <= (-1.15d+115)) then
        tmp = t_0
    else if (d <= (-3.5d-56)) then
        tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0d0 - (((1.0d0 / ((2.0d0 * d) / (d_1 * m))) ** 2.0d0) * ((h / l) * 0.5d0)))
    else if (d <= (-5.9d-238)) then
        tmp = t_0
    else if (d <= 4.6d-303) then
        tmp = sqrt((h / (l ** 3.0d0))) * (0.125d0 * (m * ((d_1 * m) / (d / d_1))))
    else
        tmp = ((sqrt(d) / sqrt(h)) * ((d / l) ** 0.5d0)) * (1.0d0 - (0.125d0 * ((h / l) * (t_1 * t_1))))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * Math.pow((D * (M / (2.0 * d))), 2.0))));
	double t_1 = M * (D / d);
	double tmp;
	if (d <= -1.15e+115) {
		tmp = t_0;
	} else if (d <= -3.5e-56) {
		tmp = (Math.sqrt((d / h)) * Math.sqrt((d / l))) * (1.0 - (Math.pow((1.0 / ((2.0 * d) / (D * M))), 2.0) * ((h / l) * 0.5)));
	} else if (d <= -5.9e-238) {
		tmp = t_0;
	} else if (d <= 4.6e-303) {
		tmp = Math.sqrt((h / Math.pow(l, 3.0))) * (0.125 * (M * ((D * M) / (d / D))));
	} else {
		tmp = ((Math.sqrt(d) / Math.sqrt(h)) * Math.pow((d / l), 0.5)) * (1.0 - (0.125 * ((h / l) * (t_1 * t_1))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * math.pow((D * (M / (2.0 * d))), 2.0))))
	t_1 = M * (D / d)
	tmp = 0
	if d <= -1.15e+115:
		tmp = t_0
	elif d <= -3.5e-56:
		tmp = (math.sqrt((d / h)) * math.sqrt((d / l))) * (1.0 - (math.pow((1.0 / ((2.0 * d) / (D * M))), 2.0) * ((h / l) * 0.5)))
	elif d <= -5.9e-238:
		tmp = t_0
	elif d <= 4.6e-303:
		tmp = math.sqrt((h / math.pow(l, 3.0))) * (0.125 * (M * ((D * M) / (d / D))))
	else:
		tmp = ((math.sqrt(d) / math.sqrt(h)) * math.pow((d / l), 0.5)) * (1.0 - (0.125 * ((h / l) * (t_1 * t_1))))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 - Float64(-0.5 * Float64(Float64(h / l) * (Float64(D * Float64(M / Float64(2.0 * d))) ^ 2.0)))))
	t_1 = Float64(M * Float64(D / d))
	tmp = 0.0
	if (d <= -1.15e+115)
		tmp = t_0;
	elseif (d <= -3.5e-56)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * Float64(1.0 - Float64((Float64(1.0 / Float64(Float64(2.0 * d) / Float64(D * M))) ^ 2.0) * Float64(Float64(h / l) * 0.5))));
	elseif (d <= -5.9e-238)
		tmp = t_0;
	elseif (d <= 4.6e-303)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(0.125 * Float64(M * Float64(Float64(D * M) / Float64(d / D)))));
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(0.125 * Float64(Float64(h / l) * Float64(t_1 * t_1)))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (d * sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * ((D * (M / (2.0 * d))) ^ 2.0))));
	t_1 = M * (D / d);
	tmp = 0.0;
	if (d <= -1.15e+115)
		tmp = t_0;
	elseif (d <= -3.5e-56)
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - (((1.0 / ((2.0 * d) / (D * M))) ^ 2.0) * ((h / l) * 0.5)));
	elseif (d <= -5.9e-238)
		tmp = t_0;
	elseif (d <= 4.6e-303)
		tmp = sqrt((h / (l ^ 3.0))) * (0.125 * (M * ((D * M) / (d / D))));
	else
		tmp = ((sqrt(d) / sqrt(h)) * ((d / l) ^ 0.5)) * (1.0 - (0.125 * ((h / l) * (t_1 * t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.15e+115], t$95$0, If[LessEqual[d, -3.5e-56], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Power[N[(1.0 / N[(N[(2.0 * d), $MachinePrecision] / N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5.9e-238], t$95$0, If[LessEqual[d, 4.6e-303], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.125 * N[(M * N[(N[(D * M), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.125 * N[(N[(h / l), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.15000000000000002e115 or -3.4999999999999998e-56 < d < -5.8999999999999998e-238

   1. Initial program 67.6%

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

   3. Applied egg-rr 22.4%

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

      1. expm1-def 27.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 56.4%

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

   5. Simplified 56.4%

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

   6. Taylor expanded in d around -inf 81.8%

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

      1. mul-1-neg 81.8%

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

      2. distribute-rgt-neg-in 81.8%

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

      3. *-commutative 81.8%

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

   8. Simplified 81.8%

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

   if -1.15000000000000002e115 < d < -3.4999999999999998e-56

   1. Initial program 91.6%

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

      1. metadata-eval 91.6%

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

      2. unpow1/2 91.6%

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

      3. metadata-eval 91.6%

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

      4. unpow1/2 91.6%

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

      5. *-commutative 91.6%

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

      6. associate-*l* 91.6%

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

      7. times-frac 91.6%

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

      8. metadata-eval 91.6%

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

   3. Simplified 91.6%

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

      1. frac-times 91.6%

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

      2. clear-num 91.6%

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

      3. *-commutative 91.6%

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

   5. Applied egg-rr 91.6%

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

   if -5.8999999999999998e-238 < d < 4.59999999999999991e-303

   1. Initial program 28.2%

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

   3. Applied egg-rr 5.6%

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

      1. expm1-def 5.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 22.7%

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

   5. Simplified 22.7%

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

   6. Taylor expanded in d around -inf 33.8%

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

      1. mul-1-neg 33.8%

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

      2. distribute-rgt-neg-in 33.8%

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

      3. *-commutative 33.8%

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

   8. Simplified 33.8%

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

   9. Taylor expanded in D around inf 61.9%

      \[\leadsto \color{blue}{0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 61.9%

          \[\leadsto 0.125 \cdot \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

       2. unpow2 61.9%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{d}\right) \]

       3. unpow2 61.9%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{d}\right) \]

       4. associate-*l/ 56.4%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)}\right) \]

       5. *-commutative 56.4%

          \[\leadsto \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)\right) \cdot 0.125} \]

       6. associate-*l* 56.4%

          \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right) \cdot 0.125\right)} \]

       7. *-commutative 56.4%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(0.125 \cdot \left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)\right)} \]

       8. associate-*r* 57.2%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \color{blue}{\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot M\right)}\right) \]

       9. *-commutative 57.2%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \color{blue}{\left(M \cdot \left(\frac{D \cdot D}{d} \cdot M\right)\right)}\right) \]

       10. associate-/l* 57.1%

           \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \left(\color{blue}{\frac{D}{\frac{d}{D}}} \cdot M\right)\right)\right) \]

       11. associate-*l/ 62.5%

           \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \color{blue}{\frac{D \cdot M}{\frac{d}{D}}}\right)\right) \]

   11. Simplified 62.5%

       \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)} \]

   if 4.59999999999999991e-303 < d

   1. Initial program 66.0%

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

      1. metadata-eval 66.0%

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

      2. pow1/2 66.0%

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

      3. sqrt-div 78.6%

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

   3. Applied egg-rr 78.6%

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

   4. Taylor expanded in M around 0 44.8%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   5. Step-by-step derivation

      1. associate-*r/ 38.0%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right) \]

      2. *-commutative 38.0%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right) \]

      3. associate-*r/ 38.0%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]

      4. associate-*r* 41.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}\right) \]

      5. *-commutative 41.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{h \cdot \left({D}^{2} \cdot {M}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. *-commutative 41.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \frac{h \cdot \left({D}^{2} \cdot {M}^{2}\right)}{\color{blue}{\ell \cdot {d}^{2}}}\right) \]

      7. times-frac 42.7%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)}\right) \]

      8. *-commutative 42.7%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}}\right)\right) \]

      9. unpow2 42.7%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot {D}^{2}}{{d}^{2}}\right)\right) \]

      10. unpow2 42.7%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot \color{blue}{\left(D \cdot D\right)}}{{d}^{2}}\right)\right) \]

      11. unswap-sqr 57.2%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot \frac{\color{blue}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}}{{d}^{2}}\right)\right) \]

      12. unpow2 57.2%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{d \cdot d}}\right)\right) \]

      13. times-frac 66.0%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot \color{blue}{\left(\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d}\right)}\right)\right) \]

      14. associate-*r/ 65.2%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot \left(\color{blue}{\left(M \cdot \frac{D}{d}\right)} \cdot \frac{M \cdot D}{d}\right)\right)\right) \]

      15. associate-*r/ 66.0%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(M \cdot \frac{D}{d}\right)}\right)\right)\right) \]

   6. Simplified 78.7%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \left(\frac{h}{\ell} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)\right)}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.15 \cdot 10^{+115}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-56}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{1}{\frac{2 \cdot d}{D \cdot M}}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.5\right)\right)\\ \mathbf{elif}\;d \leq -5.9 \cdot 10^{-238}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)\right)\right)\\ \end{array} \]

Alternative 9: 70.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{{\ell}^{3}}}\\ t_1 := \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)\\ t_2 := M \cdot \frac{D}{d}\\ t_3 := \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(t_2 \cdot t_2\right)\right) \cdot 0.25\right)\right)\right)\\ t_4 := M \cdot \frac{D \cdot M}{\frac{d}{D}}\\ \mathbf{if}\;d \leq -3.35 \cdot 10^{+112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -4.1 \cdot 10^{-56}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq -5.6 \cdot 10^{-238}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_0 \cdot \left(0.125 \cdot t_4\right)\\ \mathbf{elif}\;d \leq 7.8 \cdot 10^{-113}:\\ \;\;\;\;t_0 \cdot \left(t_4 \cdot -0.125\right)\\ \mathbf{elif}\;d \leq 1.28 \cdot 10^{+134}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h (pow l 3.0))))
        (t_1
         (*
          (* d (sqrt (/ 1.0 (* h l))))
          (- -1.0 (* -0.5 (* (/ h l) (pow (* D (/ M (* 2.0 d))) 2.0))))))
        (t_2 (* M (/ D d)))
        (t_3
         (*
          (sqrt (/ d h))
          (* (sqrt (/ d l)) (- 1.0 (* 0.5 (* (* (/ h l) (* t_2 t_2)) 0.25))))))
        (t_4 (* M (/ (* D M) (/ d D)))))
   (if (<= d -3.35e+112)
     t_1
     (if (<= d -4.1e-56)
       t_3
       (if (<= d -5.6e-238)
         t_1
         (if (<= d -2e-310)
           (* t_0 (* 0.125 t_4))
           (if (<= d 7.8e-113)
             (* t_0 (* t_4 -0.125))
             (if (<= d 1.28e+134) t_3 (* d (/ (pow h -0.5) (sqrt l)))))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / pow(l, 3.0)));
	double t_1 = (d * sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * pow((D * (M / (2.0 * d))), 2.0))));
	double t_2 = M * (D / d);
	double t_3 = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * (((h / l) * (t_2 * t_2)) * 0.25))));
	double t_4 = M * ((D * M) / (d / D));
	double tmp;
	if (d <= -3.35e+112) {
		tmp = t_1;
	} else if (d <= -4.1e-56) {
		tmp = t_3;
	} else if (d <= -5.6e-238) {
		tmp = t_1;
	} else if (d <= -2e-310) {
		tmp = t_0 * (0.125 * t_4);
	} else if (d <= 7.8e-113) {
		tmp = t_0 * (t_4 * -0.125);
	} else if (d <= 1.28e+134) {
		tmp = t_3;
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this 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) :: tmp
    t_0 = sqrt((h / (l ** 3.0d0)))
    t_1 = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) - ((-0.5d0) * ((h / l) * ((d_1 * (m / (2.0d0 * d))) ** 2.0d0))))
    t_2 = m * (d_1 / d)
    t_3 = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 - (0.5d0 * (((h / l) * (t_2 * t_2)) * 0.25d0))))
    t_4 = m * ((d_1 * m) / (d / d_1))
    if (d <= (-3.35d+112)) then
        tmp = t_1
    else if (d <= (-4.1d-56)) then
        tmp = t_3
    else if (d <= (-5.6d-238)) then
        tmp = t_1
    else if (d <= (-2d-310)) then
        tmp = t_0 * (0.125d0 * t_4)
    else if (d <= 7.8d-113) then
        tmp = t_0 * (t_4 * (-0.125d0))
    else if (d <= 1.28d+134) then
        tmp = t_3
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / Math.pow(l, 3.0)));
	double t_1 = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * Math.pow((D * (M / (2.0 * d))), 2.0))));
	double t_2 = M * (D / d);
	double t_3 = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - (0.5 * (((h / l) * (t_2 * t_2)) * 0.25))));
	double t_4 = M * ((D * M) / (d / D));
	double tmp;
	if (d <= -3.35e+112) {
		tmp = t_1;
	} else if (d <= -4.1e-56) {
		tmp = t_3;
	} else if (d <= -5.6e-238) {
		tmp = t_1;
	} else if (d <= -2e-310) {
		tmp = t_0 * (0.125 * t_4);
	} else if (d <= 7.8e-113) {
		tmp = t_0 * (t_4 * -0.125);
	} else if (d <= 1.28e+134) {
		tmp = t_3;
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt((h / math.pow(l, 3.0)))
	t_1 = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * math.pow((D * (M / (2.0 * d))), 2.0))))
	t_2 = M * (D / d)
	t_3 = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - (0.5 * (((h / l) * (t_2 * t_2)) * 0.25))))
	t_4 = M * ((D * M) / (d / D))
	tmp = 0
	if d <= -3.35e+112:
		tmp = t_1
	elif d <= -4.1e-56:
		tmp = t_3
	elif d <= -5.6e-238:
		tmp = t_1
	elif d <= -2e-310:
		tmp = t_0 * (0.125 * t_4)
	elif d <= 7.8e-113:
		tmp = t_0 * (t_4 * -0.125)
	elif d <= 1.28e+134:
		tmp = t_3
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / (l ^ 3.0)))
	t_1 = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 - Float64(-0.5 * Float64(Float64(h / l) * (Float64(D * Float64(M / Float64(2.0 * d))) ^ 2.0)))))
	t_2 = Float64(M * Float64(D / d))
	t_3 = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(0.5 * Float64(Float64(Float64(h / l) * Float64(t_2 * t_2)) * 0.25)))))
	t_4 = Float64(M * Float64(Float64(D * M) / Float64(d / D)))
	tmp = 0.0
	if (d <= -3.35e+112)
		tmp = t_1;
	elseif (d <= -4.1e-56)
		tmp = t_3;
	elseif (d <= -5.6e-238)
		tmp = t_1;
	elseif (d <= -2e-310)
		tmp = Float64(t_0 * Float64(0.125 * t_4));
	elseif (d <= 7.8e-113)
		tmp = Float64(t_0 * Float64(t_4 * -0.125));
	elseif (d <= 1.28e+134)
		tmp = t_3;
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / (l ^ 3.0)));
	t_1 = (d * sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * ((D * (M / (2.0 * d))) ^ 2.0))));
	t_2 = M * (D / d);
	t_3 = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * (((h / l) * (t_2 * t_2)) * 0.25))));
	t_4 = M * ((D * M) / (d / D));
	tmp = 0.0;
	if (d <= -3.35e+112)
		tmp = t_1;
	elseif (d <= -4.1e-56)
		tmp = t_3;
	elseif (d <= -5.6e-238)
		tmp = t_1;
	elseif (d <= -2e-310)
		tmp = t_0 * (0.125 * t_4);
	elseif (d <= 7.8e-113)
		tmp = t_0 * (t_4 * -0.125);
	elseif (d <= 1.28e+134)
		tmp = t_3;
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(N[(h / l), $MachinePrecision] * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(M * N[(N[(D * M), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.35e+112], t$95$1, If[LessEqual[d, -4.1e-56], t$95$3, If[LessEqual[d, -5.6e-238], t$95$1, If[LessEqual[d, -2e-310], N[(t$95$0 * N[(0.125 * t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.8e-113], N[(t$95$0 * N[(t$95$4 * -0.125), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.28e+134], t$95$3, N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -3.3499999999999999e112 or -4.1000000000000001e-56 < d < -5.60000000000000008e-238

   1. Initial program 67.6%

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

   3. Applied egg-rr 22.4%

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

      1. expm1-def 27.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 56.4%

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

   5. Simplified 56.4%

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

   6. Taylor expanded in d around -inf 81.8%

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

      1. mul-1-neg 81.8%

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

      2. distribute-rgt-neg-in 81.8%

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

      3. *-commutative 81.8%

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

   8. Simplified 81.8%

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

   if -3.3499999999999999e112 < d < -4.1000000000000001e-56 or 7.7999999999999997e-113 < d < 1.27999999999999999e134

   1. Initial program 87.6%

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

      1. associate-*l* 87.6%

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

      2. metadata-eval 87.6%

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

      3. unpow1/2 87.6%

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

      4. metadata-eval 87.6%

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

      5. unpow1/2 87.6%

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

      6. associate-*l* 87.6%

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

      7. metadata-eval 87.6%

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

      8. times-frac 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in M around 0 65.4%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)}\right)\right) \]
   5. Step-by-step derivation

      1. associate-*r/ 65.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right)\right) \]

      2. *-commutative 65.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right) \]

      3. associate-*r/ 65.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      4. *-commutative 65.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot 0.25\right)}\right)\right) \]

      5. associate-*r* 65.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell} \cdot 0.25\right)\right)\right) \]

      6. *-commutative 65.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{\color{blue}{h \cdot \left({D}^{2} \cdot {M}^{2}\right)}}{{d}^{2} \cdot \ell} \cdot 0.25\right)\right)\right) \]

      7. *-commutative 65.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h \cdot \left({D}^{2} \cdot {M}^{2}\right)}{\color{blue}{\ell \cdot {d}^{2}}} \cdot 0.25\right)\right)\right) \]

      8. times-frac 68.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\color{blue}{\left(\frac{h}{\ell} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)} \cdot 0.25\right)\right)\right) \]

      9. *-commutative 68.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}}\right) \cdot 0.25\right)\right)\right) \]

      10. unpow2 68.7%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot {D}^{2}}{{d}^{2}}\right) \cdot 0.25\right)\right)\right) \]

      11. unpow2 68.7%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot \color{blue}{\left(D \cdot D\right)}}{{d}^{2}}\right) \cdot 0.25\right)\right)\right) \]

      12. unswap-sqr 85.4%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{\color{blue}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}}{{d}^{2}}\right) \cdot 0.25\right)\right)\right) \]

      13. unpow2 85.4%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{d \cdot d}}\right) \cdot 0.25\right)\right)\right) \]

      14. times-frac 87.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \color{blue}{\left(\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d}\right)}\right) \cdot 0.25\right)\right)\right) \]

      15. associate-*r/ 87.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(\color{blue}{\left(M \cdot \frac{D}{d}\right)} \cdot \frac{M \cdot D}{d}\right)\right) \cdot 0.25\right)\right)\right) \]

      16. associate-*r/ 87.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(M \cdot \frac{D}{d}\right)}\right)\right) \cdot 0.25\right)\right)\right) \]

   6. Simplified 87.6%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\left(\frac{h}{\ell} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)\right) \cdot 0.25\right)}\right)\right) \]

   if -5.60000000000000008e-238 < d < -1.999999999999994e-310

   1. Initial program 29.9%

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

   3. Applied egg-rr 5.9%

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

      1. expm1-def 5.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 24.0%

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

   5. Simplified 24.0%

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

   6. Taylor expanded in d around -inf 35.8%

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

      1. mul-1-neg 35.8%

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

      2. distribute-rgt-neg-in 35.8%

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

      3. *-commutative 35.8%

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

   8. Simplified 35.8%

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

   9. Taylor expanded in D around inf 65.3%

      \[\leadsto \color{blue}{0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 65.3%

          \[\leadsto 0.125 \cdot \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

       2. unpow2 65.3%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{d}\right) \]

       3. unpow2 65.3%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{d}\right) \]

       4. associate-*l/ 59.5%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)}\right) \]

       5. *-commutative 59.5%

          \[\leadsto \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)\right) \cdot 0.125} \]

       6. associate-*l* 59.5%

          \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right) \cdot 0.125\right)} \]

       7. *-commutative 59.5%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(0.125 \cdot \left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)\right)} \]

       8. associate-*r* 60.3%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \color{blue}{\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot M\right)}\right) \]

       9. *-commutative 60.3%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \color{blue}{\left(M \cdot \left(\frac{D \cdot D}{d} \cdot M\right)\right)}\right) \]

       10. associate-/l* 60.2%

           \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \left(\color{blue}{\frac{D}{\frac{d}{D}}} \cdot M\right)\right)\right) \]

       11. associate-*l/ 65.9%

           \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \color{blue}{\frac{D \cdot M}{\frac{d}{D}}}\right)\right) \]

   11. Simplified 65.9%

       \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)} \]

   if -1.999999999999994e-310 < d < 7.7999999999999997e-113

   1. Initial program 35.1%

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

      1. metadata-eval 35.1%

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

      2. unpow1/2 35.1%

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

      3. metadata-eval 35.1%

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

      4. unpow1/2 35.1%

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

      5. *-commutative 35.1%

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

      6. associate-*l* 35.1%

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

      7. times-frac 35.2%

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

      8. metadata-eval 35.2%

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

   3. Simplified 35.2%

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

      1. frac-times 35.1%

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

      2. clear-num 35.2%

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

      3. *-commutative 35.2%

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

   5. Applied egg-rr 35.2%

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

      1. associate-*r* 35.2%

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

      2. *-commutative 35.2%

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

      3. clear-num 35.1%

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

      4. *-commutative 35.1%

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

      5. metadata-eval 35.1%

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

      6. associate-*r/ 37.4%

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

      7. metadata-eval 37.4%

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

      8. times-frac 37.5%

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

   7. Applied egg-rr 37.5%

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

   8. Taylor expanded in d around 0 40.1%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 40.1%

         \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative 40.1%

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      3. unpow2 40.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{d}\right) \]

