Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.2% → 76.2%

Time: 33.9s

Alternatives: 31

Speedup: 1.0×

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: 66.2% 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: 76.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\\ \mathbf{if}\;d \leq -1.35 \cdot 10^{-64}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot {\left({\left(\frac{-1}{\ell}\right)}^{0.25} \cdot {\left(\frac{-1}{d}\right)}^{-0.25}\right)}^{2}\right) \cdot \left(1 - t_0 \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(t_0 \cdot \frac{h}{\ell}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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 (* (/ M 2.0) (/ D d)) 2.0)))
   (if (<= d -1.35e-64)
     (*
      (*
       (sqrt (/ d h))
       (pow (* (pow (/ -1.0 l) 0.25) (pow (/ -1.0 d) -0.25)) 2.0))
      (- 1.0 (* t_0 (* 0.5 (/ h l)))))
     (if (<= d -5e-310)
       (*
        (* d (sqrt (/ 1.0 (* h l))))
        (+ -1.0 (* (* D (/ D l)) (* (* (/ M (/ d M)) (/ h d)) 0.125))))
       (*
        (/ (sqrt d) (sqrt h))
        (* (/ (sqrt d) (sqrt l)) (- 1.0 (* 0.5 (* t_0 (/ h l))))))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = pow(((M / 2.0) * (D / d)), 2.0);
	double tmp;
	if (d <= -1.35e-64) {
		tmp = (sqrt((d / h)) * pow((pow((-1.0 / l), 0.25) * pow((-1.0 / d), -0.25)), 2.0)) * (1.0 - (t_0 * (0.5 * (h / l))));
	} else if (d <= -5e-310) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125)));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) / sqrt(l)) * (1.0 - (0.5 * (t_0 * (h / l)))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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 = ((m / 2.0d0) * (d_1 / d)) ** 2.0d0
    if (d <= (-1.35d-64)) then
        tmp = (sqrt((d / h)) * (((((-1.0d0) / l) ** 0.25d0) * (((-1.0d0) / d) ** (-0.25d0))) ** 2.0d0)) * (1.0d0 - (t_0 * (0.5d0 * (h / l))))
    else if (d <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) + ((d_1 * (d_1 / l)) * (((m / (d / m)) * (h / d)) * 0.125d0)))
    else
        tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) / sqrt(l)) * (1.0d0 - (0.5d0 * (t_0 * (h / l)))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.pow(((M / 2.0) * (D / d)), 2.0);
	double tmp;
	if (d <= -1.35e-64) {
		tmp = (Math.sqrt((d / h)) * Math.pow((Math.pow((-1.0 / l), 0.25) * Math.pow((-1.0 / d), -0.25)), 2.0)) * (1.0 - (t_0 * (0.5 * (h / l))));
	} else if (d <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125)));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * ((Math.sqrt(d) / Math.sqrt(l)) * (1.0 - (0.5 * (t_0 * (h / l)))));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.pow(((M / 2.0) * (D / d)), 2.0)
	tmp = 0
	if d <= -1.35e-64:
		tmp = (math.sqrt((d / h)) * math.pow((math.pow((-1.0 / l), 0.25) * math.pow((-1.0 / d), -0.25)), 2.0)) * (1.0 - (t_0 * (0.5 * (h / l))))
	elif d <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125)))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * ((math.sqrt(d) / math.sqrt(l)) * (1.0 - (0.5 * (t_0 * (h / l)))))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0
	tmp = 0.0
	if (d <= -1.35e-64)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * (Float64((Float64(-1.0 / l) ^ 0.25) * (Float64(-1.0 / d) ^ -0.25)) ^ 2.0)) * Float64(1.0 - Float64(t_0 * Float64(0.5 * Float64(h / l)))));
	elseif (d <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 + Float64(Float64(D * Float64(D / l)) * Float64(Float64(Float64(M / Float64(d / M)) * Float64(h / d)) * 0.125))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(Float64(sqrt(d) / sqrt(l)) * Float64(1.0 - Float64(0.5 * Float64(t_0 * Float64(h / l))))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = ((M / 2.0) * (D / d)) ^ 2.0;
	tmp = 0.0;
	if (d <= -1.35e-64)
		tmp = (sqrt((d / h)) * ((((-1.0 / l) ^ 0.25) * ((-1.0 / d) ^ -0.25)) ^ 2.0)) * (1.0 - (t_0 * (0.5 * (h / l))));
	elseif (d <= -5e-310)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125)));
	else
		tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) / sqrt(l)) * (1.0 - (0.5 * (t_0 * (h / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
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[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[d, -1.35e-64], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Power[N[(N[Power[N[(-1.0 / l), $MachinePrecision], 0.25], $MachinePrecision] * N[Power[N[(-1.0 / d), $MachinePrecision], -0.25], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(t$95$0 * N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(t$95$0 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.34999999999999993e-64

   1. Initial program 80.9%

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

      1. metadata-eval 80.9%

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

      2. unpow1/2 80.9%

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

      3. metadata-eval 80.9%

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

      4. unpow1/2 80.9%

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

      5. *-commutative 80.9%

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

      6. associate-*l* 80.9%

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

      7. times-frac 80.9%

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

      8. metadata-eval 80.9%

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

   3. Simplified 80.9%

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

      1. pow1/2 80.9%

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

      2. sqr-pow 80.9%

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

      3. pow2 80.9%

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

      4. metadata-eval 80.9%

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

   5. Applied egg-rr 80.9%

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

   6. Taylor expanded in d around -inf 81.7%

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

      1. distribute-lft-in 81.7%

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

      2. exp-sum 82.0%

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

      3. *-commutative 82.0%

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

      4. exp-to-pow 83.0%

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

      5. *-commutative 83.0%

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

      6. *-commutative 83.0%

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

      7. associate-*l* 83.0%

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

      8. metadata-eval 83.0%

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

      9. metadata-eval 83.0%

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

      10. metadata-eval 83.0%

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

      11. exp-to-pow 85.3%

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

      12. metadata-eval 85.3%

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

      13. metadata-eval 85.3%

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

   8. Simplified 85.3%

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

   if -1.34999999999999993e-64 < d < -4.999999999999985e-310

   1. Initial program 53.6%

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

   3. Applied egg-rr 7.2%

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

      1. expm1-def 13.6%

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

      2. expm1-log1p 49.3%

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

      3. *-commutative 49.3%

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

      4. *-commutative 49.3%

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

      5. associate-/l* 49.3%

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

   5. Simplified 49.3%

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

   6. Taylor expanded in d around -inf 78.1%

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

      1. mul-1-neg 78.1%

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

      2. associate-/r* 78.1%

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

      3. distribute-rgt-neg-in 78.1%

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

      4. associate-/l/ 78.1%

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

   8. Simplified 78.1%

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

   9. Taylor expanded in M around 0 53.7%

      \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 53.7%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

       2. times-frac 60.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.125\right) \]

       3. unpow2 60.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{\color{blue}{d \cdot d}}\right) \cdot 0.125\right) \]

       4. times-frac 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{d} \cdot \frac{h}{d}\right)}\right) \cdot 0.125\right) \]

       5. unpow2 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \left(\frac{\color{blue}{M \cdot M}}{d} \cdot \frac{h}{d}\right)\right) \cdot 0.125\right) \]

       6. associate-*l* 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

       7. unpow2 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \frac{\color{blue}{D \cdot D}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       8. associate-*l/ 66.9%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{\ell} \cdot D\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       9. *-commutative 66.9%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       10. associate-/l* 69.1%

           \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

   11. Simplified 69.1%

       \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

   if -4.999999999999985e-310 < d

   1. Initial program 64.6%

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

      1. associate-*l* 64.0%

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

      2. metadata-eval 64.0%

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

      3. unpow1/2 64.0%

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

      4. metadata-eval 64.0%

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

      5. unpow1/2 64.0%

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

      6. associate-*l* 64.0%

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

      7. metadata-eval 64.0%

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

      8. times-frac 63.4%

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

   3. Simplified 63.4%

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

      1. sqrt-div 71.2%

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

   5. Applied egg-rr 71.2%

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

      1. sqrt-div 78.7%

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

   7. Applied egg-rr 78.7%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.35 \cdot 10^{-64}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot {\left({\left(\frac{-1}{\ell}\right)}^{0.25} \cdot {\left(\frac{-1}{d}\right)}^{-0.25}\right)}^{2}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\\ \end{array} \]

Alternative 2: 76.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -3.2 \cdot 10^{-64}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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-64)
   (*
    (* (sqrt (/ d h)) (sqrt (/ d l)))
    (- 1.0 (/ (* h (* 0.5 (pow (* M (/ D (* d 2.0))) 2.0))) l)))
   (if (<= d -5e-310)
     (*
      (* d (sqrt (/ 1.0 (* h l))))
      (+ -1.0 (* (* D (/ D l)) (* (* (/ M (/ d M)) (/ h d)) 0.125))))
     (*
      (/ (sqrt d) (sqrt h))
      (*
       (/ (sqrt d) (sqrt l))
       (- 1.0 (* 0.5 (* (pow (* (/ M 2.0) (/ D d)) 2.0) (/ h l)))))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -3.2e-64) {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - ((h * (0.5 * pow((M * (D / (d * 2.0))), 2.0))) / l));
	} else if (d <= -5e-310) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125)));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) / sqrt(l)) * (1.0 - (0.5 * (pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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-64)) then
        tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0d0 - ((h * (0.5d0 * ((m * (d_1 / (d * 2.0d0))) ** 2.0d0))) / l))
    else if (d <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) + ((d_1 * (d_1 / l)) * (((m / (d / m)) * (h / d)) * 0.125d0)))
    else
        tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) / sqrt(l)) * (1.0d0 - (0.5d0 * ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (h / l)))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -3.2e-64) {
		tmp = (Math.sqrt((d / h)) * Math.sqrt((d / l))) * (1.0 - ((h * (0.5 * Math.pow((M * (D / (d * 2.0))), 2.0))) / l));
	} else if (d <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125)));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * ((Math.sqrt(d) / Math.sqrt(l)) * (1.0 - (0.5 * (Math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if d <= -3.2e-64:
		tmp = (math.sqrt((d / h)) * math.sqrt((d / l))) * (1.0 - ((h * (0.5 * math.pow((M * (D / (d * 2.0))), 2.0))) / l))
	elif d <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125)))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * ((math.sqrt(d) / math.sqrt(l)) * (1.0 - (0.5 * (math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -3.2e-64)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(Float64(h * Float64(0.5 * (Float64(M * Float64(D / Float64(d * 2.0))) ^ 2.0))) / l)));
	elseif (d <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 + Float64(Float64(D * Float64(D / l)) * Float64(Float64(Float64(M / Float64(d / M)) * Float64(h / d)) * 0.125))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(Float64(sqrt(d) / sqrt(l)) * Float64(1.0 - Float64(0.5 * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * Float64(h / l))))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -3.2e-64)
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - ((h * (0.5 * ((M * (D / (d * 2.0))) ^ 2.0))) / l));
	elseif (d <= -5e-310)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125)));
	else
		tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) / sqrt(l)) * (1.0 - (0.5 * ((((M / 2.0) * (D / d)) ^ 2.0) * (h / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
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-64], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h * N[(0.5 * N[Power[N[(M * N[(D / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.19999999999999975e-64

   1. Initial program 80.9%

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

      1. metadata-eval 80.9%

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

      2. unpow1/2 80.9%

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

      3. metadata-eval 80.9%

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

      4. unpow1/2 80.9%

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

      5. *-commutative 80.9%

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

      6. associate-*l* 80.9%

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

      7. times-frac 80.9%

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

      8. metadata-eval 80.9%

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

   3. Simplified 80.9%

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

      1. associate-*r* 80.9%

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

      2. frac-times 80.9%

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

      3. *-commutative 80.9%

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

      4. metadata-eval 80.9%

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

      5. associate-*r/ 82.2%

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

      6. metadata-eval 82.2%

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

      7. *-commutative 82.2%

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

      8. frac-times 82.2%

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

      9. associate-*l/ 82.2%

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

      10. associate-*r/ 82.2%

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

      11. associate-/l/ 82.2%

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

      12. *-commutative 82.2%

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

   5. Applied egg-rr 82.2%

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

   if -3.19999999999999975e-64 < d < -4.999999999999985e-310

   1. Initial program 53.6%

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

   3. Applied egg-rr 7.2%

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

      1. expm1-def 13.6%

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

      2. expm1-log1p 49.3%

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

      3. *-commutative 49.3%

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

      4. *-commutative 49.3%

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

      5. associate-/l* 49.3%

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

   5. Simplified 49.3%

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

   6. Taylor expanded in d around -inf 78.1%

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

      1. mul-1-neg 78.1%

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

      2. associate-/r* 78.1%

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

      3. distribute-rgt-neg-in 78.1%

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

      4. associate-/l/ 78.1%

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

   8. Simplified 78.1%

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

   9. Taylor expanded in M around 0 53.7%

      \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 53.7%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

       2. times-frac 60.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.125\right) \]

       3. unpow2 60.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{\color{blue}{d \cdot d}}\right) \cdot 0.125\right) \]

       4. times-frac 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{d} \cdot \frac{h}{d}\right)}\right) \cdot 0.125\right) \]

       5. unpow2 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \left(\frac{\color{blue}{M \cdot M}}{d} \cdot \frac{h}{d}\right)\right) \cdot 0.125\right) \]

       6. associate-*l* 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

       7. unpow2 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \frac{\color{blue}{D \cdot D}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       8. associate-*l/ 66.9%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{\ell} \cdot D\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       9. *-commutative 66.9%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       10. associate-/l* 69.1%

           \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

   11. Simplified 69.1%

       \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

   if -4.999999999999985e-310 < d

   1. Initial program 64.6%

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

      1. associate-*l* 64.0%

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

      2. metadata-eval 64.0%

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

      3. unpow1/2 64.0%

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

      4. metadata-eval 64.0%

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

      5. unpow1/2 64.0%

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

      6. associate-*l* 64.0%

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

      7. metadata-eval 64.0%

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

      8. times-frac 63.4%

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

   3. Simplified 63.4%

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

      1. sqrt-div 71.2%

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

   5. Applied egg-rr 71.2%

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

      1. sqrt-div 78.7%

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

   7. Applied egg-rr 78.7%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.2 \cdot 10^{-64}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\\ \end{array} \]

Alternative 3: 70.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\\ t_1 := \frac{\sqrt{d}}{\sqrt{h}}\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := \frac{M}{\frac{d}{M}}\\ t_4 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -1.26 \cdot 10^{-64}:\\ \;\;\;\;\left(t_2 \cdot t_4\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(t_3 \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{elif}\;d \leq 6.4 \cdot 10^{-226}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot t_3\right)\right)\right)\\ \mathbf{elif}\;d \leq 3.4 \cdot 10^{-121}:\\ \;\;\;\;\left(1 - t_0 \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \left(t_1 \cdot t_4\right)\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{-94}:\\ \;\;\;\;t_1 \cdot \left(t_4 \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{h}{d \cdot \ell}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{+148}:\\ \;\;\;\;t_2 \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(t_0 \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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 (* (/ M 2.0) (/ D d)) 2.0))
        (t_1 (/ (sqrt d) (sqrt h)))
        (t_2 (sqrt (/ d h)))
        (t_3 (/ M (/ d M)))
        (t_4 (sqrt (/ d l))))
   (if (<= d -1.26e-64)
     (*
      (* t_2 t_4)
      (- 1.0 (/ (* h (* 0.5 (pow (* M (/ D (* d 2.0))) 2.0))) l)))
     (if (<= d -5e-310)
       (*
        (* d (sqrt (/ 1.0 (* h l))))
        (+ -1.0 (* (* D (/ D l)) (* (* t_3 (/ h d)) 0.125))))
       (if (<= d 6.4e-226)
         (* (sqrt (/ h (pow l 3.0))) (* -0.125 (* D (* D t_3))))
         (if (<= d 3.4e-121)
           (* (- 1.0 (* t_0 (* 0.5 (/ h l)))) (* t_1 t_4))
           (if (<= d 2.2e-94)
             (*
              t_1
              (*
               t_4
               (-
                1.0
                (* 0.5 (* 0.25 (* (/ (* D D) (/ d (* M M))) (/ h (* d l))))))))
             (if (<= d 2.3e+148)
               (*
                t_2
                (* (/ (sqrt d) (sqrt l)) (- 1.0 (* 0.5 (* t_0 (/ h l))))))
               (/ d (* (sqrt h) (sqrt l)))))))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = pow(((M / 2.0) * (D / d)), 2.0);
	double t_1 = sqrt(d) / sqrt(h);
	double t_2 = sqrt((d / h));
	double t_3 = M / (d / M);
	double t_4 = sqrt((d / l));
	double tmp;
	if (d <= -1.26e-64) {
		tmp = (t_2 * t_4) * (1.0 - ((h * (0.5 * pow((M * (D / (d * 2.0))), 2.0))) / l));
	} else if (d <= -5e-310) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_3 * (h / d)) * 0.125)));
	} else if (d <= 6.4e-226) {
		tmp = sqrt((h / pow(l, 3.0))) * (-0.125 * (D * (D * t_3)));
	} else if (d <= 3.4e-121) {
		tmp = (1.0 - (t_0 * (0.5 * (h / l)))) * (t_1 * t_4);
	} else if (d <= 2.2e-94) {
		tmp = t_1 * (t_4 * (1.0 - (0.5 * (0.25 * (((D * D) / (d / (M * M))) * (h / (d * l)))))));
	} else if (d <= 2.3e+148) {
		tmp = t_2 * ((sqrt(d) / sqrt(l)) * (1.0 - (0.5 * (t_0 * (h / l)))));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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 = ((m / 2.0d0) * (d_1 / d)) ** 2.0d0
    t_1 = sqrt(d) / sqrt(h)
    t_2 = sqrt((d / h))
    t_3 = m / (d / m)
    t_4 = sqrt((d / l))
    if (d <= (-1.26d-64)) then
        tmp = (t_2 * t_4) * (1.0d0 - ((h * (0.5d0 * ((m * (d_1 / (d * 2.0d0))) ** 2.0d0))) / l))
    else if (d <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) + ((d_1 * (d_1 / l)) * ((t_3 * (h / d)) * 0.125d0)))
    else if (d <= 6.4d-226) then
        tmp = sqrt((h / (l ** 3.0d0))) * ((-0.125d0) * (d_1 * (d_1 * t_3)))
    else if (d <= 3.4d-121) then
        tmp = (1.0d0 - (t_0 * (0.5d0 * (h / l)))) * (t_1 * t_4)
    else if (d <= 2.2d-94) then
        tmp = t_1 * (t_4 * (1.0d0 - (0.5d0 * (0.25d0 * (((d_1 * d_1) / (d / (m * m))) * (h / (d * l)))))))
    else if (d <= 2.3d+148) then
        tmp = t_2 * ((sqrt(d) / sqrt(l)) * (1.0d0 - (0.5d0 * (t_0 * (h / l)))))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.pow(((M / 2.0) * (D / d)), 2.0);
	double t_1 = Math.sqrt(d) / Math.sqrt(h);
	double t_2 = Math.sqrt((d / h));
	double t_3 = M / (d / M);
	double t_4 = Math.sqrt((d / l));
	double tmp;
	if (d <= -1.26e-64) {
		tmp = (t_2 * t_4) * (1.0 - ((h * (0.5 * Math.pow((M * (D / (d * 2.0))), 2.0))) / l));
	} else if (d <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_3 * (h / d)) * 0.125)));
	} else if (d <= 6.4e-226) {
		tmp = Math.sqrt((h / Math.pow(l, 3.0))) * (-0.125 * (D * (D * t_3)));
	} else if (d <= 3.4e-121) {
		tmp = (1.0 - (t_0 * (0.5 * (h / l)))) * (t_1 * t_4);
	} else if (d <= 2.2e-94) {
		tmp = t_1 * (t_4 * (1.0 - (0.5 * (0.25 * (((D * D) / (d / (M * M))) * (h / (d * l)))))));
	} else if (d <= 2.3e+148) {
		tmp = t_2 * ((Math.sqrt(d) / Math.sqrt(l)) * (1.0 - (0.5 * (t_0 * (h / l)))));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.pow(((M / 2.0) * (D / d)), 2.0)
	t_1 = math.sqrt(d) / math.sqrt(h)
	t_2 = math.sqrt((d / h))
	t_3 = M / (d / M)
	t_4 = math.sqrt((d / l))
	tmp = 0
	if d <= -1.26e-64:
		tmp = (t_2 * t_4) * (1.0 - ((h * (0.5 * math.pow((M * (D / (d * 2.0))), 2.0))) / l))
	elif d <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_3 * (h / d)) * 0.125)))
	elif d <= 6.4e-226:
		tmp = math.sqrt((h / math.pow(l, 3.0))) * (-0.125 * (D * (D * t_3)))
	elif d <= 3.4e-121:
		tmp = (1.0 - (t_0 * (0.5 * (h / l)))) * (t_1 * t_4)
	elif d <= 2.2e-94:
		tmp = t_1 * (t_4 * (1.0 - (0.5 * (0.25 * (((D * D) / (d / (M * M))) * (h / (d * l)))))))
	elif d <= 2.3e+148:
		tmp = t_2 * ((math.sqrt(d) / math.sqrt(l)) * (1.0 - (0.5 * (t_0 * (h / l)))))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0
	t_1 = Float64(sqrt(d) / sqrt(h))
	t_2 = sqrt(Float64(d / h))
	t_3 = Float64(M / Float64(d / M))
	t_4 = sqrt(Float64(d / l))
	tmp = 0.0
	if (d <= -1.26e-64)
		tmp = Float64(Float64(t_2 * t_4) * Float64(1.0 - Float64(Float64(h * Float64(0.5 * (Float64(M * Float64(D / Float64(d * 2.0))) ^ 2.0))) / l)));
	elseif (d <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 + Float64(Float64(D * Float64(D / l)) * Float64(Float64(t_3 * Float64(h / d)) * 0.125))));
	elseif (d <= 6.4e-226)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(-0.125 * Float64(D * Float64(D * t_3))));
	elseif (d <= 3.4e-121)
		tmp = Float64(Float64(1.0 - Float64(t_0 * Float64(0.5 * Float64(h / l)))) * Float64(t_1 * t_4));
	elseif (d <= 2.2e-94)
		tmp = Float64(t_1 * Float64(t_4 * Float64(1.0 - Float64(0.5 * Float64(0.25 * Float64(Float64(Float64(D * D) / Float64(d / Float64(M * M))) * Float64(h / Float64(d * l))))))));
	elseif (d <= 2.3e+148)
		tmp = Float64(t_2 * Float64(Float64(sqrt(d) / sqrt(l)) * Float64(1.0 - Float64(0.5 * Float64(t_0 * Float64(h / l))))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = ((M / 2.0) * (D / d)) ^ 2.0;
	t_1 = sqrt(d) / sqrt(h);
	t_2 = sqrt((d / h));
	t_3 = M / (d / M);
	t_4 = sqrt((d / l));
	tmp = 0.0;
	if (d <= -1.26e-64)
		tmp = (t_2 * t_4) * (1.0 - ((h * (0.5 * ((M * (D / (d * 2.0))) ^ 2.0))) / l));
	elseif (d <= -5e-310)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_3 * (h / d)) * 0.125)));
	elseif (d <= 6.4e-226)
		tmp = sqrt((h / (l ^ 3.0))) * (-0.125 * (D * (D * t_3)));
	elseif (d <= 3.4e-121)
		tmp = (1.0 - (t_0 * (0.5 * (h / l)))) * (t_1 * t_4);
	elseif (d <= 2.2e-94)
		tmp = t_1 * (t_4 * (1.0 - (0.5 * (0.25 * (((D * D) / (d / (M * M))) * (h / (d * l)))))));
	elseif (d <= 2.3e+148)
		tmp = t_2 * ((sqrt(d) / sqrt(l)) * (1.0 - (0.5 * (t_0 * (h / l)))));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
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[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.26e-64], N[(N[(t$95$2 * t$95$4), $MachinePrecision] * N[(1.0 - N[(N[(h * N[(0.5 * N[Power[N[(M * N[(D / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$3 * N[(h / d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6.4e-226], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(D * N[(D * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.4e-121], N[(N[(1.0 - N[(t$95$0 * N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.2e-94], N[(t$95$1 * N[(t$95$4 * N[(1.0 - N[(0.5 * N[(0.25 * N[(N[(N[(D * D), $MachinePrecision] / N[(d / N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.3e+148], N[(t$95$2 * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(t$95$0 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if d < -1.2599999999999999e-64

   1. Initial program 80.9%

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

      1. metadata-eval 80.9%

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

      2. unpow1/2 80.9%

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

      3. metadata-eval 80.9%

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

      4. unpow1/2 80.9%

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

      5. *-commutative 80.9%

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

      6. associate-*l* 80.9%

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

      7. times-frac 80.9%

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

      8. metadata-eval 80.9%

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

   3. Simplified 80.9%

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

      1. associate-*r* 80.9%

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

      2. frac-times 80.9%

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

      3. *-commutative 80.9%

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

      4. metadata-eval 80.9%

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

      5. associate-*r/ 82.2%

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

      6. metadata-eval 82.2%

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

      7. *-commutative 82.2%

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

      8. frac-times 82.2%

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

      9. associate-*l/ 82.2%

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

      10. associate-*r/ 82.2%

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

      11. associate-/l/ 82.2%

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

      12. *-commutative 82.2%

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

   5. Applied egg-rr 82.2%

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

   if -1.2599999999999999e-64 < d < -4.999999999999985e-310

   1. Initial program 53.6%

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

   3. Applied egg-rr 7.2%

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

      1. expm1-def 13.6%

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

      2. expm1-log1p 49.3%

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

      3. *-commutative 49.3%

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

      4. *-commutative 49.3%

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

      5. associate-/l* 49.3%

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

   5. Simplified 49.3%

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

   6. Taylor expanded in d around -inf 78.1%

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

      1. mul-1-neg 78.1%

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

      2. associate-/r* 78.1%

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

      3. distribute-rgt-neg-in 78.1%

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

      4. associate-/l/ 78.1%

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

   8. Simplified 78.1%

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

   9. Taylor expanded in M around 0 53.7%

      \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 53.7%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

       2. times-frac 60.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.125\right) \]

       3. unpow2 60.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{\color{blue}{d \cdot d}}\right) \cdot 0.125\right) \]

       4. times-frac 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{d} \cdot \frac{h}{d}\right)}\right) \cdot 0.125\right) \]

       5. unpow2 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \left(\frac{\color{blue}{M \cdot M}}{d} \cdot \frac{h}{d}\right)\right) \cdot 0.125\right) \]

       6. associate-*l* 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

       7. unpow2 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \frac{\color{blue}{D \cdot D}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       8. associate-*l/ 66.9%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{\ell} \cdot D\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       9. *-commutative 66.9%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       10. associate-/l* 69.1%

           \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

   11. Simplified 69.1%

       \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

   if -4.999999999999985e-310 < d < 6.39999999999999965e-226

   1. Initial program 39.3%

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

      1. metadata-eval 39.3%

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

      2. unpow1/2 39.3%

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

      3. metadata-eval 39.3%

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

      4. unpow1/2 39.3%

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

      5. *-commutative 39.3%

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

      6. associate-*l* 39.3%

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

      7. times-frac 36.2%

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

      8. metadata-eval 36.2%

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

   3. Simplified 36.2%

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

      1. associate-*r* 36.2%

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

      2. frac-times 39.3%

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

      3. *-commutative 39.3%

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

      4. metadata-eval 39.3%

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

      5. associate-*r/ 39.4%

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

      6. metadata-eval 39.4%

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

      7. *-commutative 39.4%

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

      8. frac-times 36.3%

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

      9. associate-*l/ 36.3%

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

      10. associate-*r/ 36.3%

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

      11. associate-/l/ 36.3%

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

      12. *-commutative 36.3%

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

   5. Applied egg-rr 36.3%

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

   6. Taylor expanded in d around 0 46.8%

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

      1. associate-*r* 46.8%

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

      2. *-commutative 46.8%

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

      3. associate-*r/ 46.8%

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

      4. unpow2 46.8%

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

      5. unpow2 46.8%

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

      6. associate-*l* 50.5%

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

      7. associate-/l* 57.8%

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

   8. Simplified 57.8%

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

   if 6.39999999999999965e-226 < d < 3.40000000000000001e-121

   1. Initial program 58.5%

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

      1. metadata-eval 58.5%

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

      2. unpow1/2 58.5%

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

      3. metadata-eval 58.5%

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

      4. unpow1/2 58.5%

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

      5. *-commutative 58.5%

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

      6. associate-*l* 58.5%

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

      7. times-frac 58.2%

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

      8. metadata-eval 58.2%

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

   3. Simplified 58.2%

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

      1. sqrt-div 66.2%

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

   5. Applied egg-rr 69.8%

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

   if 3.40000000000000001e-121 < d < 2.20000000000000001e-94

   1. Initial program 39.0%

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

      1. associate-*l* 39.0%

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

      2. metadata-eval 39.0%

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

      3. unpow1/2 39.0%

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

      4. metadata-eval 39.0%

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

      5. unpow1/2 39.0%

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

      6. associate-*l* 39.0%

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

      7. metadata-eval 39.0%

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

      8. times-frac 39.0%

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

   3. Simplified 39.0%

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

      1. sqrt-div 64.0%

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

   5. Applied egg-rr 64.0%

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

   6. Taylor expanded in M around 0 75.8%

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

      1. associate-*r/ 75.8%

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

      2. *-commutative 75.8%

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

      3. associate-*r/ 75.8%

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

      4. associate-*r* 88.2%

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

      5. unpow2 88.2%

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

      6. associate-*l* 88.2%

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

      7. times-frac 88.0%

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

      8. associate-/l* 88.2%

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

      9. unpow2 88.2%

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

      10. unpow2 88.2%

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

   8. Simplified 88.2%

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

   if 2.20000000000000001e-94 < d < 2.3000000000000001e148

   1. Initial program 80.7%

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

      1. associate-*l* 80.7%

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

      2. metadata-eval 80.7%

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

      3. unpow1/2 80.7%

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

      4. metadata-eval 80.7%

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

      5. unpow1/2 80.7%

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

      6. associate-*l* 80.7%

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

      7. metadata-eval 80.7%

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

      8. times-frac 80.7%

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

   3. Simplified 80.7%

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

      1. sqrt-div 88.7%

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

   5. Applied egg-rr 87.2%

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

   if 2.3000000000000001e148 < d

   1. Initial program 70.4%

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

      1. metadata-eval 70.4%

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

      2. unpow1/2 70.4%

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

      3. metadata-eval 70.4%

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

      4. unpow1/2 70.4%

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

      5. *-commutative 70.4%

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

      6. associate-*l* 70.4%

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

      7. times-frac 70.4%

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

      8. metadata-eval 70.4%

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

   3. Simplified 70.4%

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

      1. associate-*r* 70.4%

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

      2. frac-times 70.4%

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

      3. *-commutative 70.4%

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

      4. metadata-eval 70.4%

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

      5. associate-*r/ 74.0%

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

      6. metadata-eval 74.0%

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

      7. *-commutative 74.0%

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

      8. frac-times 74.0%

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

      9. associate-*l/ 74.0%

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

      10. associate-*r/ 74.0%

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

      11. associate-/l/ 74.0%

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

      12. *-commutative 74.0%

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

   5. Applied egg-rr 74.0%

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

   6. Taylor expanded in d around inf 70.0%

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

      1. *-commutative 70.0%

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

      2. *-commutative 70.0%

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

      3. associate-/r* 70.0%

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

   8. Simplified 70.0%

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

      1. associate-/r* 70.0%

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

      2. associate-/l/ 70.0%

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

      3. add-cbrt-cube 62.1%

         \[\leadsto d \cdot \color{blue}{\sqrt[3]{\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}} \]

      4. add-sqr-sqrt 62.0%

         \[\leadsto d \cdot \sqrt[3]{\color{blue}{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]

      5. pow1 62.0%

         \[\leadsto \color{blue}{{\left(d \cdot \sqrt[3]{\frac{\frac{1}{\ell}}{h} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}\right)}^{1}} \]

      6. add-sqr-sqrt 62.1%

         \[\leadsto {\left(d \cdot \sqrt[3]{\color{blue}{\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}\right)}^{1} \]

      7. add-cbrt-cube 70.0%

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

      8. associate-/l/ 70.0%

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

      9. sqrt-div 69.9%

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

      10. metadata-eval 69.9%

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

      11. *-commutative 69.9%

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

   10. Applied egg-rr 69.9%

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

       1. unpow1 69.9%

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

       2. associate-*r/ 70.0%

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

       3. *-rgt-identity 70.0%

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

   12. Simplified 70.0%

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

       1. sqrt-prod 84.4%

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

   14. Applied egg-rr 84.4%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.26 \cdot 10^{-64}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{elif}\;d \leq 6.4 \cdot 10^{-226}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot \frac{M}{\frac{d}{M}}\right)\right)\right)\\ \mathbf{elif}\;d \leq 3.4 \cdot 10^{-121}:\\ \;\;\;\;\left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{-94}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{h}{d \cdot \ell}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 4: 73.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{d}}{\sqrt{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -1.26 \cdot 10^{-64}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot t_1\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-226}:\\ \;\;\;\;t_0 \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{h}{d \cdot \ell}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot t_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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) (sqrt h))) (t_1 (sqrt (/ d l))))
   (if (<= d -1.26e-64)
     (*
      (* (sqrt (/ d h)) t_1)
      (- 1.0 (/ (* h (* 0.5 (pow (* M (/ D (* d 2.0))) 2.0))) l)))
     (if (<= d -5e-310)
       (*
        (* d (sqrt (/ 1.0 (* h l))))
        (+ -1.0 (* (* D (/ D l)) (* (* (/ M (/ d M)) (/ h d)) 0.125))))
       (if (<= d 7e-226)
         (*
          t_0
          (*
           (/ (sqrt d) (sqrt l))
           (-
            1.0
            (* 0.5 (* 0.25 (* (/ (* D D) (/ d (* M M))) (/ h (* d l))))))))
         (*
          t_0
          (*
           (- 1.0 (* 0.5 (* (pow (* (/ M 2.0) (/ D d)) 2.0) (/ h l))))
           t_1)))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(d) / sqrt(h);
	double t_1 = sqrt((d / l));
	double tmp;
	if (d <= -1.26e-64) {
		tmp = (sqrt((d / h)) * t_1) * (1.0 - ((h * (0.5 * pow((M * (D / (d * 2.0))), 2.0))) / l));
	} else if (d <= -5e-310) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125)));
	} else if (d <= 7e-226) {
		tmp = t_0 * ((sqrt(d) / sqrt(l)) * (1.0 - (0.5 * (0.25 * (((D * D) / (d / (M * M))) * (h / (d * l)))))));
	} else {
		tmp = t_0 * ((1.0 - (0.5 * (pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))) * t_1);
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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) / sqrt(h)
    t_1 = sqrt((d / l))
    if (d <= (-1.26d-64)) then
        tmp = (sqrt((d / h)) * t_1) * (1.0d0 - ((h * (0.5d0 * ((m * (d_1 / (d * 2.0d0))) ** 2.0d0))) / l))
    else if (d <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) + ((d_1 * (d_1 / l)) * (((m / (d / m)) * (h / d)) * 0.125d0)))
    else if (d <= 7d-226) then
        tmp = t_0 * ((sqrt(d) / sqrt(l)) * (1.0d0 - (0.5d0 * (0.25d0 * (((d_1 * d_1) / (d / (m * m))) * (h / (d * l)))))))
    else
        tmp = t_0 * ((1.0d0 - (0.5d0 * ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (h / l)))) * t_1)
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(d) / Math.sqrt(h);
	double t_1 = Math.sqrt((d / l));
	double tmp;
	if (d <= -1.26e-64) {
		tmp = (Math.sqrt((d / h)) * t_1) * (1.0 - ((h * (0.5 * Math.pow((M * (D / (d * 2.0))), 2.0))) / l));
	} else if (d <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125)));
	} else if (d <= 7e-226) {
		tmp = t_0 * ((Math.sqrt(d) / Math.sqrt(l)) * (1.0 - (0.5 * (0.25 * (((D * D) / (d / (M * M))) * (h / (d * l)))))));
	} else {
		tmp = t_0 * ((1.0 - (0.5 * (Math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))) * t_1);
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt(d) / math.sqrt(h)
	t_1 = math.sqrt((d / l))
	tmp = 0
	if d <= -1.26e-64:
		tmp = (math.sqrt((d / h)) * t_1) * (1.0 - ((h * (0.5 * math.pow((M * (D / (d * 2.0))), 2.0))) / l))
	elif d <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125)))
	elif d <= 7e-226:
		tmp = t_0 * ((math.sqrt(d) / math.sqrt(l)) * (1.0 - (0.5 * (0.25 * (((D * D) / (d / (M * M))) * (h / (d * l)))))))
	else:
		tmp = t_0 * ((1.0 - (0.5 * (math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))) * t_1)
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(sqrt(d) / sqrt(h))
	t_1 = sqrt(Float64(d / l))
	tmp = 0.0
	if (d <= -1.26e-64)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * t_1) * Float64(1.0 - Float64(Float64(h * Float64(0.5 * (Float64(M * Float64(D / Float64(d * 2.0))) ^ 2.0))) / l)));
	elseif (d <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 + Float64(Float64(D * Float64(D / l)) * Float64(Float64(Float64(M / Float64(d / M)) * Float64(h / d)) * 0.125))));
	elseif (d <= 7e-226)
		tmp = Float64(t_0 * Float64(Float64(sqrt(d) / sqrt(l)) * Float64(1.0 - Float64(0.5 * Float64(0.25 * Float64(Float64(Float64(D * D) / Float64(d / Float64(M * M))) * Float64(h / Float64(d * l))))))));
	else
		tmp = Float64(t_0 * Float64(Float64(1.0 - Float64(0.5 * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * Float64(h / l)))) * t_1));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(d) / sqrt(h);
	t_1 = sqrt((d / l));
	tmp = 0.0;
	if (d <= -1.26e-64)
		tmp = (sqrt((d / h)) * t_1) * (1.0 - ((h * (0.5 * ((M * (D / (d * 2.0))) ^ 2.0))) / l));
	elseif (d <= -5e-310)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125)));
	elseif (d <= 7e-226)
		tmp = t_0 * ((sqrt(d) / sqrt(l)) * (1.0 - (0.5 * (0.25 * (((D * D) / (d / (M * M))) * (h / (d * l)))))));
	else
		tmp = t_0 * ((1.0 - (0.5 * ((((M / 2.0) * (D / d)) ^ 2.0) * (h / l)))) * t_1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
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[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.26e-64], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(1.0 - N[(N[(h * N[(0.5 * N[Power[N[(M * N[(D / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7e-226], N[(t$95$0 * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(0.25 * N[(N[(N[(D * D), $MachinePrecision] / N[(d / N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(1.0 - N[(0.5 * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.2599999999999999e-64

