Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.0% → 78.6%

Time: 27.1s

Alternatives: 24

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.0% 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: 78.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -3.4 \cdot 10^{+111}:\\ \;\;\;\;\frac{t\_0}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D\_m \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{t\_0}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D\_m}{d \cdot \frac{2}{M\_m}}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+51}:\\ \;\;\;\;\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(D\_m \cdot \left(0.5 \cdot \frac{M\_m}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (- d))))
   (if (<= l -3.4e+111)
     (*
      (/ t_0 (sqrt (- l)))
      (*
       (sqrt (/ d h))
       (+ 1.0 (* (/ h l) (* -0.5 (pow (* D_m (/ (/ M_m 2.0) d)) 2.0))))))
     (if (<= l -2e-310)
       (*
        (sqrt (/ d l))
        (*
         (/ t_0 (sqrt (- h)))
         (+ 1.0 (* (/ h l) (* -0.5 (pow (/ D_m (* d (/ 2.0 M_m))) 2.0))))))
       (if (<= l 4e+51)
         (*
          (* d (/ 1.0 (sqrt (* h l))))
          (- 1.0 (* 0.5 (/ (* h (pow (* D_m (* 0.5 (/ M_m d))) 2.0)) l))))
         (*
          (/ d (* (sqrt l) (sqrt h)))
          (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0))))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt(-d);
	double tmp;
	if (l <= -3.4e+111) {
		tmp = (t_0 / sqrt(-l)) * (sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * pow((D_m * ((M_m / 2.0) / d)), 2.0)))));
	} else if (l <= -2e-310) {
		tmp = sqrt((d / l)) * ((t_0 / sqrt(-h)) * (1.0 + ((h / l) * (-0.5 * pow((D_m / (d * (2.0 / M_m))), 2.0)))));
	} else if (l <= 4e+51) {
		tmp = (d * (1.0 / sqrt((h * l)))) * (1.0 - (0.5 * ((h * pow((D_m * (0.5 * (M_m / d))), 2.0)) / l)));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (0.5 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(-d)
    if (l <= (-3.4d+111)) then
        tmp = (t_0 / sqrt(-l)) * (sqrt((d / h)) * (1.0d0 + ((h / l) * ((-0.5d0) * ((d_m * ((m_m / 2.0d0) / d)) ** 2.0d0)))))
    else if (l <= (-2d-310)) then
        tmp = sqrt((d / l)) * ((t_0 / sqrt(-h)) * (1.0d0 + ((h / l) * ((-0.5d0) * ((d_m / (d * (2.0d0 / m_m))) ** 2.0d0)))))
    else if (l <= 4d+51) then
        tmp = (d * (1.0d0 / sqrt((h * l)))) * (1.0d0 - (0.5d0 * ((h * ((d_m * (0.5d0 * (m_m / d))) ** 2.0d0)) / l)))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - (0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_m / d)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt(-d);
	double tmp;
	if (l <= -3.4e+111) {
		tmp = (t_0 / Math.sqrt(-l)) * (Math.sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * Math.pow((D_m * ((M_m / 2.0) / d)), 2.0)))));
	} else if (l <= -2e-310) {
		tmp = Math.sqrt((d / l)) * ((t_0 / Math.sqrt(-h)) * (1.0 + ((h / l) * (-0.5 * Math.pow((D_m / (d * (2.0 / M_m))), 2.0)))));
	} else if (l <= 4e+51) {
		tmp = (d * (1.0 / Math.sqrt((h * l)))) * (1.0 - (0.5 * ((h * Math.pow((D_m * (0.5 * (M_m / d))), 2.0)) / l)));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - (0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D_m / d)), 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt(-d)
	tmp = 0
	if l <= -3.4e+111:
		tmp = (t_0 / math.sqrt(-l)) * (math.sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * math.pow((D_m * ((M_m / 2.0) / d)), 2.0)))))
	elif l <= -2e-310:
		tmp = math.sqrt((d / l)) * ((t_0 / math.sqrt(-h)) * (1.0 + ((h / l) * (-0.5 * math.pow((D_m / (d * (2.0 / M_m))), 2.0)))))
	elif l <= 4e+51:
		tmp = (d * (1.0 / math.sqrt((h * l)))) * (1.0 - (0.5 * ((h * math.pow((D_m * (0.5 * (M_m / d))), 2.0)) / l)))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - (0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D_m / d)), 2.0))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -3.4e+111)
		tmp = Float64(Float64(t_0 / sqrt(Float64(-l))) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(D_m * Float64(Float64(M_m / 2.0) / d)) ^ 2.0))))));
	elseif (l <= -2e-310)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(D_m / Float64(d * Float64(2.0 / M_m))) ^ 2.0))))));
	elseif (l <= 4e+51)
		tmp = Float64(Float64(d * Float64(1.0 / sqrt(Float64(h * l)))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(D_m * Float64(0.5 * Float64(M_m / d))) ^ 2.0)) / l))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt(-d);
	tmp = 0.0;
	if (l <= -3.4e+111)
		tmp = (t_0 / sqrt(-l)) * (sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * ((D_m * ((M_m / 2.0) / d)) ^ 2.0)))));
	elseif (l <= -2e-310)
		tmp = sqrt((d / l)) * ((t_0 / sqrt(-h)) * (1.0 + ((h / l) * (-0.5 * ((D_m / (d * (2.0 / M_m))) ^ 2.0)))));
	elseif (l <= 4e+51)
		tmp = (d * (1.0 / sqrt((h * l)))) * (1.0 - (0.5 * ((h * ((D_m * (0.5 * (M_m / d))) ^ 2.0)) / l)));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (0.5 * ((h / l) * (((M_m / 2.0) * (D_m / d)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -3.4e+111], N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D$95$m * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D$95$m / N[(d * N[(2.0 / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4e+51], N[(N[(d * N[(1.0 / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(D$95$m * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -3.4000000000000001e111

   1. Initial program 59.3%

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

   3. Simplified 59.1%

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

      1. frac-2neg 62.0%

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

      2. sqrt-div 80.0%

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

   6. Applied egg-rr 77.2%

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

   if -3.4000000000000001e111 < l < -1.999999999999994e-310

   1. Initial program 73.9%

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

   3. Simplified 71.2%

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

      1. associate-*r/ 74.0%

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

      2. div-inv 74.0%

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

      3. associate-*r* 74.0%

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

      4. *-commutative 74.0%

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

      5. div-inv 74.0%

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

      6. associate-/r* 74.0%

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

      7. frac-times 72.0%

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

      8. clear-num 72.0%

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

      9. frac-times 71.5%

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

      10. *-un-lft-identity 71.5%

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

   6. Applied egg-rr 71.5%

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

      1. frac-2neg 71.5%

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

      2. sqrt-div 80.2%

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

   8. Applied egg-rr 80.2%

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

   if -1.999999999999994e-310 < l < 4e51

   1. Initial program 73.9%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in d around 0 75.1%

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

      1. sqrt-div 76.6%

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

      2. metadata-eval 76.6%

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

   7. Applied egg-rr 76.6%

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

      1. associate-*r/ 84.4%

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

   9. Applied egg-rr 89.1%

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

   if 4e51 < l

   1. Initial program 42.5%

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

   3. Simplified 42.5%

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

      1. *-commutative 42.5%

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

      2. sqrt-div 47.4%

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

      3. sqrt-div 68.6%

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

      4. frac-times 68.6%

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

      5. add-sqr-sqrt 68.9%

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

   6. Applied egg-rr 68.9%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.4 \cdot 10^{+111}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d \cdot \frac{2}{M}}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+51}:\\ \;\;\;\;\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left({\left({\left(\frac{d}{h}\right)}^{0.25}\right)}^{2} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D\_m \cdot \left(0.5 \cdot \frac{M\_m}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-103}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -1e-309)
   (*
    (* (pow (pow (/ d h) 0.25) 2.0) (/ (sqrt (- d)) (sqrt (- l))))
    (- 1.0 (* 0.5 (pow (* (* D_m (* 0.5 (/ M_m d))) (sqrt (/ h l))) 2.0))))
   (if (<= d 7e-103)
     (*
      d
      (*
       (pow (* h l) -0.5)
       (- 1.0 (* (/ h l) (* 0.5 (pow (* D_m (* M_m (/ 0.5 d))) 2.0))))))
     (*
      (/ d (* (sqrt l) (sqrt h)))
      (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1e-309) {
		tmp = (pow(pow((d / h), 0.25), 2.0) * (sqrt(-d) / sqrt(-l))) * (1.0 - (0.5 * pow(((D_m * (0.5 * (M_m / d))) * sqrt((h / l))), 2.0)));
	} else if (d <= 7e-103) {
		tmp = d * (pow((h * l), -0.5) * (1.0 - ((h / l) * (0.5 * pow((D_m * (M_m * (0.5 / d))), 2.0)))));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (0.5 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-1d-309)) then
        tmp = ((((d / h) ** 0.25d0) ** 2.0d0) * (sqrt(-d) / sqrt(-l))) * (1.0d0 - (0.5d0 * (((d_m * (0.5d0 * (m_m / d))) * sqrt((h / l))) ** 2.0d0)))
    else if (d <= 7d-103) then
        tmp = d * (((h * l) ** (-0.5d0)) * (1.0d0 - ((h / l) * (0.5d0 * ((d_m * (m_m * (0.5d0 / d))) ** 2.0d0)))))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - (0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_m / d)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1e-309) {
		tmp = (Math.pow(Math.pow((d / h), 0.25), 2.0) * (Math.sqrt(-d) / Math.sqrt(-l))) * (1.0 - (0.5 * Math.pow(((D_m * (0.5 * (M_m / d))) * Math.sqrt((h / l))), 2.0)));
	} else if (d <= 7e-103) {
		tmp = d * (Math.pow((h * l), -0.5) * (1.0 - ((h / l) * (0.5 * Math.pow((D_m * (M_m * (0.5 / d))), 2.0)))));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - (0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D_m / d)), 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -1e-309:
		tmp = (math.pow(math.pow((d / h), 0.25), 2.0) * (math.sqrt(-d) / math.sqrt(-l))) * (1.0 - (0.5 * math.pow(((D_m * (0.5 * (M_m / d))) * math.sqrt((h / l))), 2.0)))
	elif d <= 7e-103:
		tmp = d * (math.pow((h * l), -0.5) * (1.0 - ((h / l) * (0.5 * math.pow((D_m * (M_m * (0.5 / d))), 2.0)))))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - (0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D_m / d)), 2.0))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -1e-309)
		tmp = Float64(Float64(((Float64(d / h) ^ 0.25) ^ 2.0) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l)))) * Float64(1.0 - Float64(0.5 * (Float64(Float64(D_m * Float64(0.5 * Float64(M_m / d))) * sqrt(Float64(h / l))) ^ 2.0))));
	elseif (d <= 7e-103)
		tmp = Float64(d * Float64((Float64(h * l) ^ -0.5) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(D_m * Float64(M_m * Float64(0.5 / d))) ^ 2.0))))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -1e-309)
		tmp = ((((d / h) ^ 0.25) ^ 2.0) * (sqrt(-d) / sqrt(-l))) * (1.0 - (0.5 * (((D_m * (0.5 * (M_m / d))) * sqrt((h / l))) ^ 2.0)));
	elseif (d <= 7e-103)
		tmp = d * (((h * l) ^ -0.5) * (1.0 - ((h / l) * (0.5 * ((D_m * (M_m * (0.5 / d))) ^ 2.0)))));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (0.5 * ((h / l) * (((M_m / 2.0) * (D_m / d)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -1e-309], N[(N[(N[Power[N[Power[N[(d / h), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[Power[N[(N[(D$95$m * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7e-103], N[(d * N[(N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.000000000000002e-309

   1. Initial program 70.5%

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

   3. Simplified 69.0%

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

      1. add-sqr-sqrt 69.0%

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

      2. pow2 69.0%

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

      3. sqrt-prod 69.0%

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

      4. sqrt-pow1 71.2%

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

      5. frac-times 74.2%

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

      6. associate-/r* 74.2%

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

      7. metadata-eval 74.2%

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

      8. pow1 74.2%

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

      9. div-inv 74.2%

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

      10. *-commutative 74.2%

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

      11. associate-*r* 74.2%

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

      12. div-inv 74.2%

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

      13. associate-*r/ 70.6%

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

      14. associate-/l/ 70.6%

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

      15. *-un-lft-identity 70.6%

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

      16. *-commutative 70.6%

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

      17. times-frac 70.6%

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

      18. metadata-eval 70.6%

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

   6. Applied egg-rr 70.6%

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

      1. pow1 70.6%

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

      2. sqrt-pow2 70.6%

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

      3. sqr-pow 70.5%

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

      4. pow2 70.5%

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

      5. metadata-eval 70.5%

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

      6. metadata-eval 70.5%

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

   8. Applied egg-rr 70.5%

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

      1. frac-2neg 70.5%

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

      2. sqrt-div 77.5%

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

   10. Applied egg-rr 77.5%

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

   if -1.000000000000002e-309 < d < 7.00000000000000032e-103

   1. Initial program 41.0%

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

   3. Simplified 34.1%

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

   5. Taylor expanded in d around 0 41.7%

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

      1. pow1 41.7%

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

   7. Applied egg-rr 58.4%

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

      1. unpow1 58.4%

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

      2. *-commutative 58.4%

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

      3. *-commutative 58.4%

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

      4. associate-*r/ 56.0%

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

      5. associate-/l* 56.0%

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

   9. Simplified 56.0%

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

   if 7.00000000000000032e-103 < d

   1. Initial program 67.4%

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

   3. Simplified 67.4%

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

      1. *-commutative 67.4%

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

      2. sqrt-div 71.7%

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

      3. sqrt-div 89.6%

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

      4. frac-times 89.6%

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

      5. add-sqr-sqrt 89.9%

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

   6. Applied egg-rr 89.9%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left({\left({\left(\frac{d}{h}\right)}^{0.25}\right)}^{2} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-103}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -2.45 \cdot 10^{+111}:\\ \;\;\;\;t\_0 \cdot \left(\frac{t\_1}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{t\_1}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D\_m}{d \cdot \frac{2}{M\_m}}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{+51}:\\ \;\;\;\;\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(D\_m \cdot \left(0.5 \cdot \frac{M\_m}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0)))))
        (t_1 (sqrt (- d))))
   (if (<= l -2.45e+111)
     (* t_0 (* (/ t_1 (sqrt (- l))) (sqrt (/ d h))))
     (if (<= l -2e-310)
       (*
        (sqrt (/ d l))
        (*
         (/ t_1 (sqrt (- h)))
         (+ 1.0 (* (/ h l) (* -0.5 (pow (/ D_m (* d (/ 2.0 M_m))) 2.0))))))
       (if (<= l 7.5e+51)
         (*
          (* d (/ 1.0 (sqrt (* h l))))
          (- 1.0 (* 0.5 (/ (* h (pow (* D_m (* 0.5 (/ M_m d))) 2.0)) l))))
         (* (/ d (* (sqrt l) (sqrt h))) t_0))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 1.0 - (0.5 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0)));
	double t_1 = sqrt(-d);
	double tmp;
	if (l <= -2.45e+111) {
		tmp = t_0 * ((t_1 / sqrt(-l)) * sqrt((d / h)));
	} else if (l <= -2e-310) {
		tmp = sqrt((d / l)) * ((t_1 / sqrt(-h)) * (1.0 + ((h / l) * (-0.5 * pow((D_m / (d * (2.0 / M_m))), 2.0)))));
	} else if (l <= 7.5e+51) {
		tmp = (d * (1.0 / sqrt((h * l)))) * (1.0 - (0.5 * ((h * pow((D_m * (0.5 * (M_m / d))), 2.0)) / l)));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * t_0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_m / d)) ** 2.0d0)))
    t_1 = sqrt(-d)
    if (l <= (-2.45d+111)) then
        tmp = t_0 * ((t_1 / sqrt(-l)) * sqrt((d / h)))
    else if (l <= (-2d-310)) then
        tmp = sqrt((d / l)) * ((t_1 / sqrt(-h)) * (1.0d0 + ((h / l) * ((-0.5d0) * ((d_m / (d * (2.0d0 / m_m))) ** 2.0d0)))))
    else if (l <= 7.5d+51) then
        tmp = (d * (1.0d0 / sqrt((h * l)))) * (1.0d0 - (0.5d0 * ((h * ((d_m * (0.5d0 * (m_m / d))) ** 2.0d0)) / l)))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 1.0 - (0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D_m / d)), 2.0)));
	double t_1 = Math.sqrt(-d);
	double tmp;
	if (l <= -2.45e+111) {
		tmp = t_0 * ((t_1 / Math.sqrt(-l)) * Math.sqrt((d / h)));
	} else if (l <= -2e-310) {
		tmp = Math.sqrt((d / l)) * ((t_1 / Math.sqrt(-h)) * (1.0 + ((h / l) * (-0.5 * Math.pow((D_m / (d * (2.0 / M_m))), 2.0)))));
	} else if (l <= 7.5e+51) {
		tmp = (d * (1.0 / Math.sqrt((h * l)))) * (1.0 - (0.5 * ((h * Math.pow((D_m * (0.5 * (M_m / d))), 2.0)) / l)));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = 1.0 - (0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D_m / d)), 2.0)))
	t_1 = math.sqrt(-d)
	tmp = 0
	if l <= -2.45e+111:
		tmp = t_0 * ((t_1 / math.sqrt(-l)) * math.sqrt((d / h)))
	elif l <= -2e-310:
		tmp = math.sqrt((d / l)) * ((t_1 / math.sqrt(-h)) * (1.0 + ((h / l) * (-0.5 * math.pow((D_m / (d * (2.0 / M_m))), 2.0)))))
	elif l <= 7.5e+51:
		tmp = (d * (1.0 / math.sqrt((h * l)))) * (1.0 - (0.5 * ((h * math.pow((D_m * (0.5 * (M_m / d))), 2.0)) / l)))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * t_0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0))))
	t_1 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -2.45e+111)
		tmp = Float64(t_0 * Float64(Float64(t_1 / sqrt(Float64(-l))) * sqrt(Float64(d / h))));
	elseif (l <= -2e-310)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(t_1 / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(D_m / Float64(d * Float64(2.0 / M_m))) ^ 2.0))))));
	elseif (l <= 7.5e+51)
		tmp = Float64(Float64(d * Float64(1.0 / sqrt(Float64(h * l)))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(D_m * Float64(0.5 * Float64(M_m / d))) ^ 2.0)) / l))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * t_0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = 1.0 - (0.5 * ((h / l) * (((M_m / 2.0) * (D_m / d)) ^ 2.0)));
	t_1 = sqrt(-d);
	tmp = 0.0;
	if (l <= -2.45e+111)
		tmp = t_0 * ((t_1 / sqrt(-l)) * sqrt((d / h)));
	elseif (l <= -2e-310)
		tmp = sqrt((d / l)) * ((t_1 / sqrt(-h)) * (1.0 + ((h / l) * (-0.5 * ((D_m / (d * (2.0 / M_m))) ^ 2.0)))));
	elseif (l <= 7.5e+51)
		tmp = (d * (1.0 / sqrt((h * l)))) * (1.0 - (0.5 * ((h * ((D_m * (0.5 * (M_m / d))) ^ 2.0)) / l)));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -2.45e+111], N[(t$95$0 * N[(N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D$95$m / N[(d * N[(2.0 / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.5e+51], N[(N[(d * N[(1.0 / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(D$95$m * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.4499999999999998e111