      4. unpow2 40.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{d}\right) \]

      5. associate-*l/ 39.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)}\right) \]

      6. associate-*r* 42.7%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot M\right)}\right) \]

      7. *-commutative 42.7%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(M \cdot \left(\frac{D \cdot D}{d} \cdot M\right)\right)}\right) \]

      8. associate-/l* 48.0%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(M \cdot \left(\color{blue}{\frac{D}{\frac{d}{D}}} \cdot M\right)\right)\right) \]

      9. associate-*l/ 50.5%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(M \cdot \color{blue}{\frac{D \cdot M}{\frac{d}{D}}}\right)\right) \]

   10. Simplified 50.5%

       \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)} \]

   if 1.27999999999999999e134 < d

   1. Initial program 66.3%

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

   2. Taylor expanded in d around inf 79.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-commutative 79.7%

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

      2. *-commutative 79.7%

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

      3. associate-/r* 79.7%

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

   4. Simplified 79.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   5. Step-by-step derivation

      1. sqrt-div 90.7%

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

      2. inv-pow 90.7%

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

   6. Applied egg-rr 90.7%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}}}{\sqrt{\ell}}} \]
   7. Step-by-step derivation

      1. sqr-pow 90.7%

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

      2. rem-sqrt-square 90.7%

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

      3. metadata-eval 90.7%

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

      4. sqr-pow 90.8%

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

      5. fabs-sqr 90.8%

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

      6. sqr-pow 90.7%

         \[\leadsto d \cdot \frac{\color{blue}{{h}^{-0.5}}}{\sqrt{\ell}} \]

   8. Simplified 90.7%

      \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.35 \cdot 10^{+112}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -4.1 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)\right) \cdot 0.25\right)\right)\right)\\ \mathbf{elif}\;d \leq -5.6 \cdot 10^{-238}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)\\ \mathbf{elif}\;d \leq 7.8 \cdot 10^{-113}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right) \cdot -0.125\right)\\ \mathbf{elif}\;d \leq 1.28 \cdot 10^{+134}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)\right) \cdot 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 10: 70.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{{\ell}^{3}}}\\ t_1 := \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)\\ t_2 := M \cdot \frac{D}{d}\\ t_3 := \frac{h}{\ell} \cdot \left(t_2 \cdot t_2\right)\\ t_4 := M \cdot \frac{D \cdot M}{\frac{d}{D}}\\ \mathbf{if}\;d \leq -4.9 \cdot 10^{+115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -3.6 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(t_3 \cdot 0.25\right)\right)\right)\\ \mathbf{elif}\;d \leq -4.5 \cdot 10^{-238}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_0 \cdot \left(0.125 \cdot t_4\right)\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{-113}:\\ \;\;\;\;t_0 \cdot \left(t_4 \cdot -0.125\right)\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{+133}:\\ \;\;\;\;\left(1 - 0.125 \cdot t_3\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot {\left(\frac{d}{h}\right)}^{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h (pow l 3.0))))
        (t_1
         (*
          (* d (sqrt (/ 1.0 (* h l))))
          (- -1.0 (* -0.5 (* (/ h l) (pow (* D (/ M (* 2.0 d))) 2.0))))))
        (t_2 (* M (/ D d)))
        (t_3 (* (/ h l) (* t_2 t_2)))
        (t_4 (* M (/ (* D M) (/ d D)))))
   (if (<= d -4.9e+115)
     t_1
     (if (<= d -3.6e-56)
       (* (sqrt (/ d h)) (* (sqrt (/ d l)) (- 1.0 (* 0.5 (* t_3 0.25)))))
       (if (<= d -4.5e-238)
         t_1
         (if (<= d -2e-310)
           (* t_0 (* 0.125 t_4))
           (if (<= d 7.2e-113)
             (* t_0 (* t_4 -0.125))
             (if (<= d 2.6e+133)
               (*
                (- 1.0 (* 0.125 t_3))
                (* (pow (/ d l) 0.5) (pow (/ d h) 0.5)))
               (* d (/ (pow h -0.5) (sqrt l)))))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / pow(l, 3.0)));
	double t_1 = (d * sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * pow((D * (M / (2.0 * d))), 2.0))));
	double t_2 = M * (D / d);
	double t_3 = (h / l) * (t_2 * t_2);
	double t_4 = M * ((D * M) / (d / D));
	double tmp;
	if (d <= -4.9e+115) {
		tmp = t_1;
	} else if (d <= -3.6e-56) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * (t_3 * 0.25))));
	} else if (d <= -4.5e-238) {
		tmp = t_1;
	} else if (d <= -2e-310) {
		tmp = t_0 * (0.125 * t_4);
	} else if (d <= 7.2e-113) {
		tmp = t_0 * (t_4 * -0.125);
	} else if (d <= 2.6e+133) {
		tmp = (1.0 - (0.125 * t_3)) * (pow((d / l), 0.5) * pow((d / h), 0.5));
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this 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) :: tmp
    t_0 = sqrt((h / (l ** 3.0d0)))
    t_1 = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) - ((-0.5d0) * ((h / l) * ((d_1 * (m / (2.0d0 * d))) ** 2.0d0))))
    t_2 = m * (d_1 / d)
    t_3 = (h / l) * (t_2 * t_2)
    t_4 = m * ((d_1 * m) / (d / d_1))
    if (d <= (-4.9d+115)) then
        tmp = t_1
    else if (d <= (-3.6d-56)) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 - (0.5d0 * (t_3 * 0.25d0))))
    else if (d <= (-4.5d-238)) then
        tmp = t_1
    else if (d <= (-2d-310)) then
        tmp = t_0 * (0.125d0 * t_4)
    else if (d <= 7.2d-113) then
        tmp = t_0 * (t_4 * (-0.125d0))
    else if (d <= 2.6d+133) then
        tmp = (1.0d0 - (0.125d0 * t_3)) * (((d / l) ** 0.5d0) * ((d / h) ** 0.5d0))
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / Math.pow(l, 3.0)));
	double t_1 = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * Math.pow((D * (M / (2.0 * d))), 2.0))));
	double t_2 = M * (D / d);
	double t_3 = (h / l) * (t_2 * t_2);
	double t_4 = M * ((D * M) / (d / D));
	double tmp;
	if (d <= -4.9e+115) {
		tmp = t_1;
	} else if (d <= -3.6e-56) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - (0.5 * (t_3 * 0.25))));
	} else if (d <= -4.5e-238) {
		tmp = t_1;
	} else if (d <= -2e-310) {
		tmp = t_0 * (0.125 * t_4);
	} else if (d <= 7.2e-113) {
		tmp = t_0 * (t_4 * -0.125);
	} else if (d <= 2.6e+133) {
		tmp = (1.0 - (0.125 * t_3)) * (Math.pow((d / l), 0.5) * Math.pow((d / h), 0.5));
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt((h / math.pow(l, 3.0)))
	t_1 = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * math.pow((D * (M / (2.0 * d))), 2.0))))
	t_2 = M * (D / d)
	t_3 = (h / l) * (t_2 * t_2)
	t_4 = M * ((D * M) / (d / D))
	tmp = 0
	if d <= -4.9e+115:
		tmp = t_1
	elif d <= -3.6e-56:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - (0.5 * (t_3 * 0.25))))
	elif d <= -4.5e-238:
		tmp = t_1
	elif d <= -2e-310:
		tmp = t_0 * (0.125 * t_4)
	elif d <= 7.2e-113:
		tmp = t_0 * (t_4 * -0.125)
	elif d <= 2.6e+133:
		tmp = (1.0 - (0.125 * t_3)) * (math.pow((d / l), 0.5) * math.pow((d / h), 0.5))
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / (l ^ 3.0)))
	t_1 = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 - Float64(-0.5 * Float64(Float64(h / l) * (Float64(D * Float64(M / Float64(2.0 * d))) ^ 2.0)))))
	t_2 = Float64(M * Float64(D / d))
	t_3 = Float64(Float64(h / l) * Float64(t_2 * t_2))
	t_4 = Float64(M * Float64(Float64(D * M) / Float64(d / D)))
	tmp = 0.0
	if (d <= -4.9e+115)
		tmp = t_1;
	elseif (d <= -3.6e-56)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(0.5 * Float64(t_3 * 0.25)))));
	elseif (d <= -4.5e-238)
		tmp = t_1;
	elseif (d <= -2e-310)
		tmp = Float64(t_0 * Float64(0.125 * t_4));
	elseif (d <= 7.2e-113)
		tmp = Float64(t_0 * Float64(t_4 * -0.125));
	elseif (d <= 2.6e+133)
		tmp = Float64(Float64(1.0 - Float64(0.125 * t_3)) * Float64((Float64(d / l) ^ 0.5) * (Float64(d / h) ^ 0.5)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / (l ^ 3.0)));
	t_1 = (d * sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * ((D * (M / (2.0 * d))) ^ 2.0))));
	t_2 = M * (D / d);
	t_3 = (h / l) * (t_2 * t_2);
	t_4 = M * ((D * M) / (d / D));
	tmp = 0.0;
	if (d <= -4.9e+115)
		tmp = t_1;
	elseif (d <= -3.6e-56)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * (t_3 * 0.25))));
	elseif (d <= -4.5e-238)
		tmp = t_1;
	elseif (d <= -2e-310)
		tmp = t_0 * (0.125 * t_4);
	elseif (d <= 7.2e-113)
		tmp = t_0 * (t_4 * -0.125);
	elseif (d <= 2.6e+133)
		tmp = (1.0 - (0.125 * t_3)) * (((d / l) ^ 0.5) * ((d / h) ^ 0.5));
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(h / l), $MachinePrecision] * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(M * N[(N[(D * M), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -4.9e+115], t$95$1, If[LessEqual[d, -3.6e-56], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(t$95$3 * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4.5e-238], t$95$1, If[LessEqual[d, -2e-310], N[(t$95$0 * N[(0.125 * t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.2e-113], N[(t$95$0 * N[(t$95$4 * -0.125), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.6e+133], N[(N[(1.0 - N[(0.125 * t$95$3), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if d < -4.89999999999999964e115 or -3.59999999999999978e-56 < d < -4.49999999999999996e-238

   1. Initial program 67.6%

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

   3. Applied egg-rr 22.4%

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

      1. expm1-def 27.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 56.4%

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

   5. Simplified 56.4%

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

   6. Taylor expanded in d around -inf 81.8%

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

      1. mul-1-neg 81.8%

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

      2. distribute-rgt-neg-in 81.8%

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

      3. *-commutative 81.8%

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

   8. Simplified 81.8%

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

   if -4.89999999999999964e115 < d < -3.59999999999999978e-56

   1. Initial program 91.6%

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

      1. associate-*l* 91.6%

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

      2. metadata-eval 91.6%

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

      3. unpow1/2 91.6%

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

      4. metadata-eval 91.6%

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

      5. unpow1/2 91.6%

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

      6. associate-*l* 91.6%

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

      7. metadata-eval 91.6%

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

      8. times-frac 91.6%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in M around 0 80.1%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)}\right)\right) \]
   5. Step-by-step derivation

      1. associate-*r/ 80.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right)\right) \]

      2. *-commutative 80.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right) \]

      3. associate-*r/ 80.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      4. *-commutative 80.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot 0.25\right)}\right)\right) \]

      5. associate-*r* 77.3%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell} \cdot 0.25\right)\right)\right) \]

      6. *-commutative 77.3%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{\color{blue}{h \cdot \left({D}^{2} \cdot {M}^{2}\right)}}{{d}^{2} \cdot \ell} \cdot 0.25\right)\right)\right) \]

      7. *-commutative 77.3%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h \cdot \left({D}^{2} \cdot {M}^{2}\right)}{\color{blue}{\ell \cdot {d}^{2}}} \cdot 0.25\right)\right)\right) \]

      8. times-frac 80.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\color{blue}{\left(\frac{h}{\ell} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)} \cdot 0.25\right)\right)\right) \]

      9. *-commutative 80.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}}\right) \cdot 0.25\right)\right)\right) \]

      10. unpow2 80.2%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot {D}^{2}}{{d}^{2}}\right) \cdot 0.25\right)\right)\right) \]

      11. unpow2 80.2%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot \color{blue}{\left(D \cdot D\right)}}{{d}^{2}}\right) \cdot 0.25\right)\right)\right) \]

      12. unswap-sqr 88.7%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{\color{blue}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}}{{d}^{2}}\right) \cdot 0.25\right)\right)\right) \]

      13. unpow2 88.7%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{d \cdot d}}\right) \cdot 0.25\right)\right)\right) \]

      14. times-frac 91.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \color{blue}{\left(\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d}\right)}\right) \cdot 0.25\right)\right)\right) \]

      15. associate-*r/ 91.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(\color{blue}{\left(M \cdot \frac{D}{d}\right)} \cdot \frac{M \cdot D}{d}\right)\right) \cdot 0.25\right)\right)\right) \]

      16. associate-*r/ 91.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(M \cdot \frac{D}{d}\right)}\right)\right) \cdot 0.25\right)\right)\right) \]

   6. Simplified 91.6%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\left(\frac{h}{\ell} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)\right) \cdot 0.25\right)}\right)\right) \]

   if -4.49999999999999996e-238 < d < -1.999999999999994e-310

   1. Initial program 29.9%

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

   3. Applied egg-rr 5.9%

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

      1. expm1-def 5.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 24.0%

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

   5. Simplified 24.0%

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

   6. Taylor expanded in d around -inf 35.8%

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

      1. mul-1-neg 35.8%

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

      2. distribute-rgt-neg-in 35.8%

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

      3. *-commutative 35.8%

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

   8. Simplified 35.8%

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

   9. Taylor expanded in D around inf 65.3%

      \[\leadsto \color{blue}{0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 65.3%

          \[\leadsto 0.125 \cdot \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

       2. unpow2 65.3%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{d}\right) \]

       3. unpow2 65.3%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{d}\right) \]

       4. associate-*l/ 59.5%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)}\right) \]

       5. *-commutative 59.5%

          \[\leadsto \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)\right) \cdot 0.125} \]

       6. associate-*l* 59.5%

          \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right) \cdot 0.125\right)} \]

       7. *-commutative 59.5%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(0.125 \cdot \left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)\right)} \]

       8. associate-*r* 60.3%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \color{blue}{\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot M\right)}\right) \]

       9. *-commutative 60.3%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \color{blue}{\left(M \cdot \left(\frac{D \cdot D}{d} \cdot M\right)\right)}\right) \]

       10. associate-/l* 60.2%

           \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \left(\color{blue}{\frac{D}{\frac{d}{D}}} \cdot M\right)\right)\right) \]

       11. associate-*l/ 65.9%

           \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \color{blue}{\frac{D \cdot M}{\frac{d}{D}}}\right)\right) \]

   11. Simplified 65.9%

       \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)} \]

   if -1.999999999999994e-310 < d < 7.1999999999999995e-113

   1. Initial program 35.1%

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

      1. metadata-eval 35.1%

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

      2. unpow1/2 35.1%

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

      3. metadata-eval 35.1%

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

      4. unpow1/2 35.1%

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

      5. *-commutative 35.1%

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

      6. associate-*l* 35.1%

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

      7. times-frac 35.2%

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

      8. metadata-eval 35.2%

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

   3. Simplified 35.2%

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

      1. frac-times 35.1%

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

      2. clear-num 35.2%

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

      3. *-commutative 35.2%

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

   5. Applied egg-rr 35.2%

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

      1. associate-*r* 35.2%

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

      2. *-commutative 35.2%

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

      3. clear-num 35.1%

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

      4. *-commutative 35.1%

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

      5. metadata-eval 35.1%

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

      6. associate-*r/ 37.4%

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

      7. metadata-eval 37.4%

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

      8. times-frac 37.5%

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

   7. Applied egg-rr 37.5%

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

   8. Taylor expanded in d around 0 40.1%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 40.1%

         \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative 40.1%

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      3. unpow2 40.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{d}\right) \]

      4. unpow2 40.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{d}\right) \]

      5. associate-*l/ 39.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)}\right) \]

      6. associate-*r* 42.7%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot M\right)}\right) \]

      7. *-commutative 42.7%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(M \cdot \left(\frac{D \cdot D}{d} \cdot M\right)\right)}\right) \]

      8. associate-/l* 48.0%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(M \cdot \left(\color{blue}{\frac{D}{\frac{d}{D}}} \cdot M\right)\right)\right) \]

      9. associate-*l/ 50.5%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(M \cdot \color{blue}{\frac{D \cdot M}{\frac{d}{D}}}\right)\right) \]

   10. Simplified 50.5%

       \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)} \]

   if 7.1999999999999995e-113 < d < 2.5999999999999998e133

   1. Initial program 85.2%

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

   2. Taylor expanded in M around 0 56.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   3. Step-by-step derivation

      1. associate-*r/ 56.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right) \]

      2. *-commutative 56.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right) \]

      3. associate-*r/ 56.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]

      4. associate-*r* 58.2%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}\right) \]

      5. *-commutative 58.2%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{h \cdot \left({D}^{2} \cdot {M}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. *-commutative 58.2%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \frac{h \cdot \left({D}^{2} \cdot {M}^{2}\right)}{\color{blue}{\ell \cdot {d}^{2}}}\right) \]

      7. times-frac 61.7%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)}\right) \]

      8. *-commutative 61.7%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}}\right)\right) \]

      9. unpow2 61.7%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot {D}^{2}}{{d}^{2}}\right)\right) \]

      10. unpow2 61.7%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot \color{blue}{\left(D \cdot D\right)}}{{d}^{2}}\right)\right) \]

      11. unswap-sqr 83.5%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot \frac{\color{blue}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}}{{d}^{2}}\right)\right) \]

      12. unpow2 83.5%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\color{blue}{d \cdot d}}\right)\right) \]

      13. times-frac 85.2%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot \color{blue}{\left(\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d}\right)}\right)\right) \]

      14. associate-*r/ 85.2%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot \left(\color{blue}{\left(M \cdot \frac{D}{d}\right)} \cdot \frac{M \cdot D}{d}\right)\right)\right) \]

      15. associate-*r/ 85.2%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(M \cdot \frac{D}{d}\right)}\right)\right)\right) \]

   4. Simplified 85.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \left(\frac{h}{\ell} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)\right)}\right) \]

   if 2.5999999999999998e133 < d

   1. Initial program 66.3%

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

   2. Taylor expanded in d around inf 79.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-commutative 79.7%

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

      2. *-commutative 79.7%

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

      3. associate-/r* 79.7%

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

   4. Simplified 79.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   5. Step-by-step derivation

      1. sqrt-div 90.7%

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

      2. inv-pow 90.7%

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

   6. Applied egg-rr 90.7%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}}}{\sqrt{\ell}}} \]
   7. Step-by-step derivation

      1. sqr-pow 90.7%

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

      2. rem-sqrt-square 90.7%

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

      3. metadata-eval 90.7%

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

      4. sqr-pow 90.8%

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

      5. fabs-sqr 90.8%

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

      6. sqr-pow 90.7%

         \[\leadsto d \cdot \frac{\color{blue}{{h}^{-0.5}}}{\sqrt{\ell}} \]