   1. Initial program 80.9%

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

      1. metadata-eval 80.9%

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

      2. unpow1/2 80.9%

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

      3. metadata-eval 80.9%

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

      4. unpow1/2 80.9%

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

      5. *-commutative 80.9%

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

      6. associate-*l* 80.9%

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

      7. times-frac 80.9%

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

      8. metadata-eval 80.9%

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

   3. Simplified 80.9%

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

      1. associate-*r* 80.9%

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

      2. frac-times 80.9%

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

      3. *-commutative 80.9%

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

      4. metadata-eval 80.9%

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

      5. associate-*r/ 82.2%

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

      6. metadata-eval 82.2%

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

      7. *-commutative 82.2%

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

      8. frac-times 82.2%

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

      9. associate-*l/ 82.2%

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

      10. associate-*r/ 82.2%

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

      11. associate-/l/ 82.2%

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

      12. *-commutative 82.2%

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

   5. Applied egg-rr 82.2%

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

   if -1.2599999999999999e-64 < d < -4.999999999999985e-310

   1. Initial program 53.6%

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

   3. Applied egg-rr 7.2%

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

      1. expm1-def 13.6%

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

      2. expm1-log1p 49.3%

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

      3. *-commutative 49.3%

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

      4. *-commutative 49.3%

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

      5. associate-/l* 49.3%

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

   5. Simplified 49.3%

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

   6. Taylor expanded in d around -inf 78.1%

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

      1. mul-1-neg 78.1%

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

      2. associate-/r* 78.1%

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

      3. distribute-rgt-neg-in 78.1%

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

      4. associate-/l/ 78.1%

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

   8. Simplified 78.1%

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

   9. Taylor expanded in M around 0 53.7%

      \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 53.7%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

       2. times-frac 60.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.125\right) \]

       3. unpow2 60.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{\color{blue}{d \cdot d}}\right) \cdot 0.125\right) \]

       4. times-frac 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{d} \cdot \frac{h}{d}\right)}\right) \cdot 0.125\right) \]

       5. unpow2 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \left(\frac{\color{blue}{M \cdot M}}{d} \cdot \frac{h}{d}\right)\right) \cdot 0.125\right) \]

       6. associate-*l* 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

       7. unpow2 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \frac{\color{blue}{D \cdot D}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       8. associate-*l/ 66.9%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{\ell} \cdot D\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       9. *-commutative 66.9%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       10. associate-/l* 69.1%

           \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

   11. Simplified 69.1%

       \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

   if -4.999999999999985e-310 < d < 7e-226

   1. Initial program 39.3%

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

      1. associate-*l* 39.3%

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

      2. metadata-eval 39.3%

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

      3. unpow1/2 39.3%

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

      4. metadata-eval 39.3%

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

      5. unpow1/2 39.3%

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

      6. associate-*l* 39.3%

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

      7. metadata-eval 39.3%

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

      8. times-frac 36.2%

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

   3. Simplified 36.2%

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

      1. sqrt-div 43.8%

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

   5. Applied egg-rr 43.8%

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

   6. Taylor expanded in M around 0 21.6%

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

      1. associate-*r/ 21.6%

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

      2. *-commutative 21.6%

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

      3. associate-*r/ 21.6%

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

      4. associate-*r* 21.4%

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

      5. unpow2 21.4%

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

      6. associate-*l* 25.4%

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

      7. times-frac 39.4%

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

      8. associate-/l* 39.5%

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

      9. unpow2 39.5%

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

      10. unpow2 39.5%

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

   8. Simplified 39.5%

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

      1. sqrt-div 58.1%

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

   10. Applied egg-rr 64.1%

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

   if 7e-226 < d

   1. Initial program 70.4%

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

      1. associate-*l* 69.7%

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

      2. metadata-eval 69.7%

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

      3. unpow1/2 69.7%

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

      4. metadata-eval 69.7%

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

      5. unpow1/2 69.7%

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

      6. associate-*l* 69.7%

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

      7. metadata-eval 69.7%

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

      8. times-frac 69.6%

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

   3. Simplified 69.6%

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

      1. sqrt-div 77.4%

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

   5. Applied egg-rr 77.4%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.26 \cdot 10^{-64}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-226}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \frac{h}{d \cdot \ell}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \]

Alternative 5: 71.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \frac{M}{\frac{d}{M}}\\ \mathbf{if}\;d \leq -1.26 \cdot 10^{-64}:\\ \;\;\;\;\left(t_0 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(t_1 \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{elif}\;d \leq 9.2 \cdot 10^{-226}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot t_1\right)\right)\right)\\ \mathbf{elif}\;d \leq 5.7 \cdot 10^{+150} \lor \neg \left(d \leq 7.5 \cdot 10^{+206}\right):\\ \;\;\;\;t_0 \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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 (/ M (/ d M))))
   (if (<= d -1.26e-64)
     (*
      (* t_0 (sqrt (/ d l)))
      (- 1.0 (/ (* h (* 0.5 (pow (* M (/ D (* d 2.0))) 2.0))) l)))
     (if (<= d -5e-310)
       (*
        (* d (sqrt (/ 1.0 (* h l))))
        (+ -1.0 (* (* D (/ D l)) (* (* t_1 (/ h d)) 0.125))))
       (if (<= d 9.2e-226)
         (* (sqrt (/ h (pow l 3.0))) (* -0.125 (* D (* D t_1))))
         (if (or (<= d 5.7e+150) (not (<= d 7.5e+206)))
           (*
            t_0
            (*
             (/ (sqrt d) (sqrt l))
             (- 1.0 (* 0.5 (* (pow (* (/ M 2.0) (/ D d)) 2.0) (/ h l))))))
           (* d (/ (sqrt (/ 1.0 l)) (sqrt h)))))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = M / (d / M);
	double tmp;
	if (d <= -1.26e-64) {
		tmp = (t_0 * sqrt((d / l))) * (1.0 - ((h * (0.5 * pow((M * (D / (d * 2.0))), 2.0))) / l));
	} else if (d <= -5e-310) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_1 * (h / d)) * 0.125)));
	} else if (d <= 9.2e-226) {
		tmp = sqrt((h / pow(l, 3.0))) * (-0.125 * (D * (D * t_1)));
	} else if ((d <= 5.7e+150) || !(d <= 7.5e+206)) {
		tmp = t_0 * ((sqrt(d) / sqrt(l)) * (1.0 - (0.5 * (pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))));
	} else {
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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))
    t_1 = m / (d / m)
    if (d <= (-1.26d-64)) then
        tmp = (t_0 * sqrt((d / l))) * (1.0d0 - ((h * (0.5d0 * ((m * (d_1 / (d * 2.0d0))) ** 2.0d0))) / l))
    else if (d <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) + ((d_1 * (d_1 / l)) * ((t_1 * (h / d)) * 0.125d0)))
    else if (d <= 9.2d-226) then
        tmp = sqrt((h / (l ** 3.0d0))) * ((-0.125d0) * (d_1 * (d_1 * t_1)))
    else if ((d <= 5.7d+150) .or. (.not. (d <= 7.5d+206))) then
        tmp = t_0 * ((sqrt(d) / sqrt(l)) * (1.0d0 - (0.5d0 * ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (h / l)))))
    else
        tmp = d * (sqrt((1.0d0 / l)) / sqrt(h))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
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 = M / (d / M);
	double tmp;
	if (d <= -1.26e-64) {
		tmp = (t_0 * Math.sqrt((d / l))) * (1.0 - ((h * (0.5 * Math.pow((M * (D / (d * 2.0))), 2.0))) / l));
	} else if (d <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_1 * (h / d)) * 0.125)));
	} else if (d <= 9.2e-226) {
		tmp = Math.sqrt((h / Math.pow(l, 3.0))) * (-0.125 * (D * (D * t_1)));
	} else if ((d <= 5.7e+150) || !(d <= 7.5e+206)) {
		tmp = t_0 * ((Math.sqrt(d) / Math.sqrt(l)) * (1.0 - (0.5 * (Math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))));
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt((d / h))
	t_1 = M / (d / M)
	tmp = 0
	if d <= -1.26e-64:
		tmp = (t_0 * math.sqrt((d / l))) * (1.0 - ((h * (0.5 * math.pow((M * (D / (d * 2.0))), 2.0))) / l))
	elif d <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_1 * (h / d)) * 0.125)))
	elif d <= 9.2e-226:
		tmp = math.sqrt((h / math.pow(l, 3.0))) * (-0.125 * (D * (D * t_1)))
	elif (d <= 5.7e+150) or not (d <= 7.5e+206):
		tmp = t_0 * ((math.sqrt(d) / math.sqrt(l)) * (1.0 - (0.5 * (math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))))
	else:
		tmp = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = Float64(M / Float64(d / M))
	tmp = 0.0
	if (d <= -1.26e-64)
		tmp = Float64(Float64(t_0 * sqrt(Float64(d / l))) * Float64(1.0 - Float64(Float64(h * Float64(0.5 * (Float64(M * Float64(D / Float64(d * 2.0))) ^ 2.0))) / l)));
	elseif (d <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 + Float64(Float64(D * Float64(D / l)) * Float64(Float64(t_1 * Float64(h / d)) * 0.125))));
	elseif (d <= 9.2e-226)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(-0.125 * Float64(D * Float64(D * t_1))));
	elseif ((d <= 5.7e+150) || !(d <= 7.5e+206))
		tmp = Float64(t_0 * Float64(Float64(sqrt(d) / sqrt(l)) * Float64(1.0 - Float64(0.5 * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * Float64(h / l))))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / h));
	t_1 = M / (d / M);
	tmp = 0.0;
	if (d <= -1.26e-64)
		tmp = (t_0 * sqrt((d / l))) * (1.0 - ((h * (0.5 * ((M * (D / (d * 2.0))) ^ 2.0))) / l));
	elseif (d <= -5e-310)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_1 * (h / d)) * 0.125)));
	elseif (d <= 9.2e-226)
		tmp = sqrt((h / (l ^ 3.0))) * (-0.125 * (D * (D * t_1)));
	elseif ((d <= 5.7e+150) || ~((d <= 7.5e+206)))
		tmp = t_0 * ((sqrt(d) / sqrt(l)) * (1.0 - (0.5 * ((((M / 2.0) * (D / d)) ^ 2.0) * (h / l)))));
	else
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
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[(M / N[(d / M), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.26e-64], N[(N[(t$95$0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h * N[(0.5 * N[Power[N[(M * N[(D / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * N[(h / d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 9.2e-226], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(D * N[(D * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[d, 5.7e+150], N[Not[LessEqual[d, 7.5e+206]], $MachinePrecision]], N[(t$95$0 * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.2599999999999999e-64

   1. Initial program 80.9%

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

      1. metadata-eval 80.9%

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

      2. unpow1/2 80.9%

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

      3. metadata-eval 80.9%

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

      4. unpow1/2 80.9%

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

      5. *-commutative 80.9%

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

      6. associate-*l* 80.9%

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

      7. times-frac 80.9%

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

      8. metadata-eval 80.9%

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

   3. Simplified 80.9%

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

      1. associate-*r* 80.9%

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

      2. frac-times 80.9%

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

      3. *-commutative 80.9%

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

      4. metadata-eval 80.9%

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

      5. associate-*r/ 82.2%

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

      6. metadata-eval 82.2%

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

      7. *-commutative 82.2%

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

      8. frac-times 82.2%

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

      9. associate-*l/ 82.2%

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

      10. associate-*r/ 82.2%

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

      11. associate-/l/ 82.2%

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

      12. *-commutative 82.2%

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

   5. Applied egg-rr 82.2%

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

   if -1.2599999999999999e-64 < d < -4.999999999999985e-310

   1. Initial program 53.6%

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

   3. Applied egg-rr 7.2%

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

      1. expm1-def 13.6%

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

      2. expm1-log1p 49.3%

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

      3. *-commutative 49.3%

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

      4. *-commutative 49.3%

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

      5. associate-/l* 49.3%

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

   5. Simplified 49.3%

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

   6. Taylor expanded in d around -inf 78.1%

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

      1. mul-1-neg 78.1%

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

      2. associate-/r* 78.1%

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

      3. distribute-rgt-neg-in 78.1%

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

      4. associate-/l/ 78.1%

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

   8. Simplified 78.1%

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

   9. Taylor expanded in M around 0 53.7%

      \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 53.7%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

       2. times-frac 60.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.125\right) \]

       3. unpow2 60.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{\color{blue}{d \cdot d}}\right) \cdot 0.125\right) \]

       4. times-frac 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{d} \cdot \frac{h}{d}\right)}\right) \cdot 0.125\right) \]

       5. unpow2 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \left(\frac{\color{blue}{M \cdot M}}{d} \cdot \frac{h}{d}\right)\right) \cdot 0.125\right) \]

       6. associate-*l* 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

       7. unpow2 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \frac{\color{blue}{D \cdot D}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       8. associate-*l/ 66.9%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{\ell} \cdot D\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       9. *-commutative 66.9%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       10. associate-/l* 69.1%

           \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

   11. Simplified 69.1%

       \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

   if -4.999999999999985e-310 < d < 9.2000000000000001e-226

   1. Initial program 39.3%

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

      1. metadata-eval 39.3%

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

      2. unpow1/2 39.3%

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

      3. metadata-eval 39.3%

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

      4. unpow1/2 39.3%

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

      5. *-commutative 39.3%

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

      6. associate-*l* 39.3%

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

      7. times-frac 36.2%

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

      8. metadata-eval 36.2%

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

   3. Simplified 36.2%

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

      1. associate-*r* 36.2%

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

      2. frac-times 39.3%

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

      3. *-commutative 39.3%

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

      4. metadata-eval 39.3%

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

      5. associate-*r/ 39.4%

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

      6. metadata-eval 39.4%

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

      7. *-commutative 39.4%

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

      8. frac-times 36.3%

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

      9. associate-*l/ 36.3%

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

      10. associate-*r/ 36.3%

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

      11. associate-/l/ 36.3%

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

      12. *-commutative 36.3%

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

   5. Applied egg-rr 36.3%

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

   6. Taylor expanded in d around 0 46.8%

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

      1. associate-*r* 46.8%

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

      2. *-commutative 46.8%

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

      3. associate-*r/ 46.8%

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

      4. unpow2 46.8%

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

      5. unpow2 46.8%

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

      6. associate-*l* 50.5%

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

      7. associate-/l* 57.8%

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

   8. Simplified 57.8%

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

   if 9.2000000000000001e-226 < d < 5.7000000000000002e150 or 7.49999999999999958e206 < d

   1. Initial program 73.2%

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

      1. associate-*l* 72.3%

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

      2. metadata-eval 72.3%

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

      3. unpow1/2 72.3%

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

      4. metadata-eval 72.3%

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

      5. unpow1/2 72.3%

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

      6. associate-*l* 72.3%

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

      7. metadata-eval 72.3%

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

      8. times-frac 72.3%

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

   3. Simplified 72.3%

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

      1. sqrt-div 85.2%

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

   5. Applied egg-rr 79.1%

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

   if 5.7000000000000002e150 < d < 7.49999999999999958e206

   1. Initial program 47.3%

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

   2. Taylor expanded in d around inf 62.2%

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

      1. *-commutative 62.2%

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

      2. associate-/r* 62.2%

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

   4. Simplified 62.2%

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

      1. sqrt-div 84.1%

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

   6. Applied egg-rr 84.1%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.26 \cdot 10^{-64}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{elif}\;d \leq 9.2 \cdot 10^{-226}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot \frac{M}{\frac{d}{M}}\right)\right)\right)\\ \mathbf{elif}\;d \leq 5.7 \cdot 10^{+150} \lor \neg \left(d \leq 7.5 \cdot 10^{+206}\right):\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 6: 68.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{{\ell}^{3}}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := t_1 \cdot \left(\left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot t_2\right)\\ t_4 := \frac{M}{\frac{d}{M}}\\ \mathbf{if}\;d \leq -1.05 \cdot 10^{-63}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(t_4 \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{elif}\;d \leq 10^{-225}:\\ \;\;\;\;t_0 \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot t_4\right)\right)\right)\\ \mathbf{elif}\;d \leq 3.9 \cdot 10^{-141}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{-107}:\\ \;\;\;\;t_0 \cdot \left(-0.125 \cdot \frac{{\left(M \cdot D\right)}^{2}}{d}\right)\\ \mathbf{elif}\;d \leq 1.22 \cdot 10^{+149}:\\ \;\;\;\;\left(t_1 \cdot t_2\right) \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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 (/ d h)))
        (t_2 (sqrt (/ d l)))
        (t_3
         (*
          t_1
          (* (- 1.0 (* 0.5 (* (pow (* (/ M 2.0) (/ D d)) 2.0) (/ h l)))) t_2)))
        (t_4 (/ M (/ d M))))
   (if (<= d -1.05e-63)
     t_3
     (if (<= d -5e-310)
       (*
        (* d (sqrt (/ 1.0 (* h l))))
        (+ -1.0 (* (* D (/ D l)) (* (* t_4 (/ h d)) 0.125))))
       (if (<= d 1e-225)
         (* t_0 (* -0.125 (* D (* D t_4))))
         (if (<= d 3.9e-141)
           t_3
           (if (<= d 1.55e-107)
             (* t_0 (* -0.125 (/ (pow (* M D) 2.0) d)))
             (if (<= d 1.22e+149)
               (*
                (* t_1 t_2)
                (- 1.0 (* 0.125 (/ D (/ (/ (* d d) (/ (* h M) (/ l M))) D)))))
               (/ d (* (sqrt h) (sqrt l)))))))))))

C

M = abs(M);
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((d / h));
	double t_2 = sqrt((d / l));
	double t_3 = t_1 * ((1.0 - (0.5 * (pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))) * t_2);
	double t_4 = M / (d / M);
	double tmp;
	if (d <= -1.05e-63) {
		tmp = t_3;
	} else if (d <= -5e-310) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_4 * (h / d)) * 0.125)));
	} else if (d <= 1e-225) {
		tmp = t_0 * (-0.125 * (D * (D * t_4)));
	} else if (d <= 3.9e-141) {
		tmp = t_3;
	} else if (d <= 1.55e-107) {
		tmp = t_0 * (-0.125 * (pow((M * D), 2.0) / d));
	} else if (d <= 1.22e+149) {
		tmp = (t_1 * t_2) * (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D))));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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 = sqrt((d / h))
    t_2 = sqrt((d / l))
    t_3 = t_1 * ((1.0d0 - (0.5d0 * ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (h / l)))) * t_2)
    t_4 = m / (d / m)
    if (d <= (-1.05d-63)) then
        tmp = t_3
    else if (d <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) + ((d_1 * (d_1 / l)) * ((t_4 * (h / d)) * 0.125d0)))
    else if (d <= 1d-225) then
        tmp = t_0 * ((-0.125d0) * (d_1 * (d_1 * t_4)))
    else if (d <= 3.9d-141) then
        tmp = t_3
    else if (d <= 1.55d-107) then
        tmp = t_0 * ((-0.125d0) * (((m * d_1) ** 2.0d0) / d))
    else if (d <= 1.22d+149) then
        tmp = (t_1 * t_2) * (1.0d0 - (0.125d0 * (d_1 / (((d * d) / ((h * m) / (l / m))) / d_1))))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
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((d / h));
	double t_2 = Math.sqrt((d / l));
	double t_3 = t_1 * ((1.0 - (0.5 * (Math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))) * t_2);
	double t_4 = M / (d / M);
	double tmp;
	if (d <= -1.05e-63) {
		tmp = t_3;
	} else if (d <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_4 * (h / d)) * 0.125)));
	} else if (d <= 1e-225) {
		tmp = t_0 * (-0.125 * (D * (D * t_4)));
	} else if (d <= 3.9e-141) {
		tmp = t_3;
	} else if (d <= 1.55e-107) {
		tmp = t_0 * (-0.125 * (Math.pow((M * D), 2.0) / d));
	} else if (d <= 1.22e+149) {
		tmp = (t_1 * t_2) * (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D))));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M = abs(M)
[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((d / h))
	t_2 = math.sqrt((d / l))
	t_3 = t_1 * ((1.0 - (0.5 * (math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))) * t_2)
	t_4 = M / (d / M)
	tmp = 0
	if d <= -1.05e-63:
		tmp = t_3
	elif d <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_4 * (h / d)) * 0.125)))
	elif d <= 1e-225:
		tmp = t_0 * (-0.125 * (D * (D * t_4)))
	elif d <= 3.9e-141:
		tmp = t_3
	elif d <= 1.55e-107:
		tmp = t_0 * (-0.125 * (math.pow((M * D), 2.0) / d))
	elif d <= 1.22e+149:
		tmp = (t_1 * t_2) * (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D))))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / (l ^ 3.0)))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(d / l))
	t_3 = Float64(t_1 * Float64(Float64(1.0 - Float64(0.5 * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * Float64(h / l)))) * t_2))
	t_4 = Float64(M / Float64(d / M))
	tmp = 0.0
	if (d <= -1.05e-63)
		tmp = t_3;
	elseif (d <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 + Float64(Float64(D * Float64(D / l)) * Float64(Float64(t_4 * Float64(h / d)) * 0.125))));
	elseif (d <= 1e-225)
		tmp = Float64(t_0 * Float64(-0.125 * Float64(D * Float64(D * t_4))));
	elseif (d <= 3.9e-141)
		tmp = t_3;
	elseif (d <= 1.55e-107)
		tmp = Float64(t_0 * Float64(-0.125 * Float64((Float64(M * D) ^ 2.0) / d)));
	elseif (d <= 1.22e+149)
		tmp = Float64(Float64(t_1 * t_2) * Float64(1.0 - Float64(0.125 * Float64(D / Float64(Float64(Float64(d * d) / Float64(Float64(h * M) / Float64(l / M))) / D)))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M = abs(M)
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((d / h));
	t_2 = sqrt((d / l));
	t_3 = t_1 * ((1.0 - (0.5 * ((((M / 2.0) * (D / d)) ^ 2.0) * (h / l)))) * t_2);
	t_4 = M / (d / M);
	tmp = 0.0;
	if (d <= -1.05e-63)
		tmp = t_3;
	elseif (d <= -5e-310)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_4 * (h / d)) * 0.125)));
	elseif (d <= 1e-225)
		tmp = t_0 * (-0.125 * (D * (D * t_4)));
	elseif (d <= 3.9e-141)
		tmp = t_3;
	elseif (d <= 1.55e-107)
		tmp = t_0 * (-0.125 * (((M * D) ^ 2.0) / d));
	elseif (d <= 1.22e+149)
		tmp = (t_1 * t_2) * (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D))));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
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[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(N[(1.0 - N[(0.5 * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.05e-63], t$95$3, If[LessEqual[d, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$4 * N[(h / d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1e-225], N[(t$95$0 * N[(-0.125 * N[(D * N[(D * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.9e-141], t$95$3, If[LessEqual[d, 1.55e-107], N[(t$95$0 * N[(-0.125 * N[(N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.22e+149], N[(N[(t$95$1 * t$95$2), $MachinePrecision] * N[(1.0 - N[(0.125 * N[(D / N[(N[(N[(d * d), $MachinePrecision] / N[(N[(h * M), $MachinePrecision] / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if d < -1.05e-63 or 9.9999999999999996e-226 < d < 3.8999999999999997e-141

   1. Initial program 78.3%

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

      1. associate-*l* 77.2%

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

      2. metadata-eval 77.2%

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

      3. unpow1/2 77.2%

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

      4. metadata-eval 77.2%

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

      5. unpow1/2 77.2%

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

      6. associate-*l* 77.2%

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

      7. metadata-eval 77.2%

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

      8. times-frac 77.2%

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

   3. Simplified 77.2%

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

   if -1.05e-63 < d < -4.999999999999985e-310

   1. Initial program 53.6%

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

   3. Applied egg-rr 7.2%

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

      1. expm1-def 13.6%

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

      2. expm1-log1p 49.3%

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

      3. *-commutative 49.3%

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

      4. *-commutative 49.3%

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

      5. associate-/l* 49.3%

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

   5. Simplified 49.3%

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

   6. Taylor expanded in d around -inf 78.1%

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

      1. mul-1-neg 78.1%

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

      2. associate-/r* 78.1%

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

      3. distribute-rgt-neg-in 78.1%

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

      4. associate-/l/ 78.1%

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

   8. Simplified 78.1%

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

   9. Taylor expanded in M around 0 53.7%

      \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 53.7%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

       2. times-frac 60.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.125\right) \]

       3. unpow2 60.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{\color{blue}{d \cdot d}}\right) \cdot 0.125\right) \]

       4. times-frac 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{d} \cdot \frac{h}{d}\right)}\right) \cdot 0.125\right) \]

       5. unpow2 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \left(\frac{\color{blue}{M \cdot M}}{d} \cdot \frac{h}{d}\right)\right) \cdot 0.125\right) \]

       6. associate-*l* 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

       7. unpow2 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \frac{\color{blue}{D \cdot D}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       8. associate-*l/ 66.9%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{\ell} \cdot D\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       9. *-commutative 66.9%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       10. associate-/l* 69.1%

           \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

   11. Simplified 69.1%

       \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

   if -4.999999999999985e-310 < d < 9.9999999999999996e-226

   1. Initial program 39.3%

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

      1. metadata-eval 39.3%

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

      2. unpow1/2 39.3%

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

      3. metadata-eval 39.3%

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

      4. unpow1/2 39.3%

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

      5. *-commutative 39.3%

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

      6. associate-*l* 39.3%

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

      7. times-frac 36.2%

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

      8. metadata-eval 36.2%

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

   3. Simplified 36.2%

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

      1. associate-*r* 36.2%

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

      2. frac-times 39.3%

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

      3. *-commutative 39.3%

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

      4. metadata-eval 39.3%

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

      5. associate-*r/ 39.4%

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

      6. metadata-eval 39.4%

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

      7. *-commutative 39.4%

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

      8. frac-times 36.3%

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

      9. associate-*l/ 36.3%

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

      10. associate-*r/ 36.3%

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

      11. associate-/l/ 36.3%

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

      12. *-commutative 36.3%

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

   5. Applied egg-rr 36.3%

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

   6. Taylor expanded in d around 0 46.8%

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

      1. associate-*r* 46.8%

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

      2. *-commutative 46.8%

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

      3. associate-*r/ 46.8%

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

      4. unpow2 46.8%

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

      5. unpow2 46.8%

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

      6. associate-*l* 50.5%

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

      7. associate-/l* 57.8%

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

   8. Simplified 57.8%

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

   if 3.8999999999999997e-141 < d < 1.55000000000000011e-107

   1. Initial program 24.1%

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

      1. metadata-eval 24.1%

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

      2. unpow1/2 24.1%

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

      3. metadata-eval 24.1%

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

      4. unpow1/2 24.1%

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

      5. *-commutative 24.1%

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

      6. associate-*l* 24.1%

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

      7. times-frac 23.7%

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

      8. metadata-eval 23.7%

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

   3. Simplified 23.7%

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

      1. associate-*r* 23.7%

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

      2. frac-times 24.1%

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

      3. *-commutative 24.1%

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

      4. metadata-eval 24.1%

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

      5. associate-*r/ 24.1%

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

      6. metadata-eval 24.1%

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

      7. *-commutative 24.1%

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

      8. frac-times 23.7%

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

      9. associate-*l/ 23.7%

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

      10. associate-*r/ 23.7%

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

      11. associate-/l/ 23.7%

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

      12. *-commutative 23.7%

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

   5. Applied egg-rr 23.7%

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

   6. Taylor expanded in M around 0 12.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}}{\ell}\right) \]
   7. Step-by-step derivation

      1. associate-*r* 23.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2}}}{\ell}\right) \]

      2. associate-/l* 22.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{\frac{{d}^{2}}{h}}}}{\ell}\right) \]

      3. unpow2 22.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{\frac{{d}^{2}}{h}}}{\ell}\right) \]

      4. unpow2 22.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{\frac{{d}^{2}}{h}}}{\ell}\right) \]

      5. unswap-sqr 33.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\color{blue}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}{\frac{{d}^{2}}{h}}}{\ell}\right) \]

      6. unpow2 33.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\frac{\color{blue}{d \cdot d}}{h}}}{\ell}\right) \]

      7. associate-/l* 33.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\color{blue}{\frac{d}{\frac{h}{d}}}}}{\ell}\right) \]

   8. Simplified 33.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{0.125 \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\frac{d}{\frac{h}{d}}}}}{\ell}\right) \]

   9. Taylor expanded in d around 0 34.2%

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

       1. associate-*r* 34.2%

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

       2. *-commutative 34.2%

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

       3. unpow2 34.2%

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

       4. unpow2 34.2%

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

       5. swap-sqr 56.6%

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

       6. unpow2 56.6%

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

   11. Simplified 56.6%

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

   if 1.55000000000000011e-107 < d < 1.22e149

   1. Initial program 77.4%

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

      1. metadata-eval 77.4%

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

      2. unpow1/2 77.4%

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

      3. metadata-eval 77.4%

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

      4. unpow1/2 77.4%

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

      5. *-commutative 77.4%

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

      6. associate-*l* 77.4%

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

      7. times-frac 77.4%

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

      8. metadata-eval 77.4%

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

   3. Simplified 77.4%

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

   4. Taylor expanded in M around 0 70.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   5. Step-by-step derivation

      1. associate-*r/ 70.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right) \]

      2. *-commutative 70.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left({D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{\ell \cdot {d}^{2}}\right) \]

      3. associate-*r/ 70.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}}\right) \]

      4. *-commutative 70.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

      5. times-frac 75.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)} \cdot 0.125\right) \]

      6. unpow2 75.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right) \cdot 0.125\right) \]

      7. *-commutative 75.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right) \cdot 0.125\right) \]

      8. unpow2 75.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right) \cdot 0.125\right) \]

      9. unpow2 75.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{\color{blue}{d \cdot d}}\right) \cdot 0.125\right) \]

   6. Simplified 75.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right) \cdot 0.125}\right) \]

   7. Taylor expanded in D around 0 70.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}} \cdot 0.125\right) \]
   8. Step-by-step derivation

      1. associate-/l* 72.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2} \cdot \ell}{{M}^{2} \cdot h}}} \cdot 0.125\right) \]

      2. *-commutative 72.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{D}^{2}}{\frac{\color{blue}{\ell \cdot {d}^{2}}}{{M}^{2} \cdot h}} \cdot 0.125\right) \]

      3. *-commutative 72.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{\color{blue}{h \cdot {M}^{2}}}} \cdot 0.125\right) \]

      4. unpow2 72.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot D}}{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}} \cdot 0.125\right) \]

      5. associate-/l* 72.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{D}{\frac{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}}{D}}} \cdot 0.125\right) \]

      6. *-commutative 72.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{D}{\frac{\frac{\color{blue}{{d}^{2} \cdot \ell}}{h \cdot {M}^{2}}}{D}} \cdot 0.125\right) \]

      7. associate-/l* 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{D}{\frac{\color{blue}{\frac{{d}^{2}}{\frac{h \cdot {M}^{2}}{\ell}}}}{D}} \cdot 0.125\right) \]

      8. unpow2 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{D}{\frac{\frac{\color{blue}{d \cdot d}}{\frac{h \cdot {M}^{2}}{\ell}}}{D}} \cdot 0.125\right) \]

      9. unpow2 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{\ell}}}{D}} \cdot 0.125\right) \]

      10. associate-*r* 77.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{\ell}}}{D}} \cdot 0.125\right) \]

      11. associate-/l* 82.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{D}{\frac{\frac{d \cdot d}{\color{blue}{\frac{h \cdot M}{\frac{\ell}{M}}}}}{D}} \cdot 0.125\right) \]

      12. *-commutative 82.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{M \cdot h}}{\frac{\ell}{M}}}}{D}} \cdot 0.125\right) \]

   9. Simplified 82.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{D}{\frac{\frac{d \cdot d}{\frac{M \cdot h}{\frac{\ell}{M}}}}{D}}} \cdot 0.125\right) \]

   if 1.22e149 < d

   1. Initial program 70.4%

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

      1. metadata-eval 70.4%

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

      2. unpow1/2 70.4%

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

      3. metadata-eval 70.4%

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

      4. unpow1/2 70.4%

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

      5. *-commutative 70.4%

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

      6. associate-*l* 70.4%

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

      7. times-frac 70.4%

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

      8. metadata-eval 70.4%

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

   3. Simplified 70.4%

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

      1. associate-*r* 70.4%

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

      2. frac-times 70.4%

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

      3. *-commutative 70.4%

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

      4. metadata-eval 70.4%

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

      5. associate-*r/ 74.0%

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

      6. metadata-eval 74.0%

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

      7. *-commutative 74.0%

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

      8. frac-times 74.0%

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

      9. associate-*l/ 74.0%

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

      10. associate-*r/ 74.0%

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

      11. associate-/l/ 74.0%

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

      12. *-commutative 74.0%

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

   5. Applied egg-rr 74.0%

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

   6. Taylor expanded in d around inf 70.0%

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

      1. *-commutative 70.0%

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

      2. *-commutative 70.0%

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

      3. associate-/r* 70.0%

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

   8. Simplified 70.0%

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

      1. associate-/r* 70.0%

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

      2. associate-/l/ 70.0%

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

      3. add-cbrt-cube 62.1%

         \[\leadsto d \cdot \color{blue}{\sqrt[3]{\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}} \]

      4. add-sqr-sqrt 62.0%

         \[\leadsto d \cdot \sqrt[3]{\color{blue}{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]

      5. pow1 62.0%

         \[\leadsto \color{blue}{{\left(d \cdot \sqrt[3]{\frac{\frac{1}{\ell}}{h} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}\right)}^{1}} \]

      6. add-sqr-sqrt 62.1%

         \[\leadsto {\left(d \cdot \sqrt[3]{\color{blue}{\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}\right)}^{1} \]