   1. Initial program 59.3%

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

   3. Simplified 59.3%

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

      1. frac-2neg 62.0%

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

      2. sqrt-div 80.0%

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

   6. Applied egg-rr 77.4%

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

   if -2.4499999999999998e111 < l < -1.999999999999994e-310

   1. Initial program 73.9%

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

   3. Simplified 71.2%

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

      1. associate-*r/ 74.0%

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

      2. div-inv 74.0%

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

      3. associate-*r* 74.0%

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

      4. *-commutative 74.0%

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

      5. div-inv 74.0%

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

      6. associate-/r* 74.0%

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

      7. frac-times 72.0%

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

      8. clear-num 72.0%

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

      9. frac-times 71.5%

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

      10. *-un-lft-identity 71.5%

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

   6. Applied egg-rr 71.5%

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

      1. frac-2neg 71.5%

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

      2. sqrt-div 80.2%

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

   8. Applied egg-rr 80.2%

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

   if -1.999999999999994e-310 < l < 7.4999999999999999e51

   1. Initial program 73.9%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in d around 0 75.1%

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

      1. sqrt-div 76.6%

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

      2. metadata-eval 76.6%

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

   7. Applied egg-rr 76.6%

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

      1. associate-*r/ 84.4%

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

   9. Applied egg-rr 89.1%

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

   if 7.4999999999999999e51 < l

   1. Initial program 42.5%

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

   3. Simplified 42.5%

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

      1. *-commutative 42.5%

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

      2. sqrt-div 47.4%

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

      3. sqrt-div 68.6%

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

      4. frac-times 68.6%

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

      5. add-sqr-sqrt 68.9%

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

   6. Applied egg-rr 68.9%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.45 \cdot 10^{+111}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d \cdot \frac{2}{M}}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{+51}:\\ \;\;\;\;\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D\_m}{d \cdot \frac{2}{M\_m}}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{-106}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -1e-309)
   (*
    (sqrt (/ d l))
    (*
     (/ (sqrt (- d)) (sqrt (- h)))
     (+ 1.0 (* (/ h l) (* -0.5 (pow (/ D_m (* d (/ 2.0 M_m))) 2.0))))))
   (if (<= d 2.2e-106)
     (*
      d
      (*
       (pow (* h l) -0.5)
       (- 1.0 (* (/ h l) (* 0.5 (pow (* D_m (* M_m (/ 0.5 d))) 2.0))))))
     (*
      (/ d (* (sqrt l) (sqrt h)))
      (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1e-309) {
		tmp = sqrt((d / l)) * ((sqrt(-d) / sqrt(-h)) * (1.0 + ((h / l) * (-0.5 * pow((D_m / (d * (2.0 / M_m))), 2.0)))));
	} else if (d <= 2.2e-106) {
		tmp = d * (pow((h * l), -0.5) * (1.0 - ((h / l) * (0.5 * pow((D_m * (M_m * (0.5 / d))), 2.0)))));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (0.5 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-1d-309)) then
        tmp = sqrt((d / l)) * ((sqrt(-d) / sqrt(-h)) * (1.0d0 + ((h / l) * ((-0.5d0) * ((d_m / (d * (2.0d0 / m_m))) ** 2.0d0)))))
    else if (d <= 2.2d-106) then
        tmp = d * (((h * l) ** (-0.5d0)) * (1.0d0 - ((h / l) * (0.5d0 * ((d_m * (m_m * (0.5d0 / d))) ** 2.0d0)))))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - (0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_m / d)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1e-309) {
		tmp = Math.sqrt((d / l)) * ((Math.sqrt(-d) / Math.sqrt(-h)) * (1.0 + ((h / l) * (-0.5 * Math.pow((D_m / (d * (2.0 / M_m))), 2.0)))));
	} else if (d <= 2.2e-106) {
		tmp = d * (Math.pow((h * l), -0.5) * (1.0 - ((h / l) * (0.5 * Math.pow((D_m * (M_m * (0.5 / d))), 2.0)))));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - (0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D_m / d)), 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -1e-309:
		tmp = math.sqrt((d / l)) * ((math.sqrt(-d) / math.sqrt(-h)) * (1.0 + ((h / l) * (-0.5 * math.pow((D_m / (d * (2.0 / M_m))), 2.0)))))
	elif d <= 2.2e-106:
		tmp = d * (math.pow((h * l), -0.5) * (1.0 - ((h / l) * (0.5 * math.pow((D_m * (M_m * (0.5 / d))), 2.0)))))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - (0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D_m / d)), 2.0))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -1e-309)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(D_m / Float64(d * Float64(2.0 / M_m))) ^ 2.0))))));
	elseif (d <= 2.2e-106)
		tmp = Float64(d * Float64((Float64(h * l) ^ -0.5) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(D_m * Float64(M_m * Float64(0.5 / d))) ^ 2.0))))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -1e-309)
		tmp = sqrt((d / l)) * ((sqrt(-d) / sqrt(-h)) * (1.0 + ((h / l) * (-0.5 * ((D_m / (d * (2.0 / M_m))) ^ 2.0)))));
	elseif (d <= 2.2e-106)
		tmp = d * (((h * l) ^ -0.5) * (1.0 - ((h / l) * (0.5 * ((D_m * (M_m * (0.5 / d))) ^ 2.0)))));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (0.5 * ((h / l) * (((M_m / 2.0) * (D_m / d)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -1e-309], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D$95$m / N[(d * N[(2.0 / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.2e-106], N[(d * N[(N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.000000000000002e-309

   1. Initial program 70.5%

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

   3. Simplified 68.4%

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

      1. associate-*r/ 70.5%

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

      2. div-inv 70.5%

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

      3. associate-*r* 70.5%

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

      4. *-commutative 70.5%

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

      5. div-inv 70.5%

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

      6. associate-/r* 70.5%

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

      7. frac-times 69.0%

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

      8. clear-num 69.1%

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

      9. frac-times 68.5%

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

      10. *-un-lft-identity 68.5%

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

   6. Applied egg-rr 68.5%

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

      1. frac-2neg 68.5%

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

      2. sqrt-div 75.2%

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

   8. Applied egg-rr 75.2%

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

   if -1.000000000000002e-309 < d < 2.19999999999999994e-106

   1. Initial program 41.0%

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

   3. Simplified 34.1%

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

   5. Taylor expanded in d around 0 41.7%

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

      1. pow1 41.7%

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

   7. Applied egg-rr 58.4%

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

      1. unpow1 58.4%

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

      2. *-commutative 58.4%

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

      3. *-commutative 58.4%

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

      4. associate-*r/ 56.0%

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

      5. associate-/l* 56.0%

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

   9. Simplified 56.0%

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

   if 2.19999999999999994e-106 < d

   1. Initial program 67.4%

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

   3. Simplified 67.4%

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

      1. *-commutative 67.4%

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

      2. sqrt-div 71.7%

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

      3. sqrt-div 89.6%

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

      4. frac-times 89.6%

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

      5. add-sqr-sqrt 89.9%

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

   6. Applied egg-rr 89.9%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d \cdot \frac{2}{M}}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{-106}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D\_m \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2}\right)\right) \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-106}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -1e-309)
   (*
    (sqrt (/ d l))
    (*
     (+ 1.0 (* (/ h l) (* -0.5 (pow (* D_m (/ (/ M_m 2.0) d)) 2.0))))
     (/ (sqrt (- d)) (sqrt (- h)))))
   (if (<= d 1.9e-106)
     (*
      d
      (*
       (pow (* h l) -0.5)
       (- 1.0 (* (/ h l) (* 0.5 (pow (* D_m (* M_m (/ 0.5 d))) 2.0))))))
     (*
      (/ d (* (sqrt l) (sqrt h)))
      (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1e-309) {
		tmp = sqrt((d / l)) * ((1.0 + ((h / l) * (-0.5 * pow((D_m * ((M_m / 2.0) / d)), 2.0)))) * (sqrt(-d) / sqrt(-h)));
	} else if (d <= 1.9e-106) {
		tmp = d * (pow((h * l), -0.5) * (1.0 - ((h / l) * (0.5 * pow((D_m * (M_m * (0.5 / d))), 2.0)))));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (0.5 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-1d-309)) then
        tmp = sqrt((d / l)) * ((1.0d0 + ((h / l) * ((-0.5d0) * ((d_m * ((m_m / 2.0d0) / d)) ** 2.0d0)))) * (sqrt(-d) / sqrt(-h)))
    else if (d <= 1.9d-106) then
        tmp = d * (((h * l) ** (-0.5d0)) * (1.0d0 - ((h / l) * (0.5d0 * ((d_m * (m_m * (0.5d0 / d))) ** 2.0d0)))))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - (0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_m / d)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1e-309) {
		tmp = Math.sqrt((d / l)) * ((1.0 + ((h / l) * (-0.5 * Math.pow((D_m * ((M_m / 2.0) / d)), 2.0)))) * (Math.sqrt(-d) / Math.sqrt(-h)));
	} else if (d <= 1.9e-106) {
		tmp = d * (Math.pow((h * l), -0.5) * (1.0 - ((h / l) * (0.5 * Math.pow((D_m * (M_m * (0.5 / d))), 2.0)))));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - (0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D_m / d)), 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -1e-309:
		tmp = math.sqrt((d / l)) * ((1.0 + ((h / l) * (-0.5 * math.pow((D_m * ((M_m / 2.0) / d)), 2.0)))) * (math.sqrt(-d) / math.sqrt(-h)))
	elif d <= 1.9e-106:
		tmp = d * (math.pow((h * l), -0.5) * (1.0 - ((h / l) * (0.5 * math.pow((D_m * (M_m * (0.5 / d))), 2.0)))))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - (0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D_m / d)), 2.0))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -1e-309)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(D_m * Float64(Float64(M_m / 2.0) / d)) ^ 2.0)))) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-h)))));
	elseif (d <= 1.9e-106)
		tmp = Float64(d * Float64((Float64(h * l) ^ -0.5) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(D_m * Float64(M_m * Float64(0.5 / d))) ^ 2.0))))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -1e-309)
		tmp = sqrt((d / l)) * ((1.0 + ((h / l) * (-0.5 * ((D_m * ((M_m / 2.0) / d)) ^ 2.0)))) * (sqrt(-d) / sqrt(-h)));
	elseif (d <= 1.9e-106)
		tmp = d * (((h * l) ^ -0.5) * (1.0 - ((h / l) * (0.5 * ((D_m * (M_m * (0.5 / d))) ^ 2.0)))));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (0.5 * ((h / l) * (((M_m / 2.0) * (D_m / d)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -1e-309], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D$95$m * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.9e-106], N[(d * N[(N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.000000000000002e-309

   1. Initial program 70.5%

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

   3. Simplified 68.4%

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

      1. frac-2neg 68.5%

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

      2. sqrt-div 75.2%

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

   6. Applied egg-rr 75.0%

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

   if -1.000000000000002e-309 < d < 1.9e-106

   1. Initial program 41.0%

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

   3. Simplified 34.1%

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

   5. Taylor expanded in d around 0 41.7%

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

      1. pow1 41.7%

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

   7. Applied egg-rr 58.4%

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

      1. unpow1 58.4%

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

      2. *-commutative 58.4%

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

      3. *-commutative 58.4%

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

      4. associate-*r/ 56.0%

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

      5. associate-/l* 56.0%

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

   9. Simplified 56.0%

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

   if 1.9e-106 < d

   1. Initial program 67.4%

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

   3. Simplified 67.4%

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

      1. *-commutative 67.4%

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

      2. sqrt-div 71.7%

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

      3. sqrt-div 89.6%

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

      4. frac-times 89.6%

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

      5. add-sqr-sqrt 89.9%

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

   6. Applied egg-rr 89.9%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}\right)\right) \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-106}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\\ \mathbf{if}\;d \leq -3.9 \cdot 10^{-78}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \left(0.5 \cdot \frac{{\left(\frac{D\_m}{d} \cdot \left(0.5 \cdot M\_m\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(t\_0 + -1\right)\\ \mathbf{elif}\;d \leq 1.02 \cdot 10^{-103}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - t\_0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0)))))
   (if (<= d -3.9e-78)
     (*
      (* (sqrt (/ d h)) (sqrt (/ d l)))
      (- 1.0 (* h (* 0.5 (/ (pow (* (/ D_m d) (* 0.5 M_m)) 2.0) l)))))
     (if (<= d -1e-309)
       (* (* d (sqrt (/ (/ 1.0 h) l))) (+ t_0 -1.0))
       (if (<= d 1.02e-103)
         (*
          d
          (*
           (pow (* h l) -0.5)
           (- 1.0 (* (/ h l) (* 0.5 (pow (* D_m (* M_m (/ 0.5 d))) 2.0))))))
         (* (/ d (* (sqrt l) (sqrt h))) (- 1.0 t_0)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 0.5 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0));
	double tmp;
	if (d <= -3.9e-78) {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - (h * (0.5 * (pow(((D_m / d) * (0.5 * M_m)), 2.0) / l))));
	} else if (d <= -1e-309) {
		tmp = (d * sqrt(((1.0 / h) / l))) * (t_0 + -1.0);
	} else if (d <= 1.02e-103) {
		tmp = d * (pow((h * l), -0.5) * (1.0 - ((h / l) * (0.5 * pow((D_m * (M_m * (0.5 / d))), 2.0)))));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - t_0);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_m / d)) ** 2.0d0))
    if (d <= (-3.9d-78)) then
        tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0d0 - (h * (0.5d0 * ((((d_m / d) * (0.5d0 * m_m)) ** 2.0d0) / l))))
    else if (d <= (-1d-309)) then
        tmp = (d * sqrt(((1.0d0 / h) / l))) * (t_0 + (-1.0d0))
    else if (d <= 1.02d-103) then
        tmp = d * (((h * l) ** (-0.5d0)) * (1.0d0 - ((h / l) * (0.5d0 * ((d_m * (m_m * (0.5d0 / d))) ** 2.0d0)))))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - t_0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D_m / d)), 2.0));
	double tmp;
	if (d <= -3.9e-78) {
		tmp = (Math.sqrt((d / h)) * Math.sqrt((d / l))) * (1.0 - (h * (0.5 * (Math.pow(((D_m / d) * (0.5 * M_m)), 2.0) / l))));
	} else if (d <= -1e-309) {
		tmp = (d * Math.sqrt(((1.0 / h) / l))) * (t_0 + -1.0);
	} else if (d <= 1.02e-103) {
		tmp = d * (Math.pow((h * l), -0.5) * (1.0 - ((h / l) * (0.5 * Math.pow((D_m * (M_m * (0.5 / d))), 2.0)))));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - t_0);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = 0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D_m / d)), 2.0))
	tmp = 0
	if d <= -3.9e-78:
		tmp = (math.sqrt((d / h)) * math.sqrt((d / l))) * (1.0 - (h * (0.5 * (math.pow(((D_m / d) * (0.5 * M_m)), 2.0) / l))))
	elif d <= -1e-309:
		tmp = (d * math.sqrt(((1.0 / h) / l))) * (t_0 + -1.0)
	elif d <= 1.02e-103:
		tmp = d * (math.pow((h * l), -0.5) * (1.0 - ((h / l) * (0.5 * math.pow((D_m * (M_m * (0.5 / d))), 2.0)))))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - t_0)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0)))
	tmp = 0.0
	if (d <= -3.9e-78)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(h * Float64(0.5 * Float64((Float64(Float64(D_m / d) * Float64(0.5 * M_m)) ^ 2.0) / l)))));
	elseif (d <= -1e-309)
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / h) / l))) * Float64(t_0 + -1.0));
	elseif (d <= 1.02e-103)
		tmp = Float64(d * Float64((Float64(h * l) ^ -0.5) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(D_m * Float64(M_m * Float64(0.5 / d))) ^ 2.0))))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - t_0));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = 0.5 * ((h / l) * (((M_m / 2.0) * (D_m / d)) ^ 2.0));
	tmp = 0.0;
	if (d <= -3.9e-78)
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - (h * (0.5 * ((((D_m / d) * (0.5 * M_m)) ^ 2.0) / l))));
	elseif (d <= -1e-309)
		tmp = (d * sqrt(((1.0 / h) / l))) * (t_0 + -1.0);
	elseif (d <= 1.02e-103)
		tmp = d * (((h * l) ^ -0.5) * (1.0 - ((h / l) * (0.5 * ((D_m * (M_m * (0.5 / d))) ^ 2.0)))));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - t_0);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.9e-78], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(h * N[(0.5 * N[(N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(0.5 * M$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-309], N[(N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.02e-103], N[(d * N[(N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.9000000000000002e-78