   8. Simplified 90.7%

      \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.9 \cdot 10^{+115}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -3.6 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{h}{\ell} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)\right) \cdot 0.25\right)\right)\right)\\ \mathbf{elif}\;d \leq -4.5 \cdot 10^{-238}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{-113}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right) \cdot -0.125\right)\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{+133}:\\ \;\;\;\;\left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot {\left(\frac{d}{h}\right)}^{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 11: 61.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{{\ell}^{3}}}\\ t_1 := 0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\\ t_2 := \sqrt{\frac{1}{h \cdot \ell}}\\ t_3 := d \cdot t_2\\ t_4 := M \cdot \frac{D \cdot M}{\frac{d}{D}}\\ \mathbf{if}\;d \leq -3.1 \cdot 10^{+233}:\\ \;\;\;\;t_2 \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -2.3 \cdot 10^{+110}:\\ \;\;\;\;t_3 \cdot \left(-1 - -0.5 \cdot \left(t_1 \cdot \frac{M \cdot \left(h \cdot M\right)}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -8.2 \cdot 10^{-63}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - h \cdot \frac{0.5 \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;d \leq -1.85 \cdot 10^{-200}:\\ \;\;\;\;t_3 \cdot \left(-1 - -0.5 \cdot \left(t_1 \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_0 \cdot \left(0.125 \cdot t_4\right)\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-112}:\\ \;\;\;\;t_0 \cdot \left(t_4 \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h (pow l 3.0))))
        (t_1 (* 0.25 (* (/ D d) (/ D d))))
        (t_2 (sqrt (/ 1.0 (* h l))))
        (t_3 (* d t_2))
        (t_4 (* M (/ (* D M) (/ d D)))))
   (if (<= d -3.1e+233)
     (* t_2 (- d))
     (if (<= d -2.3e+110)
       (* t_3 (- -1.0 (* -0.5 (* t_1 (/ (* M (* h M)) l)))))
       (if (<= d -8.2e-63)
         (*
          (sqrt (* (/ d h) (/ d l)))
          (- 1.0 (* h (/ (* 0.5 (pow (* (/ M d) (/ D 2.0)) 2.0)) l))))
         (if (<= d -1.85e-200)
           (* t_3 (- -1.0 (* -0.5 (* t_1 (/ (* M M) (/ l h))))))
           (if (<= d -2e-310)
             (* t_0 (* 0.125 t_4))
             (if (<= d 1.4e-112)
               (* t_0 (* t_4 -0.125))
               (* d (/ (pow h -0.5) (sqrt l)))))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / pow(l, 3.0)));
	double t_1 = 0.25 * ((D / d) * (D / d));
	double t_2 = sqrt((1.0 / (h * l)));
	double t_3 = d * t_2;
	double t_4 = M * ((D * M) / (d / D));
	double tmp;
	if (d <= -3.1e+233) {
		tmp = t_2 * -d;
	} else if (d <= -2.3e+110) {
		tmp = t_3 * (-1.0 - (-0.5 * (t_1 * ((M * (h * M)) / l))));
	} else if (d <= -8.2e-63) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 - (h * ((0.5 * pow(((M / d) * (D / 2.0)), 2.0)) / l)));
	} else if (d <= -1.85e-200) {
		tmp = t_3 * (-1.0 - (-0.5 * (t_1 * ((M * M) / (l / h)))));
	} else if (d <= -2e-310) {
		tmp = t_0 * (0.125 * t_4);
	} else if (d <= 1.4e-112) {
		tmp = t_0 * (t_4 * -0.125);
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this 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) :: tmp
    t_0 = sqrt((h / (l ** 3.0d0)))
    t_1 = 0.25d0 * ((d_1 / d) * (d_1 / d))
    t_2 = sqrt((1.0d0 / (h * l)))
    t_3 = d * t_2
    t_4 = m * ((d_1 * m) / (d / d_1))
    if (d <= (-3.1d+233)) then
        tmp = t_2 * -d
    else if (d <= (-2.3d+110)) then
        tmp = t_3 * ((-1.0d0) - ((-0.5d0) * (t_1 * ((m * (h * m)) / l))))
    else if (d <= (-8.2d-63)) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 - (h * ((0.5d0 * (((m / d) * (d_1 / 2.0d0)) ** 2.0d0)) / l)))
    else if (d <= (-1.85d-200)) then
        tmp = t_3 * ((-1.0d0) - ((-0.5d0) * (t_1 * ((m * m) / (l / h)))))
    else if (d <= (-2d-310)) then
        tmp = t_0 * (0.125d0 * t_4)
    else if (d <= 1.4d-112) then
        tmp = t_0 * (t_4 * (-0.125d0))
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / Math.pow(l, 3.0)));
	double t_1 = 0.25 * ((D / d) * (D / d));
	double t_2 = Math.sqrt((1.0 / (h * l)));
	double t_3 = d * t_2;
	double t_4 = M * ((D * M) / (d / D));
	double tmp;
	if (d <= -3.1e+233) {
		tmp = t_2 * -d;
	} else if (d <= -2.3e+110) {
		tmp = t_3 * (-1.0 - (-0.5 * (t_1 * ((M * (h * M)) / l))));
	} else if (d <= -8.2e-63) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 - (h * ((0.5 * Math.pow(((M / d) * (D / 2.0)), 2.0)) / l)));
	} else if (d <= -1.85e-200) {
		tmp = t_3 * (-1.0 - (-0.5 * (t_1 * ((M * M) / (l / h)))));
	} else if (d <= -2e-310) {
		tmp = t_0 * (0.125 * t_4);
	} else if (d <= 1.4e-112) {
		tmp = t_0 * (t_4 * -0.125);
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt((h / math.pow(l, 3.0)))
	t_1 = 0.25 * ((D / d) * (D / d))
	t_2 = math.sqrt((1.0 / (h * l)))
	t_3 = d * t_2
	t_4 = M * ((D * M) / (d / D))
	tmp = 0
	if d <= -3.1e+233:
		tmp = t_2 * -d
	elif d <= -2.3e+110:
		tmp = t_3 * (-1.0 - (-0.5 * (t_1 * ((M * (h * M)) / l))))
	elif d <= -8.2e-63:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 - (h * ((0.5 * math.pow(((M / d) * (D / 2.0)), 2.0)) / l)))
	elif d <= -1.85e-200:
		tmp = t_3 * (-1.0 - (-0.5 * (t_1 * ((M * M) / (l / h)))))
	elif d <= -2e-310:
		tmp = t_0 * (0.125 * t_4)
	elif d <= 1.4e-112:
		tmp = t_0 * (t_4 * -0.125)
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / (l ^ 3.0)))
	t_1 = Float64(0.25 * Float64(Float64(D / d) * Float64(D / d)))
	t_2 = sqrt(Float64(1.0 / Float64(h * l)))
	t_3 = Float64(d * t_2)
	t_4 = Float64(M * Float64(Float64(D * M) / Float64(d / D)))
	tmp = 0.0
	if (d <= -3.1e+233)
		tmp = Float64(t_2 * Float64(-d));
	elseif (d <= -2.3e+110)
		tmp = Float64(t_3 * Float64(-1.0 - Float64(-0.5 * Float64(t_1 * Float64(Float64(M * Float64(h * M)) / l)))));
	elseif (d <= -8.2e-63)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 - Float64(h * Float64(Float64(0.5 * (Float64(Float64(M / d) * Float64(D / 2.0)) ^ 2.0)) / l))));
	elseif (d <= -1.85e-200)
		tmp = Float64(t_3 * Float64(-1.0 - Float64(-0.5 * Float64(t_1 * Float64(Float64(M * M) / Float64(l / h))))));
	elseif (d <= -2e-310)
		tmp = Float64(t_0 * Float64(0.125 * t_4));
	elseif (d <= 1.4e-112)
		tmp = Float64(t_0 * Float64(t_4 * -0.125));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / (l ^ 3.0)));
	t_1 = 0.25 * ((D / d) * (D / d));
	t_2 = sqrt((1.0 / (h * l)));
	t_3 = d * t_2;
	t_4 = M * ((D * M) / (d / D));
	tmp = 0.0;
	if (d <= -3.1e+233)
		tmp = t_2 * -d;
	elseif (d <= -2.3e+110)
		tmp = t_3 * (-1.0 - (-0.5 * (t_1 * ((M * (h * M)) / l))));
	elseif (d <= -8.2e-63)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 - (h * ((0.5 * (((M / d) * (D / 2.0)) ^ 2.0)) / l)));
	elseif (d <= -1.85e-200)
		tmp = t_3 * (-1.0 - (-0.5 * (t_1 * ((M * M) / (l / h)))));
	elseif (d <= -2e-310)
		tmp = t_0 * (0.125 * t_4);
	elseif (d <= 1.4e-112)
		tmp = t_0 * (t_4 * -0.125);
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(0.25 * N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(d * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(M * N[(N[(D * M), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.1e+233], N[(t$95$2 * (-d)), $MachinePrecision], If[LessEqual[d, -2.3e+110], N[(t$95$3 * N[(-1.0 - N[(-0.5 * N[(t$95$1 * N[(N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -8.2e-63], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(h * N[(N[(0.5 * N[Power[N[(N[(M / d), $MachinePrecision] * N[(D / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.85e-200], N[(t$95$3 * N[(-1.0 - N[(-0.5 * N[(t$95$1 * N[(N[(M * M), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2e-310], N[(t$95$0 * N[(0.125 * t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.4e-112], N[(t$95$0 * N[(t$95$4 * -0.125), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if d < -3.10000000000000016e233

   1. Initial program 66.8%

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

   3. Applied egg-rr 48.6%

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

      1. expm1-def 48.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 66.8%

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

   5. Simplified 66.8%

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

   6. Taylor expanded in d around -inf 88.2%

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

      1. mul-1-neg 88.2%

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

      2. distribute-rgt-neg-in 88.2%

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

      3. *-commutative 88.2%

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

   8. Simplified 88.2%

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

   9. Taylor expanded in D around 0 82.3%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   10. Step-by-step derivation

       1. associate-/l/ 82.2%

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

       2. associate-*r* 82.2%

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

       3. *-commutative 82.2%

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

       4. associate-/l/ 82.3%

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

       5. *-commutative 82.3%

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

       6. mul-1-neg 82.3%

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

   11. Simplified 82.3%

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

   if -3.10000000000000016e233 < d < -2.3e110

   1. Initial program 63.7%

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

   3. Applied egg-rr 18.6%

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

      1. expm1-def 18.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 53.5%

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

   5. Simplified 53.5%

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

   6. Taylor expanded in d around -inf 78.5%

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

      1. mul-1-neg 78.5%

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

      2. distribute-rgt-neg-in 78.5%

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

      3. *-commutative 78.5%

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

   8. Simplified 78.5%

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

   9. Taylor expanded in h around 0 55.3%

      \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]
   10. Step-by-step derivation

       1. times-frac 56.2%

          \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       2. *-commutative 56.2%

          \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right)\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       3. associate-*r* 56.2%

          \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       4. unpow2 56.2%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       5. unpow2 56.2%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       6. times-frac 84.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       7. associate-/l* 73.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{{M}^{2}}{\frac{\ell}{h}}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       8. unpow2 73.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{M \cdot M}}{\frac{\ell}{h}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   11. Simplified 73.0%

       \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   12. Taylor expanded in M around 0 84.0%

       \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{{M}^{2} \cdot h}{\ell}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]
   13. Step-by-step derivation

       1. unpow2 84.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       2. associate-*l* 94.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   14. Simplified 94.0%

       \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{M \cdot \left(M \cdot h\right)}{\ell}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   if -2.3e110 < d < -8.1999999999999995e-63

   1. Initial program 89.6%

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

      1. metadata-eval 89.6%

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

      2. unpow1/2 89.6%

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

      3. metadata-eval 89.6%

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

      4. unpow1/2 89.6%

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

      5. *-commutative 89.6%

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

      6. associate-*l* 89.6%

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

      7. times-frac 89.6%

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

      8. metadata-eval 89.6%

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

   3. Simplified 89.6%

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

      1. frac-times 89.6%

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

      2. clear-num 89.5%

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

      3. *-commutative 89.5%

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

   5. Applied egg-rr 89.5%

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

      1. associate-*r* 89.5%

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

      2. *-commutative 89.5%

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

      3. clear-num 89.6%

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

      4. *-commutative 89.6%

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

      5. metadata-eval 89.6%

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

      6. associate-*r/ 89.6%

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

      7. metadata-eval 89.6%

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

      8. times-frac 89.6%

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

   7. Applied egg-rr 89.6%

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

      1. frac-times 89.6%

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

      2. associate-/r* 89.6%

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

   9. Applied egg-rr 89.6%

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

       1. pow1 89.6%

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

       2. sqrt-unprod 76.8%

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

       3. associate-/l* 76.8%

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

       4. associate-/l/ 76.8%

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

   11. Applied egg-rr 76.8%

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

       1. unpow1 76.8%

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

       2. associate-/r/ 76.8%

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

       3. times-frac 76.8%

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

   13. Simplified 76.8%

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

   if -8.1999999999999995e-63 < d < -1.85000000000000005e-200

   1. Initial program 74.3%

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

   3. Applied egg-rr 15.4%

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

      1. expm1-def 23.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 56.0%

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

   5. Simplified 56.0%

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

   6. Taylor expanded in d around -inf 80.9%

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

      1. mul-1-neg 80.9%

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

      2. distribute-rgt-neg-in 80.9%

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

      3. *-commutative 80.9%

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

   8. Simplified 80.9%

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

   9. Taylor expanded in h around 0 65.0%

      \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]
   10. Step-by-step derivation

       1. times-frac 61.1%

          \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       2. *-commutative 61.1%

          \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right)\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       3. associate-*r* 61.1%

          \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       4. unpow2 61.1%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       5. unpow2 61.1%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       6. times-frac 73.2%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       7. associate-/l* 77.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{{M}^{2}}{\frac{\ell}{h}}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       8. unpow2 77.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{M \cdot M}}{\frac{\ell}{h}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   11. Simplified 77.0%

       \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   if -1.85000000000000005e-200 < d < -1.999999999999994e-310

   1. Initial program 42.8%

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

   3. Applied egg-rr 7.9%

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

      1. expm1-def 7.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 32.3%

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

   5. Simplified 32.3%

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

   6. Taylor expanded in d around -inf 55.6%

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

      1. mul-1-neg 55.6%

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

      2. distribute-rgt-neg-in 55.6%

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

      3. *-commutative 55.6%

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

   8. Simplified 55.6%

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

   9. Taylor expanded in D around inf 69.1%

      \[\leadsto \color{blue}{0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 69.1%

          \[\leadsto 0.125 \cdot \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

       2. unpow2 69.1%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{d}\right) \]

       3. unpow2 69.1%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{d}\right) \]

       4. associate-*l/ 62.1%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)}\right) \]

       5. *-commutative 62.1%

          \[\leadsto \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)\right) \cdot 0.125} \]

       6. associate-*l* 62.1%

          \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right) \cdot 0.125\right)} \]

       7. *-commutative 62.1%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(0.125 \cdot \left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)\right)} \]

       8. associate-*r* 63.1%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \color{blue}{\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot M\right)}\right) \]

       9. *-commutative 63.1%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \color{blue}{\left(M \cdot \left(\frac{D \cdot D}{d} \cdot M\right)\right)}\right) \]

       10. associate-/l* 63.1%

           \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \left(\color{blue}{\frac{D}{\frac{d}{D}}} \cdot M\right)\right)\right) \]

       11. associate-*l/ 66.6%

           \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \color{blue}{\frac{D \cdot M}{\frac{d}{D}}}\right)\right) \]

   11. Simplified 66.6%

       \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)} \]

   if -1.999999999999994e-310 < d < 1.40000000000000011e-112

   1. Initial program 35.1%

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

      1. metadata-eval 35.1%

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

      2. unpow1/2 35.1%

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

      3. metadata-eval 35.1%

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

      4. unpow1/2 35.1%

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

      5. *-commutative 35.1%

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

      6. associate-*l* 35.1%

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

      7. times-frac 35.2%

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

      8. metadata-eval 35.2%

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

   3. Simplified 35.2%

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

      1. frac-times 35.1%

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

      2. clear-num 35.2%

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

      3. *-commutative 35.2%

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

   5. Applied egg-rr 35.2%

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

      1. associate-*r* 35.2%

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

      2. *-commutative 35.2%

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

      3. clear-num 35.1%

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

      4. *-commutative 35.1%

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

      5. metadata-eval 35.1%

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

      6. associate-*r/ 37.4%

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

      7. metadata-eval 37.4%

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

      8. times-frac 37.5%

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

   7. Applied egg-rr 37.5%

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

   8. Taylor expanded in d around 0 40.1%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 40.1%

         \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative 40.1%

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      3. unpow2 40.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{d}\right) \]

      4. unpow2 40.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{d}\right) \]

      5. associate-*l/ 39.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)}\right) \]

      6. associate-*r* 42.7%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot M\right)}\right) \]

      7. *-commutative 42.7%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(M \cdot \left(\frac{D \cdot D}{d} \cdot M\right)\right)}\right) \]

      8. associate-/l* 48.0%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(M \cdot \left(\color{blue}{\frac{D}{\frac{d}{D}}} \cdot M\right)\right)\right) \]

      9. associate-*l/ 50.5%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(M \cdot \color{blue}{\frac{D \cdot M}{\frac{d}{D}}}\right)\right) \]

   10. Simplified 50.5%

       \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)} \]

   if 1.40000000000000011e-112 < d

   1. Initial program 78.1%

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

   2. Taylor expanded in d around inf 60.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-commutative 60.1%

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

      2. *-commutative 60.1%

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

      3. associate-/r* 60.1%

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

   4. Simplified 60.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   5. Step-by-step derivation

      1. sqrt-div 72.4%

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

      2. inv-pow 72.4%

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

   6. Applied egg-rr 72.4%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}}}{\sqrt{\ell}}} \]
   7. Step-by-step derivation

      1. sqr-pow 72.5%

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

      2. rem-sqrt-square 72.5%

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

      3. metadata-eval 72.5%

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

      4. sqr-pow 72.3%

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

      5. fabs-sqr 72.3%

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

      6. sqr-pow 72.5%

         \[\leadsto d \cdot \frac{\color{blue}{{h}^{-0.5}}}{\sqrt{\ell}} \]

   8. Simplified 72.5%

      \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.1 \cdot 10^{+233}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -2.3 \cdot 10^{+110}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot \left(h \cdot M\right)}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -8.2 \cdot 10^{-63}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - h \cdot \frac{0.5 \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;d \leq -1.85 \cdot 10^{-200}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right) \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 12: 62.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -9 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)\\ \mathbf{elif}\;\ell \leq -7.8 \cdot 10^{-131}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot \left(h \cdot M\right)}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{-39}:\\ \;\;\;\;\left(1 + -0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 6.8 \cdot 10^{+58}:\\ \;\;\;\;d \cdot \frac{1}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+96}:\\ \;\;\;\;-0.125 \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -9e+119)
   (* (sqrt (/ (/ 1.0 l) h)) (- d))
   (if (<= l -7.8e-131)
     (*
      (* d (sqrt (/ 1.0 (* h l))))
      (- -1.0 (* -0.5 (* (* 0.25 (* (/ D d) (/ D d))) (/ (* M (* h M)) l)))))
     (if (<= l 4.8e-39)
       (*
        (+ 1.0 (* -0.5 (/ (* h (pow (* (/ D d) (/ M 2.0)) 2.0)) l)))
        (sqrt (* (/ d h) (/ d l))))
       (if (<= l 6.8e+58)
         (* d (/ 1.0 (sqrt (* h l))))
         (if (<= l 4e+96)
           (* -0.125 (/ (* (sqrt (/ h (pow l 3.0))) (* D (* D (* M M)))) d))
           (* d (/ (pow h -0.5) (sqrt l)))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -9e+119) {
		tmp = sqrt(((1.0 / l) / h)) * -d;
	} else if (l <= -7.8e-131) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * (h * M)) / l))));
	} else if (l <= 4.8e-39) {
		tmp = (1.0 + (-0.5 * ((h * pow(((D / d) * (M / 2.0)), 2.0)) / l))) * sqrt(((d / h) * (d / l)));
	} else if (l <= 6.8e+58) {
		tmp = d * (1.0 / sqrt((h * l)));
	} else if (l <= 4e+96) {
		tmp = -0.125 * ((sqrt((h / pow(l, 3.0))) * (D * (D * (M * M)))) / d);
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this 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) :: tmp
    if (l <= (-9d+119)) then
        tmp = sqrt(((1.0d0 / l) / h)) * -d
    else if (l <= (-7.8d-131)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) - ((-0.5d0) * ((0.25d0 * ((d_1 / d) * (d_1 / d))) * ((m * (h * m)) / l))))
    else if (l <= 4.8d-39) then
        tmp = (1.0d0 + ((-0.5d0) * ((h * (((d_1 / d) * (m / 2.0d0)) ** 2.0d0)) / l))) * sqrt(((d / h) * (d / l)))
    else if (l <= 6.8d+58) then
        tmp = d * (1.0d0 / sqrt((h * l)))
    else if (l <= 4d+96) then
        tmp = (-0.125d0) * ((sqrt((h / (l ** 3.0d0))) * (d_1 * (d_1 * (m * m)))) / d)
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -9e+119) {
		tmp = Math.sqrt(((1.0 / l) / h)) * -d;
	} else if (l <= -7.8e-131) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * (h * M)) / l))));
	} else if (l <= 4.8e-39) {
		tmp = (1.0 + (-0.5 * ((h * Math.pow(((D / d) * (M / 2.0)), 2.0)) / l))) * Math.sqrt(((d / h) * (d / l)));
	} else if (l <= 6.8e+58) {
		tmp = d * (1.0 / Math.sqrt((h * l)));
	} else if (l <= 4e+96) {
		tmp = -0.125 * ((Math.sqrt((h / Math.pow(l, 3.0))) * (D * (D * (M * M)))) / d);
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -9e+119:
		tmp = math.sqrt(((1.0 / l) / h)) * -d
	elif l <= -7.8e-131:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * (h * M)) / l))))
	elif l <= 4.8e-39:
		tmp = (1.0 + (-0.5 * ((h * math.pow(((D / d) * (M / 2.0)), 2.0)) / l))) * math.sqrt(((d / h) * (d / l)))
	elif l <= 6.8e+58:
		tmp = d * (1.0 / math.sqrt((h * l)))
	elif l <= 4e+96:
		tmp = -0.125 * ((math.sqrt((h / math.pow(l, 3.0))) * (D * (D * (M * M)))) / d)
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -9e+119)
		tmp = Float64(sqrt(Float64(Float64(1.0 / l) / h)) * Float64(-d));
	elseif (l <= -7.8e-131)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 - Float64(-0.5 * Float64(Float64(0.25 * Float64(Float64(D / d) * Float64(D / d))) * Float64(Float64(M * Float64(h * M)) / l)))));
	elseif (l <= 4.8e-39)
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(Float64(h * (Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0)) / l))) * sqrt(Float64(Float64(d / h) * Float64(d / l))));
	elseif (l <= 6.8e+58)
		tmp = Float64(d * Float64(1.0 / sqrt(Float64(h * l))));
	elseif (l <= 4e+96)
		tmp = Float64(-0.125 * Float64(Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(D * Float64(D * Float64(M * M)))) / d));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -9e+119)
		tmp = sqrt(((1.0 / l) / h)) * -d;
	elseif (l <= -7.8e-131)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * (h * M)) / l))));
	elseif (l <= 4.8e-39)
		tmp = (1.0 + (-0.5 * ((h * (((D / d) * (M / 2.0)) ^ 2.0)) / l))) * sqrt(((d / h) * (d / l)));
	elseif (l <= 6.8e+58)
		tmp = d * (1.0 / sqrt((h * l)));
	elseif (l <= 4e+96)
		tmp = -0.125 * ((sqrt((h / (l ^ 3.0))) * (D * (D * (M * M)))) / d);
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -9e+119], N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], If[LessEqual[l, -7.8e-131], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[(0.25 * N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.8e-39], N[(N[(1.0 + N[(-0.5 * N[(N[(h * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.8e+58], N[(d * N[(1.0 / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4e+96], N[(-0.125 * N[(N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(D * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if l < -9.00000000000000039e119