      7. add-cbrt-cube 70.0%

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

      8. associate-/l/ 70.0%

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

      9. sqrt-div 69.9%

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

      10. metadata-eval 69.9%

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

      11. *-commutative 69.9%

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

   10. Applied egg-rr 69.9%

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

       1. unpow1 69.9%

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

       2. associate-*r/ 70.0%

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

       3. *-rgt-identity 70.0%

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

   12. Simplified 70.0%

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

       1. sqrt-prod 84.4%

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

   14. Applied egg-rr 84.4%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.05 \cdot 10^{-63}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{elif}\;d \leq 10^{-225}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot \frac{M}{\frac{d}{M}}\right)\right)\right)\\ \mathbf{elif}\;d \leq 3.9 \cdot 10^{-141}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{-107}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{\left(M \cdot D\right)}^{2}}{d}\right)\\ \mathbf{elif}\;d \leq 1.22 \cdot 10^{+149}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 7: 69.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\\ t_1 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ t_2 := \frac{M}{\frac{d}{M}}\\ \mathbf{if}\;d \leq -3.2 \cdot 10^{-64}:\\ \;\;\;\;t_1 \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot t_0\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(t_2 \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-226}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot t_2\right)\right)\right)\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{-94}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h \cdot \frac{\ell}{d}}} \cdot \left(1 + \frac{t_0 \cdot \left(h \cdot -0.5\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq 1.3 \cdot 10^{+150}:\\ \;\;\;\;t_1 \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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 (* M (/ D (* d 2.0))) 2.0))
        (t_1 (* (sqrt (/ d h)) (sqrt (/ d l))))
        (t_2 (/ M (/ d M))))
   (if (<= d -3.2e-64)
     (* t_1 (- 1.0 (/ (* h (* 0.5 t_0)) l)))
     (if (<= d -5e-310)
       (*
        (* d (sqrt (/ 1.0 (* h l))))
        (+ -1.0 (* (* D (/ D l)) (* (* t_2 (/ h d)) 0.125))))
       (if (<= d 7e-226)
         (* (sqrt (/ h (pow l 3.0))) (* -0.125 (* D (* D t_2))))
         (if (<= d 2.2e-94)
           (*
            (/ (sqrt d) (sqrt (* h (/ l d))))
            (+ 1.0 (/ (* t_0 (* h -0.5)) l)))
           (if (<= d 1.3e+150)
             (*
              t_1
              (- 1.0 (* 0.125 (/ D (/ (/ (* d d) (/ (* h M) (/ l M))) D)))))
             (/ d (* (sqrt h) (sqrt l))))))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = pow((M * (D / (d * 2.0))), 2.0);
	double t_1 = sqrt((d / h)) * sqrt((d / l));
	double t_2 = M / (d / M);
	double tmp;
	if (d <= -3.2e-64) {
		tmp = t_1 * (1.0 - ((h * (0.5 * t_0)) / l));
	} else if (d <= -5e-310) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_2 * (h / d)) * 0.125)));
	} else if (d <= 7e-226) {
		tmp = sqrt((h / pow(l, 3.0))) * (-0.125 * (D * (D * t_2)));
	} else if (d <= 2.2e-94) {
		tmp = (sqrt(d) / sqrt((h * (l / d)))) * (1.0 + ((t_0 * (h * -0.5)) / l));
	} else if (d <= 1.3e+150) {
		tmp = t_1 * (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D))));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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 = (m * (d_1 / (d * 2.0d0))) ** 2.0d0
    t_1 = sqrt((d / h)) * sqrt((d / l))
    t_2 = m / (d / m)
    if (d <= (-3.2d-64)) then
        tmp = t_1 * (1.0d0 - ((h * (0.5d0 * t_0)) / l))
    else if (d <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) + ((d_1 * (d_1 / l)) * ((t_2 * (h / d)) * 0.125d0)))
    else if (d <= 7d-226) then
        tmp = sqrt((h / (l ** 3.0d0))) * ((-0.125d0) * (d_1 * (d_1 * t_2)))
    else if (d <= 2.2d-94) then
        tmp = (sqrt(d) / sqrt((h * (l / d)))) * (1.0d0 + ((t_0 * (h * (-0.5d0))) / l))
    else if (d <= 1.3d+150) then
        tmp = t_1 * (1.0d0 - (0.125d0 * (d_1 / (((d * d) / ((h * m) / (l / m))) / d_1))))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.pow((M * (D / (d * 2.0))), 2.0);
	double t_1 = Math.sqrt((d / h)) * Math.sqrt((d / l));
	double t_2 = M / (d / M);
	double tmp;
	if (d <= -3.2e-64) {
		tmp = t_1 * (1.0 - ((h * (0.5 * t_0)) / l));
	} else if (d <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_2 * (h / d)) * 0.125)));
	} else if (d <= 7e-226) {
		tmp = Math.sqrt((h / Math.pow(l, 3.0))) * (-0.125 * (D * (D * t_2)));
	} else if (d <= 2.2e-94) {
		tmp = (Math.sqrt(d) / Math.sqrt((h * (l / d)))) * (1.0 + ((t_0 * (h * -0.5)) / l));
	} else if (d <= 1.3e+150) {
		tmp = t_1 * (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D))));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.pow((M * (D / (d * 2.0))), 2.0)
	t_1 = math.sqrt((d / h)) * math.sqrt((d / l))
	t_2 = M / (d / M)
	tmp = 0
	if d <= -3.2e-64:
		tmp = t_1 * (1.0 - ((h * (0.5 * t_0)) / l))
	elif d <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_2 * (h / d)) * 0.125)))
	elif d <= 7e-226:
		tmp = math.sqrt((h / math.pow(l, 3.0))) * (-0.125 * (D * (D * t_2)))
	elif d <= 2.2e-94:
		tmp = (math.sqrt(d) / math.sqrt((h * (l / d)))) * (1.0 + ((t_0 * (h * -0.5)) / l))
	elif d <= 1.3e+150:
		tmp = t_1 * (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D))))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(M * Float64(D / Float64(d * 2.0))) ^ 2.0
	t_1 = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)))
	t_2 = Float64(M / Float64(d / M))
	tmp = 0.0
	if (d <= -3.2e-64)
		tmp = Float64(t_1 * Float64(1.0 - Float64(Float64(h * Float64(0.5 * t_0)) / l)));
	elseif (d <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 + Float64(Float64(D * Float64(D / l)) * Float64(Float64(t_2 * Float64(h / d)) * 0.125))));
	elseif (d <= 7e-226)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(-0.125 * Float64(D * Float64(D * t_2))));
	elseif (d <= 2.2e-94)
		tmp = Float64(Float64(sqrt(d) / sqrt(Float64(h * Float64(l / d)))) * Float64(1.0 + Float64(Float64(t_0 * Float64(h * -0.5)) / l)));
	elseif (d <= 1.3e+150)
		tmp = Float64(t_1 * Float64(1.0 - Float64(0.125 * Float64(D / Float64(Float64(Float64(d * d) / Float64(Float64(h * M) / Float64(l / M))) / D)))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (M * (D / (d * 2.0))) ^ 2.0;
	t_1 = sqrt((d / h)) * sqrt((d / l));
	t_2 = M / (d / M);
	tmp = 0.0;
	if (d <= -3.2e-64)
		tmp = t_1 * (1.0 - ((h * (0.5 * t_0)) / l));
	elseif (d <= -5e-310)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_2 * (h / d)) * 0.125)));
	elseif (d <= 7e-226)
		tmp = sqrt((h / (l ^ 3.0))) * (-0.125 * (D * (D * t_2)));
	elseif (d <= 2.2e-94)
		tmp = (sqrt(d) / sqrt((h * (l / d)))) * (1.0 + ((t_0 * (h * -0.5)) / l));
	elseif (d <= 1.3e+150)
		tmp = t_1 * (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D))));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
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[(M * N[(D / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.2e-64], N[(t$95$1 * N[(1.0 - N[(N[(h * N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$2 * N[(h / d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7e-226], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(D * N[(D * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.2e-94], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[N[(h * N[(l / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(t$95$0 * N[(h * -0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.3e+150], N[(t$95$1 * N[(1.0 - N[(0.125 * N[(D / N[(N[(N[(d * d), $MachinePrecision] / N[(N[(h * M), $MachinePrecision] / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if d < -3.19999999999999975e-64

   1. Initial program 80.9%

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

      1. metadata-eval 80.9%

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

      2. unpow1/2 80.9%

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

      3. metadata-eval 80.9%

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

      4. unpow1/2 80.9%

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

      5. *-commutative 80.9%

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

      6. associate-*l* 80.9%

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

      7. times-frac 80.9%

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

      8. metadata-eval 80.9%

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

   3. Simplified 80.9%

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

      1. associate-*r* 80.9%

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

      2. frac-times 80.9%

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

      3. *-commutative 80.9%

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

      4. metadata-eval 80.9%

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

      5. associate-*r/ 82.2%

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

      6. metadata-eval 82.2%

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

      7. *-commutative 82.2%

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

      8. frac-times 82.2%

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

      9. associate-*l/ 82.2%

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

      10. associate-*r/ 82.2%

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

      11. associate-/l/ 82.2%

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

      12. *-commutative 82.2%

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

   5. Applied egg-rr 82.2%

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

   if -3.19999999999999975e-64 < d < -4.999999999999985e-310

   1. Initial program 53.6%

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

   3. Applied egg-rr 7.2%

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

      1. expm1-def 13.6%

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

      2. expm1-log1p 49.3%

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

      3. *-commutative 49.3%

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

      4. *-commutative 49.3%

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

      5. associate-/l* 49.3%

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

   5. Simplified 49.3%

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

   6. Taylor expanded in d around -inf 78.1%

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

      1. mul-1-neg 78.1%

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

      2. associate-/r* 78.1%

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

      3. distribute-rgt-neg-in 78.1%

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

      4. associate-/l/ 78.1%

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

   8. Simplified 78.1%

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

   9. Taylor expanded in M around 0 53.7%

      \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 53.7%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

       2. times-frac 60.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.125\right) \]

       3. unpow2 60.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{\color{blue}{d \cdot d}}\right) \cdot 0.125\right) \]

       4. times-frac 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{d} \cdot \frac{h}{d}\right)}\right) \cdot 0.125\right) \]

       5. unpow2 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \left(\frac{\color{blue}{M \cdot M}}{d} \cdot \frac{h}{d}\right)\right) \cdot 0.125\right) \]

       6. associate-*l* 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

       7. unpow2 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \frac{\color{blue}{D \cdot D}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       8. associate-*l/ 66.9%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{\ell} \cdot D\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       9. *-commutative 66.9%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       10. associate-/l* 69.1%

           \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

   11. Simplified 69.1%

       \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

   if -4.999999999999985e-310 < d < 7e-226

   1. Initial program 39.3%

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

      1. metadata-eval 39.3%

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

      2. unpow1/2 39.3%

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

      3. metadata-eval 39.3%

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

      4. unpow1/2 39.3%

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

      5. *-commutative 39.3%

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

      6. associate-*l* 39.3%

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

      7. times-frac 36.2%

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

      8. metadata-eval 36.2%

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

   3. Simplified 36.2%

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

      1. associate-*r* 36.2%

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

      2. frac-times 39.3%

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

      3. *-commutative 39.3%

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

      4. metadata-eval 39.3%

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

      5. associate-*r/ 39.4%

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

      6. metadata-eval 39.4%

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

      7. *-commutative 39.4%

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

      8. frac-times 36.3%

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

      9. associate-*l/ 36.3%

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

      10. associate-*r/ 36.3%

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

      11. associate-/l/ 36.3%

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

      12. *-commutative 36.3%

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

   5. Applied egg-rr 36.3%

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

   6. Taylor expanded in d around 0 46.8%

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

      1. associate-*r* 46.8%

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

      2. *-commutative 46.8%

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

      3. associate-*r/ 46.8%

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

      4. unpow2 46.8%

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

      5. unpow2 46.8%

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

      6. associate-*l* 50.5%

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

      7. associate-/l* 57.8%

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

   8. Simplified 57.8%

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

   if 7e-226 < d < 2.20000000000000001e-94

   1. Initial program 53.9%

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

      1. metadata-eval 53.9%

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

      2. unpow1/2 53.9%

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

      3. metadata-eval 53.9%

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

      4. unpow1/2 53.9%

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

      5. *-commutative 53.9%

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

      6. associate-*l* 53.9%

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

      7. times-frac 53.7%

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

      8. metadata-eval 53.7%

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

   3. Simplified 53.7%

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

      1. associate-*r* 53.7%

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

      2. frac-times 53.9%

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

      3. *-commutative 53.9%

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

      4. metadata-eval 53.9%

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

      5. associate-*r/ 59.6%

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

      6. metadata-eval 59.6%

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

      7. *-commutative 59.6%

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

      8. frac-times 59.4%

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

      9. associate-*l/ 59.4%

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

      10. associate-*r/ 59.4%

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

      11. associate-/l/ 59.4%

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

      12. *-commutative 59.4%

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

   5. Applied egg-rr 59.4%

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

      1. *-commutative 59.4%

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

      2. sqrt-prod 48.3%

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

      3. associate-/l* 42.5%

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

      4. associate-*r/ 42.6%

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

      5. *-commutative 42.6%

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

      6. associate-*r/ 42.5%

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

      7. associate-*r/ 42.5%

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

      8. pow1 42.5%

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

   7. Applied egg-rr 42.7%

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

      1. unpow1 42.7%

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

      2. *-commutative 42.7%

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

      3. associate-/l* 42.6%

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

      4. cancel-sign-sub-inv 42.6%

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

      5. associate-*r/ 42.6%

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

      6. associate-*r/ 48.4%

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

      7. distribute-lft-neg-in 48.4%

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

      8. distribute-rgt-neg-in 48.4%

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

      9. *-commutative 48.4%

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

      10. *-commutative 48.4%

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

      11. distribute-rgt-neg-in 48.4%

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

      12. metadata-eval 48.4%

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

   9. Simplified 48.4%

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

       1. sqrt-div 65.6%

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

       2. associate-/r/ 56.7%

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

   11. Applied egg-rr 56.7%

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

       1. associate-/r/ 65.6%

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

       2. *-rgt-identity 65.6%

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

       3. associate-*r/ 65.6%

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

       4. associate-/r/ 65.6%

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

       5. associate-*l/ 65.5%

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

       6. *-lft-identity 65.5%

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

   13. Simplified 65.5%

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

   if 2.20000000000000001e-94 < d < 1.30000000000000003e150

   1. Initial program 80.7%

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

      1. metadata-eval 80.7%

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

      2. unpow1/2 80.7%

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

      3. metadata-eval 80.7%

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

      4. unpow1/2 80.7%

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

      5. *-commutative 80.7%

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

      6. associate-*l* 80.7%

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

      7. times-frac 80.7%

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

      8. metadata-eval 80.7%

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

   3. Simplified 80.7%

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

   4. Taylor expanded in M around 0 71.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   5. Step-by-step derivation

      1. associate-*r/ 71.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right) \]

      2. *-commutative 71.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left({D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{\ell \cdot {d}^{2}}\right) \]

      3. associate-*r/ 71.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}}\right) \]

      4. *-commutative 71.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

      5. times-frac 75.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)} \cdot 0.125\right) \]

      6. unpow2 75.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right) \cdot 0.125\right) \]

      7. *-commutative 75.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right) \cdot 0.125\right) \]

      8. unpow2 75.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right) \cdot 0.125\right) \]

      9. unpow2 75.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{\color{blue}{d \cdot d}}\right) \cdot 0.125\right) \]

   6. Simplified 75.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right) \cdot 0.125}\right) \]

   7. Taylor expanded in D around 0 71.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}} \cdot 0.125\right) \]
   8. Step-by-step derivation

      1. associate-/l* 73.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2} \cdot \ell}{{M}^{2} \cdot h}}} \cdot 0.125\right) \]

      2. *-commutative 73.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{D}^{2}}{\frac{\color{blue}{\ell \cdot {d}^{2}}}{{M}^{2} \cdot h}} \cdot 0.125\right) \]

      3. *-commutative 73.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{\color{blue}{h \cdot {M}^{2}}}} \cdot 0.125\right) \]

      4. unpow2 73.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot D}}{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}} \cdot 0.125\right) \]

      5. associate-/l* 73.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{D}{\frac{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}}{D}}} \cdot 0.125\right) \]

      6. *-commutative 73.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{D}{\frac{\frac{\color{blue}{{d}^{2} \cdot \ell}}{h \cdot {M}^{2}}}{D}} \cdot 0.125\right) \]

      7. associate-/l* 77.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{D}{\frac{\color{blue}{\frac{{d}^{2}}{\frac{h \cdot {M}^{2}}{\ell}}}}{D}} \cdot 0.125\right) \]

      8. unpow2 77.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{D}{\frac{\frac{\color{blue}{d \cdot d}}{\frac{h \cdot {M}^{2}}{\ell}}}{D}} \cdot 0.125\right) \]

      9. unpow2 77.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{\ell}}}{D}} \cdot 0.125\right) \]

      10. associate-*r* 78.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{\ell}}}{D}} \cdot 0.125\right) \]

      11. associate-/l* 84.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{D}{\frac{\frac{d \cdot d}{\color{blue}{\frac{h \cdot M}{\frac{\ell}{M}}}}}{D}} \cdot 0.125\right) \]

      12. *-commutative 84.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{M \cdot h}}{\frac{\ell}{M}}}}{D}} \cdot 0.125\right) \]

   9. Simplified 84.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{D}{\frac{\frac{d \cdot d}{\frac{M \cdot h}{\frac{\ell}{M}}}}{D}}} \cdot 0.125\right) \]

   if 1.30000000000000003e150 < d

   1. Initial program 70.4%

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

      1. metadata-eval 70.4%

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

      2. unpow1/2 70.4%

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

      3. metadata-eval 70.4%

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

      4. unpow1/2 70.4%

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

      5. *-commutative 70.4%

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

      6. associate-*l* 70.4%

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

      7. times-frac 70.4%

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

      8. metadata-eval 70.4%

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

   3. Simplified 70.4%

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

      1. associate-*r* 70.4%

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

      2. frac-times 70.4%

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

      3. *-commutative 70.4%

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

      4. metadata-eval 70.4%

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

      5. associate-*r/ 74.0%

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

      6. metadata-eval 74.0%

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

      7. *-commutative 74.0%

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

      8. frac-times 74.0%

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

      9. associate-*l/ 74.0%

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

      10. associate-*r/ 74.0%

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

      11. associate-/l/ 74.0%

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

      12. *-commutative 74.0%

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

   5. Applied egg-rr 74.0%

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

   6. Taylor expanded in d around inf 70.0%

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

      1. *-commutative 70.0%

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

      2. *-commutative 70.0%

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

      3. associate-/r* 70.0%

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

   8. Simplified 70.0%

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

      1. associate-/r* 70.0%

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

      2. associate-/l/ 70.0%

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

      3. add-cbrt-cube 62.1%

         \[\leadsto d \cdot \color{blue}{\sqrt[3]{\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}} \]

      4. add-sqr-sqrt 62.0%

         \[\leadsto d \cdot \sqrt[3]{\color{blue}{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]

      5. pow1 62.0%

         \[\leadsto \color{blue}{{\left(d \cdot \sqrt[3]{\frac{\frac{1}{\ell}}{h} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}\right)}^{1}} \]

      6. add-sqr-sqrt 62.1%

         \[\leadsto {\left(d \cdot \sqrt[3]{\color{blue}{\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}\right)}^{1} \]

      7. add-cbrt-cube 70.0%

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

      8. associate-/l/ 70.0%

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

      9. sqrt-div 69.9%

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

      10. metadata-eval 69.9%

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

      11. *-commutative 69.9%

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

   10. Applied egg-rr 69.9%

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

       1. unpow1 69.9%

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

       2. associate-*r/ 70.0%

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

       3. *-rgt-identity 70.0%

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

   12. Simplified 70.0%

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

       1. sqrt-prod 84.4%

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

   14. Applied egg-rr 84.4%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.2 \cdot 10^{-64}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-226}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot \frac{M}{\frac{d}{M}}\right)\right)\right)\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{-94}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h \cdot \frac{\ell}{d}}} \cdot \left(1 + \frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \left(h \cdot -0.5\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq 1.3 \cdot 10^{+150}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 8: 70.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -2.1 \cdot 10^{+120}:\\ \;\;\;\;t_0 \cdot \left(\left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot t_1\right)\\ \mathbf{elif}\;\ell \leq -2.15 \cdot 10^{-302}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{-d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+152}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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))))
   (if (<= l -2.1e+120)
     (*
      t_0
      (* (- 1.0 (* 0.5 (* (pow (* (/ M 2.0) (/ D d)) 2.0) (/ h l)))) t_1))
     (if (<= l -2.15e-302)
       (*
        (fma (pow (* (/ M d) (/ D 2.0)) 2.0) (* (/ h l) -0.5) 1.0)
        (/ (- d) (sqrt (* h l))))
       (if (<= l 9.5e+152)
         (*
          t_0
          (* t_1 (- 1.0 (* 0.5 (/ (* h (pow (* M (/ D (* d 2.0))) 2.0)) l)))))
         (* d (/ (pow h -0.5) (sqrt l))))))))

C

M = abs(M);
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 tmp;
	if (l <= -2.1e+120) {
		tmp = t_0 * ((1.0 - (0.5 * (pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))) * t_1);
	} else if (l <= -2.15e-302) {
		tmp = fma(pow(((M / d) * (D / 2.0)), 2.0), ((h / l) * -0.5), 1.0) * (-d / sqrt((h * l)));
	} else if (l <= 9.5e+152) {
		tmp = t_0 * (t_1 * (1.0 - (0.5 * ((h * pow((M * (D / (d * 2.0))), 2.0)) / l))));
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -2.1e+120)
		tmp = Float64(t_0 * Float64(Float64(1.0 - Float64(0.5 * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * Float64(h / l)))) * t_1));
	elseif (l <= -2.15e-302)
		tmp = Float64(fma((Float64(Float64(M / d) * Float64(D / 2.0)) ^ 2.0), Float64(Float64(h / l) * -0.5), 1.0) * Float64(Float64(-d) / sqrt(Float64(h * l))));
	elseif (l <= 9.5e+152)
		tmp = Float64(t_0 * Float64(t_1 * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(M * Float64(D / Float64(d * 2.0))) ^ 2.0)) / l)))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
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]}, If[LessEqual[l, -2.1e+120], N[(t$95$0 * N[(N[(1.0 - N[(0.5 * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2.15e-302], N[(N[(N[Power[N[(N[(M / d), $MachinePrecision] * N[(D / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[((-d) / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 9.5e+152], N[(t$95$0 * N[(t$95$1 * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(M * N[(D / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $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 4 regimes

2. if l < -2.1e120

   1. Initial program 68.3%

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

      1. associate-*l* 68.4%

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

      2. metadata-eval 68.4%

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

      3. unpow1/2 68.4%

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

      4. metadata-eval 68.4%

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

      5. unpow1/2 68.4%

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

      6. associate-*l* 68.4%

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

      7. metadata-eval 68.4%

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

      8. times-frac 68.5%

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

   3. Simplified 68.5%

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

   if -2.1e120 < l < -2.1500000000000001e-302

   1. Initial program 69.3%

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

   3. Applied egg-rr 20.0%

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

      1. expm1-def 26.2%

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

      2. expm1-log1p 64.2%

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

      3. *-commutative 64.2%

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

      4. *-commutative 64.2%

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

      5. associate-/l* 64.2%

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

   5. Simplified 64.2%

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

   6. Taylor expanded in d around -inf 84.4%

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

      1. mul-1-neg 84.4%

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

      2. associate-/r* 84.4%

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

      3. distribute-rgt-neg-in 84.4%

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

      4. associate-/l/ 84.4%

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

   8. Simplified 84.4%

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

      1. pow1 84.4%

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

   10. Applied egg-rr 84.4%

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

       1. unpow1 84.4%

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

       2. associate-*r* 84.4%

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

       3. *-commutative 84.4%

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

   12. Simplified 82.9%

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

   if -2.1500000000000001e-302 < l < 9.49999999999999916e152

   1. Initial program 67.9%

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

      1. associate-*l* 67.1%

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

      2. metadata-eval 67.1%

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

      3. unpow1/2 67.1%

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

      4. metadata-eval 67.1%

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

      5. unpow1/2 67.1%

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

      6. associate-*l* 67.1%

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

      7. metadata-eval 67.1%

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

      8. times-frac 66.3%

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

   3. Simplified 66.3%

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

      1. associate-*r/ 70.9%

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

      2. associate-*l/ 70.9%

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

      3. associate-*r/ 70.9%

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

      4. associate-/l/ 70.9%

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

      5. *-commutative 70.9%

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

   5. Applied egg-rr 70.9%

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

   if 9.49999999999999916e152 < l

   1. Initial program 55.4%

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

      1. metadata-eval 55.4%

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

      2. unpow1/2 55.4%

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

      3. metadata-eval 55.4%

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

      4. unpow1/2 55.4%

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

      5. *-commutative 55.4%

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

      6. associate-*l* 55.4%

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

      7. times-frac 55.4%

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

      8. metadata-eval 55.4%

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

   3. Simplified 55.4%

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

      1. associate-*r* 55.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 55.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 55.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 55.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 50.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 50.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 50.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 50.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 50.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 50.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. associate-/l/ 50.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. *-commutative 50.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 50.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

   6. Taylor expanded in d around inf 49.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   7. Step-by-step derivation

      1. *-commutative 49.9%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. *-commutative 49.9%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. associate-/r* 49.9%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   8. Simplified 49.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   9. Step-by-step derivation

      1. sqrt-div 62.6%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

   10. Applied egg-rr 62.6%

       \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. div-inv 62.5%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{\frac{1}{h}} \cdot \frac{1}{\sqrt{\ell}}\right)} \]

       2. inv-pow 62.5%

          \[\leadsto d \cdot \left(\sqrt{\color{blue}{{h}^{-1}}} \cdot \frac{1}{\sqrt{\ell}}\right) \]

   12. Applied egg-rr 62.5%

       \[\leadsto d \cdot \color{blue}{\left(\sqrt{{h}^{-1}} \cdot \frac{1}{\sqrt{\ell}}\right)} \]
   13. Step-by-step derivation

       1. associate-*r/ 62.6%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}} \cdot 1}{\sqrt{\ell}}} \]

       2. associate-/l* 62.6%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}}}{\frac{\sqrt{\ell}}{1}}} \]

       3. sqr-pow 62.7%

          \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{\left(\frac{-1}{2}\right)} \cdot {h}^{\left(\frac{-1}{2}\right)}}}}{\frac{\sqrt{\ell}}{1}} \]

       4. rem-sqrt-square 62.7%

          \[\leadsto d \cdot \frac{\color{blue}{\left|{h}^{\left(\frac{-1}{2}\right)}\right|}}{\frac{\sqrt{\ell}}{1}} \]

       5. metadata-eval 62.7%

          \[\leadsto d \cdot \frac{\left|{h}^{\color{blue}{-0.5}}\right|}{\frac{\sqrt{\ell}}{1}} \]

       6. sqr-pow 62.7%

          \[\leadsto d \cdot \frac{\left|\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}\right|}{\frac{\sqrt{\ell}}{1}} \]

       7. fabs-sqr 62.7%

          \[\leadsto d \cdot \frac{\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}}{\frac{\sqrt{\ell}}{1}} \]

       8. sqr-pow 62.7%

          \[\leadsto d \cdot \frac{\color{blue}{{h}^{-0.5}}}{\frac{\sqrt{\ell}}{1}} \]

       9. /-rgt-identity 62.7%

          \[\leadsto d \cdot \frac{{h}^{-0.5}}{\color{blue}{\sqrt{\ell}}} \]

   14. Simplified 62.7%

       \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{+120}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq -2.15 \cdot 10^{-302}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{-d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 9: 70.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -3.7 \cdot 10^{+113}:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2}\right)\\ \mathbf{elif}\;\ell \leq -1.2 \cdot 10^{-303}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{-d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{+153}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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))))
   (if (<= l -3.7e+113)
     (*
      (* t_0 t_1)
      (- 1.0 (* (* 0.5 (/ h l)) (pow (/ (/ (* M D) 2.0) d) 2.0))))
     (if (<= l -1.2e-303)
       (*
        (fma (pow (* (/ M d) (/ D 2.0)) 2.0) (* (/ h l) -0.5) 1.0)
        (/ (- d) (sqrt (* h l))))
       (if (<= l 5.6e+153)
         (*
          t_0
          (* t_1 (- 1.0 (* 0.5 (/ (* h (pow (* M (/ D (* d 2.0))) 2.0)) l)))))
         (* d (/ (pow h -0.5) (sqrt l))))))))

C

M = abs(M);
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 tmp;
	if (l <= -3.7e+113) {
		tmp = (t_0 * t_1) * (1.0 - ((0.5 * (h / l)) * pow((((M * D) / 2.0) / d), 2.0)));
	} else if (l <= -1.2e-303) {
		tmp = fma(pow(((M / d) * (D / 2.0)), 2.0), ((h / l) * -0.5), 1.0) * (-d / sqrt((h * l)));
	} else if (l <= 5.6e+153) {
		tmp = t_0 * (t_1 * (1.0 - (0.5 * ((h * pow((M * (D / (d * 2.0))), 2.0)) / l))));
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -3.7e+113)
		tmp = Float64(Float64(t_0 * t_1) * Float64(1.0 - Float64(Float64(0.5 * Float64(h / l)) * (Float64(Float64(Float64(M * D) / 2.0) / d) ^ 2.0))));
	elseif (l <= -1.2e-303)
		tmp = Float64(fma((Float64(Float64(M / d) * Float64(D / 2.0)) ^ 2.0), Float64(Float64(h / l) * -0.5), 1.0) * Float64(Float64(-d) / sqrt(Float64(h * l))));
	elseif (l <= 5.6e+153)
		tmp = Float64(t_0 * Float64(t_1 * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(M * Float64(D / Float64(d * 2.0))) ^ 2.0)) / l)))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
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]}, If[LessEqual[l, -3.7e+113], N[(N[(t$95$0 * t$95$1), $MachinePrecision] * N[(1.0 - N[(N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(M * D), $MachinePrecision] / 2.0), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1.2e-303], N[(N[(N[Power[N[(N[(M / d), $MachinePrecision] * N[(D / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[((-d) / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.6e+153], N[(t$95$0 * N[(t$95$1 * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(M * N[(D / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $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 4 regimes

2. if l < -3.6999999999999998e113

   1. Initial program 68.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 68.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 68.3%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 68.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 68.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 68.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. frac-times 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      2. associate-/r* 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied egg-rr 68.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   if -3.6999999999999998e113 < l < -1.2e-303

   1. Initial program 69.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 20.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 26.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 64.2%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 64.2%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 64.2%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 64.2%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 64.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in d around -inf 84.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg 84.4%

         \[\leadsto \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      2. associate-/r* 84.4%

         \[\leadsto \left(-d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      3. distribute-rgt-neg-in 84.4%

         \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      4. associate-/l/ 84.4%

         \[\leadsto \left(d \cdot \left(-\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   8. Simplified 84.4%

      \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]
   9. Step-by-step derivation

      1. pow1 84.4%

         \[\leadsto \color{blue}{{\left(\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)\right)}^{1}} \]

   10. Applied egg-rr 84.4%

       \[\leadsto \color{blue}{{\left(d \cdot \left(\left(-\frac{1}{\sqrt{\ell \cdot h}}\right) \cdot \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)\right)\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 84.4%

          \[\leadsto \color{blue}{d \cdot \left(\left(-\frac{1}{\sqrt{\ell \cdot h}}\right) \cdot \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)\right)} \]

       2. associate-*r* 84.4%

          \[\leadsto \color{blue}{\left(d \cdot \left(-\frac{1}{\sqrt{\ell \cdot h}}\right)\right) \cdot \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)} \]

       3. *-commutative 84.4%

          \[\leadsto \color{blue}{\left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right) \cdot \left(d \cdot \left(-\frac{1}{\sqrt{\ell \cdot h}}\right)\right)} \]

   12. Simplified 82.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}} \]

   if -1.2e-303 < l < 5.5999999999999997e153

   1. Initial program 67.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 67.1%

         \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      2. metadata-eval 67.1%

         \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. unpow1/2 67.1%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval 67.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. unpow1/2 67.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. associate-*l* 67.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)\right) \]

      7. metadata-eval 67.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. times-frac 66.3%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   3. Simplified 66.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ 70.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right)\right) \]

      2. associate-*l/ 70.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot h}{\ell}\right)\right) \]

      3. associate-*r/ 70.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot h}{\ell}\right)\right) \]

      4. associate-/l/ 70.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)}^{2} \cdot h}{\ell}\right)\right) \]

      5. *-commutative 70.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot h}{\ell}\right)\right) \]

   5. Applied egg-rr 70.9%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}}\right)\right) \]

   if 5.5999999999999997e153 < l

   1. Initial program 55.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 55.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 55.4%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 55.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 55.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 55.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 55.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 55.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 55.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 55.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 55.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 55.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 55.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 55.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 50.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 50.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 50.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 50.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 50.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 50.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. associate-/l/ 50.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. *-commutative 50.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 50.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

   6. Taylor expanded in d around inf 49.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   7. Step-by-step derivation

      1. *-commutative 49.9%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. *-commutative 49.9%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. associate-/r* 49.9%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   8. Simplified 49.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   9. Step-by-step derivation

      1. sqrt-div 62.6%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

   10. Applied egg-rr 62.6%

       \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. div-inv 62.5%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{\frac{1}{h}} \cdot \frac{1}{\sqrt{\ell}}\right)} \]

       2. inv-pow 62.5%

          \[\leadsto d \cdot \left(\sqrt{\color{blue}{{h}^{-1}}} \cdot \frac{1}{\sqrt{\ell}}\right) \]

   12. Applied egg-rr 62.5%

       \[\leadsto d \cdot \color{blue}{\left(\sqrt{{h}^{-1}} \cdot \frac{1}{\sqrt{\ell}}\right)} \]
   13. Step-by-step derivation

       1. associate-*r/ 62.6%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}} \cdot 1}{\sqrt{\ell}}} \]

       2. associate-/l* 62.6%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}}}{\frac{\sqrt{\ell}}{1}}} \]

       3. sqr-pow 62.7%

          \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{\left(\frac{-1}{2}\right)} \cdot {h}^{\left(\frac{-1}{2}\right)}}}}{\frac{\sqrt{\ell}}{1}} \]

       4. rem-sqrt-square 62.7%

          \[\leadsto d \cdot \frac{\color{blue}{\left|{h}^{\left(\frac{-1}{2}\right)}\right|}}{\frac{\sqrt{\ell}}{1}} \]

       5. metadata-eval 62.7%

          \[\leadsto d \cdot \frac{\left|{h}^{\color{blue}{-0.5}}\right|}{\frac{\sqrt{\ell}}{1}} \]

       6. sqr-pow 62.7%

          \[\leadsto d \cdot \frac{\left|\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}\right|}{\frac{\sqrt{\ell}}{1}} \]

       7. fabs-sqr 62.7%

          \[\leadsto d \cdot \frac{\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}}{\frac{\sqrt{\ell}}{1}} \]

       8. sqr-pow 62.7%

          \[\leadsto d \cdot \frac{\color{blue}{{h}^{-0.5}}}{\frac{\sqrt{\ell}}{1}} \]

       9. /-rgt-identity 62.7%

          \[\leadsto d \cdot \frac{{h}^{-0.5}}{\color{blue}{\sqrt{\ell}}} \]

   14. Simplified 62.7%

       \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.7 \cdot 10^{+113}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2}\right)\\ \mathbf{elif}\;\ell \leq -1.2 \cdot 10^{-303}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{-d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 10: 70.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -2.65 \cdot 10^{+116}:\\ \;\;\;\;t_0 \cdot \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2}\right)\\ \mathbf{elif}\;\ell \leq -4.2 \cdot 10^{-304}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{-d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 6.3 \cdot 10^{+153}:\\ \;\;\;\;t_0 \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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)))))
   (if (<= l -2.65e+116)
     (* t_0 (- 1.0 (* (* 0.5 (/ h l)) (pow (/ (/ (* M D) 2.0) d) 2.0))))
     (if (<= l -4.2e-304)
       (*
        (fma (pow (* (/ M d) (/ D 2.0)) 2.0) (* (/ h l) -0.5) 1.0)
        (/ (- d) (sqrt (* h l))))
       (if (<= l 6.3e+153)
         (* t_0 (- 1.0 (/ (* h (* 0.5 (pow (* M (/ D (* d 2.0))) 2.0))) l)))
         (* d (/ (pow h -0.5) (sqrt l))))))))

C

M = abs(M);
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 tmp;
	if (l <= -2.65e+116) {
		tmp = t_0 * (1.0 - ((0.5 * (h / l)) * pow((((M * D) / 2.0) / d), 2.0)));
	} else if (l <= -4.2e-304) {
		tmp = fma(pow(((M / d) * (D / 2.0)), 2.0), ((h / l) * -0.5), 1.0) * (-d / sqrt((h * l)));
	} else if (l <= 6.3e+153) {
		tmp = t_0 * (1.0 - ((h * (0.5 * pow((M * (D / (d * 2.0))), 2.0))) / l));
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)))
	tmp = 0.0
	if (l <= -2.65e+116)
		tmp = Float64(t_0 * Float64(1.0 - Float64(Float64(0.5 * Float64(h / l)) * (Float64(Float64(Float64(M * D) / 2.0) / d) ^ 2.0))));
	elseif (l <= -4.2e-304)
		tmp = Float64(fma((Float64(Float64(M / d) * Float64(D / 2.0)) ^ 2.0), Float64(Float64(h / l) * -0.5), 1.0) * Float64(Float64(-d) / sqrt(Float64(h * l))));
	elseif (l <= 6.3e+153)
		tmp = Float64(t_0 * Float64(1.0 - Float64(Float64(h * Float64(0.5 * (Float64(M * Float64(D / Float64(d * 2.0))) ^ 2.0))) / l)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
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]}, If[LessEqual[l, -2.65e+116], N[(t$95$0 * N[(1.0 - N[(N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(M * D), $MachinePrecision] / 2.0), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -4.2e-304], N[(N[(N[Power[N[(N[(M / d), $MachinePrecision] * N[(D / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[((-d) / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.3e+153], N[(t$95$0 * N[(1.0 - N[(N[(h * N[(0.5 * N[Power[N[(M * N[(D / N[(d * 2.0), $MachinePrecision]), $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 4 regimes

2. if l < -2.6500000000000001e116

   1. Initial program 68.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 68.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 68.3%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 68.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 68.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 68.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. frac-times 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      2. associate-/r* 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied egg-rr 68.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   if -2.6500000000000001e116 < l < -4.20000000000000016e-304

   1. Initial program 69.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 20.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 26.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 64.2%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 64.2%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 64.2%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 64.2%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 64.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in d around -inf 84.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg 84.4%

         \[\leadsto \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      2. associate-/r* 84.4%

         \[\leadsto \left(-d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      3. distribute-rgt-neg-in 84.4%

         \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      4. associate-/l/ 84.4%

         \[\leadsto \left(d \cdot \left(-\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   8. Simplified 84.4%

      \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]
   9. Step-by-step derivation

      1. pow1 84.4%

         \[\leadsto \color{blue}{{\left(\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)\right)}^{1}} \]

   10. Applied egg-rr 84.4%

       \[\leadsto \color{blue}{{\left(d \cdot \left(\left(-\frac{1}{\sqrt{\ell \cdot h}}\right) \cdot \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)\right)\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 84.4%

          \[\leadsto \color{blue}{d \cdot \left(\left(-\frac{1}{\sqrt{\ell \cdot h}}\right) \cdot \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)\right)} \]

       2. associate-*r* 84.4%

          \[\leadsto \color{blue}{\left(d \cdot \left(-\frac{1}{\sqrt{\ell \cdot h}}\right)\right) \cdot \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)} \]

       3. *-commutative 84.4%

          \[\leadsto \color{blue}{\left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right) \cdot \left(d \cdot \left(-\frac{1}{\sqrt{\ell \cdot h}}\right)\right)} \]

   12. Simplified 82.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}} \]

   if -4.20000000000000016e-304 < l < 6.3000000000000001e153

   1. Initial program 67.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 67.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 67.9%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 67.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 67.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 67.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 67.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 67.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 67.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 67.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 67.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 67.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 67.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 67.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 73.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 73.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 73.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 72.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 72.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 72.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. associate-/l/ 72.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. *-commutative 72.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 72.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

   if 6.3000000000000001e153 < l

   1. Initial program 55.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 55.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 55.4%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 55.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 55.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 55.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 55.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 55.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 55.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 55.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 55.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 55.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 55.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 55.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 50.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 50.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 50.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 50.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 50.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 50.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. associate-/l/ 50.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. *-commutative 50.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 50.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

   6. Taylor expanded in d around inf 49.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   7. Step-by-step derivation

      1. *-commutative 49.9%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. *-commutative 49.9%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. associate-/r* 49.9%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   8. Simplified 49.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   9. Step-by-step derivation

      1. sqrt-div 62.6%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

   10. Applied egg-rr 62.6%

       \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. div-inv 62.5%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{\frac{1}{h}} \cdot \frac{1}{\sqrt{\ell}}\right)} \]

       2. inv-pow 62.5%

          \[\leadsto d \cdot \left(\sqrt{\color{blue}{{h}^{-1}}} \cdot \frac{1}{\sqrt{\ell}}\right) \]