   1. Initial program 79.4%

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

   3. Simplified 78.3%

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

      1. add-sqr-sqrt 78.2%

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

      2. pow2 78.2%

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

      3. sqrt-prod 78.2%

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

      4. sqrt-pow1 80.6%

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

      5. frac-times 83.1%

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

      6. associate-/r* 83.1%

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

      7. metadata-eval 83.1%

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

      8. pow1 83.1%

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

      9. div-inv 83.1%

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

      10. *-commutative 83.1%

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

      11. associate-*r* 83.1%

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

      12. div-inv 83.1%

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

      13. associate-*r/ 82.0%

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

      14. associate-/l/ 82.0%

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

      15. *-un-lft-identity 82.0%

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

      16. *-commutative 82.0%

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

      17. times-frac 82.0%

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

      18. metadata-eval 82.0%

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

   6. Applied egg-rr 82.0%

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

      1. pow1 82.0%

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

      2. *-commutative 82.0%

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

      3. unpow-prod-down 79.5%

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

      4. pow2 79.5%

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

      5. add-sqr-sqrt 79.6%

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

      6. associate-*r/ 79.6%

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

   8. Applied egg-rr 79.6%

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

      1. unpow1 79.6%

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

      2. *-commutative 79.6%

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

      3. associate-*r/ 79.4%

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

      4. *-commutative 79.4%

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

      5. *-commutative 79.4%

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

      6. associate-*l* 79.4%

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

      7. *-commutative 79.4%

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

      8. associate-*l/ 79.7%

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

      9. associate-*r/ 83.2%

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

      10. associate-/l* 83.2%

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

      11. associate-/l* 82.1%

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

   10. Simplified 82.1%

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

   if -3.9000000000000002e-78 < d < -1.000000000000002e-309

   1. Initial program 56.8%

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

   3. Simplified 54.8%

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

   5. Taylor expanded in h around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 71.9%

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

      6. neg-mul-1 71.9%

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

   7. Simplified 71.9%

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

   if -1.000000000000002e-309 < d < 1.01999999999999998e-103

   1. Initial program 41.0%

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

   3. Simplified 34.1%

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

   5. Taylor expanded in d around 0 41.7%

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

      1. pow1 41.7%

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

   7. Applied egg-rr 58.4%

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

      1. unpow1 58.4%

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

      2. *-commutative 58.4%

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

      3. *-commutative 58.4%

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

      4. associate-*r/ 56.0%

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

      5. associate-/l* 56.0%

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

   9. Simplified 56.0%

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

   if 1.01999999999999998e-103 < d

   1. Initial program 67.4%

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

   3. Simplified 67.4%

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

      1. *-commutative 67.4%

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

      2. sqrt-div 71.7%

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

      3. sqrt-div 89.6%

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

      4. frac-times 89.6%

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

      5. add-sqr-sqrt 89.9%

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

   6. Applied egg-rr 89.9%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.9 \cdot 10^{-78}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \left(0.5 \cdot \frac{{\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) + -1\right)\\ \mathbf{elif}\;d \leq 1.02 \cdot 10^{-103}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\\ \mathbf{if}\;d \leq -2.5 \cdot 10^{-88}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D\_m \cdot \left(0.5 \cdot M\_m\right)}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(t\_0 + -1\right)\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{-104}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - t\_0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0)))))
   (if (<= d -2.5e-88)
     (*
      (sqrt (/ d l))
      (*
       (sqrt (/ d h))
       (+ 1.0 (* (/ h l) (* -0.5 (pow (/ (* D_m (* 0.5 M_m)) d) 2.0))))))
     (if (<= d -1e-309)
       (* (* d (sqrt (/ (/ 1.0 h) l))) (+ t_0 -1.0))
       (if (<= d 2.3e-104)
         (*
          d
          (*
           (pow (* h l) -0.5)
           (- 1.0 (* (/ h l) (* 0.5 (pow (* D_m (* M_m (/ 0.5 d))) 2.0))))))
         (* (/ d (* (sqrt l) (sqrt h))) (- 1.0 t_0)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 0.5 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0));
	double tmp;
	if (d <= -2.5e-88) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * pow(((D_m * (0.5 * M_m)) / d), 2.0)))));
	} else if (d <= -1e-309) {
		tmp = (d * sqrt(((1.0 / h) / l))) * (t_0 + -1.0);
	} else if (d <= 2.3e-104) {
		tmp = d * (pow((h * l), -0.5) * (1.0 - ((h / l) * (0.5 * pow((D_m * (M_m * (0.5 / d))), 2.0)))));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - t_0);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_m / d)) ** 2.0d0))
    if (d <= (-2.5d-88)) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + ((h / l) * ((-0.5d0) * (((d_m * (0.5d0 * m_m)) / d) ** 2.0d0)))))
    else if (d <= (-1d-309)) then
        tmp = (d * sqrt(((1.0d0 / h) / l))) * (t_0 + (-1.0d0))
    else if (d <= 2.3d-104) then
        tmp = d * (((h * l) ** (-0.5d0)) * (1.0d0 - ((h / l) * (0.5d0 * ((d_m * (m_m * (0.5d0 / d))) ** 2.0d0)))))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - t_0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D_m / d)), 2.0));
	double tmp;
	if (d <= -2.5e-88) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * Math.pow(((D_m * (0.5 * M_m)) / d), 2.0)))));
	} else if (d <= -1e-309) {
		tmp = (d * Math.sqrt(((1.0 / h) / l))) * (t_0 + -1.0);
	} else if (d <= 2.3e-104) {
		tmp = d * (Math.pow((h * l), -0.5) * (1.0 - ((h / l) * (0.5 * Math.pow((D_m * (M_m * (0.5 / d))), 2.0)))));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - t_0);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = 0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D_m / d)), 2.0))
	tmp = 0
	if d <= -2.5e-88:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * math.pow(((D_m * (0.5 * M_m)) / d), 2.0)))))
	elif d <= -1e-309:
		tmp = (d * math.sqrt(((1.0 / h) / l))) * (t_0 + -1.0)
	elif d <= 2.3e-104:
		tmp = d * (math.pow((h * l), -0.5) * (1.0 - ((h / l) * (0.5 * math.pow((D_m * (M_m * (0.5 / d))), 2.0)))))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - t_0)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0)))
	tmp = 0.0
	if (d <= -2.5e-88)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(D_m * Float64(0.5 * M_m)) / d) ^ 2.0))))));
	elseif (d <= -1e-309)
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / h) / l))) * Float64(t_0 + -1.0));
	elseif (d <= 2.3e-104)
		tmp = Float64(d * Float64((Float64(h * l) ^ -0.5) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(D_m * Float64(M_m * Float64(0.5 / d))) ^ 2.0))))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - t_0));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = 0.5 * ((h / l) * (((M_m / 2.0) * (D_m / d)) ^ 2.0));
	tmp = 0.0;
	if (d <= -2.5e-88)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * (((D_m * (0.5 * M_m)) / d) ^ 2.0)))));
	elseif (d <= -1e-309)
		tmp = (d * sqrt(((1.0 / h) / l))) * (t_0 + -1.0);
	elseif (d <= 2.3e-104)
		tmp = d * (((h * l) ^ -0.5) * (1.0 - ((h / l) * (0.5 * ((D_m * (M_m * (0.5 / d))) ^ 2.0)))));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - t_0);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.5e-88], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(D$95$m * N[(0.5 * M$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-309], N[(N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.3e-104], N[(d * N[(N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.50000000000000004e-88

   1. Initial program 80.8%

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

   3. Simplified 79.9%

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

      1. associate-*r/ 80.9%

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

      2. div-inv 80.9%

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

      3. associate-*r* 80.9%

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

      4. *-commutative 80.9%

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

      5. div-inv 80.9%

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

      6. associate-/r* 80.9%

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

      7. frac-times 78.7%

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

      8. associate-*r/ 80.9%

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

      9. div-inv 80.9%

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

      10. metadata-eval 80.9%

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

   6. Applied egg-rr 80.9%

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

   if -2.50000000000000004e-88 < d < -1.000000000000002e-309

   1. Initial program 51.2%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in h around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 70.4%

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

      6. neg-mul-1 70.4%

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

   7. Simplified 70.4%

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

   if -1.000000000000002e-309 < d < 2.2999999999999999e-104

   1. Initial program 41.0%

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

   3. Simplified 34.1%

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

   5. Taylor expanded in d around 0 41.7%

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

      1. pow1 41.7%

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

   7. Applied egg-rr 58.4%

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

      1. unpow1 58.4%

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

      2. *-commutative 58.4%

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

      3. *-commutative 58.4%

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

      4. associate-*r/ 56.0%

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

      5. associate-/l* 56.0%

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

   9. Simplified 56.0%

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

   if 2.2999999999999999e-104 < d

   1. Initial program 67.4%

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

   3. Simplified 67.4%

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

      1. *-commutative 67.4%

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

      2. sqrt-div 71.7%

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

      3. sqrt-div 89.6%

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

      4. frac-times 89.6%

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

      5. add-sqr-sqrt 89.9%

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

   6. Applied egg-rr 89.9%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.5 \cdot 10^{-88}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D \cdot \left(0.5 \cdot M\right)}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) + -1\right)\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{-104}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\\ \mathbf{if}\;d \leq -1.1 \cdot 10^{-88}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D\_m}{d \cdot \frac{2}{M\_m}}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(t\_0 + -1\right)\\ \mathbf{elif}\;d \leq 3.7 \cdot 10^{-103}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - t\_0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0)))))
   (if (<= d -1.1e-88)
     (*
      (sqrt (/ d l))
      (*
       (sqrt (/ d h))
       (+ 1.0 (* (/ h l) (* -0.5 (pow (/ D_m (* d (/ 2.0 M_m))) 2.0))))))
     (if (<= d -1e-309)
       (* (* d (sqrt (/ (/ 1.0 h) l))) (+ t_0 -1.0))
       (if (<= d 3.7e-103)
         (*
          d
          (*
           (pow (* h l) -0.5)
           (- 1.0 (* (/ h l) (* 0.5 (pow (* D_m (* M_m (/ 0.5 d))) 2.0))))))
         (* (/ d (* (sqrt l) (sqrt h))) (- 1.0 t_0)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 0.5 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0));
	double tmp;
	if (d <= -1.1e-88) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * pow((D_m / (d * (2.0 / M_m))), 2.0)))));
	} else if (d <= -1e-309) {
		tmp = (d * sqrt(((1.0 / h) / l))) * (t_0 + -1.0);
	} else if (d <= 3.7e-103) {
		tmp = d * (pow((h * l), -0.5) * (1.0 - ((h / l) * (0.5 * pow((D_m * (M_m * (0.5 / d))), 2.0)))));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - t_0);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_m / d)) ** 2.0d0))
    if (d <= (-1.1d-88)) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + ((h / l) * ((-0.5d0) * ((d_m / (d * (2.0d0 / m_m))) ** 2.0d0)))))
    else if (d <= (-1d-309)) then
        tmp = (d * sqrt(((1.0d0 / h) / l))) * (t_0 + (-1.0d0))
    else if (d <= 3.7d-103) then
        tmp = d * (((h * l) ** (-0.5d0)) * (1.0d0 - ((h / l) * (0.5d0 * ((d_m * (m_m * (0.5d0 / d))) ** 2.0d0)))))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - t_0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D_m / d)), 2.0));
	double tmp;
	if (d <= -1.1e-88) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * Math.pow((D_m / (d * (2.0 / M_m))), 2.0)))));
	} else if (d <= -1e-309) {
		tmp = (d * Math.sqrt(((1.0 / h) / l))) * (t_0 + -1.0);
	} else if (d <= 3.7e-103) {
		tmp = d * (Math.pow((h * l), -0.5) * (1.0 - ((h / l) * (0.5 * Math.pow((D_m * (M_m * (0.5 / d))), 2.0)))));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - t_0);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = 0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D_m / d)), 2.0))
	tmp = 0
	if d <= -1.1e-88:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * math.pow((D_m / (d * (2.0 / M_m))), 2.0)))))
	elif d <= -1e-309:
		tmp = (d * math.sqrt(((1.0 / h) / l))) * (t_0 + -1.0)
	elif d <= 3.7e-103:
		tmp = d * (math.pow((h * l), -0.5) * (1.0 - ((h / l) * (0.5 * math.pow((D_m * (M_m * (0.5 / d))), 2.0)))))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - t_0)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0)))
	tmp = 0.0
	if (d <= -1.1e-88)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(D_m / Float64(d * Float64(2.0 / M_m))) ^ 2.0))))));
	elseif (d <= -1e-309)
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / h) / l))) * Float64(t_0 + -1.0));
	elseif (d <= 3.7e-103)
		tmp = Float64(d * Float64((Float64(h * l) ^ -0.5) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(D_m * Float64(M_m * Float64(0.5 / d))) ^ 2.0))))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - t_0));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = 0.5 * ((h / l) * (((M_m / 2.0) * (D_m / d)) ^ 2.0));
	tmp = 0.0;
	if (d <= -1.1e-88)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * ((D_m / (d * (2.0 / M_m))) ^ 2.0)))));
	elseif (d <= -1e-309)
		tmp = (d * sqrt(((1.0 / h) / l))) * (t_0 + -1.0);
	elseif (d <= 3.7e-103)
		tmp = d * (((h * l) ^ -0.5) * (1.0 - ((h / l) * (0.5 * ((D_m * (M_m * (0.5 / d))) ^ 2.0)))));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - t_0);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.1e-88], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D$95$m / N[(d * N[(2.0 / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-309], N[(N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.7e-103], N[(d * N[(N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.10000000000000002e-88

   1. Initial program 80.8%

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

   3. Simplified 79.9%

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

      1. associate-*r/ 80.9%

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

      2. div-inv 80.9%

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

      3. associate-*r* 80.9%

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

      4. *-commutative 80.9%

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

      5. div-inv 80.9%

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

      6. associate-/r* 80.9%

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

      7. frac-times 78.7%

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

      8. clear-num 78.7%

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

      9. frac-times 79.9%

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

      10. *-un-lft-identity 79.9%

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

   6. Applied egg-rr 79.9%

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

   if -1.10000000000000002e-88 < d < -1.000000000000002e-309

   1. Initial program 51.2%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in h around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 70.4%

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

      6. neg-mul-1 70.4%

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

   7. Simplified 70.4%

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

   if -1.000000000000002e-309 < d < 3.6999999999999999e-103

   1. Initial program 41.0%

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

   3. Simplified 34.1%

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

   5. Taylor expanded in d around 0 41.7%

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

      1. pow1 41.7%

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

   7. Applied egg-rr 58.4%

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

      1. unpow1 58.4%

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

      2. *-commutative 58.4%

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

      3. *-commutative 58.4%

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

      4. associate-*r/ 56.0%

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

      5. associate-/l* 56.0%

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

   9. Simplified 56.0%

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

   if 3.6999999999999999e-103 < d

   1. Initial program 67.4%

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

   3. Simplified 67.4%

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

      1. *-commutative 67.4%

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

      2. sqrt-div 71.7%

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

      3. sqrt-div 89.6%

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

      4. frac-times 89.6%

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

      5. add-sqr-sqrt 89.9%

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

   6. Applied egg-rr 89.9%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{-88}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d \cdot \frac{2}{M}}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) + -1\right)\\ \mathbf{elif}\;d \leq 3.7 \cdot 10^{-103}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\\ \mathbf{if}\;d \leq -9.8 \cdot 10^{-89}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D\_m \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2}\right)\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(t\_0 + -1\right)\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-106}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - t\_0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0)))))
   (if (<= d -9.8e-89)
     (*
      (*
       (sqrt (/ d h))
       (+ 1.0 (* (/ h l) (* -0.5 (pow (* D_m (/ (/ M_m 2.0) d)) 2.0)))))
      (sqrt (/ d l)))
     (if (<= d -1e-309)
       (* (* d (sqrt (/ (/ 1.0 h) l))) (+ t_0 -1.0))
       (if (<= d 7e-106)
         (*
          d
          (*
           (pow (* h l) -0.5)
           (- 1.0 (* (/ h l) (* 0.5 (pow (* D_m (* M_m (/ 0.5 d))) 2.0))))))
         (* (/ d (* (sqrt l) (sqrt h))) (- 1.0 t_0)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 0.5 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0));
	double tmp;
	if (d <= -9.8e-89) {
		tmp = (sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * pow((D_m * ((M_m / 2.0) / d)), 2.0))))) * sqrt((d / l));
	} else if (d <= -1e-309) {
		tmp = (d * sqrt(((1.0 / h) / l))) * (t_0 + -1.0);
	} else if (d <= 7e-106) {
		tmp = d * (pow((h * l), -0.5) * (1.0 - ((h / l) * (0.5 * pow((D_m * (M_m * (0.5 / d))), 2.0)))));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - t_0);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_m / d)) ** 2.0d0))
    if (d <= (-9.8d-89)) then
        tmp = (sqrt((d / h)) * (1.0d0 + ((h / l) * ((-0.5d0) * ((d_m * ((m_m / 2.0d0) / d)) ** 2.0d0))))) * sqrt((d / l))
    else if (d <= (-1d-309)) then
        tmp = (d * sqrt(((1.0d0 / h) / l))) * (t_0 + (-1.0d0))
    else if (d <= 7d-106) then
        tmp = d * (((h * l) ** (-0.5d0)) * (1.0d0 - ((h / l) * (0.5d0 * ((d_m * (m_m * (0.5d0 / d))) ** 2.0d0)))))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - t_0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D_m / d)), 2.0));
	double tmp;
	if (d <= -9.8e-89) {
		tmp = (Math.sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * Math.pow((D_m * ((M_m / 2.0) / d)), 2.0))))) * Math.sqrt((d / l));
	} else if (d <= -1e-309) {
		tmp = (d * Math.sqrt(((1.0 / h) / l))) * (t_0 + -1.0);
	} else if (d <= 7e-106) {
		tmp = d * (Math.pow((h * l), -0.5) * (1.0 - ((h / l) * (0.5 * Math.pow((D_m * (M_m * (0.5 / d))), 2.0)))));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - t_0);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = 0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D_m / d)), 2.0))
	tmp = 0
	if d <= -9.8e-89:
		tmp = (math.sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * math.pow((D_m * ((M_m / 2.0) / d)), 2.0))))) * math.sqrt((d / l))
	elif d <= -1e-309:
		tmp = (d * math.sqrt(((1.0 / h) / l))) * (t_0 + -1.0)
	elif d <= 7e-106:
		tmp = d * (math.pow((h * l), -0.5) * (1.0 - ((h / l) * (0.5 * math.pow((D_m * (M_m * (0.5 / d))), 2.0)))))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - t_0)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0)))
	tmp = 0.0
	if (d <= -9.8e-89)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(D_m * Float64(Float64(M_m / 2.0) / d)) ^ 2.0))))) * sqrt(Float64(d / l)));
	elseif (d <= -1e-309)
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / h) / l))) * Float64(t_0 + -1.0));
	elseif (d <= 7e-106)
		tmp = Float64(d * Float64((Float64(h * l) ^ -0.5) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(D_m * Float64(M_m * Float64(0.5 / d))) ^ 2.0))))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - t_0));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = 0.5 * ((h / l) * (((M_m / 2.0) * (D_m / d)) ^ 2.0));
	tmp = 0.0;
	if (d <= -9.8e-89)
		tmp = (sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * ((D_m * ((M_m / 2.0) / d)) ^ 2.0))))) * sqrt((d / l));
	elseif (d <= -1e-309)
		tmp = (d * sqrt(((1.0 / h) / l))) * (t_0 + -1.0);
	elseif (d <= 7e-106)
		tmp = d * (((h * l) ^ -0.5) * (1.0 - ((h / l) * (0.5 * ((D_m * (M_m * (0.5 / d))) ^ 2.0)))));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - t_0);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -9.8e-89], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D$95$m * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-309], N[(N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7e-106], N[(d * N[(N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -9.8e-89

   1. Initial program 80.8%

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

   3. Simplified 79.9%

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

   if -9.8e-89 < d < -1.000000000000002e-309

   1. Initial program 51.2%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in h around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 70.4%

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

      6. neg-mul-1 70.4%

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

   7. Simplified 70.4%

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

   if -1.000000000000002e-309 < d < 7e-106

   1. Initial program 41.0%

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

   3. Simplified 34.1%

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

   5. Taylor expanded in d around 0 41.7%

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

      1. pow1 41.7%

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

   7. Applied egg-rr 58.4%

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

      1. unpow1 58.4%

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

      2. *-commutative 58.4%

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

      3. *-commutative 58.4%

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

      4. associate-*r/ 56.0%

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

      5. associate-/l* 56.0%

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

   9. Simplified 56.0%

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

   if 7e-106 < d

   1. Initial program 67.4%

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

   3. Simplified 67.4%

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

      1. *-commutative 67.4%

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

      2. sqrt-div 71.7%

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

      3. sqrt-div 89.6%

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

      4. frac-times 89.6%

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

      5. add-sqr-sqrt 89.9%

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

   6. Applied egg-rr 89.9%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.8 \cdot 10^{-89}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}\right)\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) + -1\right)\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-106}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\\ \mathbf{if}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(t\_0 + -1\right)\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{-104}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - t\_0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0)))))
   (if (<= d -1e-309)
     (* (* d (sqrt (/ (/ 1.0 h) l))) (+ t_0 -1.0))
     (if (<= d 2.05e-104)
       (*
        d
        (*
         (pow (* h l) -0.5)
         (- 1.0 (* (/ h l) (* 0.5 (pow (* D_m (* M_m (/ 0.5 d))) 2.0))))))
       (* (/ d (* (sqrt l) (sqrt h))) (- 1.0 t_0))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 0.5 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0));
	double tmp;
	if (d <= -1e-309) {
		tmp = (d * sqrt(((1.0 / h) / l))) * (t_0 + -1.0);
	} else if (d <= 2.05e-104) {
		tmp = d * (pow((h * l), -0.5) * (1.0 - ((h / l) * (0.5 * pow((D_m * (M_m * (0.5 / d))), 2.0)))));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - t_0);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_m / d)) ** 2.0d0))
    if (d <= (-1d-309)) then
        tmp = (d * sqrt(((1.0d0 / h) / l))) * (t_0 + (-1.0d0))
    else if (d <= 2.05d-104) then
        tmp = d * (((h * l) ** (-0.5d0)) * (1.0d0 - ((h / l) * (0.5d0 * ((d_m * (m_m * (0.5d0 / d))) ** 2.0d0)))))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - t_0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D_m / d)), 2.0));
	double tmp;
	if (d <= -1e-309) {
		tmp = (d * Math.sqrt(((1.0 / h) / l))) * (t_0 + -1.0);
	} else if (d <= 2.05e-104) {
		tmp = d * (Math.pow((h * l), -0.5) * (1.0 - ((h / l) * (0.5 * Math.pow((D_m * (M_m * (0.5 / d))), 2.0)))));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - t_0);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = 0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D_m / d)), 2.0))
	tmp = 0
	if d <= -1e-309:
		tmp = (d * math.sqrt(((1.0 / h) / l))) * (t_0 + -1.0)
	elif d <= 2.05e-104:
		tmp = d * (math.pow((h * l), -0.5) * (1.0 - ((h / l) * (0.5 * math.pow((D_m * (M_m * (0.5 / d))), 2.0)))))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - t_0)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0)))
	tmp = 0.0
	if (d <= -1e-309)
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / h) / l))) * Float64(t_0 + -1.0));
	elseif (d <= 2.05e-104)
		tmp = Float64(d * Float64((Float64(h * l) ^ -0.5) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(D_m * Float64(M_m * Float64(0.5 / d))) ^ 2.0))))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - t_0));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = 0.5 * ((h / l) * (((M_m / 2.0) * (D_m / d)) ^ 2.0));
	tmp = 0.0;
	if (d <= -1e-309)
		tmp = (d * sqrt(((1.0 / h) / l))) * (t_0 + -1.0);
	elseif (d <= 2.05e-104)
		tmp = d * (((h * l) ^ -0.5) * (1.0 - ((h / l) * (0.5 * ((D_m * (M_m * (0.5 / d))) ^ 2.0)))));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - t_0);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1e-309], N[(N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.05e-104], N[(d * N[(N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.000000000000002e-309