   1. Initial program 55.2%

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

   3. Applied egg-rr 28.7%

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

      1. expm1-def 46.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 50.5%

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

   5. Simplified 50.5%

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

   6. Taylor expanded in d around -inf 64.0%

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

      1. mul-1-neg 64.0%

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

      2. distribute-rgt-neg-in 64.0%

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

      3. *-commutative 64.0%

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

   8. Simplified 64.0%

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

   9. Taylor expanded in h around 0 41.7%

      \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]
   10. Step-by-step derivation

       1. times-frac 38.9%

          \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       2. *-commutative 38.9%

          \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right)\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       3. associate-*r* 38.9%

          \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       4. unpow2 38.9%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       5. unpow2 38.9%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       6. times-frac 47.3%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       7. associate-/l* 47.3%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{{M}^{2}}{\frac{\ell}{h}}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       8. unpow2 47.3%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{M \cdot M}}{\frac{\ell}{h}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   11. Simplified 47.3%

       \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   12. Taylor expanded in D around 0 67.4%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   13. Step-by-step derivation

       1. associate-/l/ 67.4%

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

       2. unpow-1 67.4%

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

       3. associate-*r* 67.4%

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

       4. *-commutative 67.4%

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

       5. mul-1-neg 67.4%

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

       6. unpow-1 67.4%

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

       7. associate-/l/ 67.4%

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

       8. associate-/r* 67.4%

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

   14. Simplified 67.4%

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

   if -9.00000000000000039e119 < l < -7.80000000000000039e-131

   1. Initial program 71.0%

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

   3. Applied egg-rr 14.8%

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

      1. expm1-def 21.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 49.5%

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

   5. Simplified 49.5%

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

   6. Taylor expanded in d around -inf 76.9%

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

      1. mul-1-neg 76.9%

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

      2. distribute-rgt-neg-in 76.9%

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

      3. *-commutative 76.9%

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

   8. Simplified 76.9%

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

   9. Taylor expanded in h around 0 56.5%

      \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]
   10. Step-by-step derivation

       1. times-frac 55.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       2. *-commutative 55.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right)\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       3. associate-*r* 55.0%

          \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       4. unpow2 55.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       5. unpow2 55.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       6. times-frac 63.5%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       7. associate-/l* 66.8%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{{M}^{2}}{\frac{\ell}{h}}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       8. unpow2 66.8%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{M \cdot M}}{\frac{\ell}{h}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   11. Simplified 66.8%

       \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   12. Taylor expanded in M around 0 63.5%

       \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{{M}^{2} \cdot h}{\ell}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]
   13. Step-by-step derivation

       1. unpow2 63.5%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       2. associate-*l* 68.3%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   14. Simplified 68.3%

       \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{M \cdot \left(M \cdot h\right)}{\ell}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   if -7.80000000000000039e-131 < l < 4.80000000000000031e-39

   1. Initial program 76.7%

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

   3. Applied egg-rr 24.5%

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

      1. expm1-def 29.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 68.6%

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

   5. Simplified 68.6%

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

      1. associate-*l/ 72.4%

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

      2. associate-*r/ 72.4%

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

      3. *-commutative 72.4%

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

      4. times-frac 71.2%

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

   7. Applied egg-rr 71.2%

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

   if 4.80000000000000031e-39 < l < 6.8000000000000001e58

   1. Initial program 70.5%

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

   2. Taylor expanded in d around inf 62.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. sqrt-div 62.8%

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

      2. *-commutative 62.8%

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

      3. metadata-eval 62.8%

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

   4. Applied egg-rr 62.8%

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

   if 6.8000000000000001e58 < l < 4.0000000000000002e96

   1. Initial program 30.9%

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

      1. metadata-eval 30.9%

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

      2. unpow1/2 30.9%

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

      3. metadata-eval 30.9%

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

      4. unpow1/2 30.9%

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

      5. *-commutative 30.9%

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

      6. associate-*l* 30.9%

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

      7. times-frac 31.1%

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

      8. metadata-eval 31.1%

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

   3. Simplified 31.1%

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

      1. frac-times 30.9%

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

      2. clear-num 30.9%

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

      3. *-commutative 30.9%

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

   5. Applied egg-rr 30.9%

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

      1. associate-*r* 30.9%

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

      2. *-commutative 30.9%

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

      3. clear-num 30.9%

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

      4. *-commutative 30.9%

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

      5. metadata-eval 30.9%

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

      6. associate-*r/ 30.9%

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

      7. metadata-eval 30.9%

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

      8. times-frac 31.1%

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

   7. Applied egg-rr 31.1%

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

      1. frac-times 30.9%

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

      2. associate-/r* 30.9%

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

   9. Applied egg-rr 30.9%

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

   10. Taylor expanded in d around 0 72.0%

       \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   11. Step-by-step derivation

       1. associate-*l/ 72.2%

          \[\leadsto -0.125 \cdot \color{blue}{\frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}} \]

       2. unpow2 72.2%

          \[\leadsto -0.125 \cdot \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d} \]

       3. unpow2 72.2%

          \[\leadsto -0.125 \cdot \frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d} \]

       4. associate-*l* 72.6%

          \[\leadsto -0.125 \cdot \frac{\color{blue}{\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d} \]

   12. Simplified 72.6%

       \[\leadsto \color{blue}{-0.125 \cdot \frac{\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}} \]

   if 4.0000000000000002e96 < l

   1. Initial program 61.4%

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

   2. Taylor expanded in d around inf 60.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-commutative 60.7%

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

      2. *-commutative 60.7%

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

      3. associate-/r* 60.7%

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

   4. Simplified 60.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   5. Step-by-step derivation

      1. sqrt-div 75.5%

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

      2. inv-pow 75.5%

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

   6. Applied egg-rr 75.5%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}}}{\sqrt{\ell}}} \]
   7. Step-by-step derivation

      1. sqr-pow 75.6%

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

      2. rem-sqrt-square 75.6%

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

      3. metadata-eval 75.6%

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

      4. sqr-pow 75.4%

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

      5. fabs-sqr 75.4%

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

      6. sqr-pow 75.6%

         \[\leadsto d \cdot \frac{\color{blue}{{h}^{-0.5}}}{\sqrt{\ell}} \]

   8. Simplified 75.6%

      \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)\\ \mathbf{elif}\;\ell \leq -7.8 \cdot 10^{-131}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot \left(h \cdot M\right)}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{-39}:\\ \;\;\;\;\left(1 + -0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 6.8 \cdot 10^{+58}:\\ \;\;\;\;d \cdot \frac{1}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+96}:\\ \;\;\;\;-0.125 \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 13: 64.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{{\ell}^{3}}}\\ t_1 := M \cdot \frac{D \cdot M}{\frac{d}{D}}\\ \mathbf{if}\;d \leq -5.2 \cdot 10^{-238}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_0 \cdot \left(0.125 \cdot t_1\right)\\ \mathbf{elif}\;d \leq 1.3 \cdot 10^{-111}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h (pow l 3.0)))) (t_1 (* M (/ (* D M) (/ d D)))))
   (if (<= d -5.2e-238)
     (*
      (* d (sqrt (/ 1.0 (* h l))))
      (- -1.0 (* -0.5 (* (/ h l) (pow (* D (/ M (* 2.0 d))) 2.0)))))
     (if (<= d -2e-310)
       (* t_0 (* 0.125 t_1))
       (if (<= d 1.3e-111)
         (* t_0 (* t_1 -0.125))
         (* d (/ (pow h -0.5) (sqrt l))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / pow(l, 3.0)));
	double t_1 = M * ((D * M) / (d / D));
	double tmp;
	if (d <= -5.2e-238) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * pow((D * (M / (2.0 * d))), 2.0))));
	} else if (d <= -2e-310) {
		tmp = t_0 * (0.125 * t_1);
	} else if (d <= 1.3e-111) {
		tmp = t_0 * (t_1 * -0.125);
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this 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) :: tmp
    t_0 = sqrt((h / (l ** 3.0d0)))
    t_1 = m * ((d_1 * m) / (d / d_1))
    if (d <= (-5.2d-238)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) - ((-0.5d0) * ((h / l) * ((d_1 * (m / (2.0d0 * d))) ** 2.0d0))))
    else if (d <= (-2d-310)) then
        tmp = t_0 * (0.125d0 * t_1)
    else if (d <= 1.3d-111) then
        tmp = t_0 * (t_1 * (-0.125d0))
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / Math.pow(l, 3.0)));
	double t_1 = M * ((D * M) / (d / D));
	double tmp;
	if (d <= -5.2e-238) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * Math.pow((D * (M / (2.0 * d))), 2.0))));
	} else if (d <= -2e-310) {
		tmp = t_0 * (0.125 * t_1);
	} else if (d <= 1.3e-111) {
		tmp = t_0 * (t_1 * -0.125);
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt((h / math.pow(l, 3.0)))
	t_1 = M * ((D * M) / (d / D))
	tmp = 0
	if d <= -5.2e-238:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * math.pow((D * (M / (2.0 * d))), 2.0))))
	elif d <= -2e-310:
		tmp = t_0 * (0.125 * t_1)
	elif d <= 1.3e-111:
		tmp = t_0 * (t_1 * -0.125)
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / (l ^ 3.0)))
	t_1 = Float64(M * Float64(Float64(D * M) / Float64(d / D)))
	tmp = 0.0
	if (d <= -5.2e-238)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 - Float64(-0.5 * Float64(Float64(h / l) * (Float64(D * Float64(M / Float64(2.0 * d))) ^ 2.0)))));
	elseif (d <= -2e-310)
		tmp = Float64(t_0 * Float64(0.125 * t_1));
	elseif (d <= 1.3e-111)
		tmp = Float64(t_0 * Float64(t_1 * -0.125));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / (l ^ 3.0)));
	t_1 = M * ((D * M) / (d / D));
	tmp = 0.0;
	if (d <= -5.2e-238)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((h / l) * ((D * (M / (2.0 * d))) ^ 2.0))));
	elseif (d <= -2e-310)
		tmp = t_0 * (0.125 * t_1);
	elseif (d <= 1.3e-111)
		tmp = t_0 * (t_1 * -0.125);
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(M * N[(N[(D * M), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5.2e-238], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2e-310], N[(t$95$0 * N[(0.125 * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.3e-111], N[(t$95$0 * N[(t$95$1 * -0.125), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -5.2000000000000002e-238

   1. Initial program 75.2%

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

   3. Applied egg-rr 24.3%

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

      1. expm1-def 34.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 63.1%

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

   5. Simplified 63.1%

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

   6. Taylor expanded in d around -inf 79.6%

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

      1. mul-1-neg 79.6%

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

      2. distribute-rgt-neg-in 79.6%

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

      3. *-commutative 79.6%

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

   8. Simplified 79.6%

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

   if -5.2000000000000002e-238 < d < -1.999999999999994e-310

   1. Initial program 29.9%

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

   3. Applied egg-rr 5.9%

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

      1. expm1-def 5.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 24.0%

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

   5. Simplified 24.0%

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

   6. Taylor expanded in d around -inf 35.8%

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

      1. mul-1-neg 35.8%

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

      2. distribute-rgt-neg-in 35.8%

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

      3. *-commutative 35.8%

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

   8. Simplified 35.8%

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

   9. Taylor expanded in D around inf 65.3%

      \[\leadsto \color{blue}{0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 65.3%

          \[\leadsto 0.125 \cdot \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

       2. unpow2 65.3%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{d}\right) \]

       3. unpow2 65.3%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{d}\right) \]

       4. associate-*l/ 59.5%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)}\right) \]

       5. *-commutative 59.5%

          \[\leadsto \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)\right) \cdot 0.125} \]

       6. associate-*l* 59.5%

          \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right) \cdot 0.125\right)} \]

       7. *-commutative 59.5%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(0.125 \cdot \left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)\right)} \]

       8. associate-*r* 60.3%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \color{blue}{\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot M\right)}\right) \]

       9. *-commutative 60.3%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \color{blue}{\left(M \cdot \left(\frac{D \cdot D}{d} \cdot M\right)\right)}\right) \]

       10. associate-/l* 60.2%

           \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \left(\color{blue}{\frac{D}{\frac{d}{D}}} \cdot M\right)\right)\right) \]

       11. associate-*l/ 65.9%

           \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \color{blue}{\frac{D \cdot M}{\frac{d}{D}}}\right)\right) \]

   11. Simplified 65.9%

       \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)} \]

   if -1.999999999999994e-310 < d < 1.29999999999999991e-111

   1. Initial program 35.1%

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

      1. metadata-eval 35.1%

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

      2. unpow1/2 35.1%

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

      3. metadata-eval 35.1%

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

      4. unpow1/2 35.1%

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

      5. *-commutative 35.1%

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

      6. associate-*l* 35.1%

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

      7. times-frac 35.2%

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

      8. metadata-eval 35.2%

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

   3. Simplified 35.2%

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

      1. frac-times 35.1%

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

      2. clear-num 35.2%

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

      3. *-commutative 35.2%

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

   5. Applied egg-rr 35.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{1}{\frac{d \cdot 2}{M \cdot D}}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* 35.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{1}{\frac{d \cdot 2}{M \cdot D}}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative 35.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\left(\frac{1}{\frac{\color{blue}{2 \cdot d}}{M \cdot D}}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. clear-num 35.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative 35.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      5. metadata-eval 35.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*r/ 37.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      7. metadata-eval 37.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      8. times-frac 37.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(0.5 \cdot {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2}\right) \cdot h}{\ell}\right) \]

   7. Applied egg-rr 37.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(0.5 \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

   8. Taylor expanded in d around 0 40.1%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 40.1%

         \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative 40.1%

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      3. unpow2 40.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{d}\right) \]

      4. unpow2 40.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{d}\right) \]

      5. associate-*l/ 39.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)}\right) \]

      6. associate-*r* 42.7%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot M\right)}\right) \]

      7. *-commutative 42.7%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(M \cdot \left(\frac{D \cdot D}{d} \cdot M\right)\right)}\right) \]

      8. associate-/l* 48.0%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(M \cdot \left(\color{blue}{\frac{D}{\frac{d}{D}}} \cdot M\right)\right)\right) \]

      9. associate-*l/ 50.5%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(M \cdot \color{blue}{\frac{D \cdot M}{\frac{d}{D}}}\right)\right) \]

   10. Simplified 50.5%

       \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)} \]

   if 1.29999999999999991e-111 < d

   1. Initial program 78.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 60.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-commutative 60.1%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. *-commutative 60.1%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. associate-/r* 60.1%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   4. Simplified 60.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   5. Step-by-step derivation

      1. sqrt-div 72.4%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

      2. inv-pow 72.4%

         \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{-1}}}}{\sqrt{\ell}} \]

   6. Applied egg-rr 72.4%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}}}{\sqrt{\ell}}} \]
   7. Step-by-step derivation

      1. sqr-pow 72.5%

         \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{\left(\frac{-1}{2}\right)} \cdot {h}^{\left(\frac{-1}{2}\right)}}}}{\sqrt{\ell}} \]

      2. rem-sqrt-square 72.5%

         \[\leadsto d \cdot \frac{\color{blue}{\left|{h}^{\left(\frac{-1}{2}\right)}\right|}}{\sqrt{\ell}} \]

      3. metadata-eval 72.5%

         \[\leadsto d \cdot \frac{\left|{h}^{\color{blue}{-0.5}}\right|}{\sqrt{\ell}} \]

      4. sqr-pow 72.3%

         \[\leadsto d \cdot \frac{\left|\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}\right|}{\sqrt{\ell}} \]

      5. fabs-sqr 72.3%

         \[\leadsto d \cdot \frac{\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}}{\sqrt{\ell}} \]

      6. sqr-pow 72.5%

         \[\leadsto d \cdot \frac{\color{blue}{{h}^{-0.5}}}{\sqrt{\ell}} \]

   8. Simplified 72.5%

      \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.2 \cdot 10^{-238}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)\\ \mathbf{elif}\;d \leq 1.3 \cdot 10^{-111}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right) \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 14: 59.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{{\ell}^{3}}}\\ t_1 := \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;d \leq -2.85 \cdot 10^{+236}:\\ \;\;\;\;t_1 \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -1.9 \cdot 10^{-193}:\\ \;\;\;\;\left(d \cdot t_1\right) \cdot \left(-1 - -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot \left(h \cdot M\right)}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -3 \cdot 10^{-301}:\\ \;\;\;\;M \cdot \left(t_0 \cdot \frac{D \cdot D}{\frac{\frac{d}{M}}{0.125}}\right)\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-112}:\\ \;\;\;\;-0.125 \cdot \frac{t_0 \cdot \left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h (pow l 3.0)))) (t_1 (sqrt (/ 1.0 (* h l)))))
   (if (<= d -2.85e+236)
     (* t_1 (- d))
     (if (<= d -1.9e-193)
       (*
        (* d t_1)
        (- -1.0 (* -0.5 (* (* 0.25 (* (/ D d) (/ D d))) (/ (* M (* h M)) l)))))
       (if (<= d -3e-301)
         (* M (* t_0 (/ (* D D) (/ (/ d M) 0.125))))
         (if (<= d 1.4e-112)
           (* -0.125 (/ (* t_0 (* D (* D (* M M)))) d))
           (* d (/ (pow h -0.5) (sqrt l)))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / pow(l, 3.0)));
	double t_1 = sqrt((1.0 / (h * l)));
	double tmp;
	if (d <= -2.85e+236) {
		tmp = t_1 * -d;
	} else if (d <= -1.9e-193) {
		tmp = (d * t_1) * (-1.0 - (-0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * (h * M)) / l))));
	} else if (d <= -3e-301) {
		tmp = M * (t_0 * ((D * D) / ((d / M) / 0.125)));
	} else if (d <= 1.4e-112) {
		tmp = -0.125 * ((t_0 * (D * (D * (M * M)))) / d);
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this 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) :: tmp
    t_0 = sqrt((h / (l ** 3.0d0)))
    t_1 = sqrt((1.0d0 / (h * l)))
    if (d <= (-2.85d+236)) then
        tmp = t_1 * -d
    else if (d <= (-1.9d-193)) then
        tmp = (d * t_1) * ((-1.0d0) - ((-0.5d0) * ((0.25d0 * ((d_1 / d) * (d_1 / d))) * ((m * (h * m)) / l))))
    else if (d <= (-3d-301)) then
        tmp = m * (t_0 * ((d_1 * d_1) / ((d / m) / 0.125d0)))
    else if (d <= 1.4d-112) then
        tmp = (-0.125d0) * ((t_0 * (d_1 * (d_1 * (m * m)))) / d)
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / Math.pow(l, 3.0)));
	double t_1 = Math.sqrt((1.0 / (h * l)));
	double tmp;
	if (d <= -2.85e+236) {
		tmp = t_1 * -d;
	} else if (d <= -1.9e-193) {
		tmp = (d * t_1) * (-1.0 - (-0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * (h * M)) / l))));
	} else if (d <= -3e-301) {
		tmp = M * (t_0 * ((D * D) / ((d / M) / 0.125)));
	} else if (d <= 1.4e-112) {
		tmp = -0.125 * ((t_0 * (D * (D * (M * M)))) / d);
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt((h / math.pow(l, 3.0)))
	t_1 = math.sqrt((1.0 / (h * l)))
	tmp = 0
	if d <= -2.85e+236:
		tmp = t_1 * -d
	elif d <= -1.9e-193:
		tmp = (d * t_1) * (-1.0 - (-0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * (h * M)) / l))))
	elif d <= -3e-301:
		tmp = M * (t_0 * ((D * D) / ((d / M) / 0.125)))
	elif d <= 1.4e-112:
		tmp = -0.125 * ((t_0 * (D * (D * (M * M)))) / d)
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / (l ^ 3.0)))
	t_1 = sqrt(Float64(1.0 / Float64(h * l)))
	tmp = 0.0
	if (d <= -2.85e+236)
		tmp = Float64(t_1 * Float64(-d));
	elseif (d <= -1.9e-193)
		tmp = Float64(Float64(d * t_1) * Float64(-1.0 - Float64(-0.5 * Float64(Float64(0.25 * Float64(Float64(D / d) * Float64(D / d))) * Float64(Float64(M * Float64(h * M)) / l)))));
	elseif (d <= -3e-301)
		tmp = Float64(M * Float64(t_0 * Float64(Float64(D * D) / Float64(Float64(d / M) / 0.125))));
	elseif (d <= 1.4e-112)
		tmp = Float64(-0.125 * Float64(Float64(t_0 * Float64(D * Float64(D * Float64(M * M)))) / d));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / (l ^ 3.0)));
	t_1 = sqrt((1.0 / (h * l)));
	tmp = 0.0;
	if (d <= -2.85e+236)
		tmp = t_1 * -d;
	elseif (d <= -1.9e-193)
		tmp = (d * t_1) * (-1.0 - (-0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * (h * M)) / l))));
	elseif (d <= -3e-301)
		tmp = M * (t_0 * ((D * D) / ((d / M) / 0.125)));
	elseif (d <= 1.4e-112)
		tmp = -0.125 * ((t_0 * (D * (D * (M * M)))) / d);
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -2.85e+236], N[(t$95$1 * (-d)), $MachinePrecision], If[LessEqual[d, -1.9e-193], N[(N[(d * t$95$1), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[(0.25 * N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -3e-301], N[(M * N[(t$95$0 * N[(N[(D * D), $MachinePrecision] / N[(N[(d / M), $MachinePrecision] / 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.4e-112], N[(-0.125 * N[(N[(t$95$0 * N[(D * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -2.8499999999999998e236