   12. Applied egg-rr 62.5%

       \[\leadsto d \cdot \color{blue}{\left(\sqrt{{h}^{-1}} \cdot \frac{1}{\sqrt{\ell}}\right)} \]
   13. Step-by-step derivation

       1. associate-*r/ 62.6%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}} \cdot 1}{\sqrt{\ell}}} \]

       2. associate-/l* 62.6%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}}}{\frac{\sqrt{\ell}}{1}}} \]

       3. sqr-pow 62.7%

          \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{\left(\frac{-1}{2}\right)} \cdot {h}^{\left(\frac{-1}{2}\right)}}}}{\frac{\sqrt{\ell}}{1}} \]

       4. rem-sqrt-square 62.7%

          \[\leadsto d \cdot \frac{\color{blue}{\left|{h}^{\left(\frac{-1}{2}\right)}\right|}}{\frac{\sqrt{\ell}}{1}} \]

       5. metadata-eval 62.7%

          \[\leadsto d \cdot \frac{\left|{h}^{\color{blue}{-0.5}}\right|}{\frac{\sqrt{\ell}}{1}} \]

       6. sqr-pow 62.7%

          \[\leadsto d \cdot \frac{\left|\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}\right|}{\frac{\sqrt{\ell}}{1}} \]

       7. fabs-sqr 62.7%

          \[\leadsto d \cdot \frac{\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}}{\frac{\sqrt{\ell}}{1}} \]

       8. sqr-pow 62.7%

          \[\leadsto d \cdot \frac{\color{blue}{{h}^{-0.5}}}{\frac{\sqrt{\ell}}{1}} \]

       9. /-rgt-identity 62.7%

          \[\leadsto d \cdot \frac{{h}^{-0.5}}{\color{blue}{\sqrt{\ell}}} \]

   14. Simplified 62.7%

       \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.65 \cdot 10^{+116}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2}\right)\\ \mathbf{elif}\;\ell \leq -4.2 \cdot 10^{-304}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{-d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 6.3 \cdot 10^{+153}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 11: 65.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ t_1 := d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ t_2 := 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\\ t_3 := \frac{M}{\frac{d}{M}}\\ \mathbf{if}\;d \leq -6.5 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2} \cdot \left(h \cdot \frac{0.5}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -2.8 \cdot 10^{-66}:\\ \;\;\;\;t_0 \cdot \left(1 - \frac{0.125 \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\frac{d}{\frac{h}{d}}}}{\ell}\right)\\ \mathbf{elif}\;d \leq -4.5 \cdot 10^{-113}:\\ \;\;\;\;t_1 \cdot \left(-1 + t_2\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_1 \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(t_3 \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{-107}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot t_3\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.38 \cdot 10^{+147}:\\ \;\;\;\;t_0 \cdot \left(1 - t_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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)))))
        (t_2 (* 0.125 (/ D (/ (/ (* d d) (/ (* h M) (/ l M))) D))))
        (t_3 (/ M (/ d M))))
   (if (<= d -6.5e+80)
     (*
      (sqrt (/ (/ d h) (/ l d)))
      (- 1.0 (* (pow (* D (/ M (* d 2.0))) 2.0) (* h (/ 0.5 l)))))
     (if (<= d -2.8e-66)
       (* t_0 (- 1.0 (/ (* 0.125 (/ (* (* M D) (* M D)) (/ d (/ h d)))) l)))
       (if (<= d -4.5e-113)
         (* t_1 (+ -1.0 t_2))
         (if (<= d -5e-310)
           (* t_1 (+ -1.0 (* (* D (/ D l)) (* (* t_3 (/ h d)) 0.125))))
           (if (<= d 1.55e-107)
             (* (sqrt (/ h (pow l 3.0))) (* -0.125 (* D (* D t_3))))
             (if (<= d 1.38e+147)
               (* t_0 (- 1.0 t_2))
               (/ d (* (sqrt h) (sqrt l)))))))))))

C

M = abs(M);
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)));
	double t_2 = 0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D));
	double t_3 = M / (d / M);
	double tmp;
	if (d <= -6.5e+80) {
		tmp = sqrt(((d / h) / (l / d))) * (1.0 - (pow((D * (M / (d * 2.0))), 2.0) * (h * (0.5 / l))));
	} else if (d <= -2.8e-66) {
		tmp = t_0 * (1.0 - ((0.125 * (((M * D) * (M * D)) / (d / (h / d)))) / l));
	} else if (d <= -4.5e-113) {
		tmp = t_1 * (-1.0 + t_2);
	} else if (d <= -5e-310) {
		tmp = t_1 * (-1.0 + ((D * (D / l)) * ((t_3 * (h / d)) * 0.125)));
	} else if (d <= 1.55e-107) {
		tmp = sqrt((h / pow(l, 3.0))) * (-0.125 * (D * (D * t_3)));
	} else if (d <= 1.38e+147) {
		tmp = t_0 * (1.0 - t_2);
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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)) * sqrt((d / l))
    t_1 = d * sqrt((1.0d0 / (h * l)))
    t_2 = 0.125d0 * (d_1 / (((d * d) / ((h * m) / (l / m))) / d_1))
    t_3 = m / (d / m)
    if (d <= (-6.5d+80)) then
        tmp = sqrt(((d / h) / (l / d))) * (1.0d0 - (((d_1 * (m / (d * 2.0d0))) ** 2.0d0) * (h * (0.5d0 / l))))
    else if (d <= (-2.8d-66)) then
        tmp = t_0 * (1.0d0 - ((0.125d0 * (((m * d_1) * (m * d_1)) / (d / (h / d)))) / l))
    else if (d <= (-4.5d-113)) then
        tmp = t_1 * ((-1.0d0) + t_2)
    else if (d <= (-5d-310)) then
        tmp = t_1 * ((-1.0d0) + ((d_1 * (d_1 / l)) * ((t_3 * (h / d)) * 0.125d0)))
    else if (d <= 1.55d-107) then
        tmp = sqrt((h / (l ** 3.0d0))) * ((-0.125d0) * (d_1 * (d_1 * t_3)))
    else if (d <= 1.38d+147) then
        tmp = t_0 * (1.0d0 - t_2)
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
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)));
	double t_2 = 0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D));
	double t_3 = M / (d / M);
	double tmp;
	if (d <= -6.5e+80) {
		tmp = Math.sqrt(((d / h) / (l / d))) * (1.0 - (Math.pow((D * (M / (d * 2.0))), 2.0) * (h * (0.5 / l))));
	} else if (d <= -2.8e-66) {
		tmp = t_0 * (1.0 - ((0.125 * (((M * D) * (M * D)) / (d / (h / d)))) / l));
	} else if (d <= -4.5e-113) {
		tmp = t_1 * (-1.0 + t_2);
	} else if (d <= -5e-310) {
		tmp = t_1 * (-1.0 + ((D * (D / l)) * ((t_3 * (h / d)) * 0.125)));
	} else if (d <= 1.55e-107) {
		tmp = Math.sqrt((h / Math.pow(l, 3.0))) * (-0.125 * (D * (D * t_3)));
	} else if (d <= 1.38e+147) {
		tmp = t_0 * (1.0 - t_2);
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M = abs(M)
[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)))
	t_2 = 0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D))
	t_3 = M / (d / M)
	tmp = 0
	if d <= -6.5e+80:
		tmp = math.sqrt(((d / h) / (l / d))) * (1.0 - (math.pow((D * (M / (d * 2.0))), 2.0) * (h * (0.5 / l))))
	elif d <= -2.8e-66:
		tmp = t_0 * (1.0 - ((0.125 * (((M * D) * (M * D)) / (d / (h / d)))) / l))
	elif d <= -4.5e-113:
		tmp = t_1 * (-1.0 + t_2)
	elif d <= -5e-310:
		tmp = t_1 * (-1.0 + ((D * (D / l)) * ((t_3 * (h / d)) * 0.125)))
	elif d <= 1.55e-107:
		tmp = math.sqrt((h / math.pow(l, 3.0))) * (-0.125 * (D * (D * t_3)))
	elif d <= 1.38e+147:
		tmp = t_0 * (1.0 - t_2)
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M = abs(M)
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(d * sqrt(Float64(1.0 / Float64(h * l))))
	t_2 = Float64(0.125 * Float64(D / Float64(Float64(Float64(d * d) / Float64(Float64(h * M) / Float64(l / M))) / D)))
	t_3 = Float64(M / Float64(d / M))
	tmp = 0.0
	if (d <= -6.5e+80)
		tmp = Float64(sqrt(Float64(Float64(d / h) / Float64(l / d))) * Float64(1.0 - Float64((Float64(D * Float64(M / Float64(d * 2.0))) ^ 2.0) * Float64(h * Float64(0.5 / l)))));
	elseif (d <= -2.8e-66)
		tmp = Float64(t_0 * Float64(1.0 - Float64(Float64(0.125 * Float64(Float64(Float64(M * D) * Float64(M * D)) / Float64(d / Float64(h / d)))) / l)));
	elseif (d <= -4.5e-113)
		tmp = Float64(t_1 * Float64(-1.0 + t_2));
	elseif (d <= -5e-310)
		tmp = Float64(t_1 * Float64(-1.0 + Float64(Float64(D * Float64(D / l)) * Float64(Float64(t_3 * Float64(h / d)) * 0.125))));
	elseif (d <= 1.55e-107)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(-0.125 * Float64(D * Float64(D * t_3))));
	elseif (d <= 1.38e+147)
		tmp = Float64(t_0 * Float64(1.0 - t_2));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M = abs(M)
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)));
	t_2 = 0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D));
	t_3 = M / (d / M);
	tmp = 0.0;
	if (d <= -6.5e+80)
		tmp = sqrt(((d / h) / (l / d))) * (1.0 - (((D * (M / (d * 2.0))) ^ 2.0) * (h * (0.5 / l))));
	elseif (d <= -2.8e-66)
		tmp = t_0 * (1.0 - ((0.125 * (((M * D) * (M * D)) / (d / (h / d)))) / l));
	elseif (d <= -4.5e-113)
		tmp = t_1 * (-1.0 + t_2);
	elseif (d <= -5e-310)
		tmp = t_1 * (-1.0 + ((D * (D / l)) * ((t_3 * (h / d)) * 0.125)));
	elseif (d <= 1.55e-107)
		tmp = sqrt((h / (l ^ 3.0))) * (-0.125 * (D * (D * t_3)));
	elseif (d <= 1.38e+147)
		tmp = t_0 * (1.0 - t_2);
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
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[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.125 * N[(D / N[(N[(N[(d * d), $MachinePrecision] / N[(N[(h * M), $MachinePrecision] / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -6.5e+80], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] / N[(l / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[Power[N[(D * N[(M / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * N[(0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2.8e-66], N[(t$95$0 * N[(1.0 - N[(N[(0.125 * N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(d / N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4.5e-113], N[(t$95$1 * N[(-1.0 + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(t$95$1 * N[(-1.0 + N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$3 * N[(h / d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.55e-107], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(D * N[(D * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.38e+147], N[(t$95$0 * N[(1.0 - t$95$2), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if d < -6.4999999999999998e80

   1. Initial program 82.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 46.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 54.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 79.1%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. associate-*r/ 75.5%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{d}{h} \cdot d}{\ell}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. associate-/l* 79.1%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{d}{h}}{\frac{\ell}{d}}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-*r/ 79.1%

         \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - {\color{blue}{\left(\frac{M \cdot D}{d \cdot 2}\right)}}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      6. associate-*l/ 79.1%

         \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - {\color{blue}{\left(\frac{M}{d \cdot 2} \cdot D\right)}}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      7. *-commutative 79.1%

         \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - {\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)}}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      8. *-commutative 79.1%

         \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - {\left(D \cdot \frac{M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      9. associate-/l* 79.1%

         \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

      10. associate-/r/ 79.1%

          \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\left(\frac{0.5}{\ell} \cdot h\right)}\right) \]

   5. Simplified 79.1%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \left(\frac{0.5}{\ell} \cdot h\right)\right)} \]

   if -6.4999999999999998e80 < d < -2.8e-66

   1. Initial program 80.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 80.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 80.3%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 80.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 80.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 80.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 80.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 80.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 80.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 80.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 80.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 80.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 80.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 80.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 85.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 85.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 85.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 85.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 85.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 85.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. associate-/l/ 85.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. *-commutative 85.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 85.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

   6. Taylor expanded in M around 0 63.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}}{\ell}\right) \]
   7. Step-by-step derivation

      1. associate-*r* 68.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2}}}{\ell}\right) \]

      2. associate-/l* 64.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{\frac{{d}^{2}}{h}}}}{\ell}\right) \]

      3. unpow2 64.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{\frac{{d}^{2}}{h}}}{\ell}\right) \]

      4. unpow2 64.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{\frac{{d}^{2}}{h}}}{\ell}\right) \]

      5. unswap-sqr 80.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\color{blue}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}{\frac{{d}^{2}}{h}}}{\ell}\right) \]

      6. unpow2 80.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\frac{\color{blue}{d \cdot d}}{h}}}{\ell}\right) \]

      7. associate-/l* 80.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\color{blue}{\frac{d}{\frac{h}{d}}}}}{\ell}\right) \]

   8. Simplified 80.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{0.125 \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\frac{d}{\frac{h}{d}}}}}{\ell}\right) \]

   if -2.8e-66 < d < -4.5000000000000001e-113

   1. Initial program 78.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 11.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 22.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 67.1%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 67.1%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 67.1%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 67.1%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 67.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in d around -inf 78.2%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg 78.2%

         \[\leadsto \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      2. associate-/r* 78.2%

         \[\leadsto \left(-d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      3. distribute-rgt-neg-in 78.2%

         \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      4. associate-/l/ 78.2%

         \[\leadsto \left(d \cdot \left(-\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   8. Simplified 78.2%

      \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   9. Taylor expanded in M around 0 56.2%

      \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   10. Step-by-step derivation

       1. associate-/l* 45.2%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{{M}^{2} \cdot h}}}\right) \]

       2. *-commutative 45.2%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{\color{blue}{h \cdot {M}^{2}}}}\right) \]

       3. unpow2 45.2%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{D \cdot D}}{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}}\right) \]

       4. associate-/l* 55.7%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{D}{\frac{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}}{D}}}\right) \]

       5. *-commutative 55.7%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{\color{blue}{{d}^{2} \cdot \ell}}{h \cdot {M}^{2}}}{D}}\right) \]

       6. associate-/l* 66.8%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\color{blue}{\frac{{d}^{2}}{\frac{h \cdot {M}^{2}}{\ell}}}}{D}}\right) \]

       7. unpow2 66.8%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{\color{blue}{d \cdot d}}{\frac{h \cdot {M}^{2}}{\ell}}}{D}}\right) \]

       8. unpow2 66.8%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{\ell}}}{D}}\right) \]

       9. associate-*r* 66.8%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{\ell}}}{D}}\right) \]

       10. associate-/l* 66.8%

           \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\color{blue}{\frac{h \cdot M}{\frac{\ell}{M}}}}}{D}}\right) \]

       11. *-commutative 66.8%

           \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{M \cdot h}}{\frac{\ell}{M}}}}{D}}\right) \]

   11. Simplified 77.9%

       \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{M \cdot h}{\frac{\ell}{M}}}}{D}}}\right) \]

   if -4.5000000000000001e-113 < d < -4.999999999999985e-310

   1. Initial program 45.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 6.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 11.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 43.2%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 43.2%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 43.2%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 43.2%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 43.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in d around -inf 77.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg 77.4%

         \[\leadsto \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      2. associate-/r* 77.4%

         \[\leadsto \left(-d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      3. distribute-rgt-neg-in 77.4%

         \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      4. associate-/l/ 77.4%

         \[\leadsto \left(d \cdot \left(-\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   8. Simplified 77.4%

      \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   9. Taylor expanded in M around 0 51.8%

      \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 51.8%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

       2. times-frac 57.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.125\right) \]

       3. unpow2 57.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{\color{blue}{d \cdot d}}\right) \cdot 0.125\right) \]

       4. times-frac 60.2%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{d} \cdot \frac{h}{d}\right)}\right) \cdot 0.125\right) \]

       5. unpow2 60.2%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \left(\frac{\color{blue}{M \cdot M}}{d} \cdot \frac{h}{d}\right)\right) \cdot 0.125\right) \]

       6. associate-*l* 60.2%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

       7. unpow2 60.2%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \frac{\color{blue}{D \cdot D}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       8. associate-*l/ 63.2%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{\ell} \cdot D\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       9. *-commutative 63.2%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       10. associate-/l* 66.1%

           \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

   11. Simplified 66.1%

       \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

   if -4.999999999999985e-310 < d < 1.55000000000000011e-107

   1. Initial program 47.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 47.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 47.9%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 46.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 46.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 46.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 46.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 46.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 46.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 46.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. associate-/l/ 46.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. *-commutative 46.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 46.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

   6. Taylor expanded in d around 0 39.5%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 39.5%

         \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative 39.5%

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      3. associate-*r/ 37.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)}\right) \]

      4. unpow2 37.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{\color{blue}{M \cdot M}}{d}\right)\right) \]

      5. unpow2 37.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \frac{M \cdot M}{d}\right)\right) \]

      6. associate-*l* 41.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(D \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right)}\right) \]

      7. associate-/l* 48.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot \color{blue}{\frac{M}{\frac{d}{M}}}\right)\right)\right) \]

   8. Simplified 48.9%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot \frac{M}{\frac{d}{M}}\right)\right)\right)} \]

   if 1.55000000000000011e-107 < d < 1.37999999999999991e147

   1. Initial program 77.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 77.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 77.4%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 77.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 77.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 77.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 77.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 77.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 77.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 77.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in M around 0 70.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   5. Step-by-step derivation

      1. associate-*r/ 70.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right) \]

      2. *-commutative 70.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left({D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{\ell \cdot {d}^{2}}\right) \]

      3. associate-*r/ 70.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}}\right) \]

      4. *-commutative 70.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

      5. times-frac 75.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)} \cdot 0.125\right) \]

      6. unpow2 75.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right) \cdot 0.125\right) \]

      7. *-commutative 75.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right) \cdot 0.125\right) \]

      8. unpow2 75.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right) \cdot 0.125\right) \]

      9. unpow2 75.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{\color{blue}{d \cdot d}}\right) \cdot 0.125\right) \]

   6. Simplified 75.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right) \cdot 0.125}\right) \]

   7. Taylor expanded in D around 0 70.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}} \cdot 0.125\right) \]
   8. Step-by-step derivation

      1. associate-/l* 72.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2} \cdot \ell}{{M}^{2} \cdot h}}} \cdot 0.125\right) \]

      2. *-commutative 72.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{D}^{2}}{\frac{\color{blue}{\ell \cdot {d}^{2}}}{{M}^{2} \cdot h}} \cdot 0.125\right) \]

      3. *-commutative 72.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{\color{blue}{h \cdot {M}^{2}}}} \cdot 0.125\right) \]

      4. unpow2 72.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot D}}{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}} \cdot 0.125\right) \]

      5. associate-/l* 72.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{D}{\frac{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}}{D}}} \cdot 0.125\right) \]

      6. *-commutative 72.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{D}{\frac{\frac{\color{blue}{{d}^{2} \cdot \ell}}{h \cdot {M}^{2}}}{D}} \cdot 0.125\right) \]

      7. associate-/l* 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{D}{\frac{\color{blue}{\frac{{d}^{2}}{\frac{h \cdot {M}^{2}}{\ell}}}}{D}} \cdot 0.125\right) \]

      8. unpow2 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{D}{\frac{\frac{\color{blue}{d \cdot d}}{\frac{h \cdot {M}^{2}}{\ell}}}{D}} \cdot 0.125\right) \]

      9. unpow2 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{\ell}}}{D}} \cdot 0.125\right) \]

      10. associate-*r* 77.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{\ell}}}{D}} \cdot 0.125\right) \]

      11. associate-/l* 82.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{D}{\frac{\frac{d \cdot d}{\color{blue}{\frac{h \cdot M}{\frac{\ell}{M}}}}}{D}} \cdot 0.125\right) \]

      12. *-commutative 82.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{M \cdot h}}{\frac{\ell}{M}}}}{D}} \cdot 0.125\right) \]

   9. Simplified 82.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{D}{\frac{\frac{d \cdot d}{\frac{M \cdot h}{\frac{\ell}{M}}}}{D}}} \cdot 0.125\right) \]

   if 1.37999999999999991e147 < d

   1. Initial program 70.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 70.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 70.4%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 70.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 74.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 74.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 74.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 74.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 74.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 74.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. associate-/l/ 74.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. *-commutative 74.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 74.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

   6. Taylor expanded in d around inf 70.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   7. Step-by-step derivation

      1. *-commutative 70.0%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. *-commutative 70.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. associate-/r* 70.0%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   8. Simplified 70.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   9. Step-by-step derivation

      1. associate-/r* 70.0%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      2. associate-/l/ 70.0%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

      3. add-cbrt-cube 62.1%

         \[\leadsto d \cdot \color{blue}{\sqrt[3]{\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}} \]

      4. add-sqr-sqrt 62.0%

         \[\leadsto d \cdot \sqrt[3]{\color{blue}{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]

      5. pow1 62.0%

         \[\leadsto \color{blue}{{\left(d \cdot \sqrt[3]{\frac{\frac{1}{\ell}}{h} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}\right)}^{1}} \]

      6. add-sqr-sqrt 62.1%

         \[\leadsto {\left(d \cdot \sqrt[3]{\color{blue}{\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}\right)}^{1} \]

      7. add-cbrt-cube 70.0%

         \[\leadsto {\left(d \cdot \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}}}\right)}^{1} \]

      8. associate-/l/ 70.0%

         \[\leadsto {\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)}^{1} \]

      9. sqrt-div 69.9%

         \[\leadsto {\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)}^{1} \]

      10. metadata-eval 69.9%

          \[\leadsto {\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)}^{1} \]

      11. *-commutative 69.9%

          \[\leadsto {\left(d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}}\right)}^{1} \]

   10. Applied egg-rr 69.9%

       \[\leadsto \color{blue}{{\left(d \cdot \frac{1}{\sqrt{\ell \cdot h}}\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 69.9%

          \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{\ell \cdot h}}} \]

       2. associate-*r/ 70.0%

          \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{\ell \cdot h}}} \]

       3. *-rgt-identity 70.0%

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell \cdot h}} \]

   12. Simplified 70.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   13. Step-by-step derivation

       1. sqrt-prod 84.4%

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

   14. Applied egg-rr 84.4%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.5 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2} \cdot \left(h \cdot \frac{0.5}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -2.8 \cdot 10^{-66}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\frac{d}{\frac{h}{d}}}}{\ell}\right)\\ \mathbf{elif}\;d \leq -4.5 \cdot 10^{-113}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{-107}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot \frac{M}{\frac{d}{M}}\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.38 \cdot 10^{+147}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 12: 61.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \frac{M}{\frac{d}{M}}\\ \mathbf{if}\;d \leq -2.4 \cdot 10^{-63}:\\ \;\;\;\;\left(1 + \frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \left(h \cdot -0.5\right)}{\ell}\right) \cdot \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(t_0 \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{-94}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot t_0\right)\right)\right)\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+149}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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 (/ M (/ d M))))
   (if (<= d -2.4e-63)
     (*
      (+ 1.0 (/ (* (pow (* M (/ D (* d 2.0))) 2.0) (* h -0.5)) l))
      (sqrt (/ d (/ h (/ d l)))))
     (if (<= d -5e-310)
       (*
        (* d (sqrt (/ 1.0 (* h l))))
        (+ -1.0 (* (* D (/ D l)) (* (* t_0 (/ h d)) 0.125))))
       (if (<= d 2.6e-94)
         (* (sqrt (/ h (pow l 3.0))) (* -0.125 (* D (* D t_0))))
         (if (<= d 7.5e+149)
           (*
            (* (sqrt (/ d h)) (sqrt (/ d l)))
            (- 1.0 (* 0.125 (* (/ (* D D) l) (/ (* h (* M M)) (* d d))))))
           (/ d (* (sqrt h) (sqrt l)))))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = M / (d / M);
	double tmp;
	if (d <= -2.4e-63) {
		tmp = (1.0 + ((pow((M * (D / (d * 2.0))), 2.0) * (h * -0.5)) / l)) * sqrt((d / (h / (d / l))));
	} else if (d <= -5e-310) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_0 * (h / d)) * 0.125)));
	} else if (d <= 2.6e-94) {
		tmp = sqrt((h / pow(l, 3.0))) * (-0.125 * (D * (D * t_0)));
	} else if (d <= 7.5e+149) {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - (0.125 * (((D * D) / l) * ((h * (M * M)) / (d * d)))));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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 = m / (d / m)
    if (d <= (-2.4d-63)) then
        tmp = (1.0d0 + ((((m * (d_1 / (d * 2.0d0))) ** 2.0d0) * (h * (-0.5d0))) / l)) * sqrt((d / (h / (d / l))))
    else if (d <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) + ((d_1 * (d_1 / l)) * ((t_0 * (h / d)) * 0.125d0)))
    else if (d <= 2.6d-94) then
        tmp = sqrt((h / (l ** 3.0d0))) * ((-0.125d0) * (d_1 * (d_1 * t_0)))
    else if (d <= 7.5d+149) then
        tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0d0 - (0.125d0 * (((d_1 * d_1) / l) * ((h * (m * m)) / (d * d)))))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = M / (d / M);
	double tmp;
	if (d <= -2.4e-63) {
		tmp = (1.0 + ((Math.pow((M * (D / (d * 2.0))), 2.0) * (h * -0.5)) / l)) * Math.sqrt((d / (h / (d / l))));
	} else if (d <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_0 * (h / d)) * 0.125)));
	} else if (d <= 2.6e-94) {
		tmp = Math.sqrt((h / Math.pow(l, 3.0))) * (-0.125 * (D * (D * t_0)));
	} else if (d <= 7.5e+149) {
		tmp = (Math.sqrt((d / h)) * Math.sqrt((d / l))) * (1.0 - (0.125 * (((D * D) / l) * ((h * (M * M)) / (d * d)))));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = M / (d / M)
	tmp = 0
	if d <= -2.4e-63:
		tmp = (1.0 + ((math.pow((M * (D / (d * 2.0))), 2.0) * (h * -0.5)) / l)) * math.sqrt((d / (h / (d / l))))
	elif d <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_0 * (h / d)) * 0.125)))
	elif d <= 2.6e-94:
		tmp = math.sqrt((h / math.pow(l, 3.0))) * (-0.125 * (D * (D * t_0)))
	elif d <= 7.5e+149:
		tmp = (math.sqrt((d / h)) * math.sqrt((d / l))) * (1.0 - (0.125 * (((D * D) / l) * ((h * (M * M)) / (d * d)))))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(M / Float64(d / M))
	tmp = 0.0
	if (d <= -2.4e-63)
		tmp = Float64(Float64(1.0 + Float64(Float64((Float64(M * Float64(D / Float64(d * 2.0))) ^ 2.0) * Float64(h * -0.5)) / l)) * sqrt(Float64(d / Float64(h / Float64(d / l)))));
	elseif (d <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 + Float64(Float64(D * Float64(D / l)) * Float64(Float64(t_0 * Float64(h / d)) * 0.125))));
	elseif (d <= 2.6e-94)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(-0.125 * Float64(D * Float64(D * t_0))));
	elseif (d <= 7.5e+149)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(0.125 * Float64(Float64(Float64(D * D) / l) * Float64(Float64(h * Float64(M * M)) / Float64(d * d))))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = M / (d / M);
	tmp = 0.0;
	if (d <= -2.4e-63)
		tmp = (1.0 + ((((M * (D / (d * 2.0))) ^ 2.0) * (h * -0.5)) / l)) * sqrt((d / (h / (d / l))));
	elseif (d <= -5e-310)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_0 * (h / d)) * 0.125)));
	elseif (d <= 2.6e-94)
		tmp = sqrt((h / (l ^ 3.0))) * (-0.125 * (D * (D * t_0)));
	elseif (d <= 7.5e+149)
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - (0.125 * (((D * D) / l) * ((h * (M * M)) / (d * d)))));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
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[(M / N[(d / M), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.4e-63], N[(N[(1.0 + N[(N[(N[Power[N[(M * N[(D / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * -0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / N[(h / N[(d / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * N[(h / d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.6e-94], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(D * N[(D * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.5e+149], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.125 * N[(N[(N[(D * D), $MachinePrecision] / l), $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -2.4000000000000001e-63

   1. Initial program 80.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 80.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 80.9%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 80.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 80.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 80.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 80.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 80.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 80.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 80.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 80.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 80.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 80.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 80.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 82.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 82.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 82.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 82.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 82.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 82.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. associate-/l/ 82.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. *-commutative 82.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 82.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. *-commutative 82.2%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      2. sqrt-prod 71.4%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      3. associate-/l* 68.7%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \]

      4. associate-*r/ 68.7%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{d \cdot 2}\right)}}^{2} \cdot 0.5}{\frac{\ell}{h}}\right) \]

      5. *-commutative 68.7%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}\right) \]

      6. associate-*r/ 68.7%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot 0.5}{\frac{\ell}{h}}\right) \]

      7. associate-*r/ 68.7%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}}\right) \]

      8. pow1 68.7%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)\right)}^{1}} \]

   7. Applied egg-rr 68.5%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{\frac{d}{\ell} \cdot d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 68.5%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{\ell} \cdot d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

      2. *-commutative 68.5%

         \[\leadsto \sqrt{\frac{\color{blue}{d \cdot \frac{d}{\ell}}}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      3. associate-/l* 67.3%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\frac{h}{\frac{d}{\ell}}}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      4. cancel-sign-sub-inv 67.3%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right) \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

      5. associate-*r/ 67.3%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right) \cdot \color{blue}{\frac{0.5 \cdot h}{\ell}}\right) \]

      6. associate-*r/ 70.3%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 + \color{blue}{\frac{\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right) \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]

      7. distribute-lft-neg-in 70.3%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 + \frac{\color{blue}{-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \left(0.5 \cdot h\right)}}{\ell}\right) \]

      8. distribute-rgt-neg-in 70.3%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 + \frac{\color{blue}{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \left(-0.5 \cdot h\right)}}{\ell}\right) \]

      9. *-commutative 70.3%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 + \frac{{\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(-0.5 \cdot h\right)}{\ell}\right) \]

      10. *-commutative 70.3%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 + \frac{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \left(-\color{blue}{h \cdot 0.5}\right)}{\ell}\right) \]

      11. distribute-rgt-neg-in 70.3%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 + \frac{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\left(h \cdot \left(-0.5\right)\right)}}{\ell}\right) \]

      12. metadata-eval 70.3%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 + \frac{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \left(h \cdot \color{blue}{-0.5}\right)}{\ell}\right) \]

   9. Simplified 70.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 + \frac{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \left(h \cdot -0.5\right)}{\ell}\right)} \]

   if -2.4000000000000001e-63 < d < -4.999999999999985e-310

   1. Initial program 53.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 7.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 13.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 49.3%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 49.3%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 49.3%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 49.3%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 49.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in d around -inf 78.1%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg 78.1%

         \[\leadsto \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      2. associate-/r* 78.1%

         \[\leadsto \left(-d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      3. distribute-rgt-neg-in 78.1%

         \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      4. associate-/l/ 78.1%

         \[\leadsto \left(d \cdot \left(-\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   8. Simplified 78.1%

      \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   9. Taylor expanded in M around 0 53.7%

      \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 53.7%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

       2. times-frac 60.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.125\right) \]

       3. unpow2 60.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{\color{blue}{d \cdot d}}\right) \cdot 0.125\right) \]

       4. times-frac 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{d} \cdot \frac{h}{d}\right)}\right) \cdot 0.125\right) \]

       5. unpow2 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \left(\frac{\color{blue}{M \cdot M}}{d} \cdot \frac{h}{d}\right)\right) \cdot 0.125\right) \]

       6. associate-*l* 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

       7. unpow2 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \frac{\color{blue}{D \cdot D}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       8. associate-*l/ 66.9%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{\ell} \cdot D\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       9. *-commutative 66.9%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       10. associate-/l* 69.1%

           \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

   11. Simplified 69.1%

       \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

   if -4.999999999999985e-310 < d < 2.59999999999999994e-94

   1. Initial program 48.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 48.1%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 48.1%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 48.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 48.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 48.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 48.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 46.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 46.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 46.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 46.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 48.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 48.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 48.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 51.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 51.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 51.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 49.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 49.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 49.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. associate-/l/ 49.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. *-commutative 49.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 49.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

   6. Taylor expanded in d around 0 40.6%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. 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. associate-*r/ 39.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)}\right) \]

      4. unpow2 39.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{\color{blue}{M \cdot M}}{d}\right)\right) \]

      5. unpow2 39.1%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \frac{M \cdot M}{d}\right)\right) \]

      6. associate-*l* 42.7%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(D \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right)}\right) \]

      7. associate-/l* 50.7%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot \color{blue}{\frac{M}{\frac{d}{M}}}\right)\right)\right) \]

   8. Simplified 50.7%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot \frac{M}{\frac{d}{M}}\right)\right)\right)} \]

   if 2.59999999999999994e-94 < d < 7.50000000000000031e149

   1. Initial program 80.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 80.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 80.3%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 80.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 80.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 80.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 80.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 80.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 80.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 80.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in M around 0 73.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   5. Step-by-step derivation

      1. associate-*r/ 73.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right) \]

      2. *-commutative 73.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left({D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{\ell \cdot {d}^{2}}\right) \]

      3. associate-*r/ 73.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}}\right) \]

      4. *-commutative 73.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

      5. times-frac 76.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)} \cdot 0.125\right) \]

      6. unpow2 76.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right) \cdot 0.125\right) \]

      7. *-commutative 76.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right) \cdot 0.125\right) \]

      8. unpow2 76.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right) \cdot 0.125\right) \]

      9. unpow2 76.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{\color{blue}{d \cdot d}}\right) \cdot 0.125\right) \]

   6. Simplified 76.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right) \cdot 0.125}\right) \]

   if 7.50000000000000031e149 < d

   1. Initial program 70.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 70.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 70.4%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 70.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 74.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 74.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 74.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 74.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 74.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 74.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. associate-/l/ 74.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. *-commutative 74.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 74.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

   6. Taylor expanded in d around inf 70.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   7. Step-by-step derivation

      1. *-commutative 70.0%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. *-commutative 70.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. associate-/r* 70.0%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   8. Simplified 70.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   9. Step-by-step derivation

      1. associate-/r* 70.0%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      2. associate-/l/ 70.0%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

      3. add-cbrt-cube 62.1%

         \[\leadsto d \cdot \color{blue}{\sqrt[3]{\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}} \]

      4. add-sqr-sqrt 62.0%

         \[\leadsto d \cdot \sqrt[3]{\color{blue}{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]

      5. pow1 62.0%

         \[\leadsto \color{blue}{{\left(d \cdot \sqrt[3]{\frac{\frac{1}{\ell}}{h} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}\right)}^{1}} \]

      6. add-sqr-sqrt 62.1%

         \[\leadsto {\left(d \cdot \sqrt[3]{\color{blue}{\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}\right)}^{1} \]

      7. add-cbrt-cube 70.0%

         \[\leadsto {\left(d \cdot \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}}}\right)}^{1} \]

      8. associate-/l/ 70.0%

         \[\leadsto {\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)}^{1} \]

      9. sqrt-div 69.9%

         \[\leadsto {\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)}^{1} \]

      10. metadata-eval 69.9%

          \[\leadsto {\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)}^{1} \]

      11. *-commutative 69.9%

          \[\leadsto {\left(d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}}\right)}^{1} \]

   10. Applied egg-rr 69.9%

       \[\leadsto \color{blue}{{\left(d \cdot \frac{1}{\sqrt{\ell \cdot h}}\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 69.9%

          \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{\ell \cdot h}}} \]

       2. associate-*r/ 70.0%

          \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{\ell \cdot h}}} \]

       3. *-rgt-identity 70.0%

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell \cdot h}} \]