   1. Initial program 70.5%

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

   3. Simplified 69.0%

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

   5. Taylor expanded in h around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 73.3%

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

      6. neg-mul-1 73.3%

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

   7. Simplified 73.3%

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

   if -1.000000000000002e-309 < d < 2.04999999999999992e-104

   1. Initial program 41.0%

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

   3. Simplified 34.1%

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

   5. Taylor expanded in d around 0 41.7%

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

      1. pow1 41.7%

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

   7. Applied egg-rr 58.4%

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

      1. unpow1 58.4%

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

      2. *-commutative 58.4%

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

      3. *-commutative 58.4%

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

      4. associate-*r/ 56.0%

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

      5. associate-/l* 56.0%

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

   9. Simplified 56.0%

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

   if 2.04999999999999992e-104 < d

   1. Initial program 67.4%

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

   3. Simplified 67.4%

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

      1. *-commutative 67.4%

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

      2. sqrt-div 71.7%

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

      3. sqrt-div 89.6%

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

      4. frac-times 89.6%

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

      5. add-sqr-sqrt 89.9%

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

   6. Applied egg-rr 89.9%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) + -1\right)\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{-104}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{if}\;d \leq -3.3 \cdot 10^{+149}:\\ \;\;\;\;d \cdot \left(-t\_0\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{0.5}{d} \cdot \left(D\_m \cdot M\_m\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 10^{-61}:\\ \;\;\;\;d \cdot \left(t\_0 \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(D\_m \cdot \left(0.5 \cdot \frac{M\_m}{d}\right)\right)}^{2}}{\ell}\right) \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (pow (* h l) -0.5)))
   (if (<= d -3.3e+149)
     (* d (- t_0))
     (if (<= d -1e-309)
       (*
        (sqrt (* (/ d h) (/ d l)))
        (+ 1.0 (* (/ h l) (* -0.5 (pow (* (/ 0.5 d) (* D_m M_m)) 2.0)))))
       (if (<= d 1e-61)
         (*
          d
          (*
           t_0
           (- 1.0 (* (/ h l) (* 0.5 (pow (* D_m (* M_m (/ 0.5 d))) 2.0))))))
         (*
          (- 1.0 (* 0.5 (/ (* h (pow (* D_m (* 0.5 (/ M_m d))) 2.0)) l)))
          (* d (sqrt (/ 1.0 (* h l))))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = pow((h * l), -0.5);
	double tmp;
	if (d <= -3.3e+149) {
		tmp = d * -t_0;
	} else if (d <= -1e-309) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + ((h / l) * (-0.5 * pow(((0.5 / d) * (D_m * M_m)), 2.0))));
	} else if (d <= 1e-61) {
		tmp = d * (t_0 * (1.0 - ((h / l) * (0.5 * pow((D_m * (M_m * (0.5 / d))), 2.0)))));
	} else {
		tmp = (1.0 - (0.5 * ((h * pow((D_m * (0.5 * (M_m / d))), 2.0)) / l))) * (d * sqrt((1.0 / (h * l))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (h * l) ** (-0.5d0)
    if (d <= (-3.3d+149)) then
        tmp = d * -t_0
    else if (d <= (-1d-309)) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 + ((h / l) * ((-0.5d0) * (((0.5d0 / d) * (d_m * m_m)) ** 2.0d0))))
    else if (d <= 1d-61) then
        tmp = d * (t_0 * (1.0d0 - ((h / l) * (0.5d0 * ((d_m * (m_m * (0.5d0 / d))) ** 2.0d0)))))
    else
        tmp = (1.0d0 - (0.5d0 * ((h * ((d_m * (0.5d0 * (m_m / d))) ** 2.0d0)) / l))) * (d * sqrt((1.0d0 / (h * l))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.pow((h * l), -0.5);
	double tmp;
	if (d <= -3.3e+149) {
		tmp = d * -t_0;
	} else if (d <= -1e-309) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 + ((h / l) * (-0.5 * Math.pow(((0.5 / d) * (D_m * M_m)), 2.0))));
	} else if (d <= 1e-61) {
		tmp = d * (t_0 * (1.0 - ((h / l) * (0.5 * Math.pow((D_m * (M_m * (0.5 / d))), 2.0)))));
	} else {
		tmp = (1.0 - (0.5 * ((h * Math.pow((D_m * (0.5 * (M_m / d))), 2.0)) / l))) * (d * Math.sqrt((1.0 / (h * l))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.pow((h * l), -0.5)
	tmp = 0
	if d <= -3.3e+149:
		tmp = d * -t_0
	elif d <= -1e-309:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 + ((h / l) * (-0.5 * math.pow(((0.5 / d) * (D_m * M_m)), 2.0))))
	elif d <= 1e-61:
		tmp = d * (t_0 * (1.0 - ((h / l) * (0.5 * math.pow((D_m * (M_m * (0.5 / d))), 2.0)))))
	else:
		tmp = (1.0 - (0.5 * ((h * math.pow((D_m * (0.5 * (M_m / d))), 2.0)) / l))) * (d * math.sqrt((1.0 / (h * l))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(h * l) ^ -0.5
	tmp = 0.0
	if (d <= -3.3e+149)
		tmp = Float64(d * Float64(-t_0));
	elseif (d <= -1e-309)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(0.5 / d) * Float64(D_m * M_m)) ^ 2.0)))));
	elseif (d <= 1e-61)
		tmp = Float64(d * Float64(t_0 * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(D_m * Float64(M_m * Float64(0.5 / d))) ^ 2.0))))));
	else
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(D_m * Float64(0.5 * Float64(M_m / d))) ^ 2.0)) / l))) * Float64(d * sqrt(Float64(1.0 / Float64(h * l)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (h * l) ^ -0.5;
	tmp = 0.0;
	if (d <= -3.3e+149)
		tmp = d * -t_0;
	elseif (d <= -1e-309)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + ((h / l) * (-0.5 * (((0.5 / d) * (D_m * M_m)) ^ 2.0))));
	elseif (d <= 1e-61)
		tmp = d * (t_0 * (1.0 - ((h / l) * (0.5 * ((D_m * (M_m * (0.5 / d))) ^ 2.0)))));
	else
		tmp = (1.0 - (0.5 * ((h * ((D_m * (0.5 * (M_m / d))) ^ 2.0)) / l))) * (d * sqrt((1.0 / (h * l))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[d, -3.3e+149], N[(d * (-t$95$0)), $MachinePrecision], If[LessEqual[d, -1e-309], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(0.5 / d), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1e-61], N[(d * N[(t$95$0 * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(D$95$m * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.3e149

   1. Initial program 75.4%

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

   3. Simplified 75.6%

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

      1. add-sqr-sqrt 75.6%

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

      2. pow2 75.6%

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

      3. sqrt-prod 75.6%

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

      4. sqrt-pow1 75.6%

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

      5. frac-times 78.2%

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

      6. associate-/r* 78.2%

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

      7. metadata-eval 78.2%

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

      8. pow1 78.2%

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

      9. div-inv 78.2%

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

      10. *-commutative 78.2%

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

      11. associate-*r* 78.2%

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

      12. div-inv 78.2%

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

      13. associate-*r/ 78.5%

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

      14. associate-/l/ 78.5%

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

      15. *-un-lft-identity 78.5%

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

      16. *-commutative 78.5%

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

      17. times-frac 78.5%

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

      18. metadata-eval 78.5%

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

   6. Applied egg-rr 78.5%

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

      1. clear-num 78.5%

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

      2. sqrt-div 78.5%

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

      3. metadata-eval 78.5%

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

   8. Applied egg-rr 78.5%

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

   9. Taylor expanded in d around -inf 68.5%

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

       1. mul-1-neg 68.5%

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

       2. *-commutative 68.5%

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

       3. associate-/r* 68.6%

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

       4. distribute-rgt-neg-in 68.6%

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

       5. associate-/r* 68.5%

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

       6. unpow-1 68.5%

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

       7. metadata-eval 68.5%

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

       8. pow-sqr 68.6%

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

       9. rem-sqrt-square 68.6%

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

       10. metadata-eval 68.6%

           \[\leadsto \left|{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right| \cdot \left(-d\right) \]

       11. pow-sqr 68.5%

           \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}}\right| \cdot \left(-d\right) \]

       12. fabs-sqr 68.5%

           \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}\right)} \cdot \left(-d\right) \]

       13. pow-sqr 68.6%

           \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(2 \cdot -0.25\right)}} \cdot \left(-d\right) \]

       14. metadata-eval 68.6%

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

   11. Simplified 68.6%

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

   if -3.3e149 < d < -1.000000000000002e-309

   1. Initial program 68.7%

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

   3. Simplified 66.6%

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

      1. add-sqr-sqrt 66.6%

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

      2. pow2 66.6%

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

      3. sqrt-prod 66.6%

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

      4. sqrt-pow1 69.6%

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

      5. frac-times 72.7%

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

      6. associate-/r* 72.7%

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

      7. metadata-eval 72.7%

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

      8. pow1 72.7%

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

      9. div-inv 72.7%

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

      10. *-commutative 72.7%

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

      11. associate-*r* 72.7%

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

      12. div-inv 72.7%

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

      13. associate-*r/ 67.7%

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

      14. associate-/l/ 67.7%

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

      15. *-un-lft-identity 67.7%

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

      16. *-commutative 67.7%

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

      17. times-frac 67.7%

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

      18. metadata-eval 67.7%

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

   6. Applied egg-rr 67.7%

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

      1. pow1 67.7%

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

      2. sqrt-unprod 57.5%

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

      3. cancel-sign-sub-inv 57.5%

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

      4. metadata-eval 57.5%

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

      5. *-commutative 57.5%

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

      6. unpow-prod-down 56.4%

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

      7. pow2 56.4%

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

      8. add-sqr-sqrt 56.5%

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

      9. associate-*r/ 56.5%

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

   8. Applied egg-rr 56.5%

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

      1. unpow1 56.5%

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

      2. associate-*r* 56.5%

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

      3. *-commutative 56.5%

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

      4. associate-/l* 56.5%

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

   10. Simplified 56.5%

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

       1. expm1-log1p-u 19.1%

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

       2. log1p-define 19.1%

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

       3. expm1-undefine 19.1%

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

       4. add-exp-log 56.5%

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

       5. +-commutative 56.5%

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

       6. associate-*l* 56.5%

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

       7. fma-define 56.5%

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

   12. Applied egg-rr 56.5%

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

       1. fma-undefine 56.5%

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

       2. associate--l+ 56.5%

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

       3. metadata-eval 56.5%

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

       4. +-rgt-identity 56.5%

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

       5. *-commutative 56.5%

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

       6. associate-*l* 56.5%

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

       7. associate-*r* 57.6%

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

       8. *-commutative 57.6%

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

   14. Simplified 57.6%

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

   if -1.000000000000002e-309 < d < 1e-61

   1. Initial program 51.3%

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

   3. Simplified 45.6%

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

   5. Taylor expanded in d around 0 48.1%

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

      1. pow1 48.1%

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

   7. Applied egg-rr 63.7%

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

      1. unpow1 63.7%

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

      2. *-commutative 63.7%

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

      3. *-commutative 63.7%

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

      4. associate-*r/ 61.8%

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

      5. associate-/l* 61.7%

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

   9. Simplified 61.7%

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

   if 1e-61 < d

   1. Initial program 63.4%

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

   3. Simplified 63.4%

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

   5. Taylor expanded in d around 0 78.4%

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

      1. associate-*r/ 86.4%

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

   7. Applied egg-rr 86.5%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.3 \cdot 10^{+149}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{0.5}{d} \cdot \left(D \cdot M\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 10^{-61}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}\right) \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{if}\;d \leq -3.3 \cdot 10^{+149}:\\ \;\;\;\;d \cdot \left(-t\_0\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{0.5}{d} \cdot \left(D\_m \cdot M\_m\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 8.5 \cdot 10^{+118}:\\ \;\;\;\;d \cdot \left(t\_0 \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (pow (* h l) -0.5)))
   (if (<= d -3.3e+149)
     (* d (- t_0))
     (if (<= d -1e-309)
       (*
        (sqrt (* (/ d h) (/ d l)))
        (+ 1.0 (* (/ h l) (* -0.5 (pow (* (/ 0.5 d) (* D_m M_m)) 2.0)))))
       (if (<= d 8.5e+118)
         (*
          d
          (*
           t_0
           (- 1.0 (* (/ h l) (* 0.5 (pow (* D_m (* M_m (/ 0.5 d))) 2.0))))))
         (* d (* (pow l -0.5) (pow h -0.5))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = pow((h * l), -0.5);
	double tmp;
	if (d <= -3.3e+149) {
		tmp = d * -t_0;
	} else if (d <= -1e-309) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + ((h / l) * (-0.5 * pow(((0.5 / d) * (D_m * M_m)), 2.0))));
	} else if (d <= 8.5e+118) {
		tmp = d * (t_0 * (1.0 - ((h / l) * (0.5 * pow((D_m * (M_m * (0.5 / d))), 2.0)))));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (h * l) ** (-0.5d0)
    if (d <= (-3.3d+149)) then
        tmp = d * -t_0
    else if (d <= (-1d-309)) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 + ((h / l) * ((-0.5d0) * (((0.5d0 / d) * (d_m * m_m)) ** 2.0d0))))
    else if (d <= 8.5d+118) then
        tmp = d * (t_0 * (1.0d0 - ((h / l) * (0.5d0 * ((d_m * (m_m * (0.5d0 / d))) ** 2.0d0)))))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.pow((h * l), -0.5);
	double tmp;
	if (d <= -3.3e+149) {
		tmp = d * -t_0;
	} else if (d <= -1e-309) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 + ((h / l) * (-0.5 * Math.pow(((0.5 / d) * (D_m * M_m)), 2.0))));
	} else if (d <= 8.5e+118) {
		tmp = d * (t_0 * (1.0 - ((h / l) * (0.5 * Math.pow((D_m * (M_m * (0.5 / d))), 2.0)))));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.pow((h * l), -0.5)
	tmp = 0
	if d <= -3.3e+149:
		tmp = d * -t_0
	elif d <= -1e-309:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 + ((h / l) * (-0.5 * math.pow(((0.5 / d) * (D_m * M_m)), 2.0))))
	elif d <= 8.5e+118:
		tmp = d * (t_0 * (1.0 - ((h / l) * (0.5 * math.pow((D_m * (M_m * (0.5 / d))), 2.0)))))
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(h * l) ^ -0.5
	tmp = 0.0
	if (d <= -3.3e+149)
		tmp = Float64(d * Float64(-t_0));
	elseif (d <= -1e-309)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(0.5 / d) * Float64(D_m * M_m)) ^ 2.0)))));
	elseif (d <= 8.5e+118)
		tmp = Float64(d * Float64(t_0 * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(D_m * Float64(M_m * Float64(0.5 / d))) ^ 2.0))))));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (h * l) ^ -0.5;
	tmp = 0.0;
	if (d <= -3.3e+149)
		tmp = d * -t_0;
	elseif (d <= -1e-309)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + ((h / l) * (-0.5 * (((0.5 / d) * (D_m * M_m)) ^ 2.0))));
	elseif (d <= 8.5e+118)
		tmp = d * (t_0 * (1.0 - ((h / l) * (0.5 * ((D_m * (M_m * (0.5 / d))) ^ 2.0)))));
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[d, -3.3e+149], N[(d * (-t$95$0)), $MachinePrecision], If[LessEqual[d, -1e-309], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(0.5 / d), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 8.5e+118], N[(d * N[(t$95$0 * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.3e149

   1. Initial program 75.4%

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

   3. Simplified 75.6%

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

      1. add-sqr-sqrt 75.6%

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

      2. pow2 75.6%

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

      3. sqrt-prod 75.6%

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

      4. sqrt-pow1 75.6%

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

      5. frac-times 78.2%

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

      6. associate-/r* 78.2%

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

      7. metadata-eval 78.2%

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

      8. pow1 78.2%

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

      9. div-inv 78.2%

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

      10. *-commutative 78.2%

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

      11. associate-*r* 78.2%

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

      12. div-inv 78.2%

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

      13. associate-*r/ 78.5%

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

      14. associate-/l/ 78.5%

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

      15. *-un-lft-identity 78.5%

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

      16. *-commutative 78.5%

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

      17. times-frac 78.5%

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

      18. metadata-eval 78.5%

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

   6. Applied egg-rr 78.5%

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

      1. clear-num 78.5%

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

      2. sqrt-div 78.5%

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

      3. metadata-eval 78.5%

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

   8. Applied egg-rr 78.5%

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

   9. Taylor expanded in d around -inf 68.5%

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

       1. mul-1-neg 68.5%

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

       2. *-commutative 68.5%

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

       3. associate-/r* 68.6%

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

       4. distribute-rgt-neg-in 68.6%

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

       5. associate-/r* 68.5%

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

       6. unpow-1 68.5%

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

       7. metadata-eval 68.5%

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

       8. pow-sqr 68.6%

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

       9. rem-sqrt-square 68.6%

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

       10. metadata-eval 68.6%

           \[\leadsto \left|{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right| \cdot \left(-d\right) \]

       11. pow-sqr 68.5%

           \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}}\right| \cdot \left(-d\right) \]

       12. fabs-sqr 68.5%

           \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}\right)} \cdot \left(-d\right) \]

       13. pow-sqr 68.6%

           \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(2 \cdot -0.25\right)}} \cdot \left(-d\right) \]