   1. Initial program 66.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 48.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 48.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 66.8%

         \[\leadsto \color{blue}{\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   5. Simplified 66.8%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   6. Taylor expanded in d around -inf 88.2%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 88.2%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]

      2. distribute-rgt-neg-in 88.2%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]

      3. *-commutative 88.2%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right)\right) \]

   8. Simplified 88.2%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

   9. Taylor expanded in D around 0 82.3%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   10. Step-by-step derivation

       1. associate-/l/ 82.2%

          \[\leadsto -1 \cdot \left(d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}}\right) \]

       2. associate-*r* 82.2%

          \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]

       3. *-commutative 82.2%

          \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-1 \cdot d\right)} \]

       4. associate-/l/ 82.3%

          \[\leadsto \sqrt{\color{blue}{\frac{1}{\ell \cdot h}}} \cdot \left(-1 \cdot d\right) \]

       5. *-commutative 82.3%

          \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

       6. mul-1-neg 82.3%

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   11. Simplified 82.3%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if -2.8499999999999998e236 < d < -1.90000000000000002e-193

   1. Initial program 79.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 21.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 35.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 65.2%

         \[\leadsto \color{blue}{\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   5. Simplified 65.2%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   6. Taylor expanded in d around -inf 77.2%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 77.2%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]

      2. distribute-rgt-neg-in 77.2%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]

      3. *-commutative 77.2%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right)\right) \]

   8. Simplified 77.2%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

   9. Taylor expanded in h around 0 61.0%

      \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]
   10. Step-by-step derivation

       1. times-frac 60.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       2. *-commutative 60.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right)\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       3. associate-*r* 60.0%

          \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       4. unpow2 60.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       5. unpow2 60.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       6. times-frac 71.1%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       7. associate-/l* 71.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{{M}^{2}}{\frac{\ell}{h}}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       8. unpow2 71.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{M \cdot M}}{\frac{\ell}{h}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   11. Simplified 71.0%

       \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   12. Taylor expanded in M around 0 71.1%

       \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{{M}^{2} \cdot h}{\ell}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]
   13. Step-by-step derivation

       1. unpow2 71.1%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       2. associate-*l* 74.6%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   14. Simplified 74.6%

       \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{M \cdot \left(M \cdot h\right)}{\ell}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   if -1.90000000000000002e-193 < d < -2.99999999999999999e-301

   1. Initial program 42.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 8.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 8.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 34.7%

         \[\leadsto \color{blue}{\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   5. Simplified 34.7%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   6. Taylor expanded in d around -inf 59.7%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 59.7%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]

      2. distribute-rgt-neg-in 59.7%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]

      3. *-commutative 59.7%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right)\right) \]

   8. Simplified 59.7%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

   9. Taylor expanded in h around 0 44.6%

      \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]
   10. Step-by-step derivation

       1. times-frac 44.6%

          \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       2. *-commutative 44.6%

          \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right)\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       3. associate-*r* 44.6%

          \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       4. unpow2 44.6%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       5. unpow2 44.6%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       6. times-frac 48.3%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       7. associate-/l* 48.3%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{{M}^{2}}{\frac{\ell}{h}}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       8. unpow2 48.3%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{M \cdot M}}{\frac{\ell}{h}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   11. Simplified 48.3%

       \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   12. Taylor expanded in D around inf 70.4%

       \[\leadsto \color{blue}{0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   13. Step-by-step derivation

       1. associate-*r* 70.4%

          \[\leadsto \color{blue}{\left(0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

       2. unpow2 70.4%

          \[\leadsto \left(0.125 \cdot \frac{{D}^{2} \cdot \color{blue}{\left(M \cdot M\right)}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       3. associate-/l* 70.3%

          \[\leadsto \left(0.125 \cdot \color{blue}{\frac{{D}^{2}}{\frac{d}{M \cdot M}}}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       4. associate-*r/ 70.3%

          \[\leadsto \color{blue}{\frac{0.125 \cdot {D}^{2}}{\frac{d}{M \cdot M}}} \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       5. unpow2 70.3%

          \[\leadsto \frac{0.125 \cdot \color{blue}{\left(D \cdot D\right)}}{\frac{d}{M \cdot M}} \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       6. associate-/r* 71.2%

          \[\leadsto \frac{0.125 \cdot \left(D \cdot D\right)}{\color{blue}{\frac{\frac{d}{M}}{M}}} \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       7. associate-/r/ 71.2%

          \[\leadsto \color{blue}{\left(\frac{0.125 \cdot \left(D \cdot D\right)}{\frac{d}{M}} \cdot M\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       8. *-commutative 71.2%

          \[\leadsto \color{blue}{\left(M \cdot \frac{0.125 \cdot \left(D \cdot D\right)}{\frac{d}{M}}\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       9. associate-*l* 71.2%

          \[\leadsto \color{blue}{M \cdot \left(\frac{0.125 \cdot \left(D \cdot D\right)}{\frac{d}{M}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]

       10. *-commutative 71.2%

           \[\leadsto M \cdot \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{0.125 \cdot \left(D \cdot D\right)}{\frac{d}{M}}\right)} \]

       11. unpow2 71.2%

           \[\leadsto M \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{0.125 \cdot \color{blue}{{D}^{2}}}{\frac{d}{M}}\right) \]

       12. *-commutative 71.2%

           \[\leadsto M \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{{D}^{2} \cdot 0.125}}{\frac{d}{M}}\right) \]

       13. associate-/l* 71.2%

           \[\leadsto M \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\frac{{D}^{2}}{\frac{\frac{d}{M}}{0.125}}}\right) \]

       14. unpow2 71.2%

           \[\leadsto M \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{D \cdot D}}{\frac{\frac{d}{M}}{0.125}}\right) \]

   14. Simplified 71.2%

       \[\leadsto \color{blue}{M \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{D \cdot D}{\frac{\frac{d}{M}}{0.125}}\right)} \]

   if -2.99999999999999999e-301 < d < 1.40000000000000011e-112

   1. Initial program 35.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 35.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 35.8%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 35.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 35.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 35.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 35.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 35.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 35.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 35.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. frac-times 35.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      2. clear-num 35.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      3. *-commutative 35.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{1}{\frac{\color{blue}{d \cdot 2}}{M \cdot D}}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied egg-rr 35.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{1}{\frac{d \cdot 2}{M \cdot D}}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* 35.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{1}{\frac{d \cdot 2}{M \cdot D}}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative 35.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\left(\frac{1}{\frac{\color{blue}{2 \cdot d}}{M \cdot D}}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. clear-num 35.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative 35.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      5. metadata-eval 35.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*r/ 38.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      7. metadata-eval 38.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      8. times-frac 38.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(0.5 \cdot {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2}\right) \cdot h}{\ell}\right) \]

   7. Applied egg-rr 38.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(0.5 \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]
   8. Step-by-step derivation

      1. frac-times 38.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(0.5 \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot h}{\ell}\right) \]

      2. associate-/r* 38.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(0.5 \cdot {\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{2}\right) \cdot h}{\ell}\right) \]

   9. Applied egg-rr 38.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(0.5 \cdot {\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{2}\right) \cdot h}{\ell}\right) \]

   10. Taylor expanded in d around 0 40.6%

       \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   11. Step-by-step derivation

       1. associate-*l/ 42.9%

          \[\leadsto -0.125 \cdot \color{blue}{\frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}} \]

       2. unpow2 42.9%

          \[\leadsto -0.125 \cdot \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d} \]

       3. unpow2 42.9%

          \[\leadsto -0.125 \cdot \frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d} \]

       4. associate-*l* 50.5%

          \[\leadsto -0.125 \cdot \frac{\color{blue}{\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d} \]

   12. Simplified 50.5%

       \[\leadsto \color{blue}{-0.125 \cdot \frac{\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}} \]

   if 1.40000000000000011e-112 < d

   1. Initial program 78.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 60.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-commutative 60.1%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. *-commutative 60.1%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. associate-/r* 60.1%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   4. Simplified 60.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   5. Step-by-step derivation

      1. sqrt-div 72.4%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

      2. inv-pow 72.4%

         \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{-1}}}}{\sqrt{\ell}} \]

   6. Applied egg-rr 72.4%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}}}{\sqrt{\ell}}} \]
   7. Step-by-step derivation

      1. sqr-pow 72.5%

         \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{\left(\frac{-1}{2}\right)} \cdot {h}^{\left(\frac{-1}{2}\right)}}}}{\sqrt{\ell}} \]

      2. rem-sqrt-square 72.5%

         \[\leadsto d \cdot \frac{\color{blue}{\left|{h}^{\left(\frac{-1}{2}\right)}\right|}}{\sqrt{\ell}} \]

      3. metadata-eval 72.5%

         \[\leadsto d \cdot \frac{\left|{h}^{\color{blue}{-0.5}}\right|}{\sqrt{\ell}} \]

      4. sqr-pow 72.3%

         \[\leadsto d \cdot \frac{\left|\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}\right|}{\sqrt{\ell}} \]

      5. fabs-sqr 72.3%

         \[\leadsto d \cdot \frac{\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}}{\sqrt{\ell}} \]

      6. sqr-pow 72.5%

         \[\leadsto d \cdot \frac{\color{blue}{{h}^{-0.5}}}{\sqrt{\ell}} \]

   8. Simplified 72.5%

      \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.85 \cdot 10^{+236}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -1.9 \cdot 10^{-193}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot \left(h \cdot M\right)}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -3 \cdot 10^{-301}:\\ \;\;\;\;M \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{D \cdot D}{\frac{\frac{d}{M}}{0.125}}\right)\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-112}:\\ \;\;\;\;-0.125 \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 15: 60.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{{\ell}^{3}}}\\ t_1 := \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;d \leq -1.45 \cdot 10^{+234}:\\ \;\;\;\;t_1 \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -1.6 \cdot 10^{-198}:\\ \;\;\;\;\left(d \cdot t_1\right) \cdot \left(-1 - -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot \left(h \cdot M\right)}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -3 \cdot 10^{-301}:\\ \;\;\;\;M \cdot \left(t_0 \cdot \frac{D \cdot D}{\frac{\frac{d}{M}}{0.125}}\right)\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-113}:\\ \;\;\;\;t_0 \cdot \left(\left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right) \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h (pow l 3.0)))) (t_1 (sqrt (/ 1.0 (* h l)))))
   (if (<= d -1.45e+234)
     (* t_1 (- d))
     (if (<= d -1.6e-198)
       (*
        (* d t_1)
        (- -1.0 (* -0.5 (* (* 0.25 (* (/ D d) (/ D d))) (/ (* M (* h M)) l)))))
       (if (<= d -3e-301)
         (* M (* t_0 (/ (* D D) (/ (/ d M) 0.125))))
         (if (<= d 8e-113)
           (* t_0 (* (* M (/ (* D M) (/ d D))) -0.125))
           (* d (/ (pow h -0.5) (sqrt l)))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / pow(l, 3.0)));
	double t_1 = sqrt((1.0 / (h * l)));
	double tmp;
	if (d <= -1.45e+234) {
		tmp = t_1 * -d;
	} else if (d <= -1.6e-198) {
		tmp = (d * t_1) * (-1.0 - (-0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * (h * M)) / l))));
	} else if (d <= -3e-301) {
		tmp = M * (t_0 * ((D * D) / ((d / M) / 0.125)));
	} else if (d <= 8e-113) {
		tmp = t_0 * ((M * ((D * M) / (d / D))) * -0.125);
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this 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) :: tmp
    t_0 = sqrt((h / (l ** 3.0d0)))
    t_1 = sqrt((1.0d0 / (h * l)))
    if (d <= (-1.45d+234)) then
        tmp = t_1 * -d
    else if (d <= (-1.6d-198)) then
        tmp = (d * t_1) * ((-1.0d0) - ((-0.5d0) * ((0.25d0 * ((d_1 / d) * (d_1 / d))) * ((m * (h * m)) / l))))
    else if (d <= (-3d-301)) then
        tmp = m * (t_0 * ((d_1 * d_1) / ((d / m) / 0.125d0)))
    else if (d <= 8d-113) then
        tmp = t_0 * ((m * ((d_1 * m) / (d / d_1))) * (-0.125d0))
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / Math.pow(l, 3.0)));
	double t_1 = Math.sqrt((1.0 / (h * l)));
	double tmp;
	if (d <= -1.45e+234) {
		tmp = t_1 * -d;
	} else if (d <= -1.6e-198) {
		tmp = (d * t_1) * (-1.0 - (-0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * (h * M)) / l))));
	} else if (d <= -3e-301) {
		tmp = M * (t_0 * ((D * D) / ((d / M) / 0.125)));
	} else if (d <= 8e-113) {
		tmp = t_0 * ((M * ((D * M) / (d / D))) * -0.125);
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt((h / math.pow(l, 3.0)))
	t_1 = math.sqrt((1.0 / (h * l)))
	tmp = 0
	if d <= -1.45e+234:
		tmp = t_1 * -d
	elif d <= -1.6e-198:
		tmp = (d * t_1) * (-1.0 - (-0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * (h * M)) / l))))
	elif d <= -3e-301:
		tmp = M * (t_0 * ((D * D) / ((d / M) / 0.125)))
	elif d <= 8e-113:
		tmp = t_0 * ((M * ((D * M) / (d / D))) * -0.125)
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / (l ^ 3.0)))
	t_1 = sqrt(Float64(1.0 / Float64(h * l)))
	tmp = 0.0
	if (d <= -1.45e+234)
		tmp = Float64(t_1 * Float64(-d));
	elseif (d <= -1.6e-198)
		tmp = Float64(Float64(d * t_1) * Float64(-1.0 - Float64(-0.5 * Float64(Float64(0.25 * Float64(Float64(D / d) * Float64(D / d))) * Float64(Float64(M * Float64(h * M)) / l)))));
	elseif (d <= -3e-301)
		tmp = Float64(M * Float64(t_0 * Float64(Float64(D * D) / Float64(Float64(d / M) / 0.125))));
	elseif (d <= 8e-113)
		tmp = Float64(t_0 * Float64(Float64(M * Float64(Float64(D * M) / Float64(d / D))) * -0.125));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / (l ^ 3.0)));
	t_1 = sqrt((1.0 / (h * l)));
	tmp = 0.0;
	if (d <= -1.45e+234)
		tmp = t_1 * -d;
	elseif (d <= -1.6e-198)
		tmp = (d * t_1) * (-1.0 - (-0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * (h * M)) / l))));
	elseif (d <= -3e-301)
		tmp = M * (t_0 * ((D * D) / ((d / M) / 0.125)));
	elseif (d <= 8e-113)
		tmp = t_0 * ((M * ((D * M) / (d / D))) * -0.125);
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.45e+234], N[(t$95$1 * (-d)), $MachinePrecision], If[LessEqual[d, -1.6e-198], N[(N[(d * t$95$1), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[(0.25 * N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -3e-301], N[(M * N[(t$95$0 * N[(N[(D * D), $MachinePrecision] / N[(N[(d / M), $MachinePrecision] / 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 8e-113], N[(t$95$0 * N[(N[(M * N[(N[(D * M), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.44999999999999993e234

   1. Initial program 66.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 48.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 48.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 66.8%

         \[\leadsto \color{blue}{\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   5. Simplified 66.8%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   6. Taylor expanded in d around -inf 88.2%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 88.2%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]

      2. distribute-rgt-neg-in 88.2%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]

      3. *-commutative 88.2%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right)\right) \]

   8. Simplified 88.2%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

   9. Taylor expanded in D around 0 82.3%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   10. Step-by-step derivation

       1. associate-/l/ 82.2%

          \[\leadsto -1 \cdot \left(d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}}\right) \]

       2. associate-*r* 82.2%

          \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]

       3. *-commutative 82.2%

          \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-1 \cdot d\right)} \]

       4. associate-/l/ 82.3%

          \[\leadsto \sqrt{\color{blue}{\frac{1}{\ell \cdot h}}} \cdot \left(-1 \cdot d\right) \]

       5. *-commutative 82.3%

          \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

       6. mul-1-neg 82.3%

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   11. Simplified 82.3%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if -1.44999999999999993e234 < d < -1.59999999999999997e-198

   1. Initial program 79.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 21.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 35.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 65.2%

         \[\leadsto \color{blue}{\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   5. Simplified 65.2%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   6. Taylor expanded in d around -inf 77.2%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 77.2%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]

      2. distribute-rgt-neg-in 77.2%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]

      3. *-commutative 77.2%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right)\right) \]

   8. Simplified 77.2%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

   9. Taylor expanded in h around 0 61.0%

      \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]
   10. Step-by-step derivation

       1. times-frac 60.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       2. *-commutative 60.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right)\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       3. associate-*r* 60.0%

          \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       4. unpow2 60.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       5. unpow2 60.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       6. times-frac 71.1%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       7. associate-/l* 71.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{{M}^{2}}{\frac{\ell}{h}}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       8. unpow2 71.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{M \cdot M}}{\frac{\ell}{h}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   11. Simplified 71.0%

       \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   12. Taylor expanded in M around 0 71.1%

       \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{{M}^{2} \cdot h}{\ell}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]
   13. Step-by-step derivation

       1. unpow2 71.1%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       2. associate-*l* 74.6%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   14. Simplified 74.6%

       \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{M \cdot \left(M \cdot h\right)}{\ell}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   if -1.59999999999999997e-198 < d < -2.99999999999999999e-301

   1. Initial program 42.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 8.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 8.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 34.7%

         \[\leadsto \color{blue}{\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   5. Simplified 34.7%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   6. Taylor expanded in d around -inf 59.7%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 59.7%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]

      2. distribute-rgt-neg-in 59.7%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]

      3. *-commutative 59.7%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right)\right) \]

   8. Simplified 59.7%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

   9. Taylor expanded in h around 0 44.6%

      \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]
   10. Step-by-step derivation

       1. times-frac 44.6%

          \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       2. *-commutative 44.6%

          \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right)\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       3. associate-*r* 44.6%

          \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       4. unpow2 44.6%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       5. unpow2 44.6%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       6. times-frac 48.3%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       7. associate-/l* 48.3%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{{M}^{2}}{\frac{\ell}{h}}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       8. unpow2 48.3%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{M \cdot M}}{\frac{\ell}{h}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   11. Simplified 48.3%

       \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   12. Taylor expanded in D around inf 70.4%

       \[\leadsto \color{blue}{0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   13. Step-by-step derivation

       1. associate-*r* 70.4%

          \[\leadsto \color{blue}{\left(0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

       2. unpow2 70.4%

          \[\leadsto \left(0.125 \cdot \frac{{D}^{2} \cdot \color{blue}{\left(M \cdot M\right)}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       3. associate-/l* 70.3%

          \[\leadsto \left(0.125 \cdot \color{blue}{\frac{{D}^{2}}{\frac{d}{M \cdot M}}}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       4. associate-*r/ 70.3%

          \[\leadsto \color{blue}{\frac{0.125 \cdot {D}^{2}}{\frac{d}{M \cdot M}}} \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       5. unpow2 70.3%

          \[\leadsto \frac{0.125 \cdot \color{blue}{\left(D \cdot D\right)}}{\frac{d}{M \cdot M}} \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       6. associate-/r* 71.2%

          \[\leadsto \frac{0.125 \cdot \left(D \cdot D\right)}{\color{blue}{\frac{\frac{d}{M}}{M}}} \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       7. associate-/r/ 71.2%

          \[\leadsto \color{blue}{\left(\frac{0.125 \cdot \left(D \cdot D\right)}{\frac{d}{M}} \cdot M\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       8. *-commutative 71.2%

          \[\leadsto \color{blue}{\left(M \cdot \frac{0.125 \cdot \left(D \cdot D\right)}{\frac{d}{M}}\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       9. associate-*l* 71.2%

          \[\leadsto \color{blue}{M \cdot \left(\frac{0.125 \cdot \left(D \cdot D\right)}{\frac{d}{M}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]

       10. *-commutative 71.2%

           \[\leadsto M \cdot \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{0.125 \cdot \left(D \cdot D\right)}{\frac{d}{M}}\right)} \]

       11. unpow2 71.2%

           \[\leadsto M \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{0.125 \cdot \color{blue}{{D}^{2}}}{\frac{d}{M}}\right) \]

       12. *-commutative 71.2%

           \[\leadsto M \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{{D}^{2} \cdot 0.125}}{\frac{d}{M}}\right) \]

       13. associate-/l* 71.2%

           \[\leadsto M \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\frac{{D}^{2}}{\frac{\frac{d}{M}}{0.125}}}\right) \]

       14. unpow2 71.2%

           \[\leadsto M \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{D \cdot D}}{\frac{\frac{d}{M}}{0.125}}\right) \]

   14. Simplified 71.2%

       \[\leadsto \color{blue}{M \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{D \cdot D}{\frac{\frac{d}{M}}{0.125}}\right)} \]

   if -2.99999999999999999e-301 < d < 7.99999999999999983e-113

   1. Initial program 35.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 35.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 35.8%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 35.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 35.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 35.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 35.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 35.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 35.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 35.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. frac-times 35.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      2. clear-num 35.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      3. *-commutative 35.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{1}{\frac{\color{blue}{d \cdot 2}}{M \cdot D}}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied egg-rr 35.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{1}{\frac{d \cdot 2}{M \cdot D}}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* 35.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{1}{\frac{d \cdot 2}{M \cdot D}}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative 35.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\left(\frac{1}{\frac{\color{blue}{2 \cdot d}}{M \cdot D}}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. clear-num 35.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative 35.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      5. metadata-eval 35.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*r/ 38.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      7. metadata-eval 38.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      8. times-frac 38.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(0.5 \cdot {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2}\right) \cdot h}{\ell}\right) \]

   7. Applied egg-rr 38.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(0.5 \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