   12. Simplified 70.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   13. Step-by-step derivation

       1. sqrt-prod 84.4%

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

   14. Applied egg-rr 84.4%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.4 \cdot 10^{-63}:\\ \;\;\;\;\left(1 + \frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \left(h \cdot -0.5\right)}{\ell}\right) \cdot \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{-94}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot \frac{M}{\frac{d}{M}}\right)\right)\right)\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+149}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 13: 64.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right)\\ t_1 := \frac{M}{\frac{d}{M}}\\ \mathbf{if}\;d \leq -1.55 \cdot 10^{-109}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(t_1 \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{-107}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot t_1\right)\right)\right)\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{+149}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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)))
          (- 1.0 (* 0.125 (/ D (/ (/ (* d d) (/ (* h M) (/ l M))) D))))))
        (t_1 (/ M (/ d M))))
   (if (<= d -1.55e-109)
     t_0
     (if (<= d -5e-310)
       (*
        (* d (sqrt (/ 1.0 (* h l))))
        (+ -1.0 (* (* D (/ D l)) (* (* t_1 (/ h d)) 0.125))))
       (if (<= d 1.55e-107)
         (* (sqrt (/ h (pow l 3.0))) (* -0.125 (* D (* D t_1))))
         (if (<= d 6.5e+149) t_0 (/ d (* (sqrt h) (sqrt l)))))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D))));
	double t_1 = M / (d / M);
	double tmp;
	if (d <= -1.55e-109) {
		tmp = t_0;
	} else if (d <= -5e-310) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_1 * (h / d)) * 0.125)));
	} else if (d <= 1.55e-107) {
		tmp = sqrt((h / pow(l, 3.0))) * (-0.125 * (D * (D * t_1)));
	} else if (d <= 6.5e+149) {
		tmp = t_0;
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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))) * (1.0d0 - (0.125d0 * (d_1 / (((d * d) / ((h * m) / (l / m))) / d_1))))
    t_1 = m / (d / m)
    if (d <= (-1.55d-109)) then
        tmp = t_0
    else if (d <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) + ((d_1 * (d_1 / l)) * ((t_1 * (h / d)) * 0.125d0)))
    else if (d <= 1.55d-107) then
        tmp = sqrt((h / (l ** 3.0d0))) * ((-0.125d0) * (d_1 * (d_1 * t_1)))
    else if (d <= 6.5d+149) then
        tmp = t_0
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
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))) * (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D))));
	double t_1 = M / (d / M);
	double tmp;
	if (d <= -1.55e-109) {
		tmp = t_0;
	} else if (d <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_1 * (h / d)) * 0.125)));
	} else if (d <= 1.55e-107) {
		tmp = Math.sqrt((h / Math.pow(l, 3.0))) * (-0.125 * (D * (D * t_1)));
	} else if (d <= 6.5e+149) {
		tmp = t_0;
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = (math.sqrt((d / h)) * math.sqrt((d / l))) * (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D))))
	t_1 = M / (d / M)
	tmp = 0
	if d <= -1.55e-109:
		tmp = t_0
	elif d <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_1 * (h / d)) * 0.125)))
	elif d <= 1.55e-107:
		tmp = math.sqrt((h / math.pow(l, 3.0))) * (-0.125 * (D * (D * t_1)))
	elif d <= 6.5e+149:
		tmp = t_0
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(0.125 * Float64(D / Float64(Float64(Float64(d * d) / Float64(Float64(h * M) / Float64(l / M))) / D)))))
	t_1 = Float64(M / Float64(d / M))
	tmp = 0.0
	if (d <= -1.55e-109)
		tmp = t_0;
	elseif (d <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 + Float64(Float64(D * Float64(D / l)) * Float64(Float64(t_1 * Float64(h / d)) * 0.125))));
	elseif (d <= 1.55e-107)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(-0.125 * Float64(D * Float64(D * t_1))));
	elseif (d <= 6.5e+149)
		tmp = t_0;
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D))));
	t_1 = M / (d / M);
	tmp = 0.0;
	if (d <= -1.55e-109)
		tmp = t_0;
	elseif (d <= -5e-310)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_1 * (h / d)) * 0.125)));
	elseif (d <= 1.55e-107)
		tmp = sqrt((h / (l ^ 3.0))) * (-0.125 * (D * (D * t_1)));
	elseif (d <= 6.5e+149)
		tmp = t_0;
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
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[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.125 * N[(D / N[(N[(N[(d * d), $MachinePrecision] / N[(N[(h * M), $MachinePrecision] / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.55e-109], t$95$0, If[LessEqual[d, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * N[(h / d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.55e-107], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(D * N[(D * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6.5e+149], t$95$0, N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.55e-109 or 1.55000000000000011e-107 < d < 6.50000000000000015e149

   1. Initial program 79.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 79.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 79.3%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 79.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in M around 0 66.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   5. Step-by-step derivation

      1. associate-*r/ 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right) \]

      2. *-commutative 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.125 \cdot \left({D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{\ell \cdot {d}^{2}}\right) \]

      3. associate-*r/ 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}}\right) \]

      4. *-commutative 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

      5. times-frac 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)} \cdot 0.125\right) \]

      6. unpow2 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right) \cdot 0.125\right) \]

      7. *-commutative 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right) \cdot 0.125\right) \]

      8. unpow2 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right) \cdot 0.125\right) \]

      9. unpow2 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{\color{blue}{d \cdot d}}\right) \cdot 0.125\right) \]

   6. Simplified 68.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right) \cdot 0.125}\right) \]

   7. Taylor expanded in D around 0 66.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}} \cdot 0.125\right) \]
   8. Step-by-step derivation

      1. associate-/l* 66.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2} \cdot \ell}{{M}^{2} \cdot h}}} \cdot 0.125\right) \]

      2. *-commutative 66.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{D}^{2}}{\frac{\color{blue}{\ell \cdot {d}^{2}}}{{M}^{2} \cdot h}} \cdot 0.125\right) \]

      3. *-commutative 66.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{\color{blue}{h \cdot {M}^{2}}}} \cdot 0.125\right) \]

      4. unpow2 66.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot D}}{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}} \cdot 0.125\right) \]

      5. associate-/l* 67.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{D}{\frac{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}}{D}}} \cdot 0.125\right) \]

      6. *-commutative 67.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{D}{\frac{\frac{\color{blue}{{d}^{2} \cdot \ell}}{h \cdot {M}^{2}}}{D}} \cdot 0.125\right) \]

      7. associate-/l* 70.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{D}{\frac{\color{blue}{\frac{{d}^{2}}{\frac{h \cdot {M}^{2}}{\ell}}}}{D}} \cdot 0.125\right) \]

      8. unpow2 70.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{D}{\frac{\frac{\color{blue}{d \cdot d}}{\frac{h \cdot {M}^{2}}{\ell}}}{D}} \cdot 0.125\right) \]

      9. unpow2 70.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{\ell}}}{D}} \cdot 0.125\right) \]

      10. associate-*r* 72.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{\ell}}}{D}} \cdot 0.125\right) \]

      11. associate-/l* 76.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{D}{\frac{\frac{d \cdot d}{\color{blue}{\frac{h \cdot M}{\frac{\ell}{M}}}}}{D}} \cdot 0.125\right) \]

      12. *-commutative 76.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{M \cdot h}}{\frac{\ell}{M}}}}{D}} \cdot 0.125\right) \]

   9. Simplified 76.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{D}{\frac{\frac{d \cdot d}{\frac{M \cdot h}{\frac{\ell}{M}}}}{D}}} \cdot 0.125\right) \]

   if -1.55e-109 < d < -4.999999999999985e-310

   1. Initial program 45.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 6.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 11.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 43.2%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 43.2%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 43.2%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 43.2%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 43.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in d around -inf 77.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg 77.4%

         \[\leadsto \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      2. associate-/r* 77.4%

         \[\leadsto \left(-d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      3. distribute-rgt-neg-in 77.4%

         \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      4. associate-/l/ 77.4%

         \[\leadsto \left(d \cdot \left(-\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   8. Simplified 77.4%

      \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   9. Taylor expanded in M around 0 51.8%

      \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 51.8%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

       2. times-frac 57.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.125\right) \]

       3. unpow2 57.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{\color{blue}{d \cdot d}}\right) \cdot 0.125\right) \]

       4. times-frac 60.2%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{d} \cdot \frac{h}{d}\right)}\right) \cdot 0.125\right) \]

       5. unpow2 60.2%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \left(\frac{\color{blue}{M \cdot M}}{d} \cdot \frac{h}{d}\right)\right) \cdot 0.125\right) \]

       6. associate-*l* 60.2%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

       7. unpow2 60.2%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \frac{\color{blue}{D \cdot D}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       8. associate-*l/ 63.2%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{\ell} \cdot D\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       9. *-commutative 63.2%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       10. associate-/l* 66.1%

           \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

   11. Simplified 66.1%

       \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

   if -4.999999999999985e-310 < d < 1.55000000000000011e-107

   1. Initial program 47.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 47.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 47.9%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 46.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 46.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 46.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 46.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 46.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 46.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 46.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. associate-/l/ 46.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. *-commutative 46.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 46.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

   6. Taylor expanded in d around 0 39.5%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 39.5%

         \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative 39.5%

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      3. associate-*r/ 37.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)}\right) \]

      4. unpow2 37.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{\color{blue}{M \cdot M}}{d}\right)\right) \]

      5. unpow2 37.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \frac{M \cdot M}{d}\right)\right) \]

      6. associate-*l* 41.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(D \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right)}\right) \]

      7. associate-/l* 48.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot \color{blue}{\frac{M}{\frac{d}{M}}}\right)\right)\right) \]

   8. Simplified 48.9%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot \frac{M}{\frac{d}{M}}\right)\right)\right)} \]

   if 6.50000000000000015e149 < d

   1. Initial program 70.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 70.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 70.4%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 70.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 70.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 74.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 74.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 74.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 74.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 74.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 74.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. associate-/l/ 74.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. *-commutative 74.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 74.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

   6. Taylor expanded in d around inf 70.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   7. Step-by-step derivation

      1. *-commutative 70.0%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. *-commutative 70.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. associate-/r* 70.0%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   8. Simplified 70.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   9. Step-by-step derivation

      1. associate-/r* 70.0%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      2. associate-/l/ 70.0%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

      3. add-cbrt-cube 62.1%

         \[\leadsto d \cdot \color{blue}{\sqrt[3]{\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}} \]

      4. add-sqr-sqrt 62.0%

         \[\leadsto d \cdot \sqrt[3]{\color{blue}{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]

      5. pow1 62.0%

         \[\leadsto \color{blue}{{\left(d \cdot \sqrt[3]{\frac{\frac{1}{\ell}}{h} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}\right)}^{1}} \]

      6. add-sqr-sqrt 62.1%

         \[\leadsto {\left(d \cdot \sqrt[3]{\color{blue}{\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}\right)}^{1} \]

      7. add-cbrt-cube 70.0%

         \[\leadsto {\left(d \cdot \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}}}\right)}^{1} \]

      8. associate-/l/ 70.0%

         \[\leadsto {\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)}^{1} \]

      9. sqrt-div 69.9%

         \[\leadsto {\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)}^{1} \]

      10. metadata-eval 69.9%

          \[\leadsto {\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)}^{1} \]

      11. *-commutative 69.9%

          \[\leadsto {\left(d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}}\right)}^{1} \]

   10. Applied egg-rr 69.9%

       \[\leadsto \color{blue}{{\left(d \cdot \frac{1}{\sqrt{\ell \cdot h}}\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 69.9%

          \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{\ell \cdot h}}} \]

       2. associate-*r/ 70.0%

          \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{\ell \cdot h}}} \]

       3. *-rgt-identity 70.0%

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell \cdot h}} \]

   12. Simplified 70.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   13. Step-by-step derivation

       1. sqrt-prod 84.4%

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

   14. Applied egg-rr 84.4%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.55 \cdot 10^{-109}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{-107}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot \frac{M}{\frac{d}{M}}\right)\right)\right)\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{+149}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 14: 62.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \left(1 + \frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \left(h \cdot -0.5\right)}{\ell}\right) \cdot \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ t_1 := \frac{M}{\frac{d}{M}}\\ \mathbf{if}\;d \leq -1.8 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(t_1 \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{-107}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot t_1\right)\right)\right)\\ \mathbf{elif}\;d \leq 8.5 \cdot 10^{+106}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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
         (*
          (+ 1.0 (/ (* (pow (* M (/ D (* d 2.0))) 2.0) (* h -0.5)) l))
          (sqrt (/ d (/ h (/ d l))))))
        (t_1 (/ M (/ d M))))
   (if (<= d -1.8e-64)
     t_0
     (if (<= d -5e-310)
       (*
        (* d (sqrt (/ 1.0 (* h l))))
        (+ -1.0 (* (* D (/ D l)) (* (* t_1 (/ h d)) 0.125))))
       (if (<= d 1.55e-107)
         (* (sqrt (/ h (pow l 3.0))) (* -0.125 (* D (* D t_1))))
         (if (<= d 8.5e+106) t_0 (* d (/ (pow h -0.5) (sqrt l)))))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 + ((pow((M * (D / (d * 2.0))), 2.0) * (h * -0.5)) / l)) * sqrt((d / (h / (d / l))));
	double t_1 = M / (d / M);
	double tmp;
	if (d <= -1.8e-64) {
		tmp = t_0;
	} else if (d <= -5e-310) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_1 * (h / d)) * 0.125)));
	} else if (d <= 1.55e-107) {
		tmp = sqrt((h / pow(l, 3.0))) * (-0.125 * (D * (D * t_1)));
	} else if (d <= 8.5e+106) {
		tmp = t_0;
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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 = (1.0d0 + ((((m * (d_1 / (d * 2.0d0))) ** 2.0d0) * (h * (-0.5d0))) / l)) * sqrt((d / (h / (d / l))))
    t_1 = m / (d / m)
    if (d <= (-1.8d-64)) then
        tmp = t_0
    else if (d <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) + ((d_1 * (d_1 / l)) * ((t_1 * (h / d)) * 0.125d0)))
    else if (d <= 1.55d-107) then
        tmp = sqrt((h / (l ** 3.0d0))) * ((-0.125d0) * (d_1 * (d_1 * t_1)))
    else if (d <= 8.5d+106) then
        tmp = t_0
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 + ((Math.pow((M * (D / (d * 2.0))), 2.0) * (h * -0.5)) / l)) * Math.sqrt((d / (h / (d / l))));
	double t_1 = M / (d / M);
	double tmp;
	if (d <= -1.8e-64) {
		tmp = t_0;
	} else if (d <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_1 * (h / d)) * 0.125)));
	} else if (d <= 1.55e-107) {
		tmp = Math.sqrt((h / Math.pow(l, 3.0))) * (-0.125 * (D * (D * t_1)));
	} else if (d <= 8.5e+106) {
		tmp = t_0;
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = (1.0 + ((math.pow((M * (D / (d * 2.0))), 2.0) * (h * -0.5)) / l)) * math.sqrt((d / (h / (d / l))))
	t_1 = M / (d / M)
	tmp = 0
	if d <= -1.8e-64:
		tmp = t_0
	elif d <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_1 * (h / d)) * 0.125)))
	elif d <= 1.55e-107:
		tmp = math.sqrt((h / math.pow(l, 3.0))) * (-0.125 * (D * (D * t_1)))
	elif d <= 8.5e+106:
		tmp = t_0
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(1.0 + Float64(Float64((Float64(M * Float64(D / Float64(d * 2.0))) ^ 2.0) * Float64(h * -0.5)) / l)) * sqrt(Float64(d / Float64(h / Float64(d / l)))))
	t_1 = Float64(M / Float64(d / M))
	tmp = 0.0
	if (d <= -1.8e-64)
		tmp = t_0;
	elseif (d <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 + Float64(Float64(D * Float64(D / l)) * Float64(Float64(t_1 * Float64(h / d)) * 0.125))));
	elseif (d <= 1.55e-107)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(-0.125 * Float64(D * Float64(D * t_1))));
	elseif (d <= 8.5e+106)
		tmp = t_0;
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (1.0 + ((((M * (D / (d * 2.0))) ^ 2.0) * (h * -0.5)) / l)) * sqrt((d / (h / (d / l))));
	t_1 = M / (d / M);
	tmp = 0.0;
	if (d <= -1.8e-64)
		tmp = t_0;
	elseif (d <= -5e-310)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_1 * (h / d)) * 0.125)));
	elseif (d <= 1.55e-107)
		tmp = sqrt((h / (l ^ 3.0))) * (-0.125 * (D * (D * t_1)));
	elseif (d <= 8.5e+106)
		tmp = t_0;
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
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[(1.0 + N[(N[(N[Power[N[(M * N[(D / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * -0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / N[(h / N[(d / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.8e-64], t$95$0, If[LessEqual[d, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * N[(h / d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.55e-107], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(D * N[(D * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 8.5e+106], t$95$0, 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 < -1.7999999999999999e-64 or 1.55000000000000011e-107 < d < 8.4999999999999992e106

   1. Initial program 79.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 79.1%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 79.1%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 79.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 79.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 79.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 79.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 79.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 79.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 79.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 79.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 79.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 79.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 79.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 82.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 82.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 82.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 82.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 82.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 82.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. associate-/l/ 82.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. *-commutative 82.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 82.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   6. Step-by-step derivation

      1. *-commutative 82.6%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      2. sqrt-prod 71.9%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      3. associate-/l* 66.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \]

      4. associate-*r/ 66.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{d \cdot 2}\right)}}^{2} \cdot 0.5}{\frac{\ell}{h}}\right) \]

      5. *-commutative 66.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}\right) \]

      6. associate-*r/ 66.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot 0.5}{\frac{\ell}{h}}\right) \]

      7. associate-*r/ 65.8%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}}\right) \]

      8. pow1 65.8%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)\right)}^{1}} \]

   7. Applied egg-rr 64.8%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{\frac{d}{\ell} \cdot d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 64.8%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{\ell} \cdot d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

      2. *-commutative 64.8%

         \[\leadsto \sqrt{\frac{\color{blue}{d \cdot \frac{d}{\ell}}}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      3. associate-/l* 65.8%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\frac{h}{\frac{d}{\ell}}}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      4. cancel-sign-sub-inv 65.8%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right) \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

      5. associate-*r/ 65.8%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right) \cdot \color{blue}{\frac{0.5 \cdot h}{\ell}}\right) \]

      6. associate-*r/ 72.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 + \color{blue}{\frac{\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right) \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]

      7. distribute-lft-neg-in 72.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 + \frac{\color{blue}{-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \left(0.5 \cdot h\right)}}{\ell}\right) \]

      8. distribute-rgt-neg-in 72.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 + \frac{\color{blue}{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \left(-0.5 \cdot h\right)}}{\ell}\right) \]

      9. *-commutative 72.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 + \frac{{\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(-0.5 \cdot h\right)}{\ell}\right) \]

      10. *-commutative 72.0%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 + \frac{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \left(-\color{blue}{h \cdot 0.5}\right)}{\ell}\right) \]

      11. distribute-rgt-neg-in 72.0%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 + \frac{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\left(h \cdot \left(-0.5\right)\right)}}{\ell}\right) \]

      12. metadata-eval 72.0%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 + \frac{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \left(h \cdot \color{blue}{-0.5}\right)}{\ell}\right) \]

   9. Simplified 72.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 + \frac{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \left(h \cdot -0.5\right)}{\ell}\right)} \]

   if -1.7999999999999999e-64 < d < -4.999999999999985e-310

   1. Initial program 53.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 7.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 13.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 49.3%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 49.3%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 49.3%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 49.3%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 49.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in d around -inf 78.1%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg 78.1%

         \[\leadsto \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      2. associate-/r* 78.1%

         \[\leadsto \left(-d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      3. distribute-rgt-neg-in 78.1%

         \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      4. associate-/l/ 78.1%

         \[\leadsto \left(d \cdot \left(-\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   8. Simplified 78.1%

      \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   9. Taylor expanded in M around 0 53.7%

      \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 53.7%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

       2. times-frac 60.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.125\right) \]

       3. unpow2 60.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{\color{blue}{d \cdot d}}\right) \cdot 0.125\right) \]

       4. times-frac 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{d} \cdot \frac{h}{d}\right)}\right) \cdot 0.125\right) \]

       5. unpow2 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \left(\frac{\color{blue}{M \cdot M}}{d} \cdot \frac{h}{d}\right)\right) \cdot 0.125\right) \]

       6. associate-*l* 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

       7. unpow2 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \frac{\color{blue}{D \cdot D}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       8. associate-*l/ 66.9%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{\ell} \cdot D\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       9. *-commutative 66.9%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       10. associate-/l* 69.1%

           \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

   11. Simplified 69.1%

       \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

   if -4.999999999999985e-310 < d < 1.55000000000000011e-107

   1. Initial program 47.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 47.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 47.9%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 46.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 46.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 46.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 46.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 46.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 46.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 46.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. associate-/l/ 46.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. *-commutative 46.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 46.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

   6. Taylor expanded in d around 0 39.5%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 39.5%

         \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative 39.5%

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      3. associate-*r/ 37.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)}\right) \]

      4. unpow2 37.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{\color{blue}{M \cdot M}}{d}\right)\right) \]

      5. unpow2 37.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \frac{M \cdot M}{d}\right)\right) \]

      6. associate-*l* 41.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(D \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right)}\right) \]

      7. associate-/l* 48.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot \color{blue}{\frac{M}{\frac{d}{M}}}\right)\right)\right) \]

   8. Simplified 48.9%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot \frac{M}{\frac{d}{M}}\right)\right)\right)} \]

   if 8.4999999999999992e106 < d

   1. Initial program 73.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 73.2%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 73.2%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 73.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 75.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. associate-/l/ 75.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. *-commutative 75.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 75.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

   6. Taylor expanded in d around inf 66.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   7. Step-by-step derivation

      1. *-commutative 66.4%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. *-commutative 66.4%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. associate-/r* 66.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   8. Simplified 66.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   9. Step-by-step derivation

      1. sqrt-div 78.3%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

   10. Applied egg-rr 78.3%

       \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. div-inv 78.3%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{\frac{1}{h}} \cdot \frac{1}{\sqrt{\ell}}\right)} \]

       2. inv-pow 78.3%

          \[\leadsto d \cdot \left(\sqrt{\color{blue}{{h}^{-1}}} \cdot \frac{1}{\sqrt{\ell}}\right) \]

   12. Applied egg-rr 78.3%

       \[\leadsto d \cdot \color{blue}{\left(\sqrt{{h}^{-1}} \cdot \frac{1}{\sqrt{\ell}}\right)} \]
   13. Step-by-step derivation

       1. associate-*r/ 78.3%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}} \cdot 1}{\sqrt{\ell}}} \]

       2. associate-/l* 78.3%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}}}{\frac{\sqrt{\ell}}{1}}} \]

       3. sqr-pow 78.4%

          \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{\left(\frac{-1}{2}\right)} \cdot {h}^{\left(\frac{-1}{2}\right)}}}}{\frac{\sqrt{\ell}}{1}} \]

       4. rem-sqrt-square 78.4%

          \[\leadsto d \cdot \frac{\color{blue}{\left|{h}^{\left(\frac{-1}{2}\right)}\right|}}{\frac{\sqrt{\ell}}{1}} \]

       5. metadata-eval 78.4%

          \[\leadsto d \cdot \frac{\left|{h}^{\color{blue}{-0.5}}\right|}{\frac{\sqrt{\ell}}{1}} \]

       6. sqr-pow 78.2%

          \[\leadsto d \cdot \frac{\left|\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}\right|}{\frac{\sqrt{\ell}}{1}} \]

       7. fabs-sqr 78.2%

          \[\leadsto d \cdot \frac{\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}}{\frac{\sqrt{\ell}}{1}} \]

       8. sqr-pow 78.4%

          \[\leadsto d \cdot \frac{\color{blue}{{h}^{-0.5}}}{\frac{\sqrt{\ell}}{1}} \]

       9. /-rgt-identity 78.4%

          \[\leadsto d \cdot \frac{{h}^{-0.5}}{\color{blue}{\sqrt{\ell}}} \]

   14. Simplified 78.4%

       \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.8 \cdot 10^{-64}:\\ \;\;\;\;\left(1 + \frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \left(h \cdot -0.5\right)}{\ell}\right) \cdot \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{-107}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot \frac{M}{\frac{d}{M}}\right)\right)\right)\\ \mathbf{elif}\;d \leq 8.5 \cdot 10^{+106}:\\ \;\;\;\;\left(1 + \frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \left(h \cdot -0.5\right)}{\ell}\right) \cdot \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 15: 61.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ t_1 := \frac{M}{\frac{d}{M}}\\ \mathbf{if}\;d \leq -1.95 \cdot 10^{-63}:\\ \;\;\;\;t_0 \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(t_1 \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{-107}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot t_1\right)\right)\right)\\ \mathbf{elif}\;d \leq 8 \cdot 10^{+104}:\\ \;\;\;\;\left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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) (/ d l)))) (t_1 (/ M (/ d M))))
   (if (<= d -1.95e-63)
     (* t_0 (- 1.0 (* (pow (* M (/ D (* d 2.0))) 2.0) (/ 0.5 (/ l h)))))
     (if (<= d -5e-310)
       (*
        (* d (sqrt (/ 1.0 (* h l))))
        (+ -1.0 (* (* D (/ D l)) (* (* t_1 (/ h d)) 0.125))))
       (if (<= d 1.55e-107)
         (* (sqrt (/ h (pow l 3.0))) (* -0.125 (* D (* D t_1))))
         (if (<= d 8e+104)
           (*
            (- 1.0 (* 0.125 (/ D (/ (/ (* d d) (/ (* h M) (/ l M))) D))))
            t_0)
           (* d (/ (pow h -0.5) (sqrt l)))))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(((d / h) * (d / l)));
	double t_1 = M / (d / M);
	double tmp;
	if (d <= -1.95e-63) {
		tmp = t_0 * (1.0 - (pow((M * (D / (d * 2.0))), 2.0) * (0.5 / (l / h))));
	} else if (d <= -5e-310) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_1 * (h / d)) * 0.125)));
	} else if (d <= 1.55e-107) {
		tmp = sqrt((h / pow(l, 3.0))) * (-0.125 * (D * (D * t_1)));
	} else if (d <= 8e+104) {
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * t_0;
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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) * (d / l)))
    t_1 = m / (d / m)
    if (d <= (-1.95d-63)) then
        tmp = t_0 * (1.0d0 - (((m * (d_1 / (d * 2.0d0))) ** 2.0d0) * (0.5d0 / (l / h))))
    else if (d <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) + ((d_1 * (d_1 / l)) * ((t_1 * (h / d)) * 0.125d0)))
    else if (d <= 1.55d-107) then
        tmp = sqrt((h / (l ** 3.0d0))) * ((-0.125d0) * (d_1 * (d_1 * t_1)))
    else if (d <= 8d+104) then
        tmp = (1.0d0 - (0.125d0 * (d_1 / (((d * d) / ((h * m) / (l / m))) / d_1)))) * t_0
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(((d / h) * (d / l)));
	double t_1 = M / (d / M);
	double tmp;
	if (d <= -1.95e-63) {
		tmp = t_0 * (1.0 - (Math.pow((M * (D / (d * 2.0))), 2.0) * (0.5 / (l / h))));
	} else if (d <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_1 * (h / d)) * 0.125)));
	} else if (d <= 1.55e-107) {
		tmp = Math.sqrt((h / Math.pow(l, 3.0))) * (-0.125 * (D * (D * t_1)));
	} else if (d <= 8e+104) {
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * t_0;
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt(((d / h) * (d / l)))
	t_1 = M / (d / M)
	tmp = 0
	if d <= -1.95e-63:
		tmp = t_0 * (1.0 - (math.pow((M * (D / (d * 2.0))), 2.0) * (0.5 / (l / h))))
	elif d <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_1 * (h / d)) * 0.125)))
	elif d <= 1.55e-107:
		tmp = math.sqrt((h / math.pow(l, 3.0))) * (-0.125 * (D * (D * t_1)))
	elif d <= 8e+104:
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * t_0
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(Float64(d / h) * Float64(d / l)))
	t_1 = Float64(M / Float64(d / M))
	tmp = 0.0
	if (d <= -1.95e-63)
		tmp = Float64(t_0 * Float64(1.0 - Float64((Float64(M * Float64(D / Float64(d * 2.0))) ^ 2.0) * Float64(0.5 / Float64(l / h)))));
	elseif (d <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 + Float64(Float64(D * Float64(D / l)) * Float64(Float64(t_1 * Float64(h / d)) * 0.125))));
	elseif (d <= 1.55e-107)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(-0.125 * Float64(D * Float64(D * t_1))));
	elseif (d <= 8e+104)
		tmp = Float64(Float64(1.0 - Float64(0.125 * Float64(D / Float64(Float64(Float64(d * d) / Float64(Float64(h * M) / Float64(l / M))) / D)))) * t_0);
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(((d / h) * (d / l)));
	t_1 = M / (d / M);
	tmp = 0.0;
	if (d <= -1.95e-63)
		tmp = t_0 * (1.0 - (((M * (D / (d * 2.0))) ^ 2.0) * (0.5 / (l / h))));
	elseif (d <= -5e-310)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_1 * (h / d)) * 0.125)));
	elseif (d <= 1.55e-107)
		tmp = sqrt((h / (l ^ 3.0))) * (-0.125 * (D * (D * t_1)));
	elseif (d <= 8e+104)
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * t_0;
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
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[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.95e-63], N[(t$95$0 * N[(1.0 - N[(N[Power[N[(M * N[(D / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * N[(h / d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.55e-107], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(D * N[(D * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 8e+104], N[(N[(1.0 - N[(0.125 * N[(D / N[(N[(N[(d * d), $MachinePrecision] / N[(N[(h * M), $MachinePrecision] / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $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 < -1.95000000000000011e-63

   1. Initial program 80.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 28.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 39.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 68.7%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 68.7%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 68.7%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 68.7%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 68.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   if -1.95000000000000011e-63 < d < -4.999999999999985e-310

   1. Initial program 53.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 7.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 13.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 49.3%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 49.3%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 49.3%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 49.3%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 49.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in d around -inf 78.1%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg 78.1%

         \[\leadsto \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      2. associate-/r* 78.1%

         \[\leadsto \left(-d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      3. distribute-rgt-neg-in 78.1%

         \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      4. associate-/l/ 78.1%

         \[\leadsto \left(d \cdot \left(-\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   8. Simplified 78.1%

      \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   9. Taylor expanded in M around 0 53.7%

      \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 53.7%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

       2. times-frac 60.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.125\right) \]

       3. unpow2 60.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{\color{blue}{d \cdot d}}\right) \cdot 0.125\right) \]

       4. times-frac 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{d} \cdot \frac{h}{d}\right)}\right) \cdot 0.125\right) \]

       5. unpow2 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \left(\frac{\color{blue}{M \cdot M}}{d} \cdot \frac{h}{d}\right)\right) \cdot 0.125\right) \]

       6. associate-*l* 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

       7. unpow2 62.5%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \frac{\color{blue}{D \cdot D}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       8. associate-*l/ 66.9%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{\ell} \cdot D\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       9. *-commutative 66.9%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       10. associate-/l* 69.1%

           \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

   11. Simplified 69.1%

       \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

   if -4.999999999999985e-310 < d < 1.55000000000000011e-107

   1. Initial program 47.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 47.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 47.9%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 46.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 46.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 46.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 46.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 46.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 46.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 46.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. associate-/l/ 46.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. *-commutative 46.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 46.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

   6. Taylor expanded in d around 0 39.5%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 39.5%

         \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative 39.5%

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      3. associate-*r/ 37.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)}\right) \]

      4. unpow2 37.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{\color{blue}{M \cdot M}}{d}\right)\right) \]

      5. unpow2 37.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \frac{M \cdot M}{d}\right)\right) \]

      6. associate-*l* 41.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(D \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right)}\right) \]

      7. associate-/l* 48.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot \color{blue}{\frac{M}{\frac{d}{M}}}\right)\right)\right) \]

   8. Simplified 48.9%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot \frac{M}{\frac{d}{M}}\right)\right)\right)} \]

   if 1.55000000000000011e-107 < d < 8e104

   1. Initial program 76.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 7.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 22.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 62.1%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 62.1%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 62.1%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 62.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 62.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in M around 0 61.9%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   7. Step-by-step derivation

      1. associate-/l* 62.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{{M}^{2} \cdot h}}}\right) \]

      2. *-commutative 62.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{\color{blue}{h \cdot {M}^{2}}}}\right) \]

      3. unpow2 62.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{D \cdot D}}{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}}\right) \]

      4. associate-/l* 62.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{D}{\frac{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}}{D}}}\right) \]

      5. *-commutative 62.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{\color{blue}{{d}^{2} \cdot \ell}}{h \cdot {M}^{2}}}{D}}\right) \]

      6. associate-/l* 62.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\color{blue}{\frac{{d}^{2}}{\frac{h \cdot {M}^{2}}{\ell}}}}{D}}\right) \]

      7. unpow2 62.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{\color{blue}{d \cdot d}}{\frac{h \cdot {M}^{2}}{\ell}}}{D}}\right) \]

      8. unpow2 62.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{\ell}}}{D}}\right) \]

      9. associate-*r* 65.1%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{\ell}}}{D}}\right) \]

      10. associate-/l* 71.3%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\color{blue}{\frac{h \cdot M}{\frac{\ell}{M}}}}}{D}}\right) \]

      11. *-commutative 71.3%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{M \cdot h}}{\frac{\ell}{M}}}}{D}}\right) \]

   8. Simplified 71.3%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{M \cdot h}{\frac{\ell}{M}}}}{D}}}\right) \]

   if 8e104 < d

   1. Initial program 73.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 73.2%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 73.2%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 73.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 75.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. associate-/l/ 75.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. *-commutative 75.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 75.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

   6. Taylor expanded in d around inf 66.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   7. Step-by-step derivation

      1. *-commutative 66.4%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. *-commutative 66.4%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. associate-/r* 66.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   8. Simplified 66.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   9. Step-by-step derivation

      1. sqrt-div 78.3%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

   10. Applied egg-rr 78.3%

       \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. div-inv 78.3%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{\frac{1}{h}} \cdot \frac{1}{\sqrt{\ell}}\right)} \]

       2. inv-pow 78.3%

          \[\leadsto d \cdot \left(\sqrt{\color{blue}{{h}^{-1}}} \cdot \frac{1}{\sqrt{\ell}}\right) \]

   12. Applied egg-rr 78.3%

       \[\leadsto d \cdot \color{blue}{\left(\sqrt{{h}^{-1}} \cdot \frac{1}{\sqrt{\ell}}\right)} \]
   13. Step-by-step derivation

       1. associate-*r/ 78.3%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}} \cdot 1}{\sqrt{\ell}}} \]

       2. associate-/l* 78.3%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}}}{\frac{\sqrt{\ell}}{1}}} \]

       3. sqr-pow 78.4%

          \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{\left(\frac{-1}{2}\right)} \cdot {h}^{\left(\frac{-1}{2}\right)}}}}{\frac{\sqrt{\ell}}{1}} \]

       4. rem-sqrt-square 78.4%

          \[\leadsto d \cdot \frac{\color{blue}{\left|{h}^{\left(\frac{-1}{2}\right)}\right|}}{\frac{\sqrt{\ell}}{1}} \]

       5. metadata-eval 78.4%

          \[\leadsto d \cdot \frac{\left|{h}^{\color{blue}{-0.5}}\right|}{\frac{\sqrt{\ell}}{1}} \]

       6. sqr-pow 78.2%

          \[\leadsto d \cdot \frac{\left|\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}\right|}{\frac{\sqrt{\ell}}{1}} \]

       7. fabs-sqr 78.2%

          \[\leadsto d \cdot \frac{\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}}{\frac{\sqrt{\ell}}{1}} \]

       8. sqr-pow 78.4%

          \[\leadsto d \cdot \frac{\color{blue}{{h}^{-0.5}}}{\frac{\sqrt{\ell}}{1}} \]

       9. /-rgt-identity 78.4%

          \[\leadsto d \cdot \frac{{h}^{-0.5}}{\color{blue}{\sqrt{\ell}}} \]

   14. Simplified 78.4%

       \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.95 \cdot 10^{-63}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{-107}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot \frac{M}{\frac{d}{M}}\right)\right)\right)\\ \mathbf{elif}\;d \leq 8 \cdot 10^{+104}:\\ \;\;\;\;\left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 16: 61.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ t_1 := \frac{M}{\frac{d}{M}}\\ \mathbf{if}\;d \leq -3.3 \cdot 10^{-92}:\\ \;\;\;\;t_0 \cdot \left(1 - {\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(t_1 \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{-107}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot t_1\right)\right)\right)\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{+106}:\\ \;\;\;\;\left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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) (/ d l)))) (t_1 (/ M (/ d M))))
   (if (<= d -3.3e-92)
     (* t_0 (- 1.0 (* (pow (/ (* D (* M 0.5)) d) 2.0) (/ 0.5 (/ l h)))))
     (if (<= d -5e-310)
       (*
        (* d (sqrt (/ 1.0 (* h l))))
        (+ -1.0 (* (* D (/ D l)) (* (* t_1 (/ h d)) 0.125))))
       (if (<= d 1.55e-107)
         (* (sqrt (/ h (pow l 3.0))) (* -0.125 (* D (* D t_1))))
         (if (<= d 2.8e+106)
           (*
            (- 1.0 (* 0.125 (/ D (/ (/ (* d d) (/ (* h M) (/ l M))) D))))
            t_0)
           (* d (/ (pow h -0.5) (sqrt l)))))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(((d / h) * (d / l)));
	double t_1 = M / (d / M);
	double tmp;
	if (d <= -3.3e-92) {
		tmp = t_0 * (1.0 - (pow(((D * (M * 0.5)) / d), 2.0) * (0.5 / (l / h))));
	} else if (d <= -5e-310) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_1 * (h / d)) * 0.125)));
	} else if (d <= 1.55e-107) {
		tmp = sqrt((h / pow(l, 3.0))) * (-0.125 * (D * (D * t_1)));
	} else if (d <= 2.8e+106) {
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * t_0;
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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) * (d / l)))
    t_1 = m / (d / m)
    if (d <= (-3.3d-92)) then
        tmp = t_0 * (1.0d0 - ((((d_1 * (m * 0.5d0)) / d) ** 2.0d0) * (0.5d0 / (l / h))))
    else if (d <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) + ((d_1 * (d_1 / l)) * ((t_1 * (h / d)) * 0.125d0)))
    else if (d <= 1.55d-107) then
        tmp = sqrt((h / (l ** 3.0d0))) * ((-0.125d0) * (d_1 * (d_1 * t_1)))
    else if (d <= 2.8d+106) then
        tmp = (1.0d0 - (0.125d0 * (d_1 / (((d * d) / ((h * m) / (l / m))) / d_1)))) * t_0
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(((d / h) * (d / l)));
	double t_1 = M / (d / M);
	double tmp;
	if (d <= -3.3e-92) {
		tmp = t_0 * (1.0 - (Math.pow(((D * (M * 0.5)) / d), 2.0) * (0.5 / (l / h))));
	} else if (d <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_1 * (h / d)) * 0.125)));
	} else if (d <= 1.55e-107) {
		tmp = Math.sqrt((h / Math.pow(l, 3.0))) * (-0.125 * (D * (D * t_1)));
	} else if (d <= 2.8e+106) {
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * t_0;
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt(((d / h) * (d / l)))
	t_1 = M / (d / M)
	tmp = 0
	if d <= -3.3e-92:
		tmp = t_0 * (1.0 - (math.pow(((D * (M * 0.5)) / d), 2.0) * (0.5 / (l / h))))
	elif d <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_1 * (h / d)) * 0.125)))
	elif d <= 1.55e-107:
		tmp = math.sqrt((h / math.pow(l, 3.0))) * (-0.125 * (D * (D * t_1)))
	elif d <= 2.8e+106:
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * t_0
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(Float64(d / h) * Float64(d / l)))
	t_1 = Float64(M / Float64(d / M))
	tmp = 0.0
	if (d <= -3.3e-92)
		tmp = Float64(t_0 * Float64(1.0 - Float64((Float64(Float64(D * Float64(M * 0.5)) / d) ^ 2.0) * Float64(0.5 / Float64(l / h)))));
	elseif (d <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 + Float64(Float64(D * Float64(D / l)) * Float64(Float64(t_1 * Float64(h / d)) * 0.125))));
	elseif (d <= 1.55e-107)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(-0.125 * Float64(D * Float64(D * t_1))));
	elseif (d <= 2.8e+106)
		tmp = Float64(Float64(1.0 - Float64(0.125 * Float64(D / Float64(Float64(Float64(d * d) / Float64(Float64(h * M) / Float64(l / M))) / D)))) * t_0);
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(((d / h) * (d / l)));
	t_1 = M / (d / M);
	tmp = 0.0;
	if (d <= -3.3e-92)
		tmp = t_0 * (1.0 - ((((D * (M * 0.5)) / d) ^ 2.0) * (0.5 / (l / h))));
	elseif (d <= -5e-310)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_1 * (h / d)) * 0.125)));
	elseif (d <= 1.55e-107)
		tmp = sqrt((h / (l ^ 3.0))) * (-0.125 * (D * (D * t_1)));
	elseif (d <= 2.8e+106)
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * t_0;
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
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[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.3e-92], N[(t$95$0 * N[(1.0 - N[(N[Power[N[(N[(D * N[(M * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * N[(h / d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.55e-107], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(D * N[(D * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.8e+106], N[(N[(1.0 - N[(0.125 * N[(D / N[(N[(N[(d * d), $MachinePrecision] / N[(N[(h * M), $MachinePrecision] / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $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 < -3.29999999999999998e-92