       14. metadata-eval 68.6%

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

   11. Simplified 68.6%

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

   if -3.3e149 < d < -1.000000000000002e-309

   1. Initial program 68.7%

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

   3. Simplified 66.6%

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

      1. add-sqr-sqrt 66.6%

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

      2. pow2 66.6%

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

      3. sqrt-prod 66.6%

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

      4. sqrt-pow1 69.6%

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

      5. frac-times 72.7%

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

      6. associate-/r* 72.7%

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

      7. metadata-eval 72.7%

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

      8. pow1 72.7%

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

      9. div-inv 72.7%

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

      10. *-commutative 72.7%

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

      11. associate-*r* 72.7%

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

      12. div-inv 72.7%

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

      13. associate-*r/ 67.7%

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

      14. associate-/l/ 67.7%

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

      15. *-un-lft-identity 67.7%

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

      16. *-commutative 67.7%

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

      17. times-frac 67.7%

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

      18. metadata-eval 67.7%

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

   6. Applied egg-rr 67.7%

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

      1. pow1 67.7%

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

      2. sqrt-unprod 57.5%

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

      3. cancel-sign-sub-inv 57.5%

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

      4. metadata-eval 57.5%

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

      5. *-commutative 57.5%

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

      6. unpow-prod-down 56.4%

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

      7. pow2 56.4%

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

      8. add-sqr-sqrt 56.5%

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

      9. associate-*r/ 56.5%

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

   8. Applied egg-rr 56.5%

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

      1. unpow1 56.5%

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

      2. associate-*r* 56.5%

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

      3. *-commutative 56.5%

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

      4. associate-/l* 56.5%

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

   10. Simplified 56.5%

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

       1. expm1-log1p-u 19.1%

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

       2. log1p-define 19.1%

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

       3. expm1-undefine 19.1%

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

       4. add-exp-log 56.5%

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

       5. +-commutative 56.5%

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

       6. associate-*l* 56.5%

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

       7. fma-define 56.5%

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

   12. Applied egg-rr 56.5%

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

       1. fma-undefine 56.5%

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

       2. associate--l+ 56.5%

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

       3. metadata-eval 56.5%

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

       4. +-rgt-identity 56.5%

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

       5. *-commutative 56.5%

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

       6. associate-*l* 56.5%

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

       7. associate-*r* 57.6%

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

       8. *-commutative 57.6%

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

   14. Simplified 57.6%

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

   if -1.000000000000002e-309 < d < 8.50000000000000033e118

   1. Initial program 63.5%

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

   3. Simplified 59.8%

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

   5. Taylor expanded in d around 0 59.1%

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

      1. pow1 59.1%

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

   7. Applied egg-rr 68.0%

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

      1. unpow1 68.0%

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

      2. *-commutative 68.0%

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

      3. *-commutative 68.0%

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

      4. associate-*r/ 66.8%

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

      5. associate-/l* 66.8%

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

   9. Simplified 66.8%

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

   if 8.50000000000000033e118 < d

   1. Initial program 49.6%

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

   3. Simplified 49.6%

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

      1. add-sqr-sqrt 49.6%

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

      2. pow2 49.6%

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

      3. sqrt-prod 49.6%

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

      4. sqrt-pow1 49.7%

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

      5. frac-times 49.6%

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

      6. associate-/r* 49.6%

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

      7. metadata-eval 49.6%

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

      8. pow1 49.6%

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

      9. div-inv 49.6%

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

      10. *-commutative 49.6%

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

      11. associate-*r* 49.6%

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

      12. div-inv 49.6%

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

      13. associate-*r/ 49.7%

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

      14. associate-/l/ 49.7%

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

      15. *-un-lft-identity 49.7%

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

      16. *-commutative 49.7%

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

      17. times-frac 49.7%

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

      18. metadata-eval 49.7%

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

   6. Applied egg-rr 49.7%

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

   7. Taylor expanded in d around inf 82.4%

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

      1. pow1 82.4%

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

      2. pow1/2 82.4%

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

      3. inv-pow 82.4%

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

      4. *-commutative 82.4%

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

      5. pow-pow 82.3%

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

      6. *-commutative 82.3%

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

      7. metadata-eval 82.3%

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

   9. Applied egg-rr 82.3%

      \[\leadsto \color{blue}{{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 82.3%

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

   11. Simplified 82.3%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   12. Step-by-step derivation

       1. *-commutative 82.3%

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

       2. unpow-prod-down 90.6%

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

   13. Applied egg-rr 90.6%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.3 \cdot 10^{+149}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{0.5}{d} \cdot \left(D \cdot M\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 8.5 \cdot 10^{+118}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{if}\;d \leq -1.1 \cdot 10^{+150}:\\ \;\;\;\;d \cdot \left(-t\_0\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(\frac{M\_m \cdot \left(0.5 \cdot D\_m\right)}{d}\right)}^{2}\right)\\ \mathbf{elif}\;d \leq 7 \cdot 10^{+118}:\\ \;\;\;\;d \cdot \left(t\_0 \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (pow (* h l) -0.5)))
   (if (<= d -1.1e+150)
     (* d (- t_0))
     (if (<= d -1e-309)
       (*
        (sqrt (* (/ d h) (/ d l)))
        (+ 1.0 (* (* (/ h l) -0.5) (pow (/ (* M_m (* 0.5 D_m)) d) 2.0))))
       (if (<= d 7e+118)
         (*
          d
          (*
           t_0
           (- 1.0 (* (/ h l) (* 0.5 (pow (* D_m (* M_m (/ 0.5 d))) 2.0))))))
         (* d (* (pow l -0.5) (pow h -0.5))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = pow((h * l), -0.5);
	double tmp;
	if (d <= -1.1e+150) {
		tmp = d * -t_0;
	} else if (d <= -1e-309) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (((h / l) * -0.5) * pow(((M_m * (0.5 * D_m)) / d), 2.0)));
	} else if (d <= 7e+118) {
		tmp = d * (t_0 * (1.0 - ((h / l) * (0.5 * pow((D_m * (M_m * (0.5 / d))), 2.0)))));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (h * l) ** (-0.5d0)
    if (d <= (-1.1d+150)) then
        tmp = d * -t_0
    else if (d <= (-1d-309)) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 + (((h / l) * (-0.5d0)) * (((m_m * (0.5d0 * d_m)) / d) ** 2.0d0)))
    else if (d <= 7d+118) then
        tmp = d * (t_0 * (1.0d0 - ((h / l) * (0.5d0 * ((d_m * (m_m * (0.5d0 / d))) ** 2.0d0)))))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.pow((h * l), -0.5);
	double tmp;
	if (d <= -1.1e+150) {
		tmp = d * -t_0;
	} else if (d <= -1e-309) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 + (((h / l) * -0.5) * Math.pow(((M_m * (0.5 * D_m)) / d), 2.0)));
	} else if (d <= 7e+118) {
		tmp = d * (t_0 * (1.0 - ((h / l) * (0.5 * Math.pow((D_m * (M_m * (0.5 / d))), 2.0)))));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.pow((h * l), -0.5)
	tmp = 0
	if d <= -1.1e+150:
		tmp = d * -t_0
	elif d <= -1e-309:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 + (((h / l) * -0.5) * math.pow(((M_m * (0.5 * D_m)) / d), 2.0)))
	elif d <= 7e+118:
		tmp = d * (t_0 * (1.0 - ((h / l) * (0.5 * math.pow((D_m * (M_m * (0.5 / d))), 2.0)))))
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(h * l) ^ -0.5
	tmp = 0.0
	if (d <= -1.1e+150)
		tmp = Float64(d * Float64(-t_0));
	elseif (d <= -1e-309)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(Float64(Float64(h / l) * -0.5) * (Float64(Float64(M_m * Float64(0.5 * D_m)) / d) ^ 2.0))));
	elseif (d <= 7e+118)
		tmp = Float64(d * Float64(t_0 * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(D_m * Float64(M_m * Float64(0.5 / d))) ^ 2.0))))));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (h * l) ^ -0.5;
	tmp = 0.0;
	if (d <= -1.1e+150)
		tmp = d * -t_0;
	elseif (d <= -1e-309)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (((h / l) * -0.5) * (((M_m * (0.5 * D_m)) / d) ^ 2.0)));
	elseif (d <= 7e+118)
		tmp = d * (t_0 * (1.0 - ((h / l) * (0.5 * ((D_m * (M_m * (0.5 / d))) ^ 2.0)))));
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[d, -1.1e+150], N[(d * (-t$95$0)), $MachinePrecision], If[LessEqual[d, -1e-309], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(N[(M$95$m * N[(0.5 * D$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7e+118], N[(d * N[(t$95$0 * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.1e150

   1. Initial program 75.4%

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

   3. Simplified 75.6%

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

      1. add-sqr-sqrt 75.6%

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

      2. pow2 75.6%

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

      3. sqrt-prod 75.6%

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

      4. sqrt-pow1 75.6%

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

      5. frac-times 78.2%

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

      6. associate-/r* 78.2%

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

      7. metadata-eval 78.2%

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

      8. pow1 78.2%

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

      9. div-inv 78.2%

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

      10. *-commutative 78.2%

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

      11. associate-*r* 78.2%

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

      12. div-inv 78.2%

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

      13. associate-*r/ 78.5%

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

      14. associate-/l/ 78.5%

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

      15. *-un-lft-identity 78.5%

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

      16. *-commutative 78.5%

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

      17. times-frac 78.5%

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

      18. metadata-eval 78.5%

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

   6. Applied egg-rr 78.5%

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

      1. clear-num 78.5%

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

      2. sqrt-div 78.5%

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

      3. metadata-eval 78.5%

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

   8. Applied egg-rr 78.5%

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

   9. Taylor expanded in d around -inf 68.5%

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

       1. mul-1-neg 68.5%

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

       2. *-commutative 68.5%

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

       3. associate-/r* 68.6%

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

       4. distribute-rgt-neg-in 68.6%

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

       5. associate-/r* 68.5%

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

       6. unpow-1 68.5%

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

       7. metadata-eval 68.5%

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

       8. pow-sqr 68.6%

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

       9. rem-sqrt-square 68.6%

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

       10. metadata-eval 68.6%

           \[\leadsto \left|{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right| \cdot \left(-d\right) \]

       11. pow-sqr 68.5%

           \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}}\right| \cdot \left(-d\right) \]

       12. fabs-sqr 68.5%

           \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}\right)} \cdot \left(-d\right) \]

       13. pow-sqr 68.6%

           \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(2 \cdot -0.25\right)}} \cdot \left(-d\right) \]

       14. metadata-eval 68.6%

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

   11. Simplified 68.6%

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

   if -1.1e150 < d < -1.000000000000002e-309

   1. Initial program 68.7%

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

   3. Simplified 66.6%

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

      1. add-sqr-sqrt 66.6%

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

      2. pow2 66.6%

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

      3. sqrt-prod 66.6%

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

      4. sqrt-pow1 69.6%

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

      5. frac-times 72.7%

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

      6. associate-/r* 72.7%

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

      7. metadata-eval 72.7%

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

      8. pow1 72.7%

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

      9. div-inv 72.7%

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

      10. *-commutative 72.7%

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

      11. associate-*r* 72.7%

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

      12. div-inv 72.7%

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

      13. associate-*r/ 67.7%

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

      14. associate-/l/ 67.7%

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

      15. *-un-lft-identity 67.7%

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

      16. *-commutative 67.7%

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

      17. times-frac 67.7%

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

      18. metadata-eval 67.7%

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

   6. Applied egg-rr 67.7%

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

      1. pow1 67.7%

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

      2. sqrt-pow2 67.7%

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

      3. sqr-pow 67.6%

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

      4. pow2 67.6%

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

      5. metadata-eval 67.6%

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

      6. metadata-eval 67.6%

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

   8. Applied egg-rr 67.6%

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

      1. pow1 67.6%

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

   10. Applied egg-rr 56.5%

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

       1. unpow1 56.5%

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

       2. *-commutative 56.5%

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

       3. associate-*r* 56.5%

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

       4. associate-*r/ 57.6%

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

   12. Simplified 57.6%

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

   if -1.000000000000002e-309 < d < 7.00000000000000033e118

   1. Initial program 63.5%

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

   3. Simplified 59.8%

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

   5. Taylor expanded in d around 0 59.1%

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

      1. pow1 59.1%

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

   7. Applied egg-rr 68.0%

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

      1. unpow1 68.0%

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

      2. *-commutative 68.0%

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

      3. *-commutative 68.0%

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

      4. associate-*r/ 66.8%

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

      5. associate-/l* 66.8%

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

   9. Simplified 66.8%

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

   if 7.00000000000000033e118 < d

   1. Initial program 49.6%

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

   3. Simplified 49.6%

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

      1. add-sqr-sqrt 49.6%

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

      2. pow2 49.6%

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

      3. sqrt-prod 49.6%

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

      4. sqrt-pow1 49.7%

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

      5. frac-times 49.6%

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

      6. associate-/r* 49.6%

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

      7. metadata-eval 49.6%

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

      8. pow1 49.6%

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

      9. div-inv 49.6%

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

      10. *-commutative 49.6%

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

      11. associate-*r* 49.6%

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

      12. div-inv 49.6%

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

      13. associate-*r/ 49.7%

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

      14. associate-/l/ 49.7%

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

      15. *-un-lft-identity 49.7%

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

      16. *-commutative 49.7%

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

      17. times-frac 49.7%

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

      18. metadata-eval 49.7%

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

   6. Applied egg-rr 49.7%

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

   7. Taylor expanded in d around inf 82.4%

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

      1. pow1 82.4%

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

      2. pow1/2 82.4%

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

      3. inv-pow 82.4%

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

      4. *-commutative 82.4%

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

      5. pow-pow 82.3%

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

      6. *-commutative 82.3%

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

      7. metadata-eval 82.3%

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

   9. Applied egg-rr 82.3%

      \[\leadsto \color{blue}{{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 82.3%

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

   11. Simplified 82.3%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   12. Step-by-step derivation

       1. *-commutative 82.3%

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

       2. unpow-prod-down 90.6%

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

   13. Applied egg-rr 90.6%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{+150}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(\frac{M \cdot \left(0.5 \cdot D\right)}{d}\right)}^{2}\right)\\ \mathbf{elif}\;d \leq 7 \cdot 10^{+118}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := {\left(h \cdot \ell\right)}^{-0.5}\\ t_1 := {\left(D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2}\\ \mathbf{if}\;d \leq -9 \cdot 10^{+149}:\\ \;\;\;\;d \cdot \left(-t\_0\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + t\_1 \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{elif}\;d \leq 9 \cdot 10^{+118}:\\ \;\;\;\;d \cdot \left(t\_0 \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot t\_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (pow (* h l) -0.5)) (t_1 (pow (* D_m (* M_m (/ 0.5 d))) 2.0)))
   (if (<= d -9e+149)
     (* d (- t_0))
     (if (<= d -1e-309)
       (* (sqrt (* (/ d h) (/ d l))) (+ 1.0 (* t_1 (* (/ h l) -0.5))))
       (if (<= d 9e+118)
         (* d (* t_0 (- 1.0 (* (/ h l) (* 0.5 t_1)))))
         (* d (* (pow l -0.5) (pow h -0.5))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = pow((h * l), -0.5);
	double t_1 = pow((D_m * (M_m * (0.5 / d))), 2.0);
	double tmp;
	if (d <= -9e+149) {
		tmp = d * -t_0;
	} else if (d <= -1e-309) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (t_1 * ((h / l) * -0.5)));
	} else if (d <= 9e+118) {
		tmp = d * (t_0 * (1.0 - ((h / l) * (0.5 * t_1))));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (h * l) ** (-0.5d0)
    t_1 = (d_m * (m_m * (0.5d0 / d))) ** 2.0d0
    if (d <= (-9d+149)) then
        tmp = d * -t_0
    else if (d <= (-1d-309)) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 + (t_1 * ((h / l) * (-0.5d0))))
    else if (d <= 9d+118) then
        tmp = d * (t_0 * (1.0d0 - ((h / l) * (0.5d0 * t_1))))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.pow((h * l), -0.5);
	double t_1 = Math.pow((D_m * (M_m * (0.5 / d))), 2.0);
	double tmp;
	if (d <= -9e+149) {
		tmp = d * -t_0;
	} else if (d <= -1e-309) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 + (t_1 * ((h / l) * -0.5)));
	} else if (d <= 9e+118) {
		tmp = d * (t_0 * (1.0 - ((h / l) * (0.5 * t_1))));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.pow((h * l), -0.5)
	t_1 = math.pow((D_m * (M_m * (0.5 / d))), 2.0)
	tmp = 0
	if d <= -9e+149:
		tmp = d * -t_0
	elif d <= -1e-309:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 + (t_1 * ((h / l) * -0.5)))
	elif d <= 9e+118:
		tmp = d * (t_0 * (1.0 - ((h / l) * (0.5 * t_1))))
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(h * l) ^ -0.5
	t_1 = Float64(D_m * Float64(M_m * Float64(0.5 / d))) ^ 2.0
	tmp = 0.0
	if (d <= -9e+149)
		tmp = Float64(d * Float64(-t_0));
	elseif (d <= -1e-309)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(t_1 * Float64(Float64(h / l) * -0.5))));
	elseif (d <= 9e+118)
		tmp = Float64(d * Float64(t_0 * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * t_1)))));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (h * l) ^ -0.5;
	t_1 = (D_m * (M_m * (0.5 / d))) ^ 2.0;
	tmp = 0.0;
	if (d <= -9e+149)
		tmp = d * -t_0;
	elseif (d <= -1e-309)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (t_1 * ((h / l) * -0.5)));
	elseif (d <= 9e+118)
		tmp = d * (t_0 * (1.0 - ((h / l) * (0.5 * t_1))));
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[d, -9e+149], N[(d * (-t$95$0)), $MachinePrecision], If[LessEqual[d, -1e-309], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(t$95$1 * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 9e+118], N[(d * N[(t$95$0 * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -8.99999999999999965e149

   1. Initial program 75.4%

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

   3. Simplified 75.6%

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

      1. add-sqr-sqrt 75.6%

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

      2. pow2 75.6%

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

      3. sqrt-prod 75.6%

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

      4. sqrt-pow1 75.6%

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

      5. frac-times 78.2%

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

      6. associate-/r* 78.2%

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

      7. metadata-eval 78.2%

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

      8. pow1 78.2%

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

      9. div-inv 78.2%

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

      10. *-commutative 78.2%

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

      11. associate-*r* 78.2%

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

      12. div-inv 78.2%

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

      13. associate-*r/ 78.5%

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

      14. associate-/l/ 78.5%

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

      15. *-un-lft-identity 78.5%

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

      16. *-commutative 78.5%

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

      17. times-frac 78.5%

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

      18. metadata-eval 78.5%

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

   6. Applied egg-rr 78.5%

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

      1. clear-num 78.5%

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

      2. sqrt-div 78.5%

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

      3. metadata-eval 78.5%

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

   8. Applied egg-rr 78.5%

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

   9. Taylor expanded in d around -inf 68.5%

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

       1. mul-1-neg 68.5%

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

       2. *-commutative 68.5%

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

       3. associate-/r* 68.6%

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

       4. distribute-rgt-neg-in 68.6%

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

       5. associate-/r* 68.5%

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

       6. unpow-1 68.5%

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

       7. metadata-eval 68.5%

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

       8. pow-sqr 68.6%

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

       9. rem-sqrt-square 68.6%

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

       10. metadata-eval 68.6%

           \[\leadsto \left|{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right| \cdot \left(-d\right) \]

       11. pow-sqr 68.5%

           \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}}\right| \cdot \left(-d\right) \]

       12. fabs-sqr 68.5%

           \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}\right)} \cdot \left(-d\right) \]

       13. pow-sqr 68.6%

           \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(2 \cdot -0.25\right)}} \cdot \left(-d\right) \]

       14. metadata-eval 68.6%

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

   11. Simplified 68.6%

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

   if -8.99999999999999965e149 < d < -1.000000000000002e-309

   1. Initial program 68.7%

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

   3. Simplified 66.6%

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

      1. add-sqr-sqrt 66.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 66.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 66.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 69.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. frac-times 72.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-/r* 72.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. metadata-eval 72.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. pow1 72.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. div-inv 72.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. *-commutative 72.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. associate-*r* 72.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot \frac{1}{2}\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. div-inv 72.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \color{blue}{\frac{M}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. associate-*r/ 67.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{\frac{M}{2}}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      14. associate-/l/ 67.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      15. *-un-lft-identity 67.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{\color{blue}{1 \cdot M}}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      16. *-commutative 67.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      17. times-frac 67.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      18. metadata-eval 67.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 67.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. pow1 67.7%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)}^{1}} \]

      2. sqrt-unprod 57.5%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)}^{1} \]

      3. cancel-sign-sub-inv 57.5%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-0.5\right) \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)}\right)}^{1} \]

      4. metadata-eval 57.5%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \color{blue}{-0.5} \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)}^{1} \]

      5. *-commutative 57.5%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot {\color{blue}{\left(\sqrt{\frac{h}{\ell}} \cdot \left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)\right)}}^{2}\right)\right)}^{1} \]

      6. unpow-prod-down 56.4%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \color{blue}{\left({\left(\sqrt{\frac{h}{\ell}}\right)}^{2} \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}\right)}\right)\right)}^{1} \]

      7. pow2 56.4%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\color{blue}{\left(\sqrt{\frac{h}{\ell}} \cdot \sqrt{\frac{h}{\ell}}\right)} \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}\right)\right)\right)}^{1} \]

      8. add-sqr-sqrt 56.5%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\color{blue}{\frac{h}{\ell}} \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}\right)\right)\right)}^{1} \]

      9. associate-*r/ 56.5%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \color{blue}{\frac{0.5 \cdot M}{d}}\right)}^{2}\right)\right)\right)}^{1} \]