   8. Taylor expanded in d around 0 40.6%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 40.6%

         \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative 40.6%

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      3. unpow2 40.6%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{d}\right) \]

      4. unpow2 40.6%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{d}\right) \]

      5. associate-*l/ 40.4%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)}\right) \]

      6. associate-*r* 43.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot M\right)}\right) \]

      7. *-commutative 43.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(M \cdot \left(\frac{D \cdot D}{d} \cdot M\right)\right)}\right) \]

      8. associate-/l* 48.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(M \cdot \left(\color{blue}{\frac{D}{\frac{d}{D}}} \cdot M\right)\right)\right) \]

      9. associate-*l/ 50.5%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(M \cdot \color{blue}{\frac{D \cdot M}{\frac{d}{D}}}\right)\right) \]

   10. Simplified 50.5%

       \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)} \]

   if 7.99999999999999983e-113 < d

   1. Initial program 78.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 60.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-commutative 60.1%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. *-commutative 60.1%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. associate-/r* 60.1%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   4. Simplified 60.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   5. Step-by-step derivation

      1. sqrt-div 72.4%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

      2. inv-pow 72.4%

         \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{-1}}}}{\sqrt{\ell}} \]

   6. Applied egg-rr 72.4%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}}}{\sqrt{\ell}}} \]
   7. Step-by-step derivation

      1. sqr-pow 72.5%

         \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{\left(\frac{-1}{2}\right)} \cdot {h}^{\left(\frac{-1}{2}\right)}}}}{\sqrt{\ell}} \]

      2. rem-sqrt-square 72.5%

         \[\leadsto d \cdot \frac{\color{blue}{\left|{h}^{\left(\frac{-1}{2}\right)}\right|}}{\sqrt{\ell}} \]

      3. metadata-eval 72.5%

         \[\leadsto d \cdot \frac{\left|{h}^{\color{blue}{-0.5}}\right|}{\sqrt{\ell}} \]

      4. sqr-pow 72.3%

         \[\leadsto d \cdot \frac{\left|\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}\right|}{\sqrt{\ell}} \]

      5. fabs-sqr 72.3%

         \[\leadsto d \cdot \frac{\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}}{\sqrt{\ell}} \]

      6. sqr-pow 72.5%

         \[\leadsto d \cdot \frac{\color{blue}{{h}^{-0.5}}}{\sqrt{\ell}} \]

   8. Simplified 72.5%

      \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.45 \cdot 10^{+234}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -1.6 \cdot 10^{-198}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot \left(h \cdot M\right)}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -3 \cdot 10^{-301}:\\ \;\;\;\;M \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{D \cdot D}{\frac{\frac{d}{M}}{0.125}}\right)\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-113}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right) \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 16: 61.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{{\ell}^{3}}}\\ t_1 := \sqrt{\frac{1}{h \cdot \ell}}\\ t_2 := M \cdot \frac{D \cdot M}{\frac{d}{D}}\\ \mathbf{if}\;d \leq -1.4 \cdot 10^{+236}:\\ \;\;\;\;t_1 \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -2.05 \cdot 10^{-188}:\\ \;\;\;\;\left(d \cdot t_1\right) \cdot \left(-1 - -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot \left(h \cdot M\right)}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_0 \cdot \left(0.125 \cdot t_2\right)\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{-112}:\\ \;\;\;\;t_0 \cdot \left(t_2 \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h (pow l 3.0))))
        (t_1 (sqrt (/ 1.0 (* h l))))
        (t_2 (* M (/ (* D M) (/ d D)))))
   (if (<= d -1.4e+236)
     (* t_1 (- d))
     (if (<= d -2.05e-188)
       (*
        (* d t_1)
        (- -1.0 (* -0.5 (* (* 0.25 (* (/ D d) (/ D d))) (/ (* M (* h M)) l)))))
       (if (<= d -2e-310)
         (* t_0 (* 0.125 t_2))
         (if (<= d 1.5e-112)
           (* t_0 (* t_2 -0.125))
           (* d (/ (pow h -0.5) (sqrt l)))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / pow(l, 3.0)));
	double t_1 = sqrt((1.0 / (h * l)));
	double t_2 = M * ((D * M) / (d / D));
	double tmp;
	if (d <= -1.4e+236) {
		tmp = t_1 * -d;
	} else if (d <= -2.05e-188) {
		tmp = (d * t_1) * (-1.0 - (-0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * (h * M)) / l))));
	} else if (d <= -2e-310) {
		tmp = t_0 * (0.125 * t_2);
	} else if (d <= 1.5e-112) {
		tmp = t_0 * (t_2 * -0.125);
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this 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) :: tmp
    t_0 = sqrt((h / (l ** 3.0d0)))
    t_1 = sqrt((1.0d0 / (h * l)))
    t_2 = m * ((d_1 * m) / (d / d_1))
    if (d <= (-1.4d+236)) then
        tmp = t_1 * -d
    else if (d <= (-2.05d-188)) then
        tmp = (d * t_1) * ((-1.0d0) - ((-0.5d0) * ((0.25d0 * ((d_1 / d) * (d_1 / d))) * ((m * (h * m)) / l))))
    else if (d <= (-2d-310)) then
        tmp = t_0 * (0.125d0 * t_2)
    else if (d <= 1.5d-112) then
        tmp = t_0 * (t_2 * (-0.125d0))
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / Math.pow(l, 3.0)));
	double t_1 = Math.sqrt((1.0 / (h * l)));
	double t_2 = M * ((D * M) / (d / D));
	double tmp;
	if (d <= -1.4e+236) {
		tmp = t_1 * -d;
	} else if (d <= -2.05e-188) {
		tmp = (d * t_1) * (-1.0 - (-0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * (h * M)) / l))));
	} else if (d <= -2e-310) {
		tmp = t_0 * (0.125 * t_2);
	} else if (d <= 1.5e-112) {
		tmp = t_0 * (t_2 * -0.125);
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt((h / math.pow(l, 3.0)))
	t_1 = math.sqrt((1.0 / (h * l)))
	t_2 = M * ((D * M) / (d / D))
	tmp = 0
	if d <= -1.4e+236:
		tmp = t_1 * -d
	elif d <= -2.05e-188:
		tmp = (d * t_1) * (-1.0 - (-0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * (h * M)) / l))))
	elif d <= -2e-310:
		tmp = t_0 * (0.125 * t_2)
	elif d <= 1.5e-112:
		tmp = t_0 * (t_2 * -0.125)
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / (l ^ 3.0)))
	t_1 = sqrt(Float64(1.0 / Float64(h * l)))
	t_2 = Float64(M * Float64(Float64(D * M) / Float64(d / D)))
	tmp = 0.0
	if (d <= -1.4e+236)
		tmp = Float64(t_1 * Float64(-d));
	elseif (d <= -2.05e-188)
		tmp = Float64(Float64(d * t_1) * Float64(-1.0 - Float64(-0.5 * Float64(Float64(0.25 * Float64(Float64(D / d) * Float64(D / d))) * Float64(Float64(M * Float64(h * M)) / l)))));
	elseif (d <= -2e-310)
		tmp = Float64(t_0 * Float64(0.125 * t_2));
	elseif (d <= 1.5e-112)
		tmp = Float64(t_0 * Float64(t_2 * -0.125));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / (l ^ 3.0)));
	t_1 = sqrt((1.0 / (h * l)));
	t_2 = M * ((D * M) / (d / D));
	tmp = 0.0;
	if (d <= -1.4e+236)
		tmp = t_1 * -d;
	elseif (d <= -2.05e-188)
		tmp = (d * t_1) * (-1.0 - (-0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * (h * M)) / l))));
	elseif (d <= -2e-310)
		tmp = t_0 * (0.125 * t_2);
	elseif (d <= 1.5e-112)
		tmp = t_0 * (t_2 * -0.125);
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(M * N[(N[(D * M), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.4e+236], N[(t$95$1 * (-d)), $MachinePrecision], If[LessEqual[d, -2.05e-188], N[(N[(d * t$95$1), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[(0.25 * N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2e-310], N[(t$95$0 * N[(0.125 * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.5e-112], N[(t$95$0 * N[(t$95$2 * -0.125), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.39999999999999996e236

   1. Initial program 66.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 48.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 48.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 66.8%

         \[\leadsto \color{blue}{\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   5. Simplified 66.8%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   6. Taylor expanded in d around -inf 88.2%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 88.2%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]

      2. distribute-rgt-neg-in 88.2%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]

      3. *-commutative 88.2%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right)\right) \]

   8. Simplified 88.2%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

   9. Taylor expanded in D around 0 82.3%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   10. Step-by-step derivation

       1. associate-/l/ 82.2%

          \[\leadsto -1 \cdot \left(d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}}\right) \]

       2. associate-*r* 82.2%

          \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]

       3. *-commutative 82.2%

          \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-1 \cdot d\right)} \]

       4. associate-/l/ 82.3%

          \[\leadsto \sqrt{\color{blue}{\frac{1}{\ell \cdot h}}} \cdot \left(-1 \cdot d\right) \]

       5. *-commutative 82.3%

          \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

       6. mul-1-neg 82.3%

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   11. Simplified 82.3%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if -1.39999999999999996e236 < d < -2.04999999999999991e-188

   1. Initial program 79.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 21.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 35.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 65.2%

         \[\leadsto \color{blue}{\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   5. Simplified 65.2%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   6. Taylor expanded in d around -inf 77.2%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 77.2%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]

      2. distribute-rgt-neg-in 77.2%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]

      3. *-commutative 77.2%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right)\right) \]

   8. Simplified 77.2%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

   9. Taylor expanded in h around 0 61.0%

      \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]
   10. Step-by-step derivation

       1. times-frac 60.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       2. *-commutative 60.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right)\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       3. associate-*r* 60.0%

          \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       4. unpow2 60.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       5. unpow2 60.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       6. times-frac 71.1%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       7. associate-/l* 71.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{{M}^{2}}{\frac{\ell}{h}}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       8. unpow2 71.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{M \cdot M}}{\frac{\ell}{h}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   11. Simplified 71.0%

       \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   12. Taylor expanded in M around 0 71.1%

       \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{{M}^{2} \cdot h}{\ell}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]
   13. Step-by-step derivation

       1. unpow2 71.1%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       2. associate-*l* 74.6%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   14. Simplified 74.6%

       \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{M \cdot \left(M \cdot h\right)}{\ell}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   if -2.04999999999999991e-188 < d < -1.999999999999994e-310

   1. Initial program 42.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 7.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 7.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 32.3%

         \[\leadsto \color{blue}{\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   5. Simplified 32.3%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   6. Taylor expanded in d around -inf 55.6%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 55.6%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]

      2. distribute-rgt-neg-in 55.6%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]

      3. *-commutative 55.6%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right)\right) \]

   8. Simplified 55.6%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

   9. Taylor expanded in D around inf 69.1%

      \[\leadsto \color{blue}{0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 69.1%

          \[\leadsto 0.125 \cdot \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

       2. unpow2 69.1%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{d}\right) \]

       3. unpow2 69.1%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{d}\right) \]

       4. associate-*l/ 62.1%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)}\right) \]

       5. *-commutative 62.1%

          \[\leadsto \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)\right) \cdot 0.125} \]

       6. associate-*l* 62.1%

          \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right) \cdot 0.125\right)} \]

       7. *-commutative 62.1%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(0.125 \cdot \left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)\right)} \]

       8. associate-*r* 63.1%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \color{blue}{\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot M\right)}\right) \]

       9. *-commutative 63.1%

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \color{blue}{\left(M \cdot \left(\frac{D \cdot D}{d} \cdot M\right)\right)}\right) \]

       10. associate-/l* 63.1%

           \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \left(\color{blue}{\frac{D}{\frac{d}{D}}} \cdot M\right)\right)\right) \]

       11. associate-*l/ 66.6%

           \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \color{blue}{\frac{D \cdot M}{\frac{d}{D}}}\right)\right) \]

   11. Simplified 66.6%

       \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)} \]

   if -1.999999999999994e-310 < d < 1.5e-112

   1. Initial program 35.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 35.1%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 35.1%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 35.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 35.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 35.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 35.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 35.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 35.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 35.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. frac-times 35.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      2. clear-num 35.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      3. *-commutative 35.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{1}{\frac{\color{blue}{d \cdot 2}}{M \cdot D}}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied egg-rr 35.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{1}{\frac{d \cdot 2}{M \cdot D}}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* 35.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{1}{\frac{d \cdot 2}{M \cdot D}}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative 35.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\left(\frac{1}{\frac{\color{blue}{2 \cdot d}}{M \cdot D}}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. clear-num 35.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative 35.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      5. metadata-eval 35.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*r/ 37.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      7. metadata-eval 37.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      8. times-frac 37.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(0.5 \cdot {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2}\right) \cdot h}{\ell}\right) \]

   7. Applied egg-rr 37.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(0.5 \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

   8. Taylor expanded in d around 0 40.1%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 40.1%

         \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative 40.1%

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      3. unpow2 40.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{d}\right) \]

      4. unpow2 40.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{d}\right) \]

      5. associate-*l/ 39.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(\frac{D \cdot D}{d} \cdot \left(M \cdot M\right)\right)}\right) \]

      6. associate-*r* 42.7%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(\left(\frac{D \cdot D}{d} \cdot M\right) \cdot M\right)}\right) \]

      7. *-commutative 42.7%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(M \cdot \left(\frac{D \cdot D}{d} \cdot M\right)\right)}\right) \]

      8. associate-/l* 48.0%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(M \cdot \left(\color{blue}{\frac{D}{\frac{d}{D}}} \cdot M\right)\right)\right) \]

      9. associate-*l/ 50.5%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(M \cdot \color{blue}{\frac{D \cdot M}{\frac{d}{D}}}\right)\right) \]

   10. Simplified 50.5%

       \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)} \]

   if 1.5e-112 < d

   1. Initial program 78.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 60.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-commutative 60.1%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. *-commutative 60.1%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. associate-/r* 60.1%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   4. Simplified 60.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   5. Step-by-step derivation

      1. sqrt-div 72.4%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

      2. inv-pow 72.4%

         \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{-1}}}}{\sqrt{\ell}} \]

   6. Applied egg-rr 72.4%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}}}{\sqrt{\ell}}} \]
   7. Step-by-step derivation

      1. sqr-pow 72.5%

         \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{\left(\frac{-1}{2}\right)} \cdot {h}^{\left(\frac{-1}{2}\right)}}}}{\sqrt{\ell}} \]

      2. rem-sqrt-square 72.5%

         \[\leadsto d \cdot \frac{\color{blue}{\left|{h}^{\left(\frac{-1}{2}\right)}\right|}}{\sqrt{\ell}} \]

      3. metadata-eval 72.5%

         \[\leadsto d \cdot \frac{\left|{h}^{\color{blue}{-0.5}}\right|}{\sqrt{\ell}} \]

      4. sqr-pow 72.3%

         \[\leadsto d \cdot \frac{\left|\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}\right|}{\sqrt{\ell}} \]

      5. fabs-sqr 72.3%

         \[\leadsto d \cdot \frac{\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}}{\sqrt{\ell}} \]

      6. sqr-pow 72.5%

         \[\leadsto d \cdot \frac{\color{blue}{{h}^{-0.5}}}{\sqrt{\ell}} \]

   8. Simplified 72.5%

      \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.4 \cdot 10^{+236}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -2.05 \cdot 10^{-188}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot \left(h \cdot M\right)}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(0.125 \cdot \left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right)\right)\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot \frac{D \cdot M}{\frac{d}{D}}\right) \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 17: 60.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;d \leq -5.5 \cdot 10^{+231}:\\ \;\;\;\;t_0 \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -5.4 \cdot 10^{-304}:\\ \;\;\;\;\left(d \cdot t_0\right) \cdot \left(-1 - -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot \left(h \cdot M\right)}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{-113}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* h l)))))
   (if (<= d -5.5e+231)
     (* t_0 (- d))
     (if (<= d -5.4e-304)
       (*
        (* d t_0)
        (- -1.0 (* -0.5 (* (* 0.25 (* (/ D d) (/ D d))) (/ (* M (* h M)) l)))))
       (if (<= d 7.2e-113)
         (* -0.125 (* (/ (* D D) (/ d (* M M))) (/ (sqrt h) (pow l 1.5))))
         (* d (/ (pow h -0.5) (sqrt l))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (h * l)));
	double tmp;
	if (d <= -5.5e+231) {
		tmp = t_0 * -d;
	} else if (d <= -5.4e-304) {
		tmp = (d * t_0) * (-1.0 - (-0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * (h * M)) / l))));
	} else if (d <= 7.2e-113) {
		tmp = -0.125 * (((D * D) / (d / (M * M))) * (sqrt(h) / pow(l, 1.5)));
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / (h * l)))
    if (d <= (-5.5d+231)) then
        tmp = t_0 * -d
    else if (d <= (-5.4d-304)) then
        tmp = (d * t_0) * ((-1.0d0) - ((-0.5d0) * ((0.25d0 * ((d_1 / d) * (d_1 / d))) * ((m * (h * m)) / l))))
    else if (d <= 7.2d-113) then
        tmp = (-0.125d0) * (((d_1 * d_1) / (d / (m * m))) * (sqrt(h) / (l ** 1.5d0)))
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((1.0 / (h * l)));
	double tmp;
	if (d <= -5.5e+231) {
		tmp = t_0 * -d;
	} else if (d <= -5.4e-304) {
		tmp = (d * t_0) * (-1.0 - (-0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * (h * M)) / l))));
	} else if (d <= 7.2e-113) {
		tmp = -0.125 * (((D * D) / (d / (M * M))) * (Math.sqrt(h) / Math.pow(l, 1.5)));
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt((1.0 / (h * l)))
	tmp = 0
	if d <= -5.5e+231:
		tmp = t_0 * -d
	elif d <= -5.4e-304:
		tmp = (d * t_0) * (-1.0 - (-0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * (h * M)) / l))))
	elif d <= 7.2e-113:
		tmp = -0.125 * (((D * D) / (d / (M * M))) * (math.sqrt(h) / math.pow(l, 1.5)))
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(1.0 / Float64(h * l)))
	tmp = 0.0
	if (d <= -5.5e+231)
		tmp = Float64(t_0 * Float64(-d));
	elseif (d <= -5.4e-304)
		tmp = Float64(Float64(d * t_0) * Float64(-1.0 - Float64(-0.5 * Float64(Float64(0.25 * Float64(Float64(D / d) * Float64(D / d))) * Float64(Float64(M * Float64(h * M)) / l)))));
	elseif (d <= 7.2e-113)
		tmp = Float64(-0.125 * Float64(Float64(Float64(D * D) / Float64(d / Float64(M * M))) * Float64(sqrt(h) / (l ^ 1.5))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((1.0 / (h * l)));
	tmp = 0.0;
	if (d <= -5.5e+231)
		tmp = t_0 * -d;
	elseif (d <= -5.4e-304)
		tmp = (d * t_0) * (-1.0 - (-0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * (h * M)) / l))));
	elseif (d <= 7.2e-113)
		tmp = -0.125 * (((D * D) / (d / (M * M))) * (sqrt(h) / (l ^ 1.5)));
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -5.5e+231], N[(t$95$0 * (-d)), $MachinePrecision], If[LessEqual[d, -5.4e-304], N[(N[(d * t$95$0), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[(0.25 * N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.2e-113], N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] / N[(d / N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -5.5e231

   1. Initial program 66.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 48.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 48.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 66.8%

         \[\leadsto \color{blue}{\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   5. Simplified 66.8%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   6. Taylor expanded in d around -inf 88.2%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 88.2%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]

      2. distribute-rgt-neg-in 88.2%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]

      3. *-commutative 88.2%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right)\right) \]

   8. Simplified 88.2%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

   9. Taylor expanded in D around 0 82.3%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   10. Step-by-step derivation

       1. associate-/l/ 82.2%

          \[\leadsto -1 \cdot \left(d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}}\right) \]

       2. associate-*r* 82.2%

          \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]

       3. *-commutative 82.2%

          \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-1 \cdot d\right)} \]

       4. associate-/l/ 82.3%

          \[\leadsto \sqrt{\color{blue}{\frac{1}{\ell \cdot h}}} \cdot \left(-1 \cdot d\right) \]

       5. *-commutative 82.3%

          \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

       6. mul-1-neg 82.3%

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   11. Simplified 82.3%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if -5.5e231 < d < -5.40000000000000021e-304

   1. Initial program 70.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 17.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 28.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 57.1%

         \[\leadsto \color{blue}{\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   5. Simplified 57.1%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   6. Taylor expanded in d around -inf 72.1%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 72.1%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]

      2. distribute-rgt-neg-in 72.1%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]

      3. *-commutative 72.1%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right)\right) \]

   8. Simplified 72.1%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

   9. Taylor expanded in h around 0 56.4%

      \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]
   10. Step-by-step derivation

       1. times-frac 55.6%

          \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       2. *-commutative 55.6%

          \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right)\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       3. associate-*r* 55.6%

          \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       4. unpow2 55.6%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       5. unpow2 55.6%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       6. times-frac 65.7%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       7. associate-/l* 65.6%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{{M}^{2}}{\frac{\ell}{h}}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       8. unpow2 65.6%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{M \cdot M}}{\frac{\ell}{h}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   11. Simplified 65.6%

       \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   12. Taylor expanded in M around 0 65.7%

       \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{{M}^{2} \cdot h}{\ell}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]
   13. Step-by-step derivation

       1. unpow2 65.7%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       2. associate-*l* 68.3%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   14. Simplified 68.3%

       \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{M \cdot \left(M \cdot h\right)}{\ell}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   if -5.40000000000000021e-304 < d < 7.1999999999999995e-113

   1. Initial program 34.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around 0 39.1%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   3. Step-by-step derivation