   1. Initial program 81.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 26.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 37.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 70.0%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 70.0%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 70.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 70.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 70.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 70.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      2. frac-times 70.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      3. associate-*r/ 70.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\color{blue}{\left(\frac{\frac{M}{2} \cdot D}{d}\right)}}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      4. div-inv 70.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(\frac{\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot D}{d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      5. metadata-eval 70.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(\frac{\left(M \cdot \color{blue}{0.5}\right) \cdot D}{d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   7. Applied egg-rr 70.0%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\color{blue}{\left(\frac{\left(M \cdot 0.5\right) \cdot D}{d}\right)}}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   if -3.29999999999999998e-92 < d < -4.999999999999985e-310

   1. Initial program 49.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 8.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 13.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 44.0%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 44.0%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 44.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 44.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 44.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in d around -inf 77.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg 77.3%

         \[\leadsto \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      2. associate-/r* 77.3%

         \[\leadsto \left(-d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      3. distribute-rgt-neg-in 77.3%

         \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      4. associate-/l/ 77.3%

         \[\leadsto \left(d \cdot \left(-\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   8. Simplified 77.3%

      \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   9. Taylor expanded in M around 0 51.6%

      \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 51.6%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

       2. times-frac 59.2%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.125\right) \]

       3. unpow2 59.2%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{\color{blue}{d \cdot d}}\right) \cdot 0.125\right) \]

       4. times-frac 61.7%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{d} \cdot \frac{h}{d}\right)}\right) \cdot 0.125\right) \]

       5. unpow2 61.7%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \left(\frac{\color{blue}{M \cdot M}}{d} \cdot \frac{h}{d}\right)\right) \cdot 0.125\right) \]

       6. associate-*l* 61.7%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

       7. unpow2 61.7%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \frac{\color{blue}{D \cdot D}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       8. associate-*l/ 64.4%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{\ell} \cdot D\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       9. *-commutative 64.4%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       10. associate-/l* 67.0%

           \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

   11. Simplified 67.0%

       \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

   if -4.999999999999985e-310 < d < 1.55000000000000011e-107

   1. Initial program 47.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 47.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 47.9%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 46.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 46.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 46.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 46.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 46.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 46.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 46.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. associate-/l/ 46.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. *-commutative 46.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 46.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

   6. Taylor expanded in d around 0 39.5%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 39.5%

         \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative 39.5%

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      3. associate-*r/ 37.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)}\right) \]

      4. unpow2 37.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{\color{blue}{M \cdot M}}{d}\right)\right) \]

      5. unpow2 37.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \frac{M \cdot M}{d}\right)\right) \]

      6. associate-*l* 41.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(D \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right)}\right) \]

      7. associate-/l* 48.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot \color{blue}{\frac{M}{\frac{d}{M}}}\right)\right)\right) \]

   8. Simplified 48.9%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot \frac{M}{\frac{d}{M}}\right)\right)\right)} \]

   if 1.55000000000000011e-107 < d < 2.79999999999999993e106

   1. Initial program 76.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 7.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 22.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 62.1%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 62.1%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 62.1%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 62.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 62.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in M around 0 61.9%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   7. Step-by-step derivation

      1. associate-/l* 62.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{{M}^{2} \cdot h}}}\right) \]

      2. *-commutative 62.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{\color{blue}{h \cdot {M}^{2}}}}\right) \]

      3. unpow2 62.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{D \cdot D}}{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}}\right) \]

      4. associate-/l* 62.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{D}{\frac{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}}{D}}}\right) \]

      5. *-commutative 62.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{\color{blue}{{d}^{2} \cdot \ell}}{h \cdot {M}^{2}}}{D}}\right) \]

      6. associate-/l* 62.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\color{blue}{\frac{{d}^{2}}{\frac{h \cdot {M}^{2}}{\ell}}}}{D}}\right) \]

      7. unpow2 62.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{\color{blue}{d \cdot d}}{\frac{h \cdot {M}^{2}}{\ell}}}{D}}\right) \]

      8. unpow2 62.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{\ell}}}{D}}\right) \]

      9. associate-*r* 65.1%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{\ell}}}{D}}\right) \]

      10. associate-/l* 71.3%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\color{blue}{\frac{h \cdot M}{\frac{\ell}{M}}}}}{D}}\right) \]

      11. *-commutative 71.3%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{M \cdot h}}{\frac{\ell}{M}}}}{D}}\right) \]

   8. Simplified 71.3%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{M \cdot h}{\frac{\ell}{M}}}}{D}}}\right) \]

   if 2.79999999999999993e106 < d

   1. Initial program 73.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 73.2%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 73.2%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 73.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 75.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. associate-/l/ 75.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. *-commutative 75.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 75.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

   6. Taylor expanded in d around inf 66.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   7. Step-by-step derivation

      1. *-commutative 66.4%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. *-commutative 66.4%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. associate-/r* 66.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   8. Simplified 66.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   9. Step-by-step derivation

      1. sqrt-div 78.3%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

   10. Applied egg-rr 78.3%

       \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. div-inv 78.3%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{\frac{1}{h}} \cdot \frac{1}{\sqrt{\ell}}\right)} \]

       2. inv-pow 78.3%

          \[\leadsto d \cdot \left(\sqrt{\color{blue}{{h}^{-1}}} \cdot \frac{1}{\sqrt{\ell}}\right) \]

   12. Applied egg-rr 78.3%

       \[\leadsto d \cdot \color{blue}{\left(\sqrt{{h}^{-1}} \cdot \frac{1}{\sqrt{\ell}}\right)} \]
   13. Step-by-step derivation

       1. associate-*r/ 78.3%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}} \cdot 1}{\sqrt{\ell}}} \]

       2. associate-/l* 78.3%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}}}{\frac{\sqrt{\ell}}{1}}} \]

       3. sqr-pow 78.4%

          \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{\left(\frac{-1}{2}\right)} \cdot {h}^{\left(\frac{-1}{2}\right)}}}}{\frac{\sqrt{\ell}}{1}} \]

       4. rem-sqrt-square 78.4%

          \[\leadsto d \cdot \frac{\color{blue}{\left|{h}^{\left(\frac{-1}{2}\right)}\right|}}{\frac{\sqrt{\ell}}{1}} \]

       5. metadata-eval 78.4%

          \[\leadsto d \cdot \frac{\left|{h}^{\color{blue}{-0.5}}\right|}{\frac{\sqrt{\ell}}{1}} \]

       6. sqr-pow 78.2%

          \[\leadsto d \cdot \frac{\left|\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}\right|}{\frac{\sqrt{\ell}}{1}} \]

       7. fabs-sqr 78.2%

          \[\leadsto d \cdot \frac{\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}}{\frac{\sqrt{\ell}}{1}} \]

       8. sqr-pow 78.4%

          \[\leadsto d \cdot \frac{\color{blue}{{h}^{-0.5}}}{\frac{\sqrt{\ell}}{1}} \]

       9. /-rgt-identity 78.4%

          \[\leadsto d \cdot \frac{{h}^{-0.5}}{\color{blue}{\sqrt{\ell}}} \]

   14. Simplified 78.4%

       \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.3 \cdot 10^{-92}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{-107}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot \frac{M}{\frac{d}{M}}\right)\right)\right)\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{+106}:\\ \;\;\;\;\left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 17: 64.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(-1 + \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{-107}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot \frac{M}{\frac{d}{M}}\right)\right)\right)\\ \mathbf{elif}\;d \leq 9 \cdot 10^{+104}:\\ \;\;\;\;\left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d -5e-310)
   (*
    (/ d (sqrt (* h l)))
    (+ -1.0 (* (* 0.5 (/ h l)) (pow (* M (/ D (* d 2.0))) 2.0))))
   (if (<= d 1.55e-107)
     (* (sqrt (/ h (pow l 3.0))) (* -0.125 (* D (* D (/ M (/ d M))))))
     (if (<= d 9e+104)
       (*
        (- 1.0 (* 0.125 (/ D (/ (/ (* d d) (/ (* h M) (/ l M))) D))))
        (sqrt (* (/ d h) (/ d l))))
       (* d (/ (pow h -0.5) (sqrt l)))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -5e-310) {
		tmp = (d / sqrt((h * l))) * (-1.0 + ((0.5 * (h / l)) * pow((M * (D / (d * 2.0))), 2.0)));
	} else if (d <= 1.55e-107) {
		tmp = sqrt((h / pow(l, 3.0))) * (-0.125 * (D * (D * (M / (d / M)))));
	} else if (d <= 9e+104) {
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * sqrt(((d / h) * (d / l)));
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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 <= (-5d-310)) then
        tmp = (d / sqrt((h * l))) * ((-1.0d0) + ((0.5d0 * (h / l)) * ((m * (d_1 / (d * 2.0d0))) ** 2.0d0)))
    else if (d <= 1.55d-107) then
        tmp = sqrt((h / (l ** 3.0d0))) * ((-0.125d0) * (d_1 * (d_1 * (m / (d / m)))))
    else if (d <= 9d+104) then
        tmp = (1.0d0 - (0.125d0 * (d_1 / (((d * d) / ((h * m) / (l / m))) / d_1)))) * sqrt(((d / h) * (d / l)))
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -5e-310) {
		tmp = (d / Math.sqrt((h * l))) * (-1.0 + ((0.5 * (h / l)) * Math.pow((M * (D / (d * 2.0))), 2.0)));
	} else if (d <= 1.55e-107) {
		tmp = Math.sqrt((h / Math.pow(l, 3.0))) * (-0.125 * (D * (D * (M / (d / M)))));
	} else if (d <= 9e+104) {
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * Math.sqrt(((d / h) * (d / l)));
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if d <= -5e-310:
		tmp = (d / math.sqrt((h * l))) * (-1.0 + ((0.5 * (h / l)) * math.pow((M * (D / (d * 2.0))), 2.0)))
	elif d <= 1.55e-107:
		tmp = math.sqrt((h / math.pow(l, 3.0))) * (-0.125 * (D * (D * (M / (d / M)))))
	elif d <= 9e+104:
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * math.sqrt(((d / h) * (d / l)))
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -5e-310)
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * Float64(-1.0 + Float64(Float64(0.5 * Float64(h / l)) * (Float64(M * Float64(D / Float64(d * 2.0))) ^ 2.0))));
	elseif (d <= 1.55e-107)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(-0.125 * Float64(D * Float64(D * Float64(M / Float64(d / M))))));
	elseif (d <= 9e+104)
		tmp = Float64(Float64(1.0 - Float64(0.125 * Float64(D / Float64(Float64(Float64(d * d) / Float64(Float64(h * M) / Float64(l / M))) / D)))) * sqrt(Float64(Float64(d / h) * Float64(d / l))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -5e-310)
		tmp = (d / sqrt((h * l))) * (-1.0 + ((0.5 * (h / l)) * ((M * (D / (d * 2.0))) ^ 2.0)));
	elseif (d <= 1.55e-107)
		tmp = sqrt((h / (l ^ 3.0))) * (-0.125 * (D * (D * (M / (d / M)))));
	elseif (d <= 9e+104)
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * sqrt(((d / h) * (d / l)));
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, -5e-310], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(M * N[(D / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.55e-107], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(D * N[(D * N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 9e+104], N[(N[(1.0 - N[(0.125 * N[(D / N[(N[(N[(d * d), $MachinePrecision] / N[(N[(h * M), $MachinePrecision] / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $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 < -4.999999999999985e-310

   1. Initial program 69.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 19.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 28.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 60.4%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 60.4%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 60.4%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 60.4%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 60.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in d around -inf 71.6%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg 71.6%

         \[\leadsto \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      2. associate-/r* 71.5%

         \[\leadsto \left(-d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      3. distribute-rgt-neg-in 71.5%

         \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      4. associate-/l/ 71.6%

         \[\leadsto \left(d \cdot \left(-\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   8. Simplified 71.6%

      \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]
   9. Step-by-step derivation

      1. pow1 71.6%

         \[\leadsto \color{blue}{{\left(\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)\right)}^{1}} \]

   10. Applied egg-rr 71.6%

       \[\leadsto \color{blue}{{\left(d \cdot \left(\left(-\frac{1}{\sqrt{\ell \cdot h}}\right) \cdot \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)\right)\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 71.6%

          \[\leadsto \color{blue}{d \cdot \left(\left(-\frac{1}{\sqrt{\ell \cdot h}}\right) \cdot \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)\right)} \]

       2. associate-*r* 71.5%

          \[\leadsto \color{blue}{\left(d \cdot \left(-\frac{1}{\sqrt{\ell \cdot h}}\right)\right) \cdot \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)} \]

       3. *-commutative 71.5%

          \[\leadsto \color{blue}{\left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right) \cdot \left(d \cdot \left(-\frac{1}{\sqrt{\ell \cdot h}}\right)\right)} \]

       4. distribute-neg-frac 71.5%

          \[\leadsto \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right) \cdot \left(d \cdot \color{blue}{\frac{-1}{\sqrt{\ell \cdot h}}}\right) \]

       5. metadata-eval 71.5%

          \[\leadsto \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right) \cdot \left(d \cdot \frac{\color{blue}{-1}}{\sqrt{\ell \cdot h}}\right) \]

       6. associate-*r/ 71.5%

          \[\leadsto \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right) \cdot \color{blue}{\frac{d \cdot -1}{\sqrt{\ell \cdot h}}} \]

       7. *-commutative 71.5%

          \[\leadsto \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{\color{blue}{-1 \cdot d}}{\sqrt{\ell \cdot h}} \]

       8. mul-1-neg 71.5%

          \[\leadsto \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{\color{blue}{-d}}{\sqrt{\ell \cdot h}} \]

   12. Simplified 71.5%

       \[\leadsto \color{blue}{\left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}} \]

   if -4.999999999999985e-310 < d < 1.55000000000000011e-107

   1. Initial program 47.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 47.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 47.9%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 46.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 46.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 46.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 46.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 46.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 46.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 46.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. associate-/l/ 46.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. *-commutative 46.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 46.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

   6. Taylor expanded in d around 0 39.5%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 39.5%

         \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative 39.5%

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      3. associate-*r/ 37.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)}\right) \]

      4. unpow2 37.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{\color{blue}{M \cdot M}}{d}\right)\right) \]

      5. unpow2 37.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \frac{M \cdot M}{d}\right)\right) \]

      6. associate-*l* 41.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(D \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right)}\right) \]

      7. associate-/l* 48.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot \color{blue}{\frac{M}{\frac{d}{M}}}\right)\right)\right) \]

   8. Simplified 48.9%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot \frac{M}{\frac{d}{M}}\right)\right)\right)} \]

   if 1.55000000000000011e-107 < d < 8.9999999999999997e104

   1. Initial program 76.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 7.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 22.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 62.1%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 62.1%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 62.1%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 62.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 62.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in M around 0 61.9%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   7. Step-by-step derivation

      1. associate-/l* 62.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{{M}^{2} \cdot h}}}\right) \]

      2. *-commutative 62.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{\color{blue}{h \cdot {M}^{2}}}}\right) \]

      3. unpow2 62.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{D \cdot D}}{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}}\right) \]

      4. associate-/l* 62.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{D}{\frac{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}}{D}}}\right) \]

      5. *-commutative 62.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{\color{blue}{{d}^{2} \cdot \ell}}{h \cdot {M}^{2}}}{D}}\right) \]

      6. associate-/l* 62.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\color{blue}{\frac{{d}^{2}}{\frac{h \cdot {M}^{2}}{\ell}}}}{D}}\right) \]

      7. unpow2 62.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{\color{blue}{d \cdot d}}{\frac{h \cdot {M}^{2}}{\ell}}}{D}}\right) \]

      8. unpow2 62.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{\ell}}}{D}}\right) \]

      9. associate-*r* 65.1%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{\ell}}}{D}}\right) \]

      10. associate-/l* 71.3%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\color{blue}{\frac{h \cdot M}{\frac{\ell}{M}}}}}{D}}\right) \]

      11. *-commutative 71.3%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{M \cdot h}}{\frac{\ell}{M}}}}{D}}\right) \]

   8. Simplified 71.3%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{M \cdot h}{\frac{\ell}{M}}}}{D}}}\right) \]

   if 8.9999999999999997e104 < d

   1. Initial program 73.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 73.2%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 73.2%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 73.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 75.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. associate-/l/ 75.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. *-commutative 75.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 75.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

   6. Taylor expanded in d around inf 66.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   7. Step-by-step derivation

      1. *-commutative 66.4%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. *-commutative 66.4%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. associate-/r* 66.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   8. Simplified 66.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   9. Step-by-step derivation

      1. sqrt-div 78.3%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

   10. Applied egg-rr 78.3%

       \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. div-inv 78.3%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{\frac{1}{h}} \cdot \frac{1}{\sqrt{\ell}}\right)} \]

       2. inv-pow 78.3%

          \[\leadsto d \cdot \left(\sqrt{\color{blue}{{h}^{-1}}} \cdot \frac{1}{\sqrt{\ell}}\right) \]

   12. Applied egg-rr 78.3%

       \[\leadsto d \cdot \color{blue}{\left(\sqrt{{h}^{-1}} \cdot \frac{1}{\sqrt{\ell}}\right)} \]
   13. Step-by-step derivation

       1. associate-*r/ 78.3%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}} \cdot 1}{\sqrt{\ell}}} \]

       2. associate-/l* 78.3%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}}}{\frac{\sqrt{\ell}}{1}}} \]

       3. sqr-pow 78.4%

          \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{\left(\frac{-1}{2}\right)} \cdot {h}^{\left(\frac{-1}{2}\right)}}}}{\frac{\sqrt{\ell}}{1}} \]

       4. rem-sqrt-square 78.4%

          \[\leadsto d \cdot \frac{\color{blue}{\left|{h}^{\left(\frac{-1}{2}\right)}\right|}}{\frac{\sqrt{\ell}}{1}} \]

       5. metadata-eval 78.4%

          \[\leadsto d \cdot \frac{\left|{h}^{\color{blue}{-0.5}}\right|}{\frac{\sqrt{\ell}}{1}} \]

       6. sqr-pow 78.2%

          \[\leadsto d \cdot \frac{\left|\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}\right|}{\frac{\sqrt{\ell}}{1}} \]

       7. fabs-sqr 78.2%

          \[\leadsto d \cdot \frac{\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}}{\frac{\sqrt{\ell}}{1}} \]

       8. sqr-pow 78.4%

          \[\leadsto d \cdot \frac{\color{blue}{{h}^{-0.5}}}{\frac{\sqrt{\ell}}{1}} \]

       9. /-rgt-identity 78.4%

          \[\leadsto d \cdot \frac{{h}^{-0.5}}{\color{blue}{\sqrt{\ell}}} \]

   14. Simplified 78.4%

       \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(-1 + \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{-107}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot \frac{M}{\frac{d}{M}}\right)\right)\right)\\ \mathbf{elif}\;d \leq 9 \cdot 10^{+104}:\\ \;\;\;\;\left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 18: 56.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\\ t_1 := \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{if}\;d \leq -2 \cdot 10^{-92}:\\ \;\;\;\;\left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right) \cdot t_1\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + t_0\right)\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{-224}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{+106}:\\ \;\;\;\;\left(1 - t_0\right) \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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 (/ D l)) (* (* (/ M (/ d M)) (/ h d)) 0.125)))
        (t_1 (sqrt (* (/ d h) (/ d l)))))
   (if (<= d -2e-92)
     (* (- 1.0 (* 0.125 (/ D (/ (/ (* d d) (/ (* h M) (/ l M))) D)))) t_1)
     (if (<= d -1e-309)
       (* (* d (sqrt (/ 1.0 (* h l)))) (+ -1.0 t_0))
       (if (<= d 2.8e-224)
         (* -0.125 (* (/ (* D D) (/ d (* M M))) (sqrt (/ h (pow l 3.0)))))
         (if (<= d 2.3e+106)
           (* (- 1.0 t_0) t_1)
           (* d (/ (pow h -0.5) (sqrt l)))))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125);
	double t_1 = sqrt(((d / h) * (d / l)));
	double tmp;
	if (d <= -2e-92) {
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * t_1;
	} else if (d <= -1e-309) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + t_0);
	} else if (d <= 2.8e-224) {
		tmp = -0.125 * (((D * D) / (d / (M * M))) * sqrt((h / pow(l, 3.0))));
	} else if (d <= 2.3e+106) {
		tmp = (1.0 - t_0) * t_1;
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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 * (d_1 / l)) * (((m / (d / m)) * (h / d)) * 0.125d0)
    t_1 = sqrt(((d / h) * (d / l)))
    if (d <= (-2d-92)) then
        tmp = (1.0d0 - (0.125d0 * (d_1 / (((d * d) / ((h * m) / (l / m))) / d_1)))) * t_1
    else if (d <= (-1d-309)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) + t_0)
    else if (d <= 2.8d-224) then
        tmp = (-0.125d0) * (((d_1 * d_1) / (d / (m * m))) * sqrt((h / (l ** 3.0d0))))
    else if (d <= 2.3d+106) then
        tmp = (1.0d0 - t_0) * t_1
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125);
	double t_1 = Math.sqrt(((d / h) * (d / l)));
	double tmp;
	if (d <= -2e-92) {
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * t_1;
	} else if (d <= -1e-309) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 + t_0);
	} else if (d <= 2.8e-224) {
		tmp = -0.125 * (((D * D) / (d / (M * M))) * Math.sqrt((h / Math.pow(l, 3.0))));
	} else if (d <= 2.3e+106) {
		tmp = (1.0 - t_0) * t_1;
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = (D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125)
	t_1 = math.sqrt(((d / h) * (d / l)))
	tmp = 0
	if d <= -2e-92:
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * t_1
	elif d <= -1e-309:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 + t_0)
	elif d <= 2.8e-224:
		tmp = -0.125 * (((D * D) / (d / (M * M))) * math.sqrt((h / math.pow(l, 3.0))))
	elif d <= 2.3e+106:
		tmp = (1.0 - t_0) * t_1
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(D * Float64(D / l)) * Float64(Float64(Float64(M / Float64(d / M)) * Float64(h / d)) * 0.125))
	t_1 = sqrt(Float64(Float64(d / h) * Float64(d / l)))
	tmp = 0.0
	if (d <= -2e-92)
		tmp = Float64(Float64(1.0 - Float64(0.125 * Float64(D / Float64(Float64(Float64(d * d) / Float64(Float64(h * M) / Float64(l / M))) / D)))) * t_1);
	elseif (d <= -1e-309)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 + t_0));
	elseif (d <= 2.8e-224)
		tmp = Float64(-0.125 * Float64(Float64(Float64(D * D) / Float64(d / Float64(M * M))) * sqrt(Float64(h / (l ^ 3.0)))));
	elseif (d <= 2.3e+106)
		tmp = Float64(Float64(1.0 - t_0) * t_1);
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125);
	t_1 = sqrt(((d / h) * (d / l)));
	tmp = 0.0;
	if (d <= -2e-92)
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * t_1;
	elseif (d <= -1e-309)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + t_0);
	elseif (d <= 2.8e-224)
		tmp = -0.125 * (((D * D) / (d / (M * M))) * sqrt((h / (l ^ 3.0))));
	elseif (d <= 2.3e+106)
		tmp = (1.0 - t_0) * t_1;
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
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[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -2e-92], N[(N[(1.0 - N[(0.125 * N[(D / N[(N[(N[(d * d), $MachinePrecision] / N[(N[(h * M), $MachinePrecision] / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[d, -1e-309], 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[d, 2.8e-224], N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] / N[(d / N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.3e+106], N[(N[(1.0 - t$95$0), $MachinePrecision] * t$95$1), $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 < -1.99999999999999998e-92

   1. Initial program 81.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 26.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 37.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 70.0%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 70.0%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 70.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 70.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 70.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in M around 0 54.4%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   7. Step-by-step derivation

      1. associate-/l* 54.3%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{{M}^{2} \cdot h}}}\right) \]

      2. *-commutative 54.3%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{\color{blue}{h \cdot {M}^{2}}}}\right) \]

      3. unpow2 54.3%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{D \cdot D}}{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}}\right) \]

      4. associate-/l* 56.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{D}{\frac{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}}{D}}}\right) \]

      5. *-commutative 56.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{\color{blue}{{d}^{2} \cdot \ell}}{h \cdot {M}^{2}}}{D}}\right) \]

      6. associate-/l* 57.6%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\color{blue}{\frac{{d}^{2}}{\frac{h \cdot {M}^{2}}{\ell}}}}{D}}\right) \]

      7. unpow2 57.6%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{\color{blue}{d \cdot d}}{\frac{h \cdot {M}^{2}}{\ell}}}{D}}\right) \]

      8. unpow2 57.6%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{\ell}}}{D}}\right) \]

      9. associate-*r* 59.3%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{\ell}}}{D}}\right) \]

      10. associate-/l* 62.0%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\color{blue}{\frac{h \cdot M}{\frac{\ell}{M}}}}}{D}}\right) \]

      11. *-commutative 62.0%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{M \cdot h}}{\frac{\ell}{M}}}}{D}}\right) \]

   8. Simplified 62.0%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{M \cdot h}{\frac{\ell}{M}}}}{D}}}\right) \]

   if -1.99999999999999998e-92 < d < -1.000000000000002e-309

   1. Initial program 49.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 8.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 13.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 44.0%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 44.0%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 44.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 44.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 44.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in d around -inf 77.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg 77.3%

         \[\leadsto \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      2. associate-/r* 77.3%

         \[\leadsto \left(-d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      3. distribute-rgt-neg-in 77.3%

         \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      4. associate-/l/ 77.3%

         \[\leadsto \left(d \cdot \left(-\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   8. Simplified 77.3%

      \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   9. Taylor expanded in M around 0 51.6%

      \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 51.6%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

       2. times-frac 59.2%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.125\right) \]

       3. unpow2 59.2%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{\color{blue}{d \cdot d}}\right) \cdot 0.125\right) \]

       4. times-frac 61.7%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{d} \cdot \frac{h}{d}\right)}\right) \cdot 0.125\right) \]

       5. unpow2 61.7%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \left(\frac{\color{blue}{M \cdot M}}{d} \cdot \frac{h}{d}\right)\right) \cdot 0.125\right) \]

       6. associate-*l* 61.7%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

       7. unpow2 61.7%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \frac{\color{blue}{D \cdot D}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       8. associate-*l/ 64.4%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{\ell} \cdot D\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       9. *-commutative 64.4%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       10. associate-/l* 67.0%

           \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

   11. Simplified 67.0%

       \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

   if -1.000000000000002e-309 < d < 2.7999999999999998e-224

   1. Initial program 43.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around 0 47.0%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   3. Step-by-step derivation

      1. associate-/l* 47.0%

         \[\leadsto -0.125 \cdot \left(\color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      2. unpow2 47.0%

         \[\leadsto -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\frac{d}{{M}^{2}}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      3. unpow2 47.0%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{\color{blue}{M \cdot M}}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

   4. Simplified 47.0%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]

   if 2.7999999999999998e-224 < d < 2.3000000000000002e106

   1. Initial program 67.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 6.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 18.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 54.8%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 54.8%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 54.8%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 54.7%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 54.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in M around 0 41.9%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 0.7%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

      2. times-frac 0.6%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.125\right) \]

      3. unpow2 0.6%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{\color{blue}{d \cdot d}}\right) \cdot 0.125\right) \]

      4. times-frac 1.1%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{d} \cdot \frac{h}{d}\right)}\right) \cdot 0.125\right) \]

      5. unpow2 1.1%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \left(\frac{\color{blue}{M \cdot M}}{d} \cdot \frac{h}{d}\right)\right) \cdot 0.125\right) \]

      6. associate-*l* 1.1%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

      7. unpow2 1.1%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \frac{\color{blue}{D \cdot D}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

      8. associate-*l/ 1.0%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{\ell} \cdot D\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

      9. *-commutative 1.0%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

      10. associate-/l* 1.0%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

   8. Simplified 60.4%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

   if 2.3000000000000002e106 < d

   1. Initial program 73.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 73.2%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 73.2%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 73.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 75.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. associate-/l/ 75.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. *-commutative 75.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 75.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

   6. Taylor expanded in d around inf 66.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   7. Step-by-step derivation

      1. *-commutative 66.4%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. *-commutative 66.4%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. associate-/r* 66.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   8. Simplified 66.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   9. Step-by-step derivation

      1. sqrt-div 78.3%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

   10. Applied egg-rr 78.3%

       \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. div-inv 78.3%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{\frac{1}{h}} \cdot \frac{1}{\sqrt{\ell}}\right)} \]

       2. inv-pow 78.3%

          \[\leadsto d \cdot \left(\sqrt{\color{blue}{{h}^{-1}}} \cdot \frac{1}{\sqrt{\ell}}\right) \]

   12. Applied egg-rr 78.3%

       \[\leadsto d \cdot \color{blue}{\left(\sqrt{{h}^{-1}} \cdot \frac{1}{\sqrt{\ell}}\right)} \]
   13. Step-by-step derivation

       1. associate-*r/ 78.3%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}} \cdot 1}{\sqrt{\ell}}} \]

       2. associate-/l* 78.3%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}}}{\frac{\sqrt{\ell}}{1}}} \]

       3. sqr-pow 78.4%

          \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{\left(\frac{-1}{2}\right)} \cdot {h}^{\left(\frac{-1}{2}\right)}}}}{\frac{\sqrt{\ell}}{1}} \]

       4. rem-sqrt-square 78.4%

          \[\leadsto d \cdot \frac{\color{blue}{\left|{h}^{\left(\frac{-1}{2}\right)}\right|}}{\frac{\sqrt{\ell}}{1}} \]

       5. metadata-eval 78.4%

          \[\leadsto d \cdot \frac{\left|{h}^{\color{blue}{-0.5}}\right|}{\frac{\sqrt{\ell}}{1}} \]

       6. sqr-pow 78.2%

          \[\leadsto d \cdot \frac{\left|\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}\right|}{\frac{\sqrt{\ell}}{1}} \]

       7. fabs-sqr 78.2%

          \[\leadsto d \cdot \frac{\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}}{\frac{\sqrt{\ell}}{1}} \]

       8. sqr-pow 78.4%

          \[\leadsto d \cdot \frac{\color{blue}{{h}^{-0.5}}}{\frac{\sqrt{\ell}}{1}} \]

       9. /-rgt-identity 78.4%

          \[\leadsto d \cdot \frac{{h}^{-0.5}}{\color{blue}{\sqrt{\ell}}} \]

   14. Simplified 78.4%

       \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{-92}:\\ \;\;\;\;\left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{-224}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{\frac{d}{M \cdot M}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{+106}:\\ \;\;\;\;\left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 19: 56.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\\ t_1 := \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{if}\;d \leq -4.8 \cdot 10^{-92}:\\ \;\;\;\;\left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right) \cdot t_1\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + t_0\right)\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{-224}:\\ \;\;\;\;-0.125 \cdot \frac{\left(D \cdot D\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{\frac{d}{M \cdot M}}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+106}:\\ \;\;\;\;\left(1 - t_0\right) \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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 (/ D l)) (* (* (/ M (/ d M)) (/ h d)) 0.125)))
        (t_1 (sqrt (* (/ d h) (/ d l)))))
   (if (<= d -4.8e-92)
     (* (- 1.0 (* 0.125 (/ D (/ (/ (* d d) (/ (* h M) (/ l M))) D)))) t_1)
     (if (<= d -5e-310)
       (* (* d (sqrt (/ 1.0 (* h l)))) (+ -1.0 t_0))
       (if (<= d 2.05e-224)
         (* -0.125 (/ (* (* D D) (sqrt (/ h (pow l 3.0)))) (/ d (* M M))))
         (if (<= d 1.7e+106)
           (* (- 1.0 t_0) t_1)
           (* d (/ (pow h -0.5) (sqrt l)))))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125);
	double t_1 = sqrt(((d / h) * (d / l)));
	double tmp;
	if (d <= -4.8e-92) {
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * t_1;
	} else if (d <= -5e-310) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + t_0);
	} else if (d <= 2.05e-224) {
		tmp = -0.125 * (((D * D) * sqrt((h / pow(l, 3.0)))) / (d / (M * M)));
	} else if (d <= 1.7e+106) {
		tmp = (1.0 - t_0) * t_1;
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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 * (d_1 / l)) * (((m / (d / m)) * (h / d)) * 0.125d0)
    t_1 = sqrt(((d / h) * (d / l)))
    if (d <= (-4.8d-92)) then
        tmp = (1.0d0 - (0.125d0 * (d_1 / (((d * d) / ((h * m) / (l / m))) / d_1)))) * t_1
    else if (d <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) + t_0)
    else if (d <= 2.05d-224) then
        tmp = (-0.125d0) * (((d_1 * d_1) * sqrt((h / (l ** 3.0d0)))) / (d / (m * m)))
    else if (d <= 1.7d+106) then
        tmp = (1.0d0 - t_0) * t_1
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125);
	double t_1 = Math.sqrt(((d / h) * (d / l)));
	double tmp;
	if (d <= -4.8e-92) {
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * t_1;
	} else if (d <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 + t_0);
	} else if (d <= 2.05e-224) {
		tmp = -0.125 * (((D * D) * Math.sqrt((h / Math.pow(l, 3.0)))) / (d / (M * M)));
	} else if (d <= 1.7e+106) {
		tmp = (1.0 - t_0) * t_1;
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = (D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125)
	t_1 = math.sqrt(((d / h) * (d / l)))
	tmp = 0
	if d <= -4.8e-92:
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * t_1
	elif d <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 + t_0)
	elif d <= 2.05e-224:
		tmp = -0.125 * (((D * D) * math.sqrt((h / math.pow(l, 3.0)))) / (d / (M * M)))
	elif d <= 1.7e+106:
		tmp = (1.0 - t_0) * t_1
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(D * Float64(D / l)) * Float64(Float64(Float64(M / Float64(d / M)) * Float64(h / d)) * 0.125))
	t_1 = sqrt(Float64(Float64(d / h) * Float64(d / l)))
	tmp = 0.0
	if (d <= -4.8e-92)
		tmp = Float64(Float64(1.0 - Float64(0.125 * Float64(D / Float64(Float64(Float64(d * d) / Float64(Float64(h * M) / Float64(l / M))) / D)))) * t_1);
	elseif (d <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 + t_0));
	elseif (d <= 2.05e-224)
		tmp = Float64(-0.125 * Float64(Float64(Float64(D * D) * sqrt(Float64(h / (l ^ 3.0)))) / Float64(d / Float64(M * M))));
	elseif (d <= 1.7e+106)
		tmp = Float64(Float64(1.0 - t_0) * t_1);
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125);
	t_1 = sqrt(((d / h) * (d / l)));
	tmp = 0.0;
	if (d <= -4.8e-92)
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * t_1;
	elseif (d <= -5e-310)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + t_0);
	elseif (d <= 2.05e-224)
		tmp = -0.125 * (((D * D) * sqrt((h / (l ^ 3.0)))) / (d / (M * M)));
	elseif (d <= 1.7e+106)
		tmp = (1.0 - t_0) * t_1;
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
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[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -4.8e-92], N[(N[(1.0 - N[(0.125 * N[(D / N[(N[(N[(d * d), $MachinePrecision] / N[(N[(h * M), $MachinePrecision] / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[d, -5e-310], 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[d, 2.05e-224], N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] * N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(d / N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.7e+106], N[(N[(1.0 - t$95$0), $MachinePrecision] * t$95$1), $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 < -4.8000000000000002e-92

   1. Initial program 81.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 26.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 37.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 70.0%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 70.0%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 70.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 70.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 70.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in M around 0 54.4%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   7. Step-by-step derivation

      1. associate-/l* 54.3%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{{M}^{2} \cdot h}}}\right) \]

      2. *-commutative 54.3%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{\color{blue}{h \cdot {M}^{2}}}}\right) \]

      3. unpow2 54.3%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{D \cdot D}}{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}}\right) \]

      4. associate-/l* 56.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{D}{\frac{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}}{D}}}\right) \]

      5. *-commutative 56.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{\color{blue}{{d}^{2} \cdot \ell}}{h \cdot {M}^{2}}}{D}}\right) \]

      6. associate-/l* 57.6%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\color{blue}{\frac{{d}^{2}}{\frac{h \cdot {M}^{2}}{\ell}}}}{D}}\right) \]

      7. unpow2 57.6%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{\color{blue}{d \cdot d}}{\frac{h \cdot {M}^{2}}{\ell}}}{D}}\right) \]

      8. unpow2 57.6%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{\ell}}}{D}}\right) \]

      9. associate-*r* 59.3%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{\ell}}}{D}}\right) \]

      10. associate-/l* 62.0%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\color{blue}{\frac{h \cdot M}{\frac{\ell}{M}}}}}{D}}\right) \]