   8. Applied egg-rr 56.5%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{0.5 \cdot M}{d}\right)}^{2}\right)\right)\right)}^{1}} \]
   9. Step-by-step derivation

      1. unpow1 56.5%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{0.5 \cdot M}{d}\right)}^{2}\right)\right)} \]

      2. associate-*r* 56.5%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \color{blue}{\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(D \cdot \frac{0.5 \cdot M}{d}\right)}^{2}}\right) \]

      3. *-commutative 56.5%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(D \cdot \frac{\color{blue}{M \cdot 0.5}}{d}\right)}^{2}\right) \]

      4. associate-/l* 56.5%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(D \cdot \color{blue}{\left(M \cdot \frac{0.5}{d}\right)}\right)}^{2}\right) \]

   10. Simplified 56.5%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)} \]

   if -1.000000000000002e-309 < d < 9.00000000000000004e118

   1. Initial program 63.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 59.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around 0 59.1%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. pow1 59.1%

         \[\leadsto \color{blue}{{\left(\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1}} \]

   7. Applied egg-rr 68.0%

      \[\leadsto \color{blue}{{\left(d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \left(0.5 \cdot {\left(\frac{\left(M \cdot 0.5\right) \cdot D}{d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 68.0%

         \[\leadsto \color{blue}{d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \left(0.5 \cdot {\left(\frac{\left(M \cdot 0.5\right) \cdot D}{d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      2. *-commutative 68.0%

         \[\leadsto d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{\left(M \cdot 0.5\right) \cdot D}{d}\right)}^{2}\right)}\right)\right) \]

      3. *-commutative 68.0%

         \[\leadsto d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot 0.5\right)}}{d}\right)}^{2}\right)\right)\right) \]

      4. associate-*r/ 66.8%

         \[\leadsto d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\color{blue}{\left(D \cdot \frac{M \cdot 0.5}{d}\right)}}^{2}\right)\right)\right) \]

      5. associate-/l* 66.8%

         \[\leadsto d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \color{blue}{\left(M \cdot \frac{0.5}{d}\right)}\right)}^{2}\right)\right)\right) \]

   9. Simplified 66.8%

      \[\leadsto \color{blue}{d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)} \]

   if 9.00000000000000004e118 < d

   1. Initial program 49.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 49.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 49.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 49.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 49.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 49.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. frac-times 49.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-/r* 49.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. metadata-eval 49.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. pow1 49.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. div-inv 49.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. *-commutative 49.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. associate-*r* 49.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot \frac{1}{2}\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. div-inv 49.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \color{blue}{\frac{M}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. associate-*r/ 49.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{\frac{M}{2}}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      14. associate-/l/ 49.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      15. *-un-lft-identity 49.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{\color{blue}{1 \cdot M}}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      16. *-commutative 49.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      17. times-frac 49.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      18. metadata-eval 49.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 49.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 82.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. pow1 82.4%

         \[\leadsto \color{blue}{{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}^{1}} \]

      2. pow1/2 82.4%

         \[\leadsto {\left(d \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right)}^{1} \]

      3. inv-pow 82.4%

         \[\leadsto {\left(d \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right)}^{1} \]

      4. *-commutative 82.4%

         \[\leadsto {\left(d \cdot {\left({\color{blue}{\left(\ell \cdot h\right)}}^{-1}\right)}^{0.5}\right)}^{1} \]

      5. pow-pow 82.3%

         \[\leadsto {\left(d \cdot \color{blue}{{\left(\ell \cdot h\right)}^{\left(-1 \cdot 0.5\right)}}\right)}^{1} \]

      6. *-commutative 82.3%

         \[\leadsto {\left(d \cdot {\color{blue}{\left(h \cdot \ell\right)}}^{\left(-1 \cdot 0.5\right)}\right)}^{1} \]

      7. metadata-eval 82.3%

         \[\leadsto {\left(d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right)}^{1} \]

   9. Applied egg-rr 82.3%

      \[\leadsto \color{blue}{{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 82.3%

          \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   11. Simplified 82.3%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   12. Step-by-step derivation

       1. *-commutative 82.3%

          \[\leadsto d \cdot {\color{blue}{\left(\ell \cdot h\right)}}^{-0.5} \]

       2. unpow-prod-down 90.6%

          \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]

   13. Applied egg-rr 90.6%

       \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9 \cdot 10^{+149}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{elif}\;d \leq 9 \cdot 10^{+118}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{if}\;d \leq -3.6 \cdot 10^{+106}:\\ \;\;\;\;d \cdot \left(-t\_0\right)\\ \mathbf{elif}\;d \leq -1.8 \cdot 10^{-274}:\\ \;\;\;\;0.125 \cdot \left(\left(\left(D\_m \cdot D\_m\right) \cdot \left(M\_m \cdot M\_m\right)\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\right)\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{+118}:\\ \;\;\;\;d \cdot \left(t\_0 \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (pow (* h l) -0.5)))
   (if (<= d -3.6e+106)
     (* d (- t_0))
     (if (<= d -1.8e-274)
       (* 0.125 (* (* (* D_m D_m) (* M_m M_m)) (/ (sqrt (/ h (pow l 3.0))) d)))
       (if (<= d 6.5e+118)
         (*
          d
          (*
           t_0
           (- 1.0 (* (/ h l) (* 0.5 (pow (* D_m (* M_m (/ 0.5 d))) 2.0))))))
         (* d (* (pow l -0.5) (pow h -0.5))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = pow((h * l), -0.5);
	double tmp;
	if (d <= -3.6e+106) {
		tmp = d * -t_0;
	} else if (d <= -1.8e-274) {
		tmp = 0.125 * (((D_m * D_m) * (M_m * M_m)) * (sqrt((h / pow(l, 3.0))) / d));
	} else if (d <= 6.5e+118) {
		tmp = d * (t_0 * (1.0 - ((h / l) * (0.5 * pow((D_m * (M_m * (0.5 / d))), 2.0)))));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (h * l) ** (-0.5d0)
    if (d <= (-3.6d+106)) then
        tmp = d * -t_0
    else if (d <= (-1.8d-274)) then
        tmp = 0.125d0 * (((d_m * d_m) * (m_m * m_m)) * (sqrt((h / (l ** 3.0d0))) / d))
    else if (d <= 6.5d+118) then
        tmp = d * (t_0 * (1.0d0 - ((h / l) * (0.5d0 * ((d_m * (m_m * (0.5d0 / d))) ** 2.0d0)))))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.pow((h * l), -0.5);
	double tmp;
	if (d <= -3.6e+106) {
		tmp = d * -t_0;
	} else if (d <= -1.8e-274) {
		tmp = 0.125 * (((D_m * D_m) * (M_m * M_m)) * (Math.sqrt((h / Math.pow(l, 3.0))) / d));
	} else if (d <= 6.5e+118) {
		tmp = d * (t_0 * (1.0 - ((h / l) * (0.5 * Math.pow((D_m * (M_m * (0.5 / d))), 2.0)))));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.pow((h * l), -0.5)
	tmp = 0
	if d <= -3.6e+106:
		tmp = d * -t_0
	elif d <= -1.8e-274:
		tmp = 0.125 * (((D_m * D_m) * (M_m * M_m)) * (math.sqrt((h / math.pow(l, 3.0))) / d))
	elif d <= 6.5e+118:
		tmp = d * (t_0 * (1.0 - ((h / l) * (0.5 * math.pow((D_m * (M_m * (0.5 / d))), 2.0)))))
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(h * l) ^ -0.5
	tmp = 0.0
	if (d <= -3.6e+106)
		tmp = Float64(d * Float64(-t_0));
	elseif (d <= -1.8e-274)
		tmp = Float64(0.125 * Float64(Float64(Float64(D_m * D_m) * Float64(M_m * M_m)) * Float64(sqrt(Float64(h / (l ^ 3.0))) / d)));
	elseif (d <= 6.5e+118)
		tmp = Float64(d * Float64(t_0 * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(D_m * Float64(M_m * Float64(0.5 / d))) ^ 2.0))))));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (h * l) ^ -0.5;
	tmp = 0.0;
	if (d <= -3.6e+106)
		tmp = d * -t_0;
	elseif (d <= -1.8e-274)
		tmp = 0.125 * (((D_m * D_m) * (M_m * M_m)) * (sqrt((h / (l ^ 3.0))) / d));
	elseif (d <= 6.5e+118)
		tmp = d * (t_0 * (1.0 - ((h / l) * (0.5 * ((D_m * (M_m * (0.5 / d))) ^ 2.0)))));
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[d, -3.6e+106], N[(d * (-t$95$0)), $MachinePrecision], If[LessEqual[d, -1.8e-274], N[(0.125 * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6.5e+118], N[(d * N[(t$95$0 * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.6000000000000001e106

   1. Initial program 76.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 74.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 74.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 74.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 74.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 74.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. frac-times 78.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-/r* 78.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. metadata-eval 78.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. pow1 78.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. div-inv 78.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. *-commutative 78.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. associate-*r* 78.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot \frac{1}{2}\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. div-inv 78.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \color{blue}{\frac{M}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. associate-*r/ 78.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{\frac{M}{2}}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      14. associate-/l/ 78.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      15. *-un-lft-identity 78.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{\color{blue}{1 \cdot M}}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      16. *-commutative 78.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      17. times-frac 78.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      18. metadata-eval 78.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 78.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. clear-num 78.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. sqrt-div 78.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. metadata-eval 78.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   8. Applied egg-rr 78.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   9. Taylor expanded in d around -inf 66.2%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg 66.2%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

       2. *-commutative 66.2%

          \[\leadsto -\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

       3. associate-/r* 66.2%

          \[\leadsto -\sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

       4. distribute-rgt-neg-in 66.2%

          \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)} \]

       5. associate-/r* 66.2%

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-d\right) \]

       6. unpow-1 66.2%

          \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot \left(-d\right) \]

       7. metadata-eval 66.2%

          \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(-d\right) \]

       8. pow-sqr 66.2%

          \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot \left(-d\right) \]

       9. rem-sqrt-square 66.2%

          \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \cdot \left(-d\right) \]

       10. metadata-eval 66.2%

           \[\leadsto \left|{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right| \cdot \left(-d\right) \]

       11. pow-sqr 66.1%

           \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}}\right| \cdot \left(-d\right) \]

       12. fabs-sqr 66.1%

           \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}\right)} \cdot \left(-d\right) \]

       13. pow-sqr 66.2%

           \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(2 \cdot -0.25\right)}} \cdot \left(-d\right) \]

       14. metadata-eval 66.2%

           \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot \left(-d\right) \]

   11. Simplified 66.2%

       \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if -3.6000000000000001e106 < d < -1.79999999999999991e-274

   1. Initial program 68.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   4. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.125 \cdot \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      2. associate-/l* 0.0%

         \[\leadsto -0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)}\right) \]

      3. *-commutative 0.0%

         \[\leadsto -0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left({D}^{2} \cdot \frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot {M}^{2}}}{d}\right)\right) \]

      4. unpow2 0.0%

         \[\leadsto -0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left({D}^{2} \cdot \frac{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot {M}^{2}}{d}\right)\right) \]

      5. rem-square-sqrt 36.4%

         \[\leadsto -0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left({D}^{2} \cdot \frac{\color{blue}{-1} \cdot {M}^{2}}{d}\right)\right) \]

      6. associate-/l* 36.4%

         \[\leadsto -0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left({D}^{2} \cdot \color{blue}{\left(-1 \cdot \frac{{M}^{2}}{d}\right)}\right)\right) \]

   5. Simplified 36.4%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left({D}^{2} \cdot \left(-1 \cdot \frac{{M}^{2}}{d}\right)\right)\right)} \]

   6. Taylor expanded in h around 0 36.4%

      \[\leadsto \color{blue}{0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 37.6%

         \[\leadsto 0.125 \cdot \color{blue}{\frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}} \]

      2. associate-/l* 37.6%

         \[\leadsto 0.125 \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\right)} \]

   8. Simplified 37.6%

      \[\leadsto \color{blue}{0.125 \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\right)} \]
   9. Step-by-step derivation

      1. unpow2 37.6%

         \[\leadsto 0.125 \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\right) \]

   10. Applied egg-rr 37.6%

       \[\leadsto 0.125 \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\right) \]
   11. Step-by-step derivation

       1. unpow2 37.6%

          \[\leadsto 0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\right) \]

   12. Applied egg-rr 37.6%

       \[\leadsto 0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\right) \]

   if -1.79999999999999991e-274 < d < 6.5e118

   1. Initial program 62.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 59.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around 0 56.2%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. pow1 56.2%

         \[\leadsto \color{blue}{{\left(\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1}} \]

   7. Applied egg-rr 64.4%

      \[\leadsto \color{blue}{{\left(d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \left(0.5 \cdot {\left(\frac{\left(M \cdot 0.5\right) \cdot D}{d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 64.4%

         \[\leadsto \color{blue}{d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \left(0.5 \cdot {\left(\frac{\left(M \cdot 0.5\right) \cdot D}{d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      2. *-commutative 64.4%

         \[\leadsto d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{\left(M \cdot 0.5\right) \cdot D}{d}\right)}^{2}\right)}\right)\right) \]

      3. *-commutative 64.4%

         \[\leadsto d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot 0.5\right)}}{d}\right)}^{2}\right)\right)\right) \]

      4. associate-*r/ 63.3%

         \[\leadsto d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\color{blue}{\left(D \cdot \frac{M \cdot 0.5}{d}\right)}}^{2}\right)\right)\right) \]

      5. associate-/l* 63.3%

         \[\leadsto d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \color{blue}{\left(M \cdot \frac{0.5}{d}\right)}\right)}^{2}\right)\right)\right) \]

   9. Simplified 63.3%

      \[\leadsto \color{blue}{d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)} \]

   if 6.5e118 < d

   1. Initial program 49.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 49.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 49.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 49.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 49.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 49.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. frac-times 49.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-/r* 49.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. metadata-eval 49.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. pow1 49.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. div-inv 49.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. *-commutative 49.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. associate-*r* 49.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot \frac{1}{2}\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. div-inv 49.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \color{blue}{\frac{M}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. associate-*r/ 49.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{\frac{M}{2}}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      14. associate-/l/ 49.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      15. *-un-lft-identity 49.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{\color{blue}{1 \cdot M}}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      16. *-commutative 49.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      17. times-frac 49.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      18. metadata-eval 49.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 49.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 82.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. pow1 82.4%

         \[\leadsto \color{blue}{{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}^{1}} \]

      2. pow1/2 82.4%

         \[\leadsto {\left(d \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right)}^{1} \]

      3. inv-pow 82.4%

         \[\leadsto {\left(d \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right)}^{1} \]

      4. *-commutative 82.4%

         \[\leadsto {\left(d \cdot {\left({\color{blue}{\left(\ell \cdot h\right)}}^{-1}\right)}^{0.5}\right)}^{1} \]

      5. pow-pow 82.3%

         \[\leadsto {\left(d \cdot \color{blue}{{\left(\ell \cdot h\right)}^{\left(-1 \cdot 0.5\right)}}\right)}^{1} \]

      6. *-commutative 82.3%

         \[\leadsto {\left(d \cdot {\color{blue}{\left(h \cdot \ell\right)}}^{\left(-1 \cdot 0.5\right)}\right)}^{1} \]

      7. metadata-eval 82.3%

         \[\leadsto {\left(d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right)}^{1} \]

   9. Applied egg-rr 82.3%

      \[\leadsto \color{blue}{{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 82.3%

          \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   11. Simplified 82.3%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   12. Step-by-step derivation

       1. *-commutative 82.3%

          \[\leadsto d \cdot {\color{blue}{\left(\ell \cdot h\right)}}^{-0.5} \]

       2. unpow-prod-down 90.6%

          \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]

   13. Applied egg-rr 90.6%

       \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.6 \cdot 10^{+106}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -1.8 \cdot 10^{-274}:\\ \;\;\;\;0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\right)\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{+118}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 55.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{+95}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -1.8 \cdot 10^{-274}:\\ \;\;\;\;0.125 \cdot \left(\left(\left(D\_m \cdot D\_m\right) \cdot \left(M\_m \cdot M\_m\right)\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\right)\\ \mathbf{elif}\;d \leq 8 \cdot 10^{+118}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -5e+95)
   (* d (- (pow (* h l) -0.5)))
   (if (<= d -1.8e-274)
     (* 0.125 (* (* (* D_m D_m) (* M_m M_m)) (/ (sqrt (/ h (pow l 3.0))) d)))
     (if (<= d 8e+118)
       (*
        (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0))))
        (/ d (sqrt (* h l))))
       (* d (* (pow l -0.5) (pow h -0.5)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -5e+95) {
		tmp = d * -pow((h * l), -0.5);
	} else if (d <= -1.8e-274) {
		tmp = 0.125 * (((D_m * D_m) * (M_m * M_m)) * (sqrt((h / pow(l, 3.0))) / d));
	} else if (d <= 8e+118) {
		tmp = (1.0 - (0.5 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0)))) * (d / sqrt((h * l)));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-5d+95)) then
        tmp = d * -((h * l) ** (-0.5d0))
    else if (d <= (-1.8d-274)) then
        tmp = 0.125d0 * (((d_m * d_m) * (m_m * m_m)) * (sqrt((h / (l ** 3.0d0))) / d))
    else if (d <= 8d+118) then
        tmp = (1.0d0 - (0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_m / d)) ** 2.0d0)))) * (d / sqrt((h * l)))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -5e+95) {
		tmp = d * -Math.pow((h * l), -0.5);
	} else if (d <= -1.8e-274) {
		tmp = 0.125 * (((D_m * D_m) * (M_m * M_m)) * (Math.sqrt((h / Math.pow(l, 3.0))) / d));
	} else if (d <= 8e+118) {
		tmp = (1.0 - (0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D_m / d)), 2.0)))) * (d / Math.sqrt((h * l)));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -5e+95:
		tmp = d * -math.pow((h * l), -0.5)
	elif d <= -1.8e-274:
		tmp = 0.125 * (((D_m * D_m) * (M_m * M_m)) * (math.sqrt((h / math.pow(l, 3.0))) / d))
	elif d <= 8e+118:
		tmp = (1.0 - (0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D_m / d)), 2.0)))) * (d / math.sqrt((h * l)))
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -5e+95)
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	elseif (d <= -1.8e-274)
		tmp = Float64(0.125 * Float64(Float64(Float64(D_m * D_m) * Float64(M_m * M_m)) * Float64(sqrt(Float64(h / (l ^ 3.0))) / d)));
	elseif (d <= 8e+118)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0)))) * Float64(d / sqrt(Float64(h * l))));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -5e+95)
		tmp = d * -((h * l) ^ -0.5);
	elseif (d <= -1.8e-274)
		tmp = 0.125 * (((D_m * D_m) * (M_m * M_m)) * (sqrt((h / (l ^ 3.0))) / d));
	elseif (d <= 8e+118)
		tmp = (1.0 - (0.5 * ((h / l) * (((M_m / 2.0) * (D_m / d)) ^ 2.0)))) * (d / sqrt((h * l)));
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -5e+95], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], If[LessEqual[d, -1.8e-274], N[(0.125 * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 8e+118], N[(N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -5.00000000000000025e95

   1. Initial program 76.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 74.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 74.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 74.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 74.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 74.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. frac-times 78.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-/r* 78.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. metadata-eval 78.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. pow1 78.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. div-inv 78.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. *-commutative 78.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. associate-*r* 78.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot \frac{1}{2}\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. div-inv 78.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \color{blue}{\frac{M}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. associate-*r/ 78.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{\frac{M}{2}}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      14. associate-/l/ 78.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      15. *-un-lft-identity 78.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{\color{blue}{1 \cdot M}}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      16. *-commutative 78.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      17. times-frac 78.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      18. metadata-eval 78.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 78.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. clear-num 78.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. sqrt-div 78.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. metadata-eval 78.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   8. Applied egg-rr 78.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   9. Taylor expanded in d around -inf 66.2%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg 66.2%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