      1. associate-/l* 36.6%

         \[\leadsto -0.125 \cdot \left(\color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      2. unpow2 36.6%

         \[\leadsto -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\frac{d}{{M}^{2}}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      3. unpow2 36.6%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{\color{blue}{M \cdot M}}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

   4. Simplified 36.6%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   5. Step-by-step derivation

      1. sqrt-div 36.6%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}}\right) \]

   6. Applied egg-rr 36.6%

      \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}}\right) \]
   7. Step-by-step derivation

      1. sqr-pow 36.6%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{\sqrt{h}}{\sqrt{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)} \cdot {\ell}^{\left(\frac{3}{2}\right)}}}}\right) \]

      2. rem-sqrt-square 44.1%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{\sqrt{h}}{\color{blue}{\left|{\ell}^{\left(\frac{3}{2}\right)}\right|}}\right) \]

      3. sqr-pow 44.1%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{\sqrt{h}}{\left|\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}\right|}\right) \]

      4. fabs-sqr 44.1%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}}\right) \]

      5. sqr-pow 44.1%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)}}}\right) \]

      6. metadata-eval 44.1%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}}\right) \]

   8. Simplified 44.1%

      \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \color{blue}{\frac{\sqrt{h}}{{\ell}^{1.5}}}\right) \]

   if 7.1999999999999995e-113 < d

   1. Initial program 78.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 60.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-commutative 60.1%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. *-commutative 60.1%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. associate-/r* 60.1%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   4. Simplified 60.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   5. Step-by-step derivation

      1. sqrt-div 72.4%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

      2. inv-pow 72.4%

         \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{-1}}}}{\sqrt{\ell}} \]

   6. Applied egg-rr 72.4%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}}}{\sqrt{\ell}}} \]
   7. Step-by-step derivation

      1. sqr-pow 72.5%

         \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{\left(\frac{-1}{2}\right)} \cdot {h}^{\left(\frac{-1}{2}\right)}}}}{\sqrt{\ell}} \]

      2. rem-sqrt-square 72.5%

         \[\leadsto d \cdot \frac{\color{blue}{\left|{h}^{\left(\frac{-1}{2}\right)}\right|}}{\sqrt{\ell}} \]

      3. metadata-eval 72.5%

         \[\leadsto d \cdot \frac{\left|{h}^{\color{blue}{-0.5}}\right|}{\sqrt{\ell}} \]

      4. sqr-pow 72.3%

         \[\leadsto d \cdot \frac{\left|\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}\right|}{\sqrt{\ell}} \]

      5. fabs-sqr 72.3%

         \[\leadsto d \cdot \frac{\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}}{\sqrt{\ell}} \]

      6. sqr-pow 72.5%

         \[\leadsto d \cdot \frac{\color{blue}{{h}^{-0.5}}}{\sqrt{\ell}} \]

   8. Simplified 72.5%

      \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.5 \cdot 10^{+231}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -5.4 \cdot 10^{-304}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot \left(h \cdot M\right)}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{-113}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 18: 57.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot \left(h \cdot M\right)}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq 7.2 \cdot 10^{+266}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(D \cdot \frac{D \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{\frac{d}{M \cdot M}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= h -2e-310)
   (*
    (* d (sqrt (/ 1.0 (* h l))))
    (- -1.0 (* -0.5 (* (* 0.25 (* (/ D d) (/ D d))) (/ (* M (* h M)) l)))))
   (if (<= h 7.2e+266)
     (* d (/ (pow h -0.5) (sqrt l)))
     (* -0.125 (* D (/ (* D (sqrt (/ h (pow l 3.0)))) (/ d (* M M))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -2e-310) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * (h * M)) / l))));
	} else if (h <= 7.2e+266) {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	} else {
		tmp = -0.125 * (D * ((D * sqrt((h / pow(l, 3.0)))) / (d / (M * M))));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this 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) :: tmp
    if (h <= (-2d-310)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) - ((-0.5d0) * ((0.25d0 * ((d_1 / d) * (d_1 / d))) * ((m * (h * m)) / l))))
    else if (h <= 7.2d+266) then
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    else
        tmp = (-0.125d0) * (d_1 * ((d_1 * sqrt((h / (l ** 3.0d0)))) / (d / (m * m))))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -2e-310) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * (h * M)) / l))));
	} else if (h <= 7.2e+266) {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	} else {
		tmp = -0.125 * (D * ((D * Math.sqrt((h / Math.pow(l, 3.0)))) / (d / (M * M))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if h <= -2e-310:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * (h * M)) / l))))
	elif h <= 7.2e+266:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	else:
		tmp = -0.125 * (D * ((D * math.sqrt((h / math.pow(l, 3.0)))) / (d / (M * M))))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -2e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 - Float64(-0.5 * Float64(Float64(0.25 * Float64(Float64(D / d) * Float64(D / d))) * Float64(Float64(M * Float64(h * M)) / l)))));
	elseif (h <= 7.2e+266)
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	else
		tmp = Float64(-0.125 * Float64(D * Float64(Float64(D * sqrt(Float64(h / (l ^ 3.0)))) / Float64(d / Float64(M * M)))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= -2e-310)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * (h * M)) / l))));
	elseif (h <= 7.2e+266)
		tmp = d * ((h ^ -0.5) / sqrt(l));
	else
		tmp = -0.125 * (D * ((D * sqrt((h / (l ^ 3.0)))) / (d / (M * M))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[h, -2e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[(0.25 * N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 7.2e+266], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.125 * N[(D * N[(N[(D * N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(d / N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.999999999999994e-310

   1. Initial program 69.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 21.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 30.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 57.9%

         \[\leadsto \color{blue}{\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   5. Simplified 57.9%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   6. Taylor expanded in d around -inf 73.7%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 73.7%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]

      2. distribute-rgt-neg-in 73.7%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]

      3. *-commutative 73.7%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right)\right) \]

   8. Simplified 73.7%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

   9. Taylor expanded in h around 0 53.9%

      \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]
   10. Step-by-step derivation

       1. times-frac 51.6%

          \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       2. *-commutative 51.6%

          \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right)\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       3. associate-*r* 51.6%

          \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       4. unpow2 51.6%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       5. unpow2 51.6%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       6. times-frac 64.3%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       7. associate-/l* 64.2%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{{M}^{2}}{\frac{\ell}{h}}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       8. unpow2 64.2%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{M \cdot M}}{\frac{\ell}{h}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   11. Simplified 64.2%

       \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   12. Taylor expanded in M around 0 64.3%

       \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{{M}^{2} \cdot h}{\ell}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]
   13. Step-by-step derivation

       1. unpow2 64.3%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       2. associate-*l* 67.2%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   14. Simplified 67.2%

       \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{M \cdot \left(M \cdot h\right)}{\ell}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   if -1.999999999999994e-310 < h < 7.19999999999999976e266

   1. Initial program 67.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 54.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-commutative 54.1%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. *-commutative 54.1%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. associate-/r* 54.1%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   4. Simplified 54.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   5. Step-by-step derivation

      1. sqrt-div 63.4%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

      2. inv-pow 63.4%

         \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{-1}}}}{\sqrt{\ell}} \]

   6. Applied egg-rr 63.4%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}}}{\sqrt{\ell}}} \]
   7. Step-by-step derivation

      1. sqr-pow 63.5%

         \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{\left(\frac{-1}{2}\right)} \cdot {h}^{\left(\frac{-1}{2}\right)}}}}{\sqrt{\ell}} \]

      2. rem-sqrt-square 63.5%

         \[\leadsto d \cdot \frac{\color{blue}{\left|{h}^{\left(\frac{-1}{2}\right)}\right|}}{\sqrt{\ell}} \]

      3. metadata-eval 63.5%

         \[\leadsto d \cdot \frac{\left|{h}^{\color{blue}{-0.5}}\right|}{\sqrt{\ell}} \]

      4. sqr-pow 63.3%

         \[\leadsto d \cdot \frac{\left|\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}\right|}{\sqrt{\ell}} \]

      5. fabs-sqr 63.3%

         \[\leadsto d \cdot \frac{\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}}{\sqrt{\ell}} \]

      6. sqr-pow 63.5%

         \[\leadsto d \cdot \frac{\color{blue}{{h}^{-0.5}}}{\sqrt{\ell}} \]

   8. Simplified 63.5%

      \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]

   if 7.19999999999999976e266 < h

   1. Initial program 44.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around 0 57.2%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   3. Step-by-step derivation

      1. associate-/l* 57.2%

         \[\leadsto -0.125 \cdot \left(\color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      2. unpow2 57.2%

         \[\leadsto -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\frac{d}{{M}^{2}}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      3. unpow2 57.2%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{\color{blue}{M \cdot M}}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

   4. Simplified 57.2%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]

   5. Taylor expanded in D around 0 57.2%

      \[\leadsto -0.125 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   6. Step-by-step derivation

      1. unpow2 57.2%

         \[\leadsto -0.125 \cdot \left(\frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      2. unpow2 57.2%

         \[\leadsto -0.125 \cdot \left(\frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      3. associate-/l* 57.2%

         \[\leadsto -0.125 \cdot \left(\color{blue}{\frac{D \cdot D}{\frac{d}{M \cdot M}}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      4. associate-*l/ 57.2%

         \[\leadsto -0.125 \cdot \color{blue}{\frac{\left(D \cdot D\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{\frac{d}{M \cdot M}}} \]

      5. *-lft-identity 57.2%

         \[\leadsto -0.125 \cdot \frac{\left(D \cdot D\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{\color{blue}{1 \cdot \frac{d}{M \cdot M}}} \]

      6. associate-*l* 57.2%

         \[\leadsto -0.125 \cdot \frac{\color{blue}{D \cdot \left(D \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}}{1 \cdot \frac{d}{M \cdot M}} \]

      7. times-frac 57.2%

         \[\leadsto -0.125 \cdot \color{blue}{\left(\frac{D}{1} \cdot \frac{D \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{\frac{d}{M \cdot M}}\right)} \]

      8. /-rgt-identity 57.2%

         \[\leadsto -0.125 \cdot \left(\color{blue}{D} \cdot \frac{D \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{\frac{d}{M \cdot M}}\right) \]

   7. Simplified 57.2%

      \[\leadsto -0.125 \cdot \color{blue}{\left(D \cdot \frac{D \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{\frac{d}{M \cdot M}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot \left(h \cdot M\right)}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq 7.2 \cdot 10^{+266}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(D \cdot \frac{D \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{\frac{d}{M \cdot M}}\right)\\ \end{array} \]

Alternative 19: 56.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;d \leq -4.7 \cdot 10^{+236}:\\ \;\;\;\;t_0 \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -5.4 \cdot 10^{-304}:\\ \;\;\;\;\left(d \cdot t_0\right) \cdot \left(-1 - -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot \left(h \cdot M\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* h l)))))
   (if (<= d -4.7e+236)
     (* t_0 (- d))
     (if (<= d -5.4e-304)
       (*
        (* d t_0)
        (- -1.0 (* -0.5 (* (* 0.25 (* (/ D d) (/ D d))) (/ (* M (* h M)) l)))))
       (* d (/ (pow h -0.5) (sqrt l)))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (h * l)));
	double tmp;
	if (d <= -4.7e+236) {
		tmp = t_0 * -d;
	} else if (d <= -5.4e-304) {
		tmp = (d * t_0) * (-1.0 - (-0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * (h * M)) / l))));
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / (h * l)))
    if (d <= (-4.7d+236)) then
        tmp = t_0 * -d
    else if (d <= (-5.4d-304)) then
        tmp = (d * t_0) * ((-1.0d0) - ((-0.5d0) * ((0.25d0 * ((d_1 / d) * (d_1 / d))) * ((m * (h * m)) / l))))
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((1.0 / (h * l)));
	double tmp;
	if (d <= -4.7e+236) {
		tmp = t_0 * -d;
	} else if (d <= -5.4e-304) {
		tmp = (d * t_0) * (-1.0 - (-0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * (h * M)) / l))));
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt((1.0 / (h * l)))
	tmp = 0
	if d <= -4.7e+236:
		tmp = t_0 * -d
	elif d <= -5.4e-304:
		tmp = (d * t_0) * (-1.0 - (-0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * (h * M)) / l))))
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(1.0 / Float64(h * l)))
	tmp = 0.0
	if (d <= -4.7e+236)
		tmp = Float64(t_0 * Float64(-d));
	elseif (d <= -5.4e-304)
		tmp = Float64(Float64(d * t_0) * Float64(-1.0 - Float64(-0.5 * Float64(Float64(0.25 * Float64(Float64(D / d) * Float64(D / d))) * Float64(Float64(M * Float64(h * M)) / l)))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((1.0 / (h * l)));
	tmp = 0.0;
	if (d <= -4.7e+236)
		tmp = t_0 * -d;
	elseif (d <= -5.4e-304)
		tmp = (d * t_0) * (-1.0 - (-0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * (h * M)) / l))));
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -4.7e+236], N[(t$95$0 * (-d)), $MachinePrecision], If[LessEqual[d, -5.4e-304], N[(N[(d * t$95$0), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(N[(0.25 * N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.69999999999999963e236

   1. Initial program 66.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 48.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 48.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 66.8%

         \[\leadsto \color{blue}{\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   5. Simplified 66.8%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   6. Taylor expanded in d around -inf 88.2%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 88.2%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]

      2. distribute-rgt-neg-in 88.2%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]

      3. *-commutative 88.2%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right)\right) \]

   8. Simplified 88.2%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

   9. Taylor expanded in D around 0 82.3%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   10. Step-by-step derivation

       1. associate-/l/ 82.2%

          \[\leadsto -1 \cdot \left(d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}}\right) \]

       2. associate-*r* 82.2%

          \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]

       3. *-commutative 82.2%

          \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-1 \cdot d\right)} \]

       4. associate-/l/ 82.3%

          \[\leadsto \sqrt{\color{blue}{\frac{1}{\ell \cdot h}}} \cdot \left(-1 \cdot d\right) \]

       5. *-commutative 82.3%

          \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

       6. mul-1-neg 82.3%

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   11. Simplified 82.3%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if -4.69999999999999963e236 < d < -5.40000000000000021e-304

   1. Initial program 70.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 17.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 28.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 57.1%

         \[\leadsto \color{blue}{\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   5. Simplified 57.1%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   6. Taylor expanded in d around -inf 72.1%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 72.1%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]

      2. distribute-rgt-neg-in 72.1%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]

      3. *-commutative 72.1%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right)\right) \]

   8. Simplified 72.1%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

   9. Taylor expanded in h around 0 56.4%

      \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]
   10. Step-by-step derivation

       1. times-frac 55.6%

          \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       2. *-commutative 55.6%

          \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right)\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       3. associate-*r* 55.6%

          \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       4. unpow2 55.6%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       5. unpow2 55.6%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       6. times-frac 65.7%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       7. associate-/l* 65.6%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{{M}^{2}}{\frac{\ell}{h}}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       8. unpow2 65.6%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{M \cdot M}}{\frac{\ell}{h}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   11. Simplified 65.6%

       \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   12. Taylor expanded in M around 0 65.7%

       \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{{M}^{2} \cdot h}{\ell}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]
   13. Step-by-step derivation

       1. unpow2 65.7%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       2. associate-*l* 68.3%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   14. Simplified 68.3%

       \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{M \cdot \left(M \cdot h\right)}{\ell}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   if -5.40000000000000021e-304 < d

   1. Initial program 65.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 50.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. *-commutative 50.1%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. *-commutative 50.1%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. associate-/r* 50.1%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   4. Simplified 50.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   5. Step-by-step derivation

      1. sqrt-div 59.4%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

      2. inv-pow 59.4%

         \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{-1}}}}{\sqrt{\ell}} \]

   6. Applied egg-rr 59.4%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}}}{\sqrt{\ell}}} \]
   7. Step-by-step derivation

      1. sqr-pow 59.5%

         \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{\left(\frac{-1}{2}\right)} \cdot {h}^{\left(\frac{-1}{2}\right)}}}}{\sqrt{\ell}} \]

      2. rem-sqrt-square 59.5%

         \[\leadsto d \cdot \frac{\color{blue}{\left|{h}^{\left(\frac{-1}{2}\right)}\right|}}{\sqrt{\ell}} \]

      3. metadata-eval 59.5%

         \[\leadsto d \cdot \frac{\left|{h}^{\color{blue}{-0.5}}\right|}{\sqrt{\ell}} \]

      4. sqr-pow 59.4%

         \[\leadsto d \cdot \frac{\left|\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}\right|}{\sqrt{\ell}} \]

      5. fabs-sqr 59.4%

         \[\leadsto d \cdot \frac{\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}}{\sqrt{\ell}} \]

      6. sqr-pow 59.5%

         \[\leadsto d \cdot \frac{\color{blue}{{h}^{-0.5}}}{\sqrt{\ell}} \]

   8. Simplified 59.5%

      \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.7 \cdot 10^{+236}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -5.4 \cdot 10^{-304}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot \left(h \cdot M\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 20: 52.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\\ \mathbf{if}\;\ell \leq -2.8 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)\\ \mathbf{elif}\;\ell \leq -3.2 \cdot 10^{-134}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - t_0\right)\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-166}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* -0.5 (* (* 0.25 (* (/ D d) (/ D d))) (/ (* M M) (/ l h))))))
   (if (<= l -2.8e+119)
     (* (sqrt (/ (/ 1.0 l) h)) (- d))
     (if (<= l -3.2e-134)
       (* (* d (sqrt (/ 1.0 (* h l)))) (- -1.0 t_0))
       (if (<= l 1.5e-166)
         (* (sqrt (* (/ d h) (/ d l))) (+ 1.0 t_0))
         (* d (/ 1.0 (sqrt (* h l)))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = -0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * M) / (l / h)));
	double tmp;
	if (l <= -2.8e+119) {
		tmp = sqrt(((1.0 / l) / h)) * -d;
	} else if (l <= -3.2e-134) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 - t_0);
	} else if (l <= 1.5e-166) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + t_0);
	} else {
		tmp = d * (1.0 / sqrt((h * l)));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-0.5d0) * ((0.25d0 * ((d_1 / d) * (d_1 / d))) * ((m * m) / (l / h)))
    if (l <= (-2.8d+119)) then
        tmp = sqrt(((1.0d0 / l) / h)) * -d
    else if (l <= (-3.2d-134)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) - t_0)
    else if (l <= 1.5d-166) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 + t_0)
    else
        tmp = d * (1.0d0 / sqrt((h * l)))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = -0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * M) / (l / h)));
	double tmp;
	if (l <= -2.8e+119) {
		tmp = Math.sqrt(((1.0 / l) / h)) * -d;
	} else if (l <= -3.2e-134) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 - t_0);
	} else if (l <= 1.5e-166) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 + t_0);
	} else {
		tmp = d * (1.0 / Math.sqrt((h * l)));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = -0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * M) / (l / h)))
	tmp = 0
	if l <= -2.8e+119:
		tmp = math.sqrt(((1.0 / l) / h)) * -d
	elif l <= -3.2e-134:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 - t_0)
	elif l <= 1.5e-166:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 + t_0)
	else:
		tmp = d * (1.0 / math.sqrt((h * l)))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(-0.5 * Float64(Float64(0.25 * Float64(Float64(D / d) * Float64(D / d))) * Float64(Float64(M * M) / Float64(l / h))))
	tmp = 0.0
	if (l <= -2.8e+119)
		tmp = Float64(sqrt(Float64(Float64(1.0 / l) / h)) * Float64(-d));
	elseif (l <= -3.2e-134)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 - t_0));
	elseif (l <= 1.5e-166)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + t_0));
	else
		tmp = Float64(d * Float64(1.0 / sqrt(Float64(h * l))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = -0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * M) / (l / h)));
	tmp = 0.0;
	if (l <= -2.8e+119)
		tmp = sqrt(((1.0 / l) / h)) * -d;
	elseif (l <= -3.2e-134)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 - t_0);
	elseif (l <= 1.5e-166)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + t_0);
	else
		tmp = d * (1.0 / sqrt((h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(-0.5 * N[(N[(0.25 * N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.8e+119], N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], If[LessEqual[l, -3.2e-134], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.5e-166], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(d * N[(1.0 / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.80000000000000013e119

   1. Initial program 55.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 28.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 46.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 50.5%

         \[\leadsto \color{blue}{\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   5. Simplified 50.5%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   6. Taylor expanded in d around -inf 64.0%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 64.0%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]

      2. distribute-rgt-neg-in 64.0%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]

      3. *-commutative 64.0%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right)\right) \]

   8. Simplified 64.0%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

   9. Taylor expanded in h around 0 41.7%

      \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]
   10. Step-by-step derivation

       1. times-frac 38.9%

          \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       2. *-commutative 38.9%

          \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right)\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       3. associate-*r* 38.9%

          \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       4. unpow2 38.9%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       5. unpow2 38.9%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       6. times-frac 47.3%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       7. associate-/l* 47.3%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{{M}^{2}}{\frac{\ell}{h}}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       8. unpow2 47.3%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{M \cdot M}}{\frac{\ell}{h}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   11. Simplified 47.3%

       \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   12. Taylor expanded in D around 0 67.4%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   13. Step-by-step derivation

       1. associate-/l/ 67.4%

          \[\leadsto -1 \cdot \left(d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}}\right) \]

       2. unpow-1 67.4%

          \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{\color{blue}{{h}^{-1}}}{\ell}}\right) \]

       3. associate-*r* 67.4%

          \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{{h}^{-1}}{\ell}}} \]

       4. *-commutative 67.4%

          \[\leadsto \color{blue}{\sqrt{\frac{{h}^{-1}}{\ell}} \cdot \left(-1 \cdot d\right)} \]

       5. mul-1-neg 67.4%

          \[\leadsto \sqrt{\frac{{h}^{-1}}{\ell}} \cdot \color{blue}{\left(-d\right)} \]