      11. *-commutative 62.0%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{M \cdot h}}{\frac{\ell}{M}}}}{D}}\right) \]

   8. Simplified 62.0%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{M \cdot h}{\frac{\ell}{M}}}}{D}}}\right) \]

   if -4.8000000000000002e-92 < d < -4.999999999999985e-310

   1. Initial program 49.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 8.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 13.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 44.0%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 44.0%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 44.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 44.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 44.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in d around -inf 77.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg 77.3%

         \[\leadsto \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      2. associate-/r* 77.3%

         \[\leadsto \left(-d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      3. distribute-rgt-neg-in 77.3%

         \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      4. associate-/l/ 77.3%

         \[\leadsto \left(d \cdot \left(-\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   8. Simplified 77.3%

      \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   9. Taylor expanded in M around 0 51.6%

      \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 51.6%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

       2. times-frac 59.2%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.125\right) \]

       3. unpow2 59.2%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{\color{blue}{d \cdot d}}\right) \cdot 0.125\right) \]

       4. times-frac 61.7%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{d} \cdot \frac{h}{d}\right)}\right) \cdot 0.125\right) \]

       5. unpow2 61.7%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \left(\frac{\color{blue}{M \cdot M}}{d} \cdot \frac{h}{d}\right)\right) \cdot 0.125\right) \]

       6. associate-*l* 61.7%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

       7. unpow2 61.7%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \frac{\color{blue}{D \cdot D}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       8. associate-*l/ 64.4%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{\ell} \cdot D\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       9. *-commutative 64.4%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       10. associate-/l* 67.0%

           \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

   11. Simplified 67.0%

       \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

   if -4.999999999999985e-310 < d < 2.04999999999999993e-224

   1. Initial program 43.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around 0 47.0%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   3. Step-by-step derivation

      1. associate-/l* 47.0%

         \[\leadsto -0.125 \cdot \left(\color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      2. associate-*l/ 43.7%

         \[\leadsto -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{\frac{d}{{M}^{2}}}} \]

      3. unpow2 43.7%

         \[\leadsto -0.125 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{\frac{d}{{M}^{2}}} \]

      4. unpow2 43.7%

         \[\leadsto -0.125 \cdot \frac{\left(D \cdot D\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{\frac{d}{\color{blue}{M \cdot M}}} \]

   4. Simplified 43.7%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{\left(D \cdot D\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{\frac{d}{M \cdot M}}} \]

   if 2.04999999999999993e-224 < d < 1.69999999999999997e106

   1. Initial program 67.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 6.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 18.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 54.8%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 54.8%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 54.8%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 54.7%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 54.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in M around 0 41.9%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 0.7%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

      2. times-frac 0.6%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.125\right) \]

      3. unpow2 0.6%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{\color{blue}{d \cdot d}}\right) \cdot 0.125\right) \]

      4. times-frac 1.1%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{d} \cdot \frac{h}{d}\right)}\right) \cdot 0.125\right) \]

      5. unpow2 1.1%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \left(\frac{\color{blue}{M \cdot M}}{d} \cdot \frac{h}{d}\right)\right) \cdot 0.125\right) \]

      6. associate-*l* 1.1%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

      7. unpow2 1.1%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \frac{\color{blue}{D \cdot D}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

      8. associate-*l/ 1.0%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{\ell} \cdot D\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

      9. *-commutative 1.0%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

      10. associate-/l* 1.0%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

   8. Simplified 60.4%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

   if 1.69999999999999997e106 < d

   1. Initial program 73.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 73.2%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 73.2%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 73.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 75.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. associate-/l/ 75.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. *-commutative 75.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 75.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

   6. Taylor expanded in d around inf 66.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   7. Step-by-step derivation

      1. *-commutative 66.4%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. *-commutative 66.4%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. associate-/r* 66.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   8. Simplified 66.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   9. Step-by-step derivation

      1. sqrt-div 78.3%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

   10. Applied egg-rr 78.3%

       \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. div-inv 78.3%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{\frac{1}{h}} \cdot \frac{1}{\sqrt{\ell}}\right)} \]

       2. inv-pow 78.3%

          \[\leadsto d \cdot \left(\sqrt{\color{blue}{{h}^{-1}}} \cdot \frac{1}{\sqrt{\ell}}\right) \]

   12. Applied egg-rr 78.3%

       \[\leadsto d \cdot \color{blue}{\left(\sqrt{{h}^{-1}} \cdot \frac{1}{\sqrt{\ell}}\right)} \]
   13. Step-by-step derivation

       1. associate-*r/ 78.3%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}} \cdot 1}{\sqrt{\ell}}} \]

       2. associate-/l* 78.3%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}}}{\frac{\sqrt{\ell}}{1}}} \]

       3. sqr-pow 78.4%

          \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{\left(\frac{-1}{2}\right)} \cdot {h}^{\left(\frac{-1}{2}\right)}}}}{\frac{\sqrt{\ell}}{1}} \]

       4. rem-sqrt-square 78.4%

          \[\leadsto d \cdot \frac{\color{blue}{\left|{h}^{\left(\frac{-1}{2}\right)}\right|}}{\frac{\sqrt{\ell}}{1}} \]

       5. metadata-eval 78.4%

          \[\leadsto d \cdot \frac{\left|{h}^{\color{blue}{-0.5}}\right|}{\frac{\sqrt{\ell}}{1}} \]

       6. sqr-pow 78.2%

          \[\leadsto d \cdot \frac{\left|\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}\right|}{\frac{\sqrt{\ell}}{1}} \]

       7. fabs-sqr 78.2%

          \[\leadsto d \cdot \frac{\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}}{\frac{\sqrt{\ell}}{1}} \]

       8. sqr-pow 78.4%

          \[\leadsto d \cdot \frac{\color{blue}{{h}^{-0.5}}}{\frac{\sqrt{\ell}}{1}} \]

       9. /-rgt-identity 78.4%

          \[\leadsto d \cdot \frac{{h}^{-0.5}}{\color{blue}{\sqrt{\ell}}} \]

   14. Simplified 78.4%

       \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.8 \cdot 10^{-92}:\\ \;\;\;\;\left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{-224}:\\ \;\;\;\;-0.125 \cdot \frac{\left(D \cdot D\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{\frac{d}{M \cdot M}}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+106}:\\ \;\;\;\;\left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 20: 59.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ t_1 := \frac{M}{\frac{d}{M}}\\ \mathbf{if}\;d \leq -3.4 \cdot 10^{-91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(t_1 \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{-107}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot t_1\right)\right)\right)\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{+105}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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
         (*
          (- 1.0 (* 0.125 (/ D (/ (/ (* d d) (/ (* h M) (/ l M))) D))))
          (sqrt (* (/ d h) (/ d l)))))
        (t_1 (/ M (/ d M))))
   (if (<= d -3.4e-91)
     t_0
     (if (<= d -5e-310)
       (*
        (* d (sqrt (/ 1.0 (* h l))))
        (+ -1.0 (* (* D (/ D l)) (* (* t_1 (/ h d)) 0.125))))
       (if (<= d 1.55e-107)
         (* (sqrt (/ h (pow l 3.0))) (* -0.125 (* D (* D t_1))))
         (if (<= d 9.5e+105) t_0 (* d (/ (pow h -0.5) (sqrt l)))))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * sqrt(((d / h) * (d / l)));
	double t_1 = M / (d / M);
	double tmp;
	if (d <= -3.4e-91) {
		tmp = t_0;
	} else if (d <= -5e-310) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_1 * (h / d)) * 0.125)));
	} else if (d <= 1.55e-107) {
		tmp = sqrt((h / pow(l, 3.0))) * (-0.125 * (D * (D * t_1)));
	} else if (d <= 9.5e+105) {
		tmp = t_0;
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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 = (1.0d0 - (0.125d0 * (d_1 / (((d * d) / ((h * m) / (l / m))) / d_1)))) * sqrt(((d / h) * (d / l)))
    t_1 = m / (d / m)
    if (d <= (-3.4d-91)) then
        tmp = t_0
    else if (d <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) + ((d_1 * (d_1 / l)) * ((t_1 * (h / d)) * 0.125d0)))
    else if (d <= 1.55d-107) then
        tmp = sqrt((h / (l ** 3.0d0))) * ((-0.125d0) * (d_1 * (d_1 * t_1)))
    else if (d <= 9.5d+105) then
        tmp = t_0
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * Math.sqrt(((d / h) * (d / l)));
	double t_1 = M / (d / M);
	double tmp;
	if (d <= -3.4e-91) {
		tmp = t_0;
	} else if (d <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_1 * (h / d)) * 0.125)));
	} else if (d <= 1.55e-107) {
		tmp = Math.sqrt((h / Math.pow(l, 3.0))) * (-0.125 * (D * (D * t_1)));
	} else if (d <= 9.5e+105) {
		tmp = t_0;
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * math.sqrt(((d / h) * (d / l)))
	t_1 = M / (d / M)
	tmp = 0
	if d <= -3.4e-91:
		tmp = t_0
	elif d <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_1 * (h / d)) * 0.125)))
	elif d <= 1.55e-107:
		tmp = math.sqrt((h / math.pow(l, 3.0))) * (-0.125 * (D * (D * t_1)))
	elif d <= 9.5e+105:
		tmp = t_0
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(1.0 - Float64(0.125 * Float64(D / Float64(Float64(Float64(d * d) / Float64(Float64(h * M) / Float64(l / M))) / D)))) * sqrt(Float64(Float64(d / h) * Float64(d / l))))
	t_1 = Float64(M / Float64(d / M))
	tmp = 0.0
	if (d <= -3.4e-91)
		tmp = t_0;
	elseif (d <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 + Float64(Float64(D * Float64(D / l)) * Float64(Float64(t_1 * Float64(h / d)) * 0.125))));
	elseif (d <= 1.55e-107)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(-0.125 * Float64(D * Float64(D * t_1))));
	elseif (d <= 9.5e+105)
		tmp = t_0;
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * sqrt(((d / h) * (d / l)));
	t_1 = M / (d / M);
	tmp = 0.0;
	if (d <= -3.4e-91)
		tmp = t_0;
	elseif (d <= -5e-310)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + ((D * (D / l)) * ((t_1 * (h / d)) * 0.125)));
	elseif (d <= 1.55e-107)
		tmp = sqrt((h / (l ^ 3.0))) * (-0.125 * (D * (D * t_1)));
	elseif (d <= 9.5e+105)
		tmp = t_0;
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
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[(1.0 - N[(0.125 * N[(D / N[(N[(N[(d * d), $MachinePrecision] / N[(N[(h * M), $MachinePrecision] / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.4e-91], t$95$0, If[LessEqual[d, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * N[(h / d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.55e-107], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(D * N[(D * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 9.5e+105], t$95$0, 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 < -3.40000000000000027e-91 or 1.55000000000000011e-107 < d < 9.4999999999999995e105

   1. Initial program 79.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 18.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 31.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 66.8%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 66.8%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 66.8%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 66.7%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 66.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in M around 0 57.5%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   7. Step-by-step derivation

      1. associate-/l* 57.8%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{{M}^{2} \cdot h}}}\right) \]

      2. *-commutative 57.8%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{\color{blue}{h \cdot {M}^{2}}}}\right) \]

      3. unpow2 57.8%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{D \cdot D}}{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}}\right) \]

      4. associate-/l* 58.8%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{D}{\frac{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}}{D}}}\right) \]

      5. *-commutative 58.8%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{\color{blue}{{d}^{2} \cdot \ell}}{h \cdot {M}^{2}}}{D}}\right) \]

      6. associate-/l* 59.8%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\color{blue}{\frac{{d}^{2}}{\frac{h \cdot {M}^{2}}{\ell}}}}{D}}\right) \]

      7. unpow2 59.8%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{\color{blue}{d \cdot d}}{\frac{h \cdot {M}^{2}}{\ell}}}{D}}\right) \]

      8. unpow2 59.8%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{\ell}}}{D}}\right) \]

      9. associate-*r* 61.7%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{\ell}}}{D}}\right) \]

      10. associate-/l* 65.8%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\color{blue}{\frac{h \cdot M}{\frac{\ell}{M}}}}}{D}}\right) \]

      11. *-commutative 65.8%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{M \cdot h}}{\frac{\ell}{M}}}}{D}}\right) \]

   8. Simplified 65.8%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{M \cdot h}{\frac{\ell}{M}}}}{D}}}\right) \]

   if -3.40000000000000027e-91 < d < -4.999999999999985e-310

   1. Initial program 49.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 8.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 13.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 44.0%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 44.0%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 44.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 44.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 44.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in d around -inf 77.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg 77.3%

         \[\leadsto \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      2. associate-/r* 77.3%

         \[\leadsto \left(-d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      3. distribute-rgt-neg-in 77.3%

         \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      4. associate-/l/ 77.3%

         \[\leadsto \left(d \cdot \left(-\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   8. Simplified 77.3%

      \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   9. Taylor expanded in M around 0 51.6%

      \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 51.6%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

       2. times-frac 59.2%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.125\right) \]

       3. unpow2 59.2%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{\color{blue}{d \cdot d}}\right) \cdot 0.125\right) \]

       4. times-frac 61.7%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{d} \cdot \frac{h}{d}\right)}\right) \cdot 0.125\right) \]

       5. unpow2 61.7%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \left(\frac{\color{blue}{M \cdot M}}{d} \cdot \frac{h}{d}\right)\right) \cdot 0.125\right) \]

       6. associate-*l* 61.7%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

       7. unpow2 61.7%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \frac{\color{blue}{D \cdot D}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       8. associate-*l/ 64.4%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{\ell} \cdot D\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       9. *-commutative 64.4%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       10. associate-/l* 67.0%

           \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

   11. Simplified 67.0%

       \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

   if -4.999999999999985e-310 < d < 1.55000000000000011e-107

   1. Initial program 47.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 47.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 47.9%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 46.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 46.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 46.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 46.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 47.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 46.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 46.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 46.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. associate-/l/ 46.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. *-commutative 46.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 46.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

   6. Taylor expanded in d around 0 39.5%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 39.5%

         \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative 39.5%

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      3. associate-*r/ 37.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)}\right) \]

      4. unpow2 37.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left({D}^{2} \cdot \frac{\color{blue}{M \cdot M}}{d}\right)\right) \]

      5. unpow2 37.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \frac{M \cdot M}{d}\right)\right) \]

      6. associate-*l* 41.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \color{blue}{\left(D \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right)}\right) \]

      7. associate-/l* 48.9%

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot \color{blue}{\frac{M}{\frac{d}{M}}}\right)\right)\right) \]

   8. Simplified 48.9%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot \frac{M}{\frac{d}{M}}\right)\right)\right)} \]

   if 9.4999999999999995e105 < d

   1. Initial program 73.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 73.2%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 73.2%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 73.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 75.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. associate-/l/ 75.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. *-commutative 75.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 75.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

   6. Taylor expanded in d around inf 66.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   7. Step-by-step derivation

      1. *-commutative 66.4%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. *-commutative 66.4%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. associate-/r* 66.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   8. Simplified 66.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   9. Step-by-step derivation

      1. sqrt-div 78.3%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

   10. Applied egg-rr 78.3%

       \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. div-inv 78.3%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{\frac{1}{h}} \cdot \frac{1}{\sqrt{\ell}}\right)} \]

       2. inv-pow 78.3%

          \[\leadsto d \cdot \left(\sqrt{\color{blue}{{h}^{-1}}} \cdot \frac{1}{\sqrt{\ell}}\right) \]

   12. Applied egg-rr 78.3%

       \[\leadsto d \cdot \color{blue}{\left(\sqrt{{h}^{-1}} \cdot \frac{1}{\sqrt{\ell}}\right)} \]
   13. Step-by-step derivation

       1. associate-*r/ 78.3%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}} \cdot 1}{\sqrt{\ell}}} \]

       2. associate-/l* 78.3%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}}}{\frac{\sqrt{\ell}}{1}}} \]

       3. sqr-pow 78.4%

          \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{\left(\frac{-1}{2}\right)} \cdot {h}^{\left(\frac{-1}{2}\right)}}}}{\frac{\sqrt{\ell}}{1}} \]

       4. rem-sqrt-square 78.4%

          \[\leadsto d \cdot \frac{\color{blue}{\left|{h}^{\left(\frac{-1}{2}\right)}\right|}}{\frac{\sqrt{\ell}}{1}} \]

       5. metadata-eval 78.4%

          \[\leadsto d \cdot \frac{\left|{h}^{\color{blue}{-0.5}}\right|}{\frac{\sqrt{\ell}}{1}} \]

       6. sqr-pow 78.2%

          \[\leadsto d \cdot \frac{\left|\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}\right|}{\frac{\sqrt{\ell}}{1}} \]

       7. fabs-sqr 78.2%

          \[\leadsto d \cdot \frac{\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}}{\frac{\sqrt{\ell}}{1}} \]

       8. sqr-pow 78.4%

          \[\leadsto d \cdot \frac{\color{blue}{{h}^{-0.5}}}{\frac{\sqrt{\ell}}{1}} \]

       9. /-rgt-identity 78.4%

          \[\leadsto d \cdot \frac{{h}^{-0.5}}{\color{blue}{\sqrt{\ell}}} \]

   14. Simplified 78.4%

       \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.4 \cdot 10^{-91}:\\ \;\;\;\;\left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{-107}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(D \cdot \left(D \cdot \frac{M}{\frac{d}{M}}\right)\right)\right)\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{+105}:\\ \;\;\;\;\left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 21: 54.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\\ t_1 := \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{if}\;d \leq -4.3 \cdot 10^{-90}:\\ \;\;\;\;\left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right) \cdot t_1\\ \mathbf{elif}\;d \leq 3.1 \cdot 10^{-301}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + t_0\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{+106}:\\ \;\;\;\;\left(1 - t_0\right) \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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 (/ D l)) (* (* (/ M (/ d M)) (/ h d)) 0.125)))
        (t_1 (sqrt (* (/ d h) (/ d l)))))
   (if (<= d -4.3e-90)
     (* (- 1.0 (* 0.125 (/ D (/ (/ (* d d) (/ (* h M) (/ l M))) D)))) t_1)
     (if (<= d 3.1e-301)
       (* (* d (sqrt (/ 1.0 (* h l)))) (+ -1.0 t_0))
       (if (<= d 3.8e+106)
         (* (- 1.0 t_0) t_1)
         (* d (/ (pow h -0.5) (sqrt l))))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125);
	double t_1 = sqrt(((d / h) * (d / l)));
	double tmp;
	if (d <= -4.3e-90) {
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * t_1;
	} else if (d <= 3.1e-301) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + t_0);
	} else if (d <= 3.8e+106) {
		tmp = (1.0 - t_0) * t_1;
	} else {
		tmp = d * (pow(h, -0.5) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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 * (d_1 / l)) * (((m / (d / m)) * (h / d)) * 0.125d0)
    t_1 = sqrt(((d / h) * (d / l)))
    if (d <= (-4.3d-90)) then
        tmp = (1.0d0 - (0.125d0 * (d_1 / (((d * d) / ((h * m) / (l / m))) / d_1)))) * t_1
    else if (d <= 3.1d-301) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) + t_0)
    else if (d <= 3.8d+106) then
        tmp = (1.0d0 - t_0) * t_1
    else
        tmp = d * ((h ** (-0.5d0)) / sqrt(l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125);
	double t_1 = Math.sqrt(((d / h) * (d / l)));
	double tmp;
	if (d <= -4.3e-90) {
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * t_1;
	} else if (d <= 3.1e-301) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 + t_0);
	} else if (d <= 3.8e+106) {
		tmp = (1.0 - t_0) * t_1;
	} else {
		tmp = d * (Math.pow(h, -0.5) / Math.sqrt(l));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = (D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125)
	t_1 = math.sqrt(((d / h) * (d / l)))
	tmp = 0
	if d <= -4.3e-90:
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * t_1
	elif d <= 3.1e-301:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 + t_0)
	elif d <= 3.8e+106:
		tmp = (1.0 - t_0) * t_1
	else:
		tmp = d * (math.pow(h, -0.5) / math.sqrt(l))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(D * Float64(D / l)) * Float64(Float64(Float64(M / Float64(d / M)) * Float64(h / d)) * 0.125))
	t_1 = sqrt(Float64(Float64(d / h) * Float64(d / l)))
	tmp = 0.0
	if (d <= -4.3e-90)
		tmp = Float64(Float64(1.0 - Float64(0.125 * Float64(D / Float64(Float64(Float64(d * d) / Float64(Float64(h * M) / Float64(l / M))) / D)))) * t_1);
	elseif (d <= 3.1e-301)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 + t_0));
	elseif (d <= 3.8e+106)
		tmp = Float64(Float64(1.0 - t_0) * t_1);
	else
		tmp = Float64(d * Float64((h ^ -0.5) / sqrt(l)));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125);
	t_1 = sqrt(((d / h) * (d / l)));
	tmp = 0.0;
	if (d <= -4.3e-90)
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * t_1;
	elseif (d <= 3.1e-301)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + t_0);
	elseif (d <= 3.8e+106)
		tmp = (1.0 - t_0) * t_1;
	else
		tmp = d * ((h ^ -0.5) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
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[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -4.3e-90], N[(N[(1.0 - N[(0.125 * N[(D / N[(N[(N[(d * d), $MachinePrecision] / N[(N[(h * M), $MachinePrecision] / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[d, 3.1e-301], 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[d, 3.8e+106], N[(N[(1.0 - t$95$0), $MachinePrecision] * t$95$1), $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 < -4.3000000000000002e-90

   1. Initial program 81.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 26.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 37.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 70.0%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 70.0%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 70.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 70.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 70.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in M around 0 54.4%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   7. Step-by-step derivation

      1. associate-/l* 54.3%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{{M}^{2} \cdot h}}}\right) \]

      2. *-commutative 54.3%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{\color{blue}{h \cdot {M}^{2}}}}\right) \]

      3. unpow2 54.3%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{D \cdot D}}{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}}\right) \]

      4. associate-/l* 56.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{D}{\frac{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}}{D}}}\right) \]

      5. *-commutative 56.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{\color{blue}{{d}^{2} \cdot \ell}}{h \cdot {M}^{2}}}{D}}\right) \]

      6. associate-/l* 57.6%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\color{blue}{\frac{{d}^{2}}{\frac{h \cdot {M}^{2}}{\ell}}}}{D}}\right) \]

      7. unpow2 57.6%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{\color{blue}{d \cdot d}}{\frac{h \cdot {M}^{2}}{\ell}}}{D}}\right) \]

      8. unpow2 57.6%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{\ell}}}{D}}\right) \]

      9. associate-*r* 59.3%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{\ell}}}{D}}\right) \]

      10. associate-/l* 62.0%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\color{blue}{\frac{h \cdot M}{\frac{\ell}{M}}}}}{D}}\right) \]

      11. *-commutative 62.0%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{M \cdot h}}{\frac{\ell}{M}}}}{D}}\right) \]

   8. Simplified 62.0%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{M \cdot h}{\frac{\ell}{M}}}}{D}}}\right) \]

   if -4.3000000000000002e-90 < d < 3.10000000000000014e-301

   1. Initial program 47.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 7.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 12.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 40.9%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 40.9%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 40.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 40.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 40.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in d around -inf 71.8%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg 71.8%

         \[\leadsto \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      2. associate-/r* 71.8%

         \[\leadsto \left(-d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      3. distribute-rgt-neg-in 71.8%

         \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      4. associate-/l/ 71.8%

         \[\leadsto \left(d \cdot \left(-\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   8. Simplified 71.8%

      \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   9. Taylor expanded in M around 0 47.9%

      \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 47.9%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

       2. times-frac 54.9%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.125\right) \]

       3. unpow2 54.9%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{\color{blue}{d \cdot d}}\right) \cdot 0.125\right) \]

       4. times-frac 57.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{d} \cdot \frac{h}{d}\right)}\right) \cdot 0.125\right) \]

       5. unpow2 57.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \left(\frac{\color{blue}{M \cdot M}}{d} \cdot \frac{h}{d}\right)\right) \cdot 0.125\right) \]

       6. associate-*l* 57.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

       7. unpow2 57.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \frac{\color{blue}{D \cdot D}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       8. associate-*l/ 59.8%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{\ell} \cdot D\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       9. *-commutative 59.8%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       10. associate-/l* 62.2%

           \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

   11. Simplified 62.2%

       \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

   if 3.10000000000000014e-301 < d < 3.7999999999999998e106

   1. Initial program 61.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 8.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 16.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 50.1%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 50.1%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 50.1%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 50.1%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 50.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in M around 0 34.6%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 0.5%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

      2. times-frac 0.5%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.125\right) \]

      3. unpow2 0.5%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{\color{blue}{d \cdot d}}\right) \cdot 0.125\right) \]

      4. times-frac 3.0%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{d} \cdot \frac{h}{d}\right)}\right) \cdot 0.125\right) \]

      5. unpow2 3.0%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \left(\frac{\color{blue}{M \cdot M}}{d} \cdot \frac{h}{d}\right)\right) \cdot 0.125\right) \]

      6. associate-*l* 3.0%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

      7. unpow2 3.0%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \frac{\color{blue}{D \cdot D}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

      8. associate-*l/ 2.8%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{\ell} \cdot D\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

      9. *-commutative 2.8%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

      10. associate-/l* 2.9%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

   8. Simplified 53.2%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

   if 3.7999999999999998e106 < d

   1. Initial program 73.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 73.2%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 73.2%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 73.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 75.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. associate-/l/ 75.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. *-commutative 75.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 75.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

   6. Taylor expanded in d around inf 66.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   7. Step-by-step derivation

      1. *-commutative 66.4%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. *-commutative 66.4%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. associate-/r* 66.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   8. Simplified 66.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   9. Step-by-step derivation

      1. sqrt-div 78.3%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

   10. Applied egg-rr 78.3%

       \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. div-inv 78.3%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{\frac{1}{h}} \cdot \frac{1}{\sqrt{\ell}}\right)} \]

       2. inv-pow 78.3%

          \[\leadsto d \cdot \left(\sqrt{\color{blue}{{h}^{-1}}} \cdot \frac{1}{\sqrt{\ell}}\right) \]

   12. Applied egg-rr 78.3%

       \[\leadsto d \cdot \color{blue}{\left(\sqrt{{h}^{-1}} \cdot \frac{1}{\sqrt{\ell}}\right)} \]
   13. Step-by-step derivation

       1. associate-*r/ 78.3%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}} \cdot 1}{\sqrt{\ell}}} \]

       2. associate-/l* 78.3%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{{h}^{-1}}}{\frac{\sqrt{\ell}}{1}}} \]

       3. sqr-pow 78.4%

          \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{\left(\frac{-1}{2}\right)} \cdot {h}^{\left(\frac{-1}{2}\right)}}}}{\frac{\sqrt{\ell}}{1}} \]

       4. rem-sqrt-square 78.4%

          \[\leadsto d \cdot \frac{\color{blue}{\left|{h}^{\left(\frac{-1}{2}\right)}\right|}}{\frac{\sqrt{\ell}}{1}} \]

       5. metadata-eval 78.4%

          \[\leadsto d \cdot \frac{\left|{h}^{\color{blue}{-0.5}}\right|}{\frac{\sqrt{\ell}}{1}} \]

       6. sqr-pow 78.2%

          \[\leadsto d \cdot \frac{\left|\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}\right|}{\frac{\sqrt{\ell}}{1}} \]

       7. fabs-sqr 78.2%

          \[\leadsto d \cdot \frac{\color{blue}{{h}^{\left(\frac{-0.5}{2}\right)} \cdot {h}^{\left(\frac{-0.5}{2}\right)}}}{\frac{\sqrt{\ell}}{1}} \]

       8. sqr-pow 78.4%

          \[\leadsto d \cdot \frac{\color{blue}{{h}^{-0.5}}}{\frac{\sqrt{\ell}}{1}} \]

       9. /-rgt-identity 78.4%

          \[\leadsto d \cdot \frac{{h}^{-0.5}}{\color{blue}{\sqrt{\ell}}} \]

   14. Simplified 78.4%

       \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.3 \cdot 10^{-90}:\\ \;\;\;\;\left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 3.1 \cdot 10^{-301}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{+106}:\\ \;\;\;\;\left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 22: 54.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\\ t_1 := \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{if}\;d \leq -3.75 \cdot 10^{-90}:\\ \;\;\;\;\left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right) \cdot t_1\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{-301}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + t_0\right)\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{+106}:\\ \;\;\;\;\left(1 - t_0\right) \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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 (/ D l)) (* (* (/ M (/ d M)) (/ h d)) 0.125)))
        (t_1 (sqrt (* (/ d h) (/ d l)))))
   (if (<= d -3.75e-90)
     (* (- 1.0 (* 0.125 (/ D (/ (/ (* d d) (/ (* h M) (/ l M))) D)))) t_1)
     (if (<= d 3.3e-301)
       (* (* d (sqrt (/ 1.0 (* h l)))) (+ -1.0 t_0))
       (if (<= d 5.5e+106) (* (- 1.0 t_0) t_1) (/ d (* (sqrt h) (sqrt l))))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125);
	double t_1 = sqrt(((d / h) * (d / l)));
	double tmp;
	if (d <= -3.75e-90) {
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * t_1;
	} else if (d <= 3.3e-301) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + t_0);
	} else if (d <= 5.5e+106) {
		tmp = (1.0 - t_0) * t_1;
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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 * (d_1 / l)) * (((m / (d / m)) * (h / d)) * 0.125d0)
    t_1 = sqrt(((d / h) * (d / l)))
    if (d <= (-3.75d-90)) then
        tmp = (1.0d0 - (0.125d0 * (d_1 / (((d * d) / ((h * m) / (l / m))) / d_1)))) * t_1
    else if (d <= 3.3d-301) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) + t_0)
    else if (d <= 5.5d+106) then
        tmp = (1.0d0 - t_0) * t_1
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125);
	double t_1 = Math.sqrt(((d / h) * (d / l)));
	double tmp;
	if (d <= -3.75e-90) {
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * t_1;
	} else if (d <= 3.3e-301) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 + t_0);
	} else if (d <= 5.5e+106) {
		tmp = (1.0 - t_0) * t_1;
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = (D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125)
	t_1 = math.sqrt(((d / h) * (d / l)))
	tmp = 0
	if d <= -3.75e-90:
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * t_1
	elif d <= 3.3e-301:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 + t_0)
	elif d <= 5.5e+106:
		tmp = (1.0 - t_0) * t_1
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(D * Float64(D / l)) * Float64(Float64(Float64(M / Float64(d / M)) * Float64(h / d)) * 0.125))
	t_1 = sqrt(Float64(Float64(d / h) * Float64(d / l)))
	tmp = 0.0
	if (d <= -3.75e-90)
		tmp = Float64(Float64(1.0 - Float64(0.125 * Float64(D / Float64(Float64(Float64(d * d) / Float64(Float64(h * M) / Float64(l / M))) / D)))) * t_1);
	elseif (d <= 3.3e-301)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 + t_0));
	elseif (d <= 5.5e+106)
		tmp = Float64(Float64(1.0 - t_0) * t_1);
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125);
	t_1 = sqrt(((d / h) * (d / l)));
	tmp = 0.0;
	if (d <= -3.75e-90)
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * t_1;
	elseif (d <= 3.3e-301)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + t_0);
	elseif (d <= 5.5e+106)
		tmp = (1.0 - t_0) * t_1;
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
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[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -3.75e-90], N[(N[(1.0 - N[(0.125 * N[(D / N[(N[(N[(d * d), $MachinePrecision] / N[(N[(h * M), $MachinePrecision] / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[d, 3.3e-301], 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[d, 5.5e+106], N[(N[(1.0 - t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.7499999999999999e-90

   1. Initial program 81.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 26.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 37.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 70.0%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 70.0%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 70.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 70.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 70.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in M around 0 54.4%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   7. Step-by-step derivation

      1. associate-/l* 54.3%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{{M}^{2} \cdot h}}}\right) \]

      2. *-commutative 54.3%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{\color{blue}{h \cdot {M}^{2}}}}\right) \]

      3. unpow2 54.3%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{D \cdot D}}{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}}\right) \]

      4. associate-/l* 56.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{D}{\frac{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}}{D}}}\right) \]

      5. *-commutative 56.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{\color{blue}{{d}^{2} \cdot \ell}}{h \cdot {M}^{2}}}{D}}\right) \]

      6. associate-/l* 57.6%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\color{blue}{\frac{{d}^{2}}{\frac{h \cdot {M}^{2}}{\ell}}}}{D}}\right) \]

      7. unpow2 57.6%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{\color{blue}{d \cdot d}}{\frac{h \cdot {M}^{2}}{\ell}}}{D}}\right) \]

      8. unpow2 57.6%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{\ell}}}{D}}\right) \]

      9. associate-*r* 59.3%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{\ell}}}{D}}\right) \]

      10. associate-/l* 62.0%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\color{blue}{\frac{h \cdot M}{\frac{\ell}{M}}}}}{D}}\right) \]

      11. *-commutative 62.0%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{M \cdot h}}{\frac{\ell}{M}}}}{D}}\right) \]

   8. Simplified 62.0%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{M \cdot h}{\frac{\ell}{M}}}}{D}}}\right) \]

   if -3.7499999999999999e-90 < d < 3.3e-301

   1. Initial program 47.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 7.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 12.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 40.9%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 40.9%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 40.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 40.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 40.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in d around -inf 71.8%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg 71.8%

         \[\leadsto \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      2. associate-/r* 71.8%

         \[\leadsto \left(-d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      3. distribute-rgt-neg-in 71.8%

         \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      4. associate-/l/ 71.8%

         \[\leadsto \left(d \cdot \left(-\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   8. Simplified 71.8%

      \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   9. Taylor expanded in M around 0 47.9%

      \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 47.9%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

       2. times-frac 54.9%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.125\right) \]

       3. unpow2 54.9%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{\color{blue}{d \cdot d}}\right) \cdot 0.125\right) \]

       4. times-frac 57.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{d} \cdot \frac{h}{d}\right)}\right) \cdot 0.125\right) \]

       5. unpow2 57.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \left(\frac{\color{blue}{M \cdot M}}{d} \cdot \frac{h}{d}\right)\right) \cdot 0.125\right) \]

       6. associate-*l* 57.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

       7. unpow2 57.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \frac{\color{blue}{D \cdot D}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       8. associate-*l/ 59.8%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{\ell} \cdot D\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       9. *-commutative 59.8%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       10. associate-/l* 62.2%

           \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

   11. Simplified 62.2%

       \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

   if 3.3e-301 < d < 5.5e106

   1. Initial program 61.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 8.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 16.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 50.1%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 50.1%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 50.1%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 50.1%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 50.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in M around 0 34.6%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 0.5%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

      2. times-frac 0.5%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.125\right) \]

      3. unpow2 0.5%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{\color{blue}{d \cdot d}}\right) \cdot 0.125\right) \]

      4. times-frac 3.0%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{d} \cdot \frac{h}{d}\right)}\right) \cdot 0.125\right) \]

      5. unpow2 3.0%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \left(\frac{\color{blue}{M \cdot M}}{d} \cdot \frac{h}{d}\right)\right) \cdot 0.125\right) \]

      6. associate-*l* 3.0%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

      7. unpow2 3.0%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \frac{\color{blue}{D \cdot D}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

      8. associate-*l/ 2.8%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{\ell} \cdot D\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

      9. *-commutative 2.8%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

      10. associate-/l* 2.9%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

   8. Simplified 53.2%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

   if 5.5e106 < d

   1. Initial program 73.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 73.2%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 73.2%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 73.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. associate-*l/ 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. associate-*r/ 75.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      11. associate-/l/ 75.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      12. *-commutative 75.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 75.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

   6. Taylor expanded in d around inf 66.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   7. Step-by-step derivation

      1. *-commutative 66.4%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. *-commutative 66.4%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. associate-/r* 66.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   8. Simplified 66.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
   9. Step-by-step derivation

      1. associate-/r* 66.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      2. associate-/l/ 66.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

      3. add-cbrt-cube 57.6%

         \[\leadsto d \cdot \color{blue}{\sqrt[3]{\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}} \]

      4. add-sqr-sqrt 57.6%

         \[\leadsto d \cdot \sqrt[3]{\color{blue}{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]

      5. pow1 57.6%

         \[\leadsto \color{blue}{{\left(d \cdot \sqrt[3]{\frac{\frac{1}{\ell}}{h} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}\right)}^{1}} \]

      6. add-sqr-sqrt 57.6%

         \[\leadsto {\left(d \cdot \sqrt[3]{\color{blue}{\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}\right)}^{1} \]

      7. add-cbrt-cube 66.4%

         \[\leadsto {\left(d \cdot \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}}}\right)}^{1} \]

      8. associate-/l/ 66.4%

         \[\leadsto {\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)}^{1} \]

      9. sqrt-div 66.3%

         \[\leadsto {\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)}^{1} \]

      10. metadata-eval 66.3%

          \[\leadsto {\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)}^{1} \]

      11. *-commutative 66.3%

          \[\leadsto {\left(d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}}\right)}^{1} \]

   10. Applied egg-rr 66.3%

       \[\leadsto \color{blue}{{\left(d \cdot \frac{1}{\sqrt{\ell \cdot h}}\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 66.3%

          \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{\ell \cdot h}}} \]

       2. associate-*r/ 66.4%

          \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{\ell \cdot h}}} \]

       3. *-rgt-identity 66.4%

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell \cdot h}} \]