       2. *-commutative 66.2%

          \[\leadsto -\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

       3. associate-/r* 66.2%

          \[\leadsto -\sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

       4. distribute-rgt-neg-in 66.2%

          \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)} \]

       5. associate-/r* 66.2%

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-d\right) \]

       6. unpow-1 66.2%

          \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot \left(-d\right) \]

       7. metadata-eval 66.2%

          \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(-d\right) \]

       8. pow-sqr 66.2%

          \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot \left(-d\right) \]

       9. rem-sqrt-square 66.2%

          \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \cdot \left(-d\right) \]

       10. metadata-eval 66.2%

           \[\leadsto \left|{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right| \cdot \left(-d\right) \]

       11. pow-sqr 66.1%

           \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}}\right| \cdot \left(-d\right) \]

       12. fabs-sqr 66.1%

           \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}\right)} \cdot \left(-d\right) \]

       13. pow-sqr 66.2%

           \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(2 \cdot -0.25\right)}} \cdot \left(-d\right) \]

       14. metadata-eval 66.2%

           \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot \left(-d\right) \]

   11. Simplified 66.2%

       \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if -5.00000000000000025e95 < d < -1.79999999999999991e-274

   1. Initial program 68.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   4. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.125 \cdot \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      2. associate-/l* 0.0%

         \[\leadsto -0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)}\right) \]

      3. *-commutative 0.0%

         \[\leadsto -0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left({D}^{2} \cdot \frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot {M}^{2}}}{d}\right)\right) \]

      4. unpow2 0.0%

         \[\leadsto -0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left({D}^{2} \cdot \frac{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot {M}^{2}}{d}\right)\right) \]

      5. rem-square-sqrt 36.4%

         \[\leadsto -0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left({D}^{2} \cdot \frac{\color{blue}{-1} \cdot {M}^{2}}{d}\right)\right) \]

      6. associate-/l* 36.4%

         \[\leadsto -0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left({D}^{2} \cdot \color{blue}{\left(-1 \cdot \frac{{M}^{2}}{d}\right)}\right)\right) \]

   5. Simplified 36.4%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left({D}^{2} \cdot \left(-1 \cdot \frac{{M}^{2}}{d}\right)\right)\right)} \]

   6. Taylor expanded in h around 0 36.4%

      \[\leadsto \color{blue}{0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 37.6%

         \[\leadsto 0.125 \cdot \color{blue}{\frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}} \]

      2. associate-/l* 37.6%

         \[\leadsto 0.125 \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\right)} \]

   8. Simplified 37.6%

      \[\leadsto \color{blue}{0.125 \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\right)} \]
   9. Step-by-step derivation

      1. unpow2 37.6%

         \[\leadsto 0.125 \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\right) \]

   10. Applied egg-rr 37.6%

       \[\leadsto 0.125 \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\right) \]
   11. Step-by-step derivation

       1. unpow2 37.6%

          \[\leadsto 0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\right) \]

   12. Applied egg-rr 37.6%

       \[\leadsto 0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\right) \]

   if -1.79999999999999991e-274 < d < 7.99999999999999973e118

   1. Initial program 62.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 59.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around 0 56.2%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. sqrt-div 57.3%

         \[\leadsto \left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. metadata-eval 57.3%

         \[\leadsto \left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Applied egg-rr 57.3%

      \[\leadsto \left(d \cdot \color{blue}{\frac{1}{\sqrt{h \cdot \ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   8. Step-by-step derivation

      1. un-div-inv 57.3%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   9. Applied egg-rr 57.3%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   if 7.99999999999999973e118 < d

   1. Initial program 49.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 49.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 49.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 49.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 49.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 49.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. frac-times 49.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-/r* 49.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. metadata-eval 49.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. pow1 49.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. div-inv 49.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. *-commutative 49.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. associate-*r* 49.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot \frac{1}{2}\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. div-inv 49.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \color{blue}{\frac{M}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. associate-*r/ 49.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{\frac{M}{2}}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      14. associate-/l/ 49.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      15. *-un-lft-identity 49.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{\color{blue}{1 \cdot M}}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      16. *-commutative 49.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      17. times-frac 49.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      18. metadata-eval 49.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 49.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 82.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. pow1 82.4%

         \[\leadsto \color{blue}{{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}^{1}} \]

      2. pow1/2 82.4%

         \[\leadsto {\left(d \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right)}^{1} \]

      3. inv-pow 82.4%

         \[\leadsto {\left(d \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right)}^{1} \]

      4. *-commutative 82.4%

         \[\leadsto {\left(d \cdot {\left({\color{blue}{\left(\ell \cdot h\right)}}^{-1}\right)}^{0.5}\right)}^{1} \]

      5. pow-pow 82.3%

         \[\leadsto {\left(d \cdot \color{blue}{{\left(\ell \cdot h\right)}^{\left(-1 \cdot 0.5\right)}}\right)}^{1} \]

      6. *-commutative 82.3%

         \[\leadsto {\left(d \cdot {\color{blue}{\left(h \cdot \ell\right)}}^{\left(-1 \cdot 0.5\right)}\right)}^{1} \]

      7. metadata-eval 82.3%

         \[\leadsto {\left(d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right)}^{1} \]

   9. Applied egg-rr 82.3%

      \[\leadsto \color{blue}{{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 82.3%

          \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   11. Simplified 82.3%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   12. Step-by-step derivation

       1. *-commutative 82.3%

          \[\leadsto d \cdot {\color{blue}{\left(\ell \cdot h\right)}}^{-0.5} \]

       2. unpow-prod-down 90.6%

          \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]

   13. Applied egg-rr 90.6%

       \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{+95}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -1.8 \cdot 10^{-274}:\\ \;\;\;\;0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\right)\\ \mathbf{elif}\;d \leq 8 \cdot 10^{+118}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 72.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right) + -1\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-65}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(D\_m \cdot \left(0.5 \cdot \frac{M\_m}{d}\right)\right)}^{2}}{\ell}\right) \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -1e-309)
   (*
    (* d (sqrt (/ (/ 1.0 h) l)))
    (+ (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0))) -1.0))
   (if (<= d 3.8e-65)
     (*
      d
      (*
       (pow (* h l) -0.5)
       (- 1.0 (* (/ h l) (* 0.5 (pow (* D_m (* M_m (/ 0.5 d))) 2.0))))))
     (*
      (- 1.0 (* 0.5 (/ (* h (pow (* D_m (* 0.5 (/ M_m d))) 2.0)) l)))
      (* d (sqrt (/ 1.0 (* h l))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1e-309) {
		tmp = (d * sqrt(((1.0 / h) / l))) * ((0.5 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0))) + -1.0);
	} else if (d <= 3.8e-65) {
		tmp = d * (pow((h * l), -0.5) * (1.0 - ((h / l) * (0.5 * pow((D_m * (M_m * (0.5 / d))), 2.0)))));
	} else {
		tmp = (1.0 - (0.5 * ((h * pow((D_m * (0.5 * (M_m / d))), 2.0)) / l))) * (d * sqrt((1.0 / (h * l))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-1d-309)) then
        tmp = (d * sqrt(((1.0d0 / h) / l))) * ((0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_m / d)) ** 2.0d0))) + (-1.0d0))
    else if (d <= 3.8d-65) then
        tmp = d * (((h * l) ** (-0.5d0)) * (1.0d0 - ((h / l) * (0.5d0 * ((d_m * (m_m * (0.5d0 / d))) ** 2.0d0)))))
    else
        tmp = (1.0d0 - (0.5d0 * ((h * ((d_m * (0.5d0 * (m_m / d))) ** 2.0d0)) / l))) * (d * sqrt((1.0d0 / (h * l))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1e-309) {
		tmp = (d * Math.sqrt(((1.0 / h) / l))) * ((0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D_m / d)), 2.0))) + -1.0);
	} else if (d <= 3.8e-65) {
		tmp = d * (Math.pow((h * l), -0.5) * (1.0 - ((h / l) * (0.5 * Math.pow((D_m * (M_m * (0.5 / d))), 2.0)))));
	} else {
		tmp = (1.0 - (0.5 * ((h * Math.pow((D_m * (0.5 * (M_m / d))), 2.0)) / l))) * (d * Math.sqrt((1.0 / (h * l))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -1e-309:
		tmp = (d * math.sqrt(((1.0 / h) / l))) * ((0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D_m / d)), 2.0))) + -1.0)
	elif d <= 3.8e-65:
		tmp = d * (math.pow((h * l), -0.5) * (1.0 - ((h / l) * (0.5 * math.pow((D_m * (M_m * (0.5 / d))), 2.0)))))
	else:
		tmp = (1.0 - (0.5 * ((h * math.pow((D_m * (0.5 * (M_m / d))), 2.0)) / l))) * (d * math.sqrt((1.0 / (h * l))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -1e-309)
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / h) / l))) * Float64(Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0))) + -1.0));
	elseif (d <= 3.8e-65)
		tmp = Float64(d * Float64((Float64(h * l) ^ -0.5) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(D_m * Float64(M_m * Float64(0.5 / d))) ^ 2.0))))));
	else
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(D_m * Float64(0.5 * Float64(M_m / d))) ^ 2.0)) / l))) * Float64(d * sqrt(Float64(1.0 / Float64(h * l)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -1e-309)
		tmp = (d * sqrt(((1.0 / h) / l))) * ((0.5 * ((h / l) * (((M_m / 2.0) * (D_m / d)) ^ 2.0))) + -1.0);
	elseif (d <= 3.8e-65)
		tmp = d * (((h * l) ^ -0.5) * (1.0 - ((h / l) * (0.5 * ((D_m * (M_m * (0.5 / d))) ^ 2.0)))));
	else
		tmp = (1.0 - (0.5 * ((h * ((D_m * (0.5 * (M_m / d))) ^ 2.0)) / l))) * (d * sqrt((1.0 / (h * l))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -1e-309], N[(N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.8e-65], N[(d * N[(N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(D$95$m * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.000000000000002e-309

   1. Initial program 70.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 69.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. associate-/r* 0.0%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. *-commutative 0.0%

         \[\leadsto \left(\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. unpow2 0.0%

         \[\leadsto \left(\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right)\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. rem-square-sqrt 73.3%

         \[\leadsto \left(\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(\color{blue}{-1} \cdot d\right)\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. neg-mul-1 73.3%

         \[\leadsto \left(\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \color{blue}{\left(-d\right)}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Simplified 73.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   if -1.000000000000002e-309 < d < 3.8000000000000002e-65

   1. Initial program 51.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 45.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around 0 48.1%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. pow1 48.1%

         \[\leadsto \color{blue}{{\left(\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1}} \]

   7. Applied egg-rr 63.7%

      \[\leadsto \color{blue}{{\left(d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \left(0.5 \cdot {\left(\frac{\left(M \cdot 0.5\right) \cdot D}{d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 63.7%

         \[\leadsto \color{blue}{d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \left(0.5 \cdot {\left(\frac{\left(M \cdot 0.5\right) \cdot D}{d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      2. *-commutative 63.7%

         \[\leadsto d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{\left(M \cdot 0.5\right) \cdot D}{d}\right)}^{2}\right)}\right)\right) \]

      3. *-commutative 63.7%

         \[\leadsto d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot 0.5\right)}}{d}\right)}^{2}\right)\right)\right) \]

      4. associate-*r/ 61.8%

         \[\leadsto d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\color{blue}{\left(D \cdot \frac{M \cdot 0.5}{d}\right)}}^{2}\right)\right)\right) \]

      5. associate-/l* 61.7%

         \[\leadsto d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \color{blue}{\left(M \cdot \frac{0.5}{d}\right)}\right)}^{2}\right)\right)\right) \]

   9. Simplified 61.7%

      \[\leadsto \color{blue}{d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)} \]

   if 3.8000000000000002e-65 < d

   1. Initial program 63.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 63.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around 0 78.4%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ 86.4%

         \[\leadsto \left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

   7. Applied egg-rr 86.5%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot h}{\ell}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) + -1\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-65}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}\right) \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 45.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{if}\;d \leq -5 \cdot 10^{+95}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -1.8 \cdot 10^{-274}:\\ \;\;\;\;0.125 \cdot \left(\left(\left(D\_m \cdot D\_m\right) \cdot \left(M\_m \cdot M\_m\right)\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\right)\\ \mathbf{elif}\;d \leq 1.22 \cdot 10^{-223}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* d (- (pow (* h l) -0.5)))))
   (if (<= d -5e+95)
     t_0
     (if (<= d -1.8e-274)
       (* 0.125 (* (* (* D_m D_m) (* M_m M_m)) (/ (sqrt (/ h (pow l 3.0))) d)))
       (if (<= d 1.22e-223) t_0 (* d (* (pow l -0.5) (pow h -0.5))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = d * -pow((h * l), -0.5);
	double tmp;
	if (d <= -5e+95) {
		tmp = t_0;
	} else if (d <= -1.8e-274) {
		tmp = 0.125 * (((D_m * D_m) * (M_m * M_m)) * (sqrt((h / pow(l, 3.0))) / d));
	} else if (d <= 1.22e-223) {
		tmp = t_0;
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d * -((h * l) ** (-0.5d0))
    if (d <= (-5d+95)) then
        tmp = t_0
    else if (d <= (-1.8d-274)) then
        tmp = 0.125d0 * (((d_m * d_m) * (m_m * m_m)) * (sqrt((h / (l ** 3.0d0))) / d))
    else if (d <= 1.22d-223) then
        tmp = t_0
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = d * -Math.pow((h * l), -0.5);
	double tmp;
	if (d <= -5e+95) {
		tmp = t_0;
	} else if (d <= -1.8e-274) {
		tmp = 0.125 * (((D_m * D_m) * (M_m * M_m)) * (Math.sqrt((h / Math.pow(l, 3.0))) / d));
	} else if (d <= 1.22e-223) {
		tmp = t_0;
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = d * -math.pow((h * l), -0.5)
	tmp = 0
	if d <= -5e+95:
		tmp = t_0
	elif d <= -1.8e-274:
		tmp = 0.125 * (((D_m * D_m) * (M_m * M_m)) * (math.sqrt((h / math.pow(l, 3.0))) / d))
	elif d <= 1.22e-223:
		tmp = t_0
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(d * Float64(-(Float64(h * l) ^ -0.5)))
	tmp = 0.0
	if (d <= -5e+95)
		tmp = t_0;
	elseif (d <= -1.8e-274)
		tmp = Float64(0.125 * Float64(Float64(Float64(D_m * D_m) * Float64(M_m * M_m)) * Float64(sqrt(Float64(h / (l ^ 3.0))) / d)));
	elseif (d <= 1.22e-223)
		tmp = t_0;
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = d * -((h * l) ^ -0.5);
	tmp = 0.0;
	if (d <= -5e+95)
		tmp = t_0;
	elseif (d <= -1.8e-274)
		tmp = 0.125 * (((D_m * D_m) * (M_m * M_m)) * (sqrt((h / (l ^ 3.0))) / d));
	elseif (d <= 1.22e-223)
		tmp = t_0;
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[d, -5e+95], t$95$0, If[LessEqual[d, -1.8e-274], N[(0.125 * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.22e-223], t$95$0, N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.00000000000000025e95 or -1.79999999999999991e-274 < d < 1.21999999999999998e-223

   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. Simplified 59.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 59.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 59.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 59.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. frac-times 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-/r* 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. metadata-eval 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. pow1 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. div-inv 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. *-commutative 63.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. associate-*r* 63.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot \frac{1}{2}\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. div-inv 63.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \color{blue}{\frac{M}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. associate-*r/ 59.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{\frac{M}{2}}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      14. associate-/l/ 59.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      15. *-un-lft-identity 59.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{\color{blue}{1 \cdot M}}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      16. *-commutative 59.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      17. times-frac 59.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      18. metadata-eval 59.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 59.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. clear-num 59.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. sqrt-div 59.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. metadata-eval 59.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   8. Applied egg-rr 59.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   9. Taylor expanded in d around -inf 50.9%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg 50.9%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

       2. *-commutative 50.9%

          \[\leadsto -\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

       3. associate-/r* 50.9%

          \[\leadsto -\sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

       4. distribute-rgt-neg-in 50.9%

          \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)} \]

       5. associate-/r* 50.9%

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-d\right) \]

       6. unpow-1 50.9%

          \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot \left(-d\right) \]

       7. metadata-eval 50.9%

          \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(-d\right) \]

       8. pow-sqr 50.9%

          \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot \left(-d\right) \]

       9. rem-sqrt-square 50.9%

          \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \cdot \left(-d\right) \]

       10. metadata-eval 50.9%

           \[\leadsto \left|{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right| \cdot \left(-d\right) \]

       11. pow-sqr 50.8%

           \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}}\right| \cdot \left(-d\right) \]

       12. fabs-sqr 50.8%

           \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}\right)} \cdot \left(-d\right) \]

       13. pow-sqr 50.9%

           \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(2 \cdot -0.25\right)}} \cdot \left(-d\right) \]

       14. metadata-eval 50.9%

           \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot \left(-d\right) \]

   11. Simplified 50.9%

       \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if -5.00000000000000025e95 < d < -1.79999999999999991e-274

   1. Initial program 68.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   4. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.125 \cdot \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      2. associate-/l* 0.0%

         \[\leadsto -0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)}\right) \]

      3. *-commutative 0.0%

         \[\leadsto -0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left({D}^{2} \cdot \frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot {M}^{2}}}{d}\right)\right) \]

      4. unpow2 0.0%

         \[\leadsto -0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left({D}^{2} \cdot \frac{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot {M}^{2}}{d}\right)\right) \]

      5. rem-square-sqrt 36.4%

         \[\leadsto -0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left({D}^{2} \cdot \frac{\color{blue}{-1} \cdot {M}^{2}}{d}\right)\right) \]

      6. associate-/l* 36.4%

         \[\leadsto -0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left({D}^{2} \cdot \color{blue}{\left(-1 \cdot \frac{{M}^{2}}{d}\right)}\right)\right) \]

   5. Simplified 36.4%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left({D}^{2} \cdot \left(-1 \cdot \frac{{M}^{2}}{d}\right)\right)\right)} \]

   6. Taylor expanded in h around 0 36.4%

      \[\leadsto \color{blue}{0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 37.6%

         \[\leadsto 0.125 \cdot \color{blue}{\frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}} \]

      2. associate-/l* 37.6%

         \[\leadsto 0.125 \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\right)} \]

   8. Simplified 37.6%

      \[\leadsto \color{blue}{0.125 \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\right)} \]
   9. Step-by-step derivation

      1. unpow2 37.6%

         \[\leadsto 0.125 \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\right) \]

   10. Applied egg-rr 37.6%

       \[\leadsto 0.125 \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\right) \]
   11. Step-by-step derivation

       1. unpow2 37.6%

          \[\leadsto 0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\right) \]

   12. Applied egg-rr 37.6%

       \[\leadsto 0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\right) \]

   if 1.21999999999999998e-223 < d

   1. Initial program 63.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 61.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 61.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 61.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. frac-times 63.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-/r* 63.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. metadata-eval 63.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. pow1 63.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. div-inv 63.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. *-commutative 63.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. associate-*r* 63.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot \frac{1}{2}\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. div-inv 63.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \color{blue}{\frac{M}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. associate-*r/ 64.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{\frac{M}{2}}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      14. associate-/l/ 64.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      15. *-un-lft-identity 64.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{\color{blue}{1 \cdot M}}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      16. *-commutative 64.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      17. times-frac 64.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      18. metadata-eval 64.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 64.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 61.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. pow1 61.1%

         \[\leadsto \color{blue}{{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}^{1}} \]

      2. pow1/2 61.1%

         \[\leadsto {\left(d \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right)}^{1} \]