       6. unpow-1 67.4%

          \[\leadsto \sqrt{\frac{\color{blue}{\frac{1}{h}}}{\ell}} \cdot \left(-d\right) \]

       7. associate-/l/ 67.4%

          \[\leadsto \sqrt{\color{blue}{\frac{1}{\ell \cdot h}}} \cdot \left(-d\right) \]

       8. associate-/r* 67.4%

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \cdot \left(-d\right) \]

   14. Simplified 67.4%

       \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)} \]

   if -2.80000000000000013e119 < l < -3.2000000000000001e-134

   1. Initial program 71.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 14.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 21.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 49.5%

         \[\leadsto \color{blue}{\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   5. Simplified 49.5%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   6. Taylor expanded in d around -inf 76.9%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 76.9%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]

      2. distribute-rgt-neg-in 76.9%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]

      3. *-commutative 76.9%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right)\right) \]

   8. Simplified 76.9%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

   9. Taylor expanded in h around 0 56.5%

      \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]
   10. Step-by-step derivation

       1. times-frac 55.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       2. *-commutative 55.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right)\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       3. associate-*r* 55.0%

          \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       4. unpow2 55.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       5. unpow2 55.0%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       6. times-frac 63.5%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       7. associate-/l* 66.8%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{{M}^{2}}{\frac{\ell}{h}}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       8. unpow2 66.8%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{M \cdot M}}{\frac{\ell}{h}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   11. Simplified 66.8%

       \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   if -3.2000000000000001e-134 < l < 1.5000000000000001e-166

   1. Initial program 76.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 23.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 26.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 71.9%

         \[\leadsto \color{blue}{\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   5. Simplified 71.9%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   6. Taylor expanded in h around 0 46.7%

      \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}\right)}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]
   7. Step-by-step derivation

      1. times-frac 30.5%

         \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

      2. *-commutative 30.5%

         \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right)\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

      3. associate-*r* 30.5%

         \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

      4. unpow2 30.5%

         \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

      5. unpow2 30.5%

         \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

      6. times-frac 43.3%

         \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

      7. associate-/l* 39.9%

         \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{{M}^{2}}{\frac{\ell}{h}}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

      8. unpow2 39.9%

         \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{M \cdot M}}{\frac{\ell}{h}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   8. Simplified 65.3%

      \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

   if 1.5000000000000001e-166 < l

   1. Initial program 63.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 56.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. sqrt-div 56.0%

         \[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \cdot d \]

      2. *-commutative 56.0%

         \[\leadsto \frac{\sqrt{1}}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot d \]

      3. metadata-eval 56.0%

         \[\leadsto \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \cdot d \]

   4. Applied egg-rr 56.0%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{h \cdot \ell}}} \cdot d \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.8 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)\\ \mathbf{elif}\;\ell \leq -3.2 \cdot 10^{-134}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-166}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 21: 49.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{-167}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -2.4e+67)
   (* (sqrt (/ (/ 1.0 l) h)) (- d))
   (if (<= l 2.7e-167)
     (*
      (sqrt (* (/ d h) (/ d l)))
      (+ 1.0 (* -0.5 (* (* 0.25 (* (/ D d) (/ D d))) (/ (* M M) (/ l h))))))
     (* d (/ 1.0 (sqrt (* h l)))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2.4e+67) {
		tmp = sqrt(((1.0 / l) / h)) * -d;
	} else if (l <= 2.7e-167) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * M) / (l / h)))));
	} else {
		tmp = d * (1.0 / sqrt((h * l)));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this 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) :: tmp
    if (l <= (-2.4d+67)) then
        tmp = sqrt(((1.0d0 / l) / h)) * -d
    else if (l <= 2.7d-167) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 + ((-0.5d0) * ((0.25d0 * ((d_1 / d) * (d_1 / d))) * ((m * m) / (l / h)))))
    else
        tmp = d * (1.0d0 / sqrt((h * l)))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2.4e+67) {
		tmp = Math.sqrt(((1.0 / l) / h)) * -d;
	} else if (l <= 2.7e-167) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * M) / (l / h)))));
	} else {
		tmp = d * (1.0 / Math.sqrt((h * l)));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -2.4e+67:
		tmp = math.sqrt(((1.0 / l) / h)) * -d
	elif l <= 2.7e-167:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * M) / (l / h)))))
	else:
		tmp = d * (1.0 / math.sqrt((h * l)))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -2.4e+67)
		tmp = Float64(sqrt(Float64(Float64(1.0 / l) / h)) * Float64(-d));
	elseif (l <= 2.7e-167)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(0.25 * Float64(Float64(D / d) * Float64(D / d))) * Float64(Float64(M * M) / Float64(l / h))))));
	else
		tmp = Float64(d * Float64(1.0 / sqrt(Float64(h * l))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -2.4e+67)
		tmp = sqrt(((1.0 / l) / h)) * -d;
	elseif (l <= 2.7e-167)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((0.25 * ((D / d) * (D / d))) * ((M * M) / (l / h)))));
	else
		tmp = d * (1.0 / sqrt((h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -2.4e+67], N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], If[LessEqual[l, 2.7e-167], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(0.25 * N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(1.0 / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.40000000000000002e67

   1. Initial program 59.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 25.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 38.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 44.0%

         \[\leadsto \color{blue}{\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   5. Simplified 44.0%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   6. Taylor expanded in d around -inf 67.4%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 67.4%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]

      2. distribute-rgt-neg-in 67.4%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]

      3. *-commutative 67.4%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right)\right) \]

   8. Simplified 67.4%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

   9. Taylor expanded in h around 0 47.1%

      \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]
   10. Step-by-step derivation

       1. times-frac 45.1%

          \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       2. *-commutative 45.1%

          \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right)\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       3. associate-*r* 45.1%

          \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       4. unpow2 45.1%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       5. unpow2 45.1%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       6. times-frac 53.3%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       7. associate-/l* 55.2%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{{M}^{2}}{\frac{\ell}{h}}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       8. unpow2 55.2%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{M \cdot M}}{\frac{\ell}{h}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   11. Simplified 55.2%

       \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   12. Taylor expanded in D around 0 63.9%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   13. Step-by-step derivation

       1. associate-/l/ 63.9%

          \[\leadsto -1 \cdot \left(d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}}\right) \]

       2. unpow-1 63.9%

          \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{\color{blue}{{h}^{-1}}}{\ell}}\right) \]

       3. associate-*r* 63.9%

          \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{{h}^{-1}}{\ell}}} \]

       4. *-commutative 63.9%

          \[\leadsto \color{blue}{\sqrt{\frac{{h}^{-1}}{\ell}} \cdot \left(-1 \cdot d\right)} \]

       5. mul-1-neg 63.9%

          \[\leadsto \sqrt{\frac{{h}^{-1}}{\ell}} \cdot \color{blue}{\left(-d\right)} \]

       6. unpow-1 63.9%

          \[\leadsto \sqrt{\frac{\color{blue}{\frac{1}{h}}}{\ell}} \cdot \left(-d\right) \]

       7. associate-/l/ 63.9%

          \[\leadsto \sqrt{\color{blue}{\frac{1}{\ell \cdot h}}} \cdot \left(-d\right) \]

       8. associate-/r* 63.9%

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \cdot \left(-d\right) \]

   14. Simplified 63.9%

       \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)} \]

   if -2.40000000000000002e67 < l < 2.7000000000000001e-167

   1. Initial program 74.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 19.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 24.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 65.3%

         \[\leadsto \color{blue}{\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   5. Simplified 65.3%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   6. Taylor expanded in h around 0 43.0%

      \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}\right)}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]
   7. Step-by-step derivation

      1. times-frac 40.0%

         \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

      2. *-commutative 40.0%

         \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right)\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

      3. associate-*r* 40.0%

         \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

      4. unpow2 40.0%

         \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

      5. unpow2 40.0%

         \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

      6. times-frac 51.1%

         \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

      7. associate-/l* 50.0%

         \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{{M}^{2}}{\frac{\ell}{h}}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

      8. unpow2 50.0%

         \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{M \cdot M}}{\frac{\ell}{h}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   8. Simplified 58.4%

      \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

   if 2.7000000000000001e-167 < l

   1. Initial program 63.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 56.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. sqrt-div 56.0%

         \[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \cdot d \]

      2. *-commutative 56.0%

         \[\leadsto \frac{\sqrt{1}}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot d \]

      3. metadata-eval 56.0%

         \[\leadsto \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \cdot d \]

   4. Applied egg-rr 56.0%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{h \cdot \ell}}} \cdot d \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{-167}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 22: 53.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := 0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\\ \mathbf{if}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(t_0 \cdot \frac{M \cdot \left(h \cdot M\right)}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq 3.6 \cdot 10^{+196}:\\ \;\;\;\;d \cdot \frac{1}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(t_0 \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* 0.25 (* (/ D d) (/ D d)))))
   (if (<= h -2e-310)
     (*
      (* d (sqrt (/ 1.0 (* h l))))
      (- -1.0 (* -0.5 (* t_0 (/ (* M (* h M)) l)))))
     (if (<= h 3.6e+196)
       (* d (/ 1.0 (sqrt (* h l))))
       (*
        (sqrt (* (/ d h) (/ d l)))
        (+ 1.0 (* -0.5 (* t_0 (/ (* M M) (/ l h))))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = 0.25 * ((D / d) * (D / d));
	double tmp;
	if (h <= -2e-310) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * (t_0 * ((M * (h * M)) / l))));
	} else if (h <= 3.6e+196) {
		tmp = d * (1.0 / sqrt((h * l)));
	} else {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * (t_0 * ((M * M) / (l / h)))));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.25d0 * ((d_1 / d) * (d_1 / d))
    if (h <= (-2d-310)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) - ((-0.5d0) * (t_0 * ((m * (h * m)) / l))))
    else if (h <= 3.6d+196) then
        tmp = d * (1.0d0 / sqrt((h * l)))
    else
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 + ((-0.5d0) * (t_0 * ((m * m) / (l / h)))))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 0.25 * ((D / d) * (D / d));
	double tmp;
	if (h <= -2e-310) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * (t_0 * ((M * (h * M)) / l))));
	} else if (h <= 3.6e+196) {
		tmp = d * (1.0 / Math.sqrt((h * l)));
	} else {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * (t_0 * ((M * M) / (l / h)))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = 0.25 * ((D / d) * (D / d))
	tmp = 0
	if h <= -2e-310:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * (t_0 * ((M * (h * M)) / l))))
	elif h <= 3.6e+196:
		tmp = d * (1.0 / math.sqrt((h * l)))
	else:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * (t_0 * ((M * M) / (l / h)))))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(0.25 * Float64(Float64(D / d) * Float64(D / d)))
	tmp = 0.0
	if (h <= -2e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 - Float64(-0.5 * Float64(t_0 * Float64(Float64(M * Float64(h * M)) / l)))));
	elseif (h <= 3.6e+196)
		tmp = Float64(d * Float64(1.0 / sqrt(Float64(h * l))));
	else
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(-0.5 * Float64(t_0 * Float64(Float64(M * M) / Float64(l / h))))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = 0.25 * ((D / d) * (D / d));
	tmp = 0.0;
	if (h <= -2e-310)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 - (-0.5 * (t_0 * ((M * (h * M)) / l))));
	elseif (h <= 3.6e+196)
		tmp = d * (1.0 / sqrt((h * l)));
	else
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * (t_0 * ((M * M) / (l / h)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(0.25 * N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -2e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(-0.5 * N[(t$95$0 * N[(N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 3.6e+196], N[(d * N[(1.0 / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(t$95$0 * N[(N[(M * M), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.999999999999994e-310

   1. Initial program 69.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 21.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 30.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 57.9%

         \[\leadsto \color{blue}{\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   5. Simplified 57.9%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   6. Taylor expanded in d around -inf 73.7%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 73.7%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]

      2. distribute-rgt-neg-in 73.7%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]

      3. *-commutative 73.7%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right)\right) \]

   8. Simplified 73.7%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

   9. Taylor expanded in h around 0 53.9%

      \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]
   10. Step-by-step derivation

       1. times-frac 51.6%

          \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       2. *-commutative 51.6%

          \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right)\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       3. associate-*r* 51.6%

          \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       4. unpow2 51.6%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       5. unpow2 51.6%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       6. times-frac 64.3%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       7. associate-/l* 64.2%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{{M}^{2}}{\frac{\ell}{h}}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       8. unpow2 64.2%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{M \cdot M}}{\frac{\ell}{h}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   11. Simplified 64.2%

       \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   12. Taylor expanded in M around 0 64.3%

       \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{{M}^{2} \cdot h}{\ell}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]
   13. Step-by-step derivation

       1. unpow2 64.3%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

       2. associate-*l* 67.2%

          \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   14. Simplified 67.2%

       \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{M \cdot \left(M \cdot h\right)}{\ell}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   if -1.999999999999994e-310 < h < 3.60000000000000007e196

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 60.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. sqrt-div 60.3%

         \[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \cdot d \]

      2. *-commutative 60.3%

         \[\leadsto \frac{\sqrt{1}}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot d \]

      3. metadata-eval 60.3%

         \[\leadsto \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \cdot d \]

   4. Applied egg-rr 60.3%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{h \cdot \ell}}} \cdot d \]

   if 3.60000000000000007e196 < h

   1. Initial program 62.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 18.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 32.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 52.2%

         \[\leadsto \color{blue}{\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   5. Simplified 52.2%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   6. Taylor expanded in h around 0 22.9%

      \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}\right)}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]
   7. Step-by-step derivation

      1. times-frac 0.8%

         \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

      2. *-commutative 0.8%

         \[\leadsto \left(1 + -0.5 \cdot \left(0.25 \cdot \left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right)\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

      3. associate-*r* 0.8%

         \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

      4. unpow2 0.8%

         \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

      5. unpow2 0.8%

         \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

      6. times-frac 1.1%

         \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

      7. associate-/l* 1.3%

         \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \color{blue}{\frac{{M}^{2}}{\frac{\ell}{h}}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

      8. unpow2 1.3%

         \[\leadsto \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{\color{blue}{M \cdot M}}{\frac{\ell}{h}}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

   8. Simplified 30.4%

      \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot \left(h \cdot M\right)}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq 3.6 \cdot 10^{+196}:\\ \;\;\;\;d \cdot \frac{1}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\\ \end{array} \]

Alternative 23: 42.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -3.2 \cdot 10^{-212}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d -3.2e-212)
   (* d (- (pow (* h l) -0.5)))
   (* d (sqrt (/ (/ 1.0 l) h)))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -3.2e-212) {
		tmp = d * -pow((h * l), -0.5);
	} else {
		tmp = d * sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this 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) :: tmp
    if (d <= (-3.2d-212)) then
        tmp = d * -((h * l) ** (-0.5d0))
    else
        tmp = d * sqrt(((1.0d0 / l) / h))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -3.2e-212) {
		tmp = d * -Math.pow((h * l), -0.5);
	} else {
		tmp = d * Math.sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if d <= -3.2e-212:
		tmp = d * -math.pow((h * l), -0.5)
	else:
		tmp = d * math.sqrt(((1.0 / l) / h))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -3.2e-212)
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) / h)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -3.2e-212)
		tmp = d * -((h * l) ^ -0.5);
	else
		tmp = d * sqrt(((1.0 / l) / h));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, -3.2e-212], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -3.1999999999999999e-212

   1. Initial program 75.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 25.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 36.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      2. expm1-log1p 63.8%

         \[\leadsto \color{blue}{\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   5. Simplified 63.8%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   6. Taylor expanded in d around -inf 78.7%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 78.7%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]

      2. distribute-rgt-neg-in 78.7%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]

      3. *-commutative 78.7%

         \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \left(d \cdot \left(-\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right)\right) \]

   8. Simplified 78.7%

      \[\leadsto \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

   9. Taylor expanded in D around 0 52.0%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   10. Step-by-step derivation

       1. associate-/l/ 52.0%

          \[\leadsto -1 \cdot \left(d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}}\right) \]

       2. associate-*r* 52.0%

          \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]

       3. *-commutative 52.0%

          \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-1 \cdot d\right)} \]

       4. mul-1-neg 52.0%

          \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \color{blue}{\left(-d\right)} \]

       5. associate-/l/ 52.0%

          \[\leadsto \sqrt{\color{blue}{\frac{1}{\ell \cdot h}}} \cdot \left(-d\right) \]

       6. *-commutative 52.0%

          \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-d\right) \]

       7. unpow-1 52.0%

          \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot \left(-d\right) \]

       8. sqr-pow 52.0%

          \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot \left(-d\right) \]

       9. rem-sqrt-square 52.3%

          \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot \left(-d\right) \]

       10. metadata-eval 52.3%

           \[\leadsto \left|{\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right| \cdot \left(-d\right) \]

       11. sqr-pow 52.3%

           \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}}\right| \cdot \left(-d\right) \]

       12. fabs-sqr 52.3%

           \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \cdot \left(-d\right) \]

       13. sqr-pow 52.3%

           \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \left(-d\right) \]

   11. Simplified 52.3%

       \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if -3.1999999999999999e-212 < d

   1. Initial program 62.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 45.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   3. Step-by-step derivation

      1. associate-/l/ 45.5%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      2. add-cbrt-cube 34.9%

         \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\frac{\frac{1}{h}}{\ell} \cdot \frac{\frac{1}{h}}{\ell}\right) \cdot \frac{\frac{1}{h}}{\ell}}}} \cdot d \]

   4. Applied egg-rr 34.9%

      \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\frac{\frac{1}{h}}{\ell} \cdot \frac{\frac{1}{h}}{\ell}\right) \cdot \frac{\frac{1}{h}}{\ell}}}} \cdot d \]

   5. Taylor expanded in h around 0 45.5%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{\ell \cdot h}}} \cdot d \]
   6. Step-by-step derivation

      1. associate-/r* 45.6%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \cdot d \]

   7. Simplified 45.6%

      \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \cdot d \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.2 \cdot 10^{-212}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]

Alternative 24: 25.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D) :precision binary64 (* d (sqrt (/ (/ 1.0 h) l))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	return d * sqrt(((1.0 / h) / l));
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this 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
    code = d * sqrt(((1.0d0 / h) / l))
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	return d * Math.sqrt(((1.0 / h) / l));
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	return d * math.sqrt(((1.0 / h) / l))

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	return Float64(d * sqrt(Float64(Float64(1.0 / h) / l)))
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp = code(d, h, l, M, D)
	tmp = d * sqrt(((1.0 / h) / l));
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.3%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

2. Taylor expanded in d around inf 30.8%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
3. Step-by-step derivation

   1. *-commutative 30.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   2. *-commutative 30.8%

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   3. associate-/r* 30.8%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

4. Simplified 30.8%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]

5. Final simplification 30.8%

   \[\leadsto d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \]

Alternative 25: 25.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D) :precision binary64 (* d (sqrt (/ (/ 1.0 l) h))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	return d * sqrt(((1.0 / l) / h));
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this 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
    code = d * sqrt(((1.0d0 / l) / h))
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	return d * Math.sqrt(((1.0 / l) / h));
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	return d * math.sqrt(((1.0 / l) / h))

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	return Float64(d * sqrt(Float64(Float64(1.0 / l) / h)))
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp = code(d, h, l, M, D)
	tmp = d * sqrt(((1.0 / l) / h));
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.3%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

2. Taylor expanded in d around inf 30.8%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
3. Step-by-step derivation

   1. associate-/l/ 30.8%

      \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

   2. add-cbrt-cube 26.6%

      \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\frac{\frac{1}{h}}{\ell} \cdot \frac{\frac{1}{h}}{\ell}\right) \cdot \frac{\frac{1}{h}}{\ell}}}} \cdot d \]

4. Applied egg-rr 26.6%

   \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\frac{\frac{1}{h}}{\ell} \cdot \frac{\frac{1}{h}}{\ell}\right) \cdot \frac{\frac{1}{h}}{\ell}}}} \cdot d \]

5. Taylor expanded in h around 0 30.8%

   \[\leadsto \sqrt{\color{blue}{\frac{1}{\ell \cdot h}}} \cdot d \]
6. Step-by-step derivation

   1. associate-/r* 30.8%

      \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \cdot d \]

7. Simplified 30.8%

   \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \cdot d \]

8. Final simplification 30.8%

   \[\leadsto d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}} \]

Alternative 26: 25.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ d \cdot {\left(h \cdot \ell\right)}^{-0.5} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D) :precision binary64 (* d (pow (* h l) -0.5)))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	return d * pow((h * l), -0.5);
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this 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
    code = d * ((h * l) ** (-0.5d0))
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	return d * Math.pow((h * l), -0.5);
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	return d * math.pow((h * l), -0.5)

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	return Float64(d * (Float64(h * l) ^ -0.5))
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp = code(d, h, l, M, D)
	tmp = d * ((h * l) ^ -0.5);
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.3%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

2. Taylor expanded in d around inf 30.8%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
3. Step-by-step derivation

   1. associate-/l/ 30.8%

      \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

   2. *-un-lft-identity 30.8%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right)} \cdot d \]

4. Applied egg-rr 30.8%

   \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right)} \cdot d \]
5. Step-by-step derivation

   1. *-lft-identity 30.8%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

   2. associate-/l/ 30.8%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{\ell \cdot h}}} \cdot d \]

   3. *-commutative 30.8%

      \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot d \]

   4. unpow-1 30.8%

      \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

   5. sqr-pow 30.8%

      \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

   6. rem-sqrt-square 30.0%

      \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

   7. metadata-eval 30.0%

      \[\leadsto \left|{\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right| \cdot d \]

   8. sqr-pow 29.9%

      \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}}\right| \cdot d \]

   9. fabs-sqr 29.9%

      \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \cdot d \]

   10. sqr-pow 30.0%

       \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

6. Simplified 30.0%

   \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

7. Final simplification 30.0%

   \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{-0.5} \]

Reproduce

herbie shell --seed 2023189 
(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)))))