   12. Simplified 66.4%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   13. Step-by-step derivation

       1. sqrt-prod 78.3%

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

   14. Applied egg-rr 78.3%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.75 \cdot 10^{-90}:\\ \;\;\;\;\left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{-301}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{+106}:\\ \;\;\;\;\left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 23: 53.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\\ t_1 := \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{if}\;d \leq -5.2 \cdot 10^{-92}:\\ \;\;\;\;\left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right) \cdot t_1\\ \mathbf{elif}\;d \leq 3.1 \cdot 10^{-301}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + t_0\right)\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{+104}:\\ \;\;\;\;\left(1 - t_0\right) \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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 (/ D l)) (* (* (/ M (/ d M)) (/ h d)) 0.125)))
        (t_1 (sqrt (* (/ d h) (/ d l)))))
   (if (<= d -5.2e-92)
     (* (- 1.0 (* 0.125 (/ D (/ (/ (* d d) (/ (* h M) (/ l M))) D)))) t_1)
     (if (<= d 3.1e-301)
       (* (* d (sqrt (/ 1.0 (* h l)))) (+ -1.0 t_0))
       (if (<= d 2.1e+104) (* (- 1.0 t_0) t_1) (* d (pow (* h l) -0.5)))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125);
	double t_1 = sqrt(((d / h) * (d / l)));
	double tmp;
	if (d <= -5.2e-92) {
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * t_1;
	} else if (d <= 3.1e-301) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + t_0);
	} else if (d <= 2.1e+104) {
		tmp = (1.0 - t_0) * t_1;
	} else {
		tmp = d * pow((h * l), -0.5);
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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 * (d_1 / l)) * (((m / (d / m)) * (h / d)) * 0.125d0)
    t_1 = sqrt(((d / h) * (d / l)))
    if (d <= (-5.2d-92)) then
        tmp = (1.0d0 - (0.125d0 * (d_1 / (((d * d) / ((h * m) / (l / m))) / d_1)))) * t_1
    else if (d <= 3.1d-301) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) + t_0)
    else if (d <= 2.1d+104) then
        tmp = (1.0d0 - t_0) * t_1
    else
        tmp = d * ((h * l) ** (-0.5d0))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125);
	double t_1 = Math.sqrt(((d / h) * (d / l)));
	double tmp;
	if (d <= -5.2e-92) {
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * t_1;
	} else if (d <= 3.1e-301) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 + t_0);
	} else if (d <= 2.1e+104) {
		tmp = (1.0 - t_0) * t_1;
	} else {
		tmp = d * Math.pow((h * l), -0.5);
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = (D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125)
	t_1 = math.sqrt(((d / h) * (d / l)))
	tmp = 0
	if d <= -5.2e-92:
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * t_1
	elif d <= 3.1e-301:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 + t_0)
	elif d <= 2.1e+104:
		tmp = (1.0 - t_0) * t_1
	else:
		tmp = d * math.pow((h * l), -0.5)
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(D * Float64(D / l)) * Float64(Float64(Float64(M / Float64(d / M)) * Float64(h / d)) * 0.125))
	t_1 = sqrt(Float64(Float64(d / h) * Float64(d / l)))
	tmp = 0.0
	if (d <= -5.2e-92)
		tmp = Float64(Float64(1.0 - Float64(0.125 * Float64(D / Float64(Float64(Float64(d * d) / Float64(Float64(h * M) / Float64(l / M))) / D)))) * t_1);
	elseif (d <= 3.1e-301)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 + t_0));
	elseif (d <= 2.1e+104)
		tmp = Float64(Float64(1.0 - t_0) * t_1);
	else
		tmp = Float64(d * (Float64(h * l) ^ -0.5));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125);
	t_1 = sqrt(((d / h) * (d / l)));
	tmp = 0.0;
	if (d <= -5.2e-92)
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * t_1;
	elseif (d <= 3.1e-301)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + t_0);
	elseif (d <= 2.1e+104)
		tmp = (1.0 - t_0) * t_1;
	else
		tmp = d * ((h * l) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
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[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -5.2e-92], N[(N[(1.0 - N[(0.125 * N[(D / N[(N[(N[(d * d), $MachinePrecision] / N[(N[(h * M), $MachinePrecision] / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[d, 3.1e-301], 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[d, 2.1e+104], N[(N[(1.0 - t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -5.2e-92

   1. Initial program 81.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 26.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 37.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 70.0%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 70.0%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 70.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 70.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 70.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in M around 0 54.4%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   7. Step-by-step derivation

      1. associate-/l* 54.3%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{{M}^{2} \cdot h}}}\right) \]

      2. *-commutative 54.3%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{\color{blue}{h \cdot {M}^{2}}}}\right) \]

      3. unpow2 54.3%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{D \cdot D}}{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}}\right) \]

      4. associate-/l* 56.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{D}{\frac{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}}{D}}}\right) \]

      5. *-commutative 56.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{\color{blue}{{d}^{2} \cdot \ell}}{h \cdot {M}^{2}}}{D}}\right) \]

      6. associate-/l* 57.6%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\color{blue}{\frac{{d}^{2}}{\frac{h \cdot {M}^{2}}{\ell}}}}{D}}\right) \]

      7. unpow2 57.6%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{\color{blue}{d \cdot d}}{\frac{h \cdot {M}^{2}}{\ell}}}{D}}\right) \]

      8. unpow2 57.6%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{\ell}}}{D}}\right) \]

      9. associate-*r* 59.3%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{\ell}}}{D}}\right) \]

      10. associate-/l* 62.0%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\color{blue}{\frac{h \cdot M}{\frac{\ell}{M}}}}}{D}}\right) \]

      11. *-commutative 62.0%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{M \cdot h}}{\frac{\ell}{M}}}}{D}}\right) \]

   8. Simplified 62.0%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{M \cdot h}{\frac{\ell}{M}}}}{D}}}\right) \]

   if -5.2e-92 < d < 3.10000000000000014e-301

   1. Initial program 47.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 7.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 12.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 40.9%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 40.9%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 40.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 40.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 40.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in d around -inf 71.8%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg 71.8%

         \[\leadsto \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      2. associate-/r* 71.8%

         \[\leadsto \left(-d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      3. distribute-rgt-neg-in 71.8%

         \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      4. associate-/l/ 71.8%

         \[\leadsto \left(d \cdot \left(-\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   8. Simplified 71.8%

      \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   9. Taylor expanded in M around 0 47.9%

      \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 47.9%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

       2. times-frac 54.9%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.125\right) \]

       3. unpow2 54.9%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{\color{blue}{d \cdot d}}\right) \cdot 0.125\right) \]

       4. times-frac 57.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{d} \cdot \frac{h}{d}\right)}\right) \cdot 0.125\right) \]

       5. unpow2 57.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \left(\frac{\color{blue}{M \cdot M}}{d} \cdot \frac{h}{d}\right)\right) \cdot 0.125\right) \]

       6. associate-*l* 57.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

       7. unpow2 57.3%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \frac{\color{blue}{D \cdot D}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       8. associate-*l/ 59.8%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{\ell} \cdot D\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       9. *-commutative 59.8%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

       10. associate-/l* 62.2%

           \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

   11. Simplified 62.2%

       \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

   if 3.10000000000000014e-301 < d < 2.0999999999999998e104

   1. Initial program 61.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 8.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 16.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 50.1%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 50.1%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 50.1%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 50.1%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 50.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in M around 0 34.6%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 0.5%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

      2. times-frac 0.5%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.125\right) \]

      3. unpow2 0.5%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{\color{blue}{d \cdot d}}\right) \cdot 0.125\right) \]

      4. times-frac 3.0%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{d} \cdot \frac{h}{d}\right)}\right) \cdot 0.125\right) \]

      5. unpow2 3.0%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \left(\frac{\color{blue}{M \cdot M}}{d} \cdot \frac{h}{d}\right)\right) \cdot 0.125\right) \]

      6. associate-*l* 3.0%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

      7. unpow2 3.0%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \frac{\color{blue}{D \cdot D}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

      8. associate-*l/ 2.8%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{\ell} \cdot D\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

      9. *-commutative 2.8%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

      10. associate-/l* 2.9%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

   8. Simplified 53.2%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

   if 2.0999999999999998e104 < d

   1. Initial program 73.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 73.2%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 73.2%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 73.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. pow1/2 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      2. sqr-pow 73.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      3. pow2 73.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval 73.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.25}}\right)}^{2}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied egg-rr 73.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{{\left({\left(\frac{d}{\ell}\right)}^{0.25}\right)}^{2}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   6. Taylor expanded in d around inf 66.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   7. Step-by-step derivation

      1. *-commutative 66.4%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. unpow-1 66.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \]

      3. sqr-pow 66.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}}} \]

      4. rem-sqrt-square 66.5%

         \[\leadsto d \cdot \color{blue}{\left|{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}\right|} \]

      5. sqr-pow 66.1%

         \[\leadsto d \cdot \left|\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \]

      6. fabs-sqr 66.1%

         \[\leadsto d \cdot \color{blue}{\left({\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \]

      7. sqr-pow 66.5%

         \[\leadsto d \cdot \color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}} \]

      8. metadata-eval 66.5%

         \[\leadsto d \cdot {\left(\ell \cdot h\right)}^{\color{blue}{-0.5}} \]

   8. Simplified 66.5%

      \[\leadsto \color{blue}{d \cdot {\left(\ell \cdot h\right)}^{-0.5}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.2 \cdot 10^{-92}:\\ \;\;\;\;\left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 3.1 \cdot 10^{-301}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right)\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{+104}:\\ \;\;\;\;\left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 24: 48.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{if}\;d \leq -9.8 \cdot 10^{-110}:\\ \;\;\;\;\left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right) \cdot t_0\\ \mathbf{elif}\;d \leq 4.8 \cdot 10^{+105}:\\ \;\;\;\;\left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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) (/ d l)))))
   (if (<= d -9.8e-110)
     (* (- 1.0 (* 0.125 (/ D (/ (/ (* d d) (/ (* h M) (/ l M))) D)))) t_0)
     (if (<= d 4.8e+105)
       (* (- 1.0 (* (* D (/ D l)) (* (* (/ M (/ d M)) (/ h d)) 0.125))) t_0)
       (* d (pow (* h l) -0.5))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(((d / h) * (d / l)));
	double tmp;
	if (d <= -9.8e-110) {
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * t_0;
	} else if (d <= 4.8e+105) {
		tmp = (1.0 - ((D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125))) * t_0;
	} else {
		tmp = d * pow((h * l), -0.5);
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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(((d / h) * (d / l)))
    if (d <= (-9.8d-110)) then
        tmp = (1.0d0 - (0.125d0 * (d_1 / (((d * d) / ((h * m) / (l / m))) / d_1)))) * t_0
    else if (d <= 4.8d+105) then
        tmp = (1.0d0 - ((d_1 * (d_1 / l)) * (((m / (d / m)) * (h / d)) * 0.125d0))) * t_0
    else
        tmp = d * ((h * l) ** (-0.5d0))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(((d / h) * (d / l)));
	double tmp;
	if (d <= -9.8e-110) {
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * t_0;
	} else if (d <= 4.8e+105) {
		tmp = (1.0 - ((D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125))) * t_0;
	} else {
		tmp = d * Math.pow((h * l), -0.5);
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt(((d / h) * (d / l)))
	tmp = 0
	if d <= -9.8e-110:
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * t_0
	elif d <= 4.8e+105:
		tmp = (1.0 - ((D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125))) * t_0
	else:
		tmp = d * math.pow((h * l), -0.5)
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(Float64(d / h) * Float64(d / l)))
	tmp = 0.0
	if (d <= -9.8e-110)
		tmp = Float64(Float64(1.0 - Float64(0.125 * Float64(D / Float64(Float64(Float64(d * d) / Float64(Float64(h * M) / Float64(l / M))) / D)))) * t_0);
	elseif (d <= 4.8e+105)
		tmp = Float64(Float64(1.0 - Float64(Float64(D * Float64(D / l)) * Float64(Float64(Float64(M / Float64(d / M)) * Float64(h / d)) * 0.125))) * t_0);
	else
		tmp = Float64(d * (Float64(h * l) ^ -0.5));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(((d / h) * (d / l)));
	tmp = 0.0;
	if (d <= -9.8e-110)
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * t_0;
	elseif (d <= 4.8e+105)
		tmp = (1.0 - ((D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125))) * t_0;
	else
		tmp = d * ((h * l) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
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[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -9.8e-110], N[(N[(1.0 - N[(0.125 * N[(D / N[(N[(N[(d * d), $MachinePrecision] / N[(N[(h * M), $MachinePrecision] / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[d, 4.8e+105], N[(N[(1.0 - N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -9.7999999999999995e-110

   1. Initial program 80.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 26.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 36.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 68.9%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 68.9%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 68.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 68.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 68.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in M around 0 52.7%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   7. Step-by-step derivation

      1. associate-/l* 52.7%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{{M}^{2} \cdot h}}}\right) \]

      2. *-commutative 52.7%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{\color{blue}{h \cdot {M}^{2}}}}\right) \]

      3. unpow2 52.7%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{D \cdot D}}{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}}\right) \]

      4. associate-/l* 54.3%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{D}{\frac{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}}{D}}}\right) \]

      5. *-commutative 54.3%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{\color{blue}{{d}^{2} \cdot \ell}}{h \cdot {M}^{2}}}{D}}\right) \]

      6. associate-/l* 57.2%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\color{blue}{\frac{{d}^{2}}{\frac{h \cdot {M}^{2}}{\ell}}}}{D}}\right) \]

      7. unpow2 57.2%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{\color{blue}{d \cdot d}}{\frac{h \cdot {M}^{2}}{\ell}}}{D}}\right) \]

      8. unpow2 57.2%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{\ell}}}{D}}\right) \]

      9. associate-*r* 58.8%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{\ell}}}{D}}\right) \]

      10. associate-/l* 61.3%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\color{blue}{\frac{h \cdot M}{\frac{\ell}{M}}}}}{D}}\right) \]

      11. *-commutative 61.3%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{M \cdot h}}{\frac{\ell}{M}}}}{D}}\right) \]

   8. Simplified 61.3%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{M \cdot h}{\frac{\ell}{M}}}}{D}}}\right) \]

   if -9.7999999999999995e-110 < d < 4.7999999999999995e105

   1. Initial program 57.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 7.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 15.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 47.3%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 47.3%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 47.3%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 47.2%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 47.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in M around 0 31.6%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 13.5%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

      2. times-frac 14.9%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.125\right) \]

      3. unpow2 14.9%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{\color{blue}{d \cdot d}}\right) \cdot 0.125\right) \]

      4. times-frac 17.4%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{d} \cdot \frac{h}{d}\right)}\right) \cdot 0.125\right) \]

      5. unpow2 17.4%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \left(\frac{\color{blue}{M \cdot M}}{d} \cdot \frac{h}{d}\right)\right) \cdot 0.125\right) \]

      6. associate-*l* 17.4%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

      7. unpow2 17.4%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \frac{\color{blue}{D \cdot D}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

      8. associate-*l/ 18.1%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{\ell} \cdot D\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

      9. *-commutative 18.1%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

      10. associate-/l* 18.8%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

   8. Simplified 47.3%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

   if 4.7999999999999995e105 < d

   1. Initial program 73.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 73.2%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 73.2%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 73.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. pow1/2 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      2. sqr-pow 73.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      3. pow2 73.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval 73.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.25}}\right)}^{2}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied egg-rr 73.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{{\left({\left(\frac{d}{\ell}\right)}^{0.25}\right)}^{2}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   6. Taylor expanded in d around inf 66.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   7. Step-by-step derivation

      1. *-commutative 66.4%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. unpow-1 66.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \]

      3. sqr-pow 66.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}}} \]

      4. rem-sqrt-square 66.5%

         \[\leadsto d \cdot \color{blue}{\left|{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}\right|} \]

      5. sqr-pow 66.1%

         \[\leadsto d \cdot \left|\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \]

      6. fabs-sqr 66.1%

         \[\leadsto d \cdot \color{blue}{\left({\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \]

      7. sqr-pow 66.5%

         \[\leadsto d \cdot \color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}} \]

      8. metadata-eval 66.5%

         \[\leadsto d \cdot {\left(\ell \cdot h\right)}^{\color{blue}{-0.5}} \]

   8. Simplified 66.5%

      \[\leadsto \color{blue}{d \cdot {\left(\ell \cdot h\right)}^{-0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.8 \cdot 10^{-110}:\\ \;\;\;\;\left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 4.8 \cdot 10^{+105}:\\ \;\;\;\;\left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 25: 55.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 2.45 \cdot 10^{-301}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right)\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{+106}:\\ \;\;\;\;\left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d 2.45e-301)
   (*
    (* d (sqrt (/ 1.0 (* h l))))
    (+ -1.0 (* 0.125 (/ D (/ (/ (* d d) (/ (* h M) (/ l M))) D)))))
   (if (<= d 4.5e+106)
     (*
      (- 1.0 (* (* D (/ D l)) (* (* (/ M (/ d M)) (/ h d)) 0.125)))
      (sqrt (* (/ d h) (/ d l))))
     (* d (pow (* h l) -0.5)))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 2.45e-301) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D))));
	} else if (d <= 4.5e+106) {
		tmp = (1.0 - ((D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125))) * sqrt(((d / h) * (d / l)));
	} else {
		tmp = d * pow((h * l), -0.5);
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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 <= 2.45d-301) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) + (0.125d0 * (d_1 / (((d * d) / ((h * m) / (l / m))) / d_1))))
    else if (d <= 4.5d+106) then
        tmp = (1.0d0 - ((d_1 * (d_1 / l)) * (((m / (d / m)) * (h / d)) * 0.125d0))) * sqrt(((d / h) * (d / l)))
    else
        tmp = d * ((h * l) ** (-0.5d0))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 2.45e-301) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 + (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D))));
	} else if (d <= 4.5e+106) {
		tmp = (1.0 - ((D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125))) * Math.sqrt(((d / h) * (d / l)));
	} else {
		tmp = d * Math.pow((h * l), -0.5);
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if d <= 2.45e-301:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 + (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D))))
	elif d <= 4.5e+106:
		tmp = (1.0 - ((D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125))) * math.sqrt(((d / h) * (d / l)))
	else:
		tmp = d * math.pow((h * l), -0.5)
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= 2.45e-301)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 + Float64(0.125 * Float64(D / Float64(Float64(Float64(d * d) / Float64(Float64(h * M) / Float64(l / M))) / D)))));
	elseif (d <= 4.5e+106)
		tmp = Float64(Float64(1.0 - Float64(Float64(D * Float64(D / l)) * Float64(Float64(Float64(M / Float64(d / M)) * Float64(h / d)) * 0.125))) * sqrt(Float64(Float64(d / h) * Float64(d / l))));
	else
		tmp = Float64(d * (Float64(h * l) ^ -0.5));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= 2.45e-301)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D))));
	elseif (d <= 4.5e+106)
		tmp = (1.0 - ((D * (D / l)) * (((M / (d / M)) * (h / d)) * 0.125))) * sqrt(((d / h) * (d / l)));
	else
		tmp = d * ((h * l) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, 2.45e-301], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(0.125 * N[(D / N[(N[(N[(d * d), $MachinePrecision] / N[(N[(h * M), $MachinePrecision] / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.5e+106], N[(N[(1.0 - N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < 2.45e-301

   1. Initial program 69.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 19.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 27.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 59.3%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 59.3%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 59.3%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 59.3%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 59.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in d around -inf 70.2%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg 70.2%

         \[\leadsto \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      2. associate-/r* 70.2%

         \[\leadsto \left(-d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      3. distribute-rgt-neg-in 70.2%

         \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      4. associate-/l/ 70.2%

         \[\leadsto \left(d \cdot \left(-\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   8. Simplified 70.2%

      \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   9. Taylor expanded in M around 0 52.4%

      \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   10. Step-by-step derivation

       1. associate-/l* 43.9%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{{M}^{2} \cdot h}}}\right) \]

       2. *-commutative 43.9%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{\color{blue}{h \cdot {M}^{2}}}}\right) \]

       3. unpow2 43.9%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{D \cdot D}}{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}}\right) \]

       4. associate-/l* 45.0%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{D}{\frac{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}}{D}}}\right) \]

       5. *-commutative 45.0%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{\color{blue}{{d}^{2} \cdot \ell}}{h \cdot {M}^{2}}}{D}}\right) \]

       6. associate-/l* 47.8%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\color{blue}{\frac{{d}^{2}}{\frac{h \cdot {M}^{2}}{\ell}}}}{D}}\right) \]

       7. unpow2 47.8%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{\color{blue}{d \cdot d}}{\frac{h \cdot {M}^{2}}{\ell}}}{D}}\right) \]

       8. unpow2 47.8%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{\ell}}}{D}}\right) \]

       9. associate-*r* 48.9%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{\ell}}}{D}}\right) \]

       10. associate-/l* 50.5%

           \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\color{blue}{\frac{h \cdot M}{\frac{\ell}{M}}}}}{D}}\right) \]

       11. *-commutative 50.5%

           \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{M \cdot h}}{\frac{\ell}{M}}}}{D}}\right) \]

   11. Simplified 61.1%

       \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{M \cdot h}{\frac{\ell}{M}}}}{D}}}\right) \]

   if 2.45e-301 < d < 4.4999999999999997e106

   1. Initial program 60.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 7.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 16.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 49.6%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 49.6%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 49.6%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 49.6%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 49.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in M around 0 34.2%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 0.5%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.125}\right) \]

      2. times-frac 0.5%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \cdot 0.125\right) \]

      3. unpow2 0.5%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{\color{blue}{d \cdot d}}\right) \cdot 0.125\right) \]

      4. times-frac 2.9%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{d} \cdot \frac{h}{d}\right)}\right) \cdot 0.125\right) \]

      5. unpow2 2.9%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(\frac{{D}^{2}}{\ell} \cdot \left(\frac{\color{blue}{M \cdot M}}{d} \cdot \frac{h}{d}\right)\right) \cdot 0.125\right) \]

      6. associate-*l* 2.9%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\frac{{D}^{2}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

      7. unpow2 2.9%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \frac{\color{blue}{D \cdot D}}{\ell} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

      8. associate-*l/ 2.8%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(\frac{D}{\ell} \cdot D\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

      9. *-commutative 2.8%

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right)} \cdot \left(\left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

      10. associate-/l* 2.8%

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{M}}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \]

   8. Simplified 52.6%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)}\right) \]

   if 4.4999999999999997e106 < d

   1. Initial program 73.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 73.2%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 73.2%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 73.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. pow1/2 73.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      2. sqr-pow 73.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      3. pow2 73.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval 73.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.25}}\right)}^{2}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied egg-rr 73.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{{\left({\left(\frac{d}{\ell}\right)}^{0.25}\right)}^{2}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   6. Taylor expanded in d around inf 66.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   7. Step-by-step derivation

      1. *-commutative 66.4%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. unpow-1 66.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \]

      3. sqr-pow 66.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}}} \]

      4. rem-sqrt-square 66.5%

         \[\leadsto d \cdot \color{blue}{\left|{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}\right|} \]

      5. sqr-pow 66.1%

         \[\leadsto d \cdot \left|\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \]

      6. fabs-sqr 66.1%

         \[\leadsto d \cdot \color{blue}{\left({\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \]

      7. sqr-pow 66.5%

         \[\leadsto d \cdot \color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}} \]

      8. metadata-eval 66.5%

         \[\leadsto d \cdot {\left(\ell \cdot h\right)}^{\color{blue}{-0.5}} \]

   8. Simplified 66.5%

      \[\leadsto \color{blue}{d \cdot {\left(\ell \cdot h\right)}^{-0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2.45 \cdot 10^{-301}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right)\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{+106}:\\ \;\;\;\;\left(1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right) \cdot 0.125\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 26: 47.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 9.5 \cdot 10^{+29}:\\ \;\;\;\;\left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l 9.5e+29)
   (*
    (- 1.0 (* 0.125 (/ D (/ (/ (* d d) (/ (* h M) (/ l M))) D))))
    (sqrt (* (/ d h) (/ d l))))
   (* d (pow (* h l) -0.5))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 9.5e+29) {
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * sqrt(((d / h) * (d / l)));
	} else {
		tmp = d * pow((h * l), -0.5);
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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 <= 9.5d+29) then
        tmp = (1.0d0 - (0.125d0 * (d_1 / (((d * d) / ((h * m) / (l / m))) / d_1)))) * sqrt(((d / h) * (d / l)))
    else
        tmp = d * ((h * l) ** (-0.5d0))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 9.5e+29) {
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * Math.sqrt(((d / h) * (d / l)));
	} else {
		tmp = d * Math.pow((h * l), -0.5);
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= 9.5e+29:
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * math.sqrt(((d / h) * (d / l)))
	else:
		tmp = d * math.pow((h * l), -0.5)
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 9.5e+29)
		tmp = Float64(Float64(1.0 - Float64(0.125 * Float64(D / Float64(Float64(Float64(d * d) / Float64(Float64(h * M) / Float64(l / M))) / D)))) * sqrt(Float64(Float64(d / h) * Float64(d / l))));
	else
		tmp = Float64(d * (Float64(h * l) ^ -0.5));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 9.5e+29)
		tmp = (1.0 - (0.125 * (D / (((d * d) / ((h * M) / (l / M))) / D)))) * sqrt(((d / h) * (d / l)));
	else
		tmp = d * ((h * l) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, 9.5e+29], N[(N[(1.0 - N[(0.125 * N[(D / N[(N[(N[(d * d), $MachinePrecision] / N[(N[(h * M), $MachinePrecision] / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 9.5000000000000003e29

   1. Initial program 69.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 16.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 22.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 59.6%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 59.6%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 59.6%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 59.5%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 59.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in M around 0 41.8%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   7. Step-by-step derivation

      1. associate-/l* 42.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{{M}^{2} \cdot h}}}\right) \]

      2. *-commutative 42.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{\color{blue}{h \cdot {M}^{2}}}}\right) \]

      3. unpow2 42.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{D \cdot D}}{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}}\right) \]

      4. associate-/l* 44.1%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{D}{\frac{\frac{\ell \cdot {d}^{2}}{h \cdot {M}^{2}}}{D}}}\right) \]

      5. *-commutative 44.1%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{\color{blue}{{d}^{2} \cdot \ell}}{h \cdot {M}^{2}}}{D}}\right) \]

      6. associate-/l* 46.1%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\color{blue}{\frac{{d}^{2}}{\frac{h \cdot {M}^{2}}{\ell}}}}{D}}\right) \]

      7. unpow2 46.1%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{\color{blue}{d \cdot d}}{\frac{h \cdot {M}^{2}}{\ell}}}{D}}\right) \]

      8. unpow2 46.1%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{\ell}}}{D}}\right) \]

      9. associate-*r* 48.8%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{\ell}}}{D}}\right) \]

      10. associate-/l* 51.6%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\color{blue}{\frac{h \cdot M}{\frac{\ell}{M}}}}}{D}}\right) \]

      11. *-commutative 51.6%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{\color{blue}{M \cdot h}}{\frac{\ell}{M}}}}{D}}\right) \]

   8. Simplified 51.6%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{M \cdot h}{\frac{\ell}{M}}}}{D}}}\right) \]

   if 9.5000000000000003e29 < l

   1. Initial program 56.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 56.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 56.4%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 56.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 56.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 56.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 56.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 54.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 54.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 54.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. pow1/2 54.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      2. sqr-pow 54.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      3. pow2 54.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval 54.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.25}}\right)}^{2}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied egg-rr 54.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{{\left({\left(\frac{d}{\ell}\right)}^{0.25}\right)}^{2}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   6. Taylor expanded in d around inf 48.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   7. Step-by-step derivation

      1. *-commutative 48.5%

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. unpow-1 48.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \]

      3. sqr-pow 48.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}}} \]

      4. rem-sqrt-square 48.5%

         \[\leadsto d \cdot \color{blue}{\left|{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}\right|} \]

      5. sqr-pow 48.3%

         \[\leadsto d \cdot \left|\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \]

      6. fabs-sqr 48.3%

         \[\leadsto d \cdot \color{blue}{\left({\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \]

      7. sqr-pow 48.5%

         \[\leadsto d \cdot \color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}} \]

      8. metadata-eval 48.5%

         \[\leadsto d \cdot {\left(\ell \cdot h\right)}^{\color{blue}{-0.5}} \]

   8. Simplified 48.5%

      \[\leadsto \color{blue}{d \cdot {\left(\ell \cdot h\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9.5 \cdot 10^{+29}:\\ \;\;\;\;\left(1 - 0.125 \cdot \frac{D}{\frac{\frac{d \cdot d}{\frac{h \cdot M}{\frac{\ell}{M}}}}{D}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 27: 38.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -7 \cdot 10^{-179}:\\ \;\;\;\;\left(-d\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= h -7e-179) (* (- d) (pow (* h l) -0.5)) (* d (sqrt (/ 1.0 (* h l))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -7e-179) {
		tmp = -d * pow((h * l), -0.5);
	} else {
		tmp = d * sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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 <= (-7d-179)) then
        tmp = -d * ((h * l) ** (-0.5d0))
    else
        tmp = d * sqrt((1.0d0 / (h * l)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -7e-179) {
		tmp = -d * Math.pow((h * l), -0.5);
	} else {
		tmp = d * Math.sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if h <= -7e-179:
		tmp = -d * math.pow((h * l), -0.5)
	else:
		tmp = d * math.sqrt((1.0 / (h * l)))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -7e-179)
		tmp = Float64(Float64(-d) * (Float64(h * l) ^ -0.5));
	else
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(h * l))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= -7e-179)
		tmp = -d * ((h * l) ^ -0.5);
	else
		tmp = d * sqrt((1.0 / (h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[h, -7e-179], N[((-d) * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -7.00000000000000049e-179

   1. Initial program 68.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied egg-rr 19.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 31.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 56.9%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. *-commutative 56.9%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      4. *-commutative 56.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

      5. associate-/l* 56.9%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   5. Simplified 56.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)} \]

   6. Taylor expanded in d around -inf 69.8%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg 69.8%

         \[\leadsto \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      2. associate-/r* 69.8%

         \[\leadsto \left(-d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      3. distribute-rgt-neg-in 69.8%

         \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

      4. associate-/l/ 69.8%

         \[\leadsto \left(d \cdot \left(-\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)\right) \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   8. Simplified 69.8%

      \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

   9. Taylor expanded in d around inf 33.6%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg 33.6%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

       2. *-commutative 33.6%

          \[\leadsto -\color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

       3. distribute-rgt-neg-in 33.6%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]

       4. unpow-1 33.6%

          \[\leadsto \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \cdot \left(-d\right) \]

       5. sqr-pow 33.6%

          \[\leadsto \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}}} \cdot \left(-d\right) \]

       6. rem-sqrt-square 33.6%

          \[\leadsto \color{blue}{\left|{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot \left(-d\right) \]

       7. sqr-pow 33.5%

          \[\leadsto \left|\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot \left(-d\right) \]

       8. fabs-sqr 33.5%

          \[\leadsto \color{blue}{\left({\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot \left(-d\right) \]

       9. sqr-pow 33.6%

          \[\leadsto \color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}} \cdot \left(-d\right) \]

       10. metadata-eval 33.6%

           \[\leadsto {\left(\ell \cdot h\right)}^{\color{blue}{-0.5}} \cdot \left(-d\right) \]

   11. Simplified 33.6%

       \[\leadsto \color{blue}{{\left(\ell \cdot h\right)}^{-0.5} \cdot \left(-d\right)} \]

   if -7.00000000000000049e-179 < h

   1. Initial program 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 38.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -7 \cdot 10^{-179}:\\ \;\;\;\;\left(-d\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]

Alternative 28: 25.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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

M = abs(M);
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 should be positive before calling this function
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

M = Math.abs(M);
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 = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	return d * math.sqrt(((1.0 / h) / l))

Julia

M = abs(M)
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 = abs(M)
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 should be positive before calling this function
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 66.6%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Step-by-step derivation

   1. metadata-eval 66.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. unpow1/2 66.6%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. metadata-eval 66.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. unpow1/2 66.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. *-commutative 66.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

   6. associate-*l* 66.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

   7. times-frac 66.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. metadata-eval 66.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

3. Simplified 66.2%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* 66.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

   2. frac-times 66.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

   3. *-commutative 66.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

   4. metadata-eval 66.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. associate-*r/ 68.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

   6. metadata-eval 68.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

   7. *-commutative 68.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

   8. frac-times 68.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   9. associate-*l/ 68.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   10. associate-*r/ 68.2%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   11. associate-/l/ 68.2%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   12. *-commutative 68.2%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

5. Applied egg-rr 68.2%

   \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

6. Taylor expanded in d around inf 28.4%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
7. Step-by-step derivation

   1. *-commutative 28.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   2. *-commutative 28.4%

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   3. associate-/r* 28.4%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

8. Simplified 28.4%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]

9. Final simplification 28.4%

   \[\leadsto d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \]

Alternative 29: 25.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ d \cdot \sqrt{\frac{1}{h \cdot \ell}} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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

M = abs(M);
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 should be positive before calling this function
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

M = Math.abs(M);
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 = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	return d * math.sqrt((1.0 / (h * l)))

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	return Float64(d * sqrt(Float64(1.0 / Float64(h * l))))
end

MATLAB

M = abs(M)
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 should be positive before calling this function
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[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.6%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

2. Taylor expanded in d around inf 28.4%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

3. Final simplification 28.4%

   \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

Alternative 30: 25.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ d \cdot {\left(h \cdot \ell\right)}^{-0.5} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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

M = abs(M);
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 should be positive before calling this function
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

M = Math.abs(M);
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 = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	return d * math.pow((h * l), -0.5)

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	return Float64(d * (Float64(h * l) ^ -0.5))
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp = code(d, h, l, M, D)
	tmp = d * ((h * l) ^ -0.5);
end

Wolfram

NOTE: M should be positive before calling this function
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 66.6%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Step-by-step derivation

   1. metadata-eval 66.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. unpow1/2 66.6%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. metadata-eval 66.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. unpow1/2 66.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. *-commutative 66.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

   6. associate-*l* 66.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

   7. times-frac 66.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. metadata-eval 66.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

3. Simplified 66.2%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
4. Step-by-step derivation

   1. pow1/2 66.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   2. sqr-pow 66.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   3. pow2 66.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   4. metadata-eval 66.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.25}}\right)}^{2}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

5. Applied egg-rr 66.2%

   \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{{\left({\left(\frac{d}{\ell}\right)}^{0.25}\right)}^{2}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

6. Taylor expanded in d around inf 28.4%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
7. Step-by-step derivation

   1. *-commutative 28.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   2. unpow-1 28.4%

      \[\leadsto d \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \]

   3. sqr-pow 28.4%

      \[\leadsto d \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}}} \]

   4. rem-sqrt-square 28.1%

      \[\leadsto d \cdot \color{blue}{\left|{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}\right|} \]

   5. sqr-pow 28.0%

      \[\leadsto d \cdot \left|\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \]

   6. fabs-sqr 28.0%

      \[\leadsto d \cdot \color{blue}{\left({\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \]

   7. sqr-pow 28.1%

      \[\leadsto d \cdot \color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}} \]

   8. metadata-eval 28.1%

      \[\leadsto d \cdot {\left(\ell \cdot h\right)}^{\color{blue}{-0.5}} \]

8. Simplified 28.1%

   \[\leadsto \color{blue}{d \cdot {\left(\ell \cdot h\right)}^{-0.5}} \]

9. Final simplification 28.1%

   \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{-0.5} \]

Alternative 31: 25.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \frac{d}{\sqrt{h \cdot \ell}} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D) :precision binary64 (/ d (sqrt (* h l))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	return d / sqrt((h * l));
}

Fortran

NOTE: M should be positive before calling this function
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((h * l))
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	return d / Math.sqrt((h * l));
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	return d / math.sqrt((h * l))

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	return Float64(d / sqrt(Float64(h * l)))
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp = code(d, h, l, M, D)
	tmp = d / sqrt((h * l));
end

Wolfram

NOTE: M should be positive before calling this function
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[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.6%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Step-by-step derivation

   1. metadata-eval 66.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. unpow1/2 66.6%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. metadata-eval 66.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. unpow1/2 66.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. *-commutative 66.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

   6. associate-*l* 66.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

   7. times-frac 66.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. metadata-eval 66.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

3. Simplified 66.2%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* 66.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

   2. frac-times 66.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

   3. *-commutative 66.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

   4. metadata-eval 66.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. associate-*r/ 68.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

   6. metadata-eval 68.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

   7. *-commutative 68.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

   8. frac-times 68.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   9. associate-*l/ 68.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   10. associate-*r/ 68.2%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   11. associate-/l/ 68.2%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   12. *-commutative 68.2%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

5. Applied egg-rr 68.2%

   \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

6. Taylor expanded in d around inf 28.4%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
7. Step-by-step derivation

   1. *-commutative 28.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   2. *-commutative 28.4%

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   3. associate-/r* 28.4%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

8. Simplified 28.4%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
9. Step-by-step derivation

   1. associate-/r* 28.4%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

   2. associate-/l/ 28.4%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   3. add-cbrt-cube 26.3%

      \[\leadsto d \cdot \color{blue}{\sqrt[3]{\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}} \]

   4. add-sqr-sqrt 26.3%

      \[\leadsto d \cdot \sqrt[3]{\color{blue}{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]

   5. pow1 26.3%

      \[\leadsto \color{blue}{{\left(d \cdot \sqrt[3]{\frac{\frac{1}{\ell}}{h} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}\right)}^{1}} \]

   6. add-sqr-sqrt 26.3%

      \[\leadsto {\left(d \cdot \sqrt[3]{\color{blue}{\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}\right)}^{1} \]

   7. add-cbrt-cube 28.4%

      \[\leadsto {\left(d \cdot \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}}}\right)}^{1} \]

   8. associate-/l/ 28.4%

      \[\leadsto {\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)}^{1} \]

   9. sqrt-div 28.1%

      \[\leadsto {\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)}^{1} \]

   10. metadata-eval 28.1%

       \[\leadsto {\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)}^{1} \]

   11. *-commutative 28.1%

       \[\leadsto {\left(d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}}\right)}^{1} \]

10. Applied egg-rr 28.1%

    \[\leadsto \color{blue}{{\left(d \cdot \frac{1}{\sqrt{\ell \cdot h}}\right)}^{1}} \]
11. Step-by-step derivation

    1. unpow1 28.1%

       \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{\ell \cdot h}}} \]

    2. associate-*r/ 28.1%

       \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{\ell \cdot h}}} \]

    3. *-rgt-identity 28.1%

       \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell \cdot h}} \]

12. Simplified 28.1%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

13. Final simplification 28.1%

    \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

Reproduce

herbie shell --seed 2023207 
(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)))))