      3. inv-pow 61.1%

         \[\leadsto {\left(d \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right)}^{1} \]

      4. *-commutative 61.1%

         \[\leadsto {\left(d \cdot {\left({\color{blue}{\left(\ell \cdot h\right)}}^{-1}\right)}^{0.5}\right)}^{1} \]

      5. pow-pow 62.0%

         \[\leadsto {\left(d \cdot \color{blue}{{\left(\ell \cdot h\right)}^{\left(-1 \cdot 0.5\right)}}\right)}^{1} \]

      6. *-commutative 62.0%

         \[\leadsto {\left(d \cdot {\color{blue}{\left(h \cdot \ell\right)}}^{\left(-1 \cdot 0.5\right)}\right)}^{1} \]

      7. metadata-eval 62.0%

         \[\leadsto {\left(d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right)}^{1} \]

   9. Applied egg-rr 62.0%

      \[\leadsto \color{blue}{{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 62.0%

          \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   11. Simplified 62.0%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   12. Step-by-step derivation

       1. *-commutative 62.0%

          \[\leadsto d \cdot {\color{blue}{\left(\ell \cdot h\right)}}^{-0.5} \]

       2. unpow-prod-down 67.8%

          \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]

   13. Applied egg-rr 67.8%

       \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{+95}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -1.8 \cdot 10^{-274}:\\ \;\;\;\;0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\right)\\ \mathbf{elif}\;d \leq 1.22 \cdot 10^{-223}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 46.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.05 \cdot 10^{-226}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -1.05e-226)
   (* d (- (pow (* h l) -0.5)))
   (if (<= l -2e-310)
     (* d (cbrt (pow (/ 1.0 (* h l)) 1.5)))
     (* d (* (pow l -0.5) (pow h -0.5))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1.05e-226) {
		tmp = d * -pow((h * l), -0.5);
	} else if (l <= -2e-310) {
		tmp = d * cbrt(pow((1.0 / (h * l)), 1.5));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1.05e-226) {
		tmp = d * -Math.pow((h * l), -0.5);
	} else if (l <= -2e-310) {
		tmp = d * Math.cbrt(Math.pow((1.0 / (h * l)), 1.5));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -1.05e-226)
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	elseif (l <= -2e-310)
		tmp = Float64(d * cbrt((Float64(1.0 / Float64(h * l)) ^ 1.5)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -1.05e-226], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, -2e-310], N[(d * N[Power[N[Power[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.0500000000000001e-226

   1. Initial program 68.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 67.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 67.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 67.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 67.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 69.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. frac-times 71.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-/r* 71.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. metadata-eval 71.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. pow1 71.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. div-inv 71.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. *-commutative 71.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. associate-*r* 71.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot \frac{1}{2}\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. div-inv 71.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \color{blue}{\frac{M}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. associate-*r/ 67.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{\frac{M}{2}}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      14. associate-/l/ 67.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      15. *-un-lft-identity 67.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{\color{blue}{1 \cdot M}}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      16. *-commutative 67.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      17. times-frac 67.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      18. metadata-eval 67.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 67.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. clear-num 66.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. sqrt-div 66.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. metadata-eval 66.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   8. Applied egg-rr 66.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   9. Taylor expanded in d around -inf 41.1%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg 41.1%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

       2. *-commutative 41.1%

          \[\leadsto -\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

       3. associate-/r* 41.1%

          \[\leadsto -\sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

       4. distribute-rgt-neg-in 41.1%

          \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)} \]

       5. associate-/r* 41.1%

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-d\right) \]

       6. unpow-1 41.1%

          \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot \left(-d\right) \]

       7. metadata-eval 41.1%

          \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(-d\right) \]

       8. pow-sqr 41.1%

          \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot \left(-d\right) \]

       9. rem-sqrt-square 41.1%

          \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \cdot \left(-d\right) \]

       10. metadata-eval 41.1%

           \[\leadsto \left|{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right| \cdot \left(-d\right) \]

       11. pow-sqr 41.1%

           \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}}\right| \cdot \left(-d\right) \]

       12. fabs-sqr 41.1%

           \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}\right)} \cdot \left(-d\right) \]

       13. pow-sqr 41.1%

           \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(2 \cdot -0.25\right)}} \cdot \left(-d\right) \]

       14. metadata-eval 41.1%

           \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot \left(-d\right) \]

   11. Simplified 41.1%

       \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if -1.0500000000000001e-226 < l < -1.999999999999994e-310

   1. Initial program 82.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 77.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 77.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 77.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 77.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 77.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. frac-times 86.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-/r* 86.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. metadata-eval 86.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. pow1 86.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. div-inv 86.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. *-commutative 86.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. associate-*r* 86.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot \frac{1}{2}\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. div-inv 86.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \color{blue}{\frac{M}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. associate-*r/ 86.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{\frac{M}{2}}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      14. associate-/l/ 86.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      15. *-un-lft-identity 86.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{\color{blue}{1 \cdot M}}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      16. *-commutative 86.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      17. times-frac 86.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      18. metadata-eval 86.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 86.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 38.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. add-cbrt-cube 42.5%

         \[\leadsto d \cdot \color{blue}{\sqrt[3]{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}}} \]

      2. pow1/3 42.5%

         \[\leadsto d \cdot \color{blue}{{\left(\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 42.5%

         \[\leadsto d \cdot {\left(\color{blue}{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}^{0.3333333333333333} \]

      4. pow1 42.5%

         \[\leadsto d \cdot {\left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{1}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}^{0.3333333333333333} \]

      5. pow1/2 42.5%

         \[\leadsto d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 42.5%

         \[\leadsto d \cdot {\color{blue}{\left({\left(\frac{1}{h \cdot \ell}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. metadata-eval 42.5%

         \[\leadsto d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   9. Applied egg-rr 42.5%

      \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   10. Step-by-step derivation

       1. unpow1/3 42.5%

          \[\leadsto d \cdot \color{blue}{\sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{1.5}}} \]

   11. Simplified 42.5%

       \[\leadsto d \cdot \color{blue}{\sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{1.5}}} \]

   if -1.999999999999994e-310 < l

   1. Initial program 58.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 56.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 56.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 56.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 56.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 56.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. frac-times 58.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-/r* 58.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. metadata-eval 58.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. pow1 58.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. div-inv 58.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. *-commutative 58.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. associate-*r* 58.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot \frac{1}{2}\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. div-inv 58.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \color{blue}{\frac{M}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. associate-*r/ 57.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{\frac{M}{2}}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      14. associate-/l/ 57.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      15. *-un-lft-identity 57.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{\color{blue}{1 \cdot M}}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      16. *-commutative 57.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      17. times-frac 57.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      18. metadata-eval 57.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 57.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 52.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. pow1 52.4%

         \[\leadsto \color{blue}{{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}^{1}} \]

      2. pow1/2 52.4%

         \[\leadsto {\left(d \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right)}^{1} \]

      3. inv-pow 52.4%

         \[\leadsto {\left(d \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right)}^{1} \]

      4. *-commutative 52.4%

         \[\leadsto {\left(d \cdot {\left({\color{blue}{\left(\ell \cdot h\right)}}^{-1}\right)}^{0.5}\right)}^{1} \]

      5. pow-pow 53.1%

         \[\leadsto {\left(d \cdot \color{blue}{{\left(\ell \cdot h\right)}^{\left(-1 \cdot 0.5\right)}}\right)}^{1} \]

      6. *-commutative 53.1%

         \[\leadsto {\left(d \cdot {\color{blue}{\left(h \cdot \ell\right)}}^{\left(-1 \cdot 0.5\right)}\right)}^{1} \]

      7. metadata-eval 53.1%

         \[\leadsto {\left(d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right)}^{1} \]

   9. Applied egg-rr 53.1%

      \[\leadsto \color{blue}{{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 53.1%

          \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   11. Simplified 53.1%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   12. Step-by-step derivation

       1. *-commutative 53.1%

          \[\leadsto d \cdot {\color{blue}{\left(\ell \cdot h\right)}}^{-0.5} \]

       2. unpow-prod-down 58.0%

          \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]

   13. Applied egg-rr 58.0%

       \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.05 \cdot 10^{-226}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 46.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.22 \cdot 10^{-223}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d 1.22e-223)
   (* d (- (pow (* h l) -0.5)))
   (* d (* (pow l -0.5) (pow h -0.5)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 1.22e-223) {
		tmp = d * -pow((h * l), -0.5);
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= 1.22d-223) then
        tmp = d * -((h * l) ** (-0.5d0))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 1.22e-223) {
		tmp = d * -Math.pow((h * l), -0.5);
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= 1.22e-223:
		tmp = d * -math.pow((h * l), -0.5)
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= 1.22e-223)
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= 1.22e-223)
		tmp = d * -((h * l) ^ -0.5);
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, 1.22e-223], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.21999999999999998e-223

   1. Initial program 65.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 63.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 63.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 63.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 63.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 65.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. frac-times 68.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-/r* 68.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. metadata-eval 68.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. pow1 68.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. div-inv 68.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. *-commutative 68.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. associate-*r* 68.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot \frac{1}{2}\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. div-inv 68.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \color{blue}{\frac{M}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. associate-*r/ 64.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{\frac{M}{2}}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      14. associate-/l/ 64.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      15. *-un-lft-identity 64.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{\color{blue}{1 \cdot M}}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      16. *-commutative 64.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      17. times-frac 64.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      18. metadata-eval 64.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 64.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. clear-num 64.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. sqrt-div 63.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. metadata-eval 63.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   8. Applied egg-rr 63.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   9. Taylor expanded in d around -inf 34.1%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg 34.1%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

       2. *-commutative 34.1%

          \[\leadsto -\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

       3. associate-/r* 34.2%

          \[\leadsto -\sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

       4. distribute-rgt-neg-in 34.2%

          \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)} \]

       5. associate-/r* 34.1%

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-d\right) \]

       6. unpow-1 34.1%

          \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot \left(-d\right) \]

       7. metadata-eval 34.1%

          \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(-d\right) \]

       8. pow-sqr 34.1%

          \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot \left(-d\right) \]

       9. rem-sqrt-square 34.2%

          \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \cdot \left(-d\right) \]

       10. metadata-eval 34.2%

           \[\leadsto \left|{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right| \cdot \left(-d\right) \]

       11. pow-sqr 34.1%

           \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}}\right| \cdot \left(-d\right) \]

       12. fabs-sqr 34.1%

           \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}\right)} \cdot \left(-d\right) \]

       13. pow-sqr 34.2%

           \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(2 \cdot -0.25\right)}} \cdot \left(-d\right) \]

       14. metadata-eval 34.2%

           \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot \left(-d\right) \]

   11. Simplified 34.2%

       \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if 1.21999999999999998e-223 < d

   1. Initial program 63.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 61.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 61.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 61.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. frac-times 63.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-/r* 63.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. metadata-eval 63.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. pow1 63.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. div-inv 63.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. *-commutative 63.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. associate-*r* 63.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot \frac{1}{2}\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. div-inv 63.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \color{blue}{\frac{M}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. associate-*r/ 64.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{\frac{M}{2}}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      14. associate-/l/ 64.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      15. *-un-lft-identity 64.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{\color{blue}{1 \cdot M}}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      16. *-commutative 64.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      17. times-frac 64.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      18. metadata-eval 64.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 64.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 61.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. pow1 61.1%

         \[\leadsto \color{blue}{{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}^{1}} \]

      2. pow1/2 61.1%

         \[\leadsto {\left(d \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right)}^{1} \]

      3. inv-pow 61.1%

         \[\leadsto {\left(d \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right)}^{1} \]

      4. *-commutative 61.1%

         \[\leadsto {\left(d \cdot {\left({\color{blue}{\left(\ell \cdot h\right)}}^{-1}\right)}^{0.5}\right)}^{1} \]

      5. pow-pow 62.0%

         \[\leadsto {\left(d \cdot \color{blue}{{\left(\ell \cdot h\right)}^{\left(-1 \cdot 0.5\right)}}\right)}^{1} \]

      6. *-commutative 62.0%

         \[\leadsto {\left(d \cdot {\color{blue}{\left(h \cdot \ell\right)}}^{\left(-1 \cdot 0.5\right)}\right)}^{1} \]

      7. metadata-eval 62.0%

         \[\leadsto {\left(d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right)}^{1} \]

   9. Applied egg-rr 62.0%

      \[\leadsto \color{blue}{{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 62.0%

          \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   11. Simplified 62.0%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   12. Step-by-step derivation

       1. *-commutative 62.0%

          \[\leadsto d \cdot {\color{blue}{\left(\ell \cdot h\right)}}^{-0.5} \]

       2. unpow-prod-down 67.8%

          \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]

   13. Applied egg-rr 67.8%

       \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.22 \cdot 10^{-223}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 42.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.8 \cdot 10^{-228}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h} \cdot \frac{1}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -6.8e-228)
   (* d (- (pow (* h l) -0.5)))
   (* d (sqrt (* (/ 1.0 h) (/ 1.0 l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -6.8e-228) {
		tmp = d * -pow((h * l), -0.5);
	} else {
		tmp = d * sqrt(((1.0 / h) * (1.0 / l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-6.8d-228)) then
        tmp = d * -((h * l) ** (-0.5d0))
    else
        tmp = d * sqrt(((1.0d0 / h) * (1.0d0 / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -6.8e-228) {
		tmp = d * -Math.pow((h * l), -0.5);
	} else {
		tmp = d * Math.sqrt(((1.0 / h) * (1.0 / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -6.8e-228:
		tmp = d * -math.pow((h * l), -0.5)
	else:
		tmp = d * math.sqrt(((1.0 / h) * (1.0 / l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -6.8e-228)
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) * Float64(1.0 / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -6.8e-228)
		tmp = d * -((h * l) ^ -0.5);
	else
		tmp = d * sqrt(((1.0 / h) * (1.0 / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -6.8e-228], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -6.79999999999999981e-228

   1. Initial program 68.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 67.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 67.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 67.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 67.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 69.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. frac-times 71.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-/r* 71.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. metadata-eval 71.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. pow1 71.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. div-inv 71.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. *-commutative 71.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. associate-*r* 71.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot \frac{1}{2}\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. div-inv 71.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \color{blue}{\frac{M}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. associate-*r/ 67.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{\frac{M}{2}}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      14. associate-/l/ 67.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      15. *-un-lft-identity 67.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{\color{blue}{1 \cdot M}}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      16. *-commutative 67.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      17. times-frac 67.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      18. metadata-eval 67.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 67.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. clear-num 66.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. sqrt-div 66.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. metadata-eval 66.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   8. Applied egg-rr 66.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   9. Taylor expanded in d around -inf 41.1%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg 41.1%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

       2. *-commutative 41.1%

          \[\leadsto -\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

       3. associate-/r* 41.1%

          \[\leadsto -\sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

       4. distribute-rgt-neg-in 41.1%

          \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)} \]

       5. associate-/r* 41.1%

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-d\right) \]

       6. unpow-1 41.1%

          \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot \left(-d\right) \]

       7. metadata-eval 41.1%

          \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(-d\right) \]

       8. pow-sqr 41.1%

          \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot \left(-d\right) \]

       9. rem-sqrt-square 41.1%

          \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \cdot \left(-d\right) \]

       10. metadata-eval 41.1%

           \[\leadsto \left|{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right| \cdot \left(-d\right) \]

       11. pow-sqr 41.1%

           \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}}\right| \cdot \left(-d\right) \]

       12. fabs-sqr 41.1%

           \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}\right)} \cdot \left(-d\right) \]

       13. pow-sqr 41.1%

           \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(2 \cdot -0.25\right)}} \cdot \left(-d\right) \]

       14. metadata-eval 41.1%

           \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot \left(-d\right) \]

   11. Simplified 41.1%

       \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if -6.79999999999999981e-228 < l

   1. Initial program 62.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 59.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 59.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 59.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 59.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 59.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. frac-times 62.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-/r* 62.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. metadata-eval 62.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. pow1 62.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. div-inv 62.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. *-commutative 62.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. associate-*r* 62.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot \frac{1}{2}\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. div-inv 62.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \color{blue}{\frac{M}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. associate-*r/ 62.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{\frac{M}{2}}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      14. associate-/l/ 62.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      15. *-un-lft-identity 62.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{\color{blue}{1 \cdot M}}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      16. *-commutative 62.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      17. times-frac 62.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      18. metadata-eval 62.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 62.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 50.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. inv-pow 50.2%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. *-commutative 50.2%

         \[\leadsto d \cdot \sqrt{{\color{blue}{\left(\ell \cdot h\right)}}^{-1}} \]

      3. unpow-prod-down 50.6%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\ell}^{-1} \cdot {h}^{-1}}} \]

      4. inv-pow 50.6%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{\ell}} \cdot {h}^{-1}} \]

      5. inv-pow 50.6%

         \[\leadsto d \cdot \sqrt{\frac{1}{\ell} \cdot \color{blue}{\frac{1}{h}}} \]

   9. Applied egg-rr 50.6%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{\ell} \cdot \frac{1}{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.8 \cdot 10^{-228}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h} \cdot \frac{1}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 42.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7.2 \cdot 10^{-227}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -7.2e-227)
   (* d (- (pow (* h l) -0.5)))
   (* d (sqrt (/ 1.0 (* h l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -7.2e-227) {
		tmp = d * -pow((h * l), -0.5);
	} else {
		tmp = d * sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-7.2d-227)) then
        tmp = d * -((h * l) ** (-0.5d0))
    else
        tmp = d * sqrt((1.0d0 / (h * l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -7.2e-227) {
		tmp = d * -Math.pow((h * l), -0.5);
	} else {
		tmp = d * Math.sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -7.2e-227:
		tmp = d * -math.pow((h * l), -0.5)
	else:
		tmp = d * math.sqrt((1.0 / (h * l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -7.2e-227)
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	else
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(h * l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -7.2e-227)
		tmp = d * -((h * l) ^ -0.5);
	else
		tmp = d * sqrt((1.0 / (h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -7.2e-227], 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 l < -7.1999999999999999e-227

   1. Initial program 68.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 67.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 67.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 67.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 67.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 69.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. frac-times 71.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-/r* 71.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. metadata-eval 71.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. pow1 71.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. div-inv 71.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. *-commutative 71.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. associate-*r* 71.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot \frac{1}{2}\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. div-inv 71.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \color{blue}{\frac{M}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. associate-*r/ 67.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{\frac{M}{2}}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      14. associate-/l/ 67.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      15. *-un-lft-identity 67.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{\color{blue}{1 \cdot M}}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      16. *-commutative 67.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      17. times-frac 67.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      18. metadata-eval 67.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 67.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. clear-num 66.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. sqrt-div 66.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. metadata-eval 66.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   8. Applied egg-rr 66.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   9. Taylor expanded in d around -inf 41.1%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg 41.1%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

       2. *-commutative 41.1%

          \[\leadsto -\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

       3. associate-/r* 41.1%

          \[\leadsto -\sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

       4. distribute-rgt-neg-in 41.1%

          \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)} \]

       5. associate-/r* 41.1%

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-d\right) \]

       6. unpow-1 41.1%

          \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot \left(-d\right) \]

       7. metadata-eval 41.1%

          \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(-d\right) \]

       8. pow-sqr 41.1%

          \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot \left(-d\right) \]

       9. rem-sqrt-square 41.1%

          \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \cdot \left(-d\right) \]

       10. metadata-eval 41.1%

           \[\leadsto \left|{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right| \cdot \left(-d\right) \]

       11. pow-sqr 41.1%

           \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}}\right| \cdot \left(-d\right) \]

       12. fabs-sqr 41.1%

           \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}\right)} \cdot \left(-d\right) \]

       13. pow-sqr 41.1%

           \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(2 \cdot -0.25\right)}} \cdot \left(-d\right) \]

       14. metadata-eval 41.1%

           \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot \left(-d\right) \]

   11. Simplified 41.1%

       \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if -7.1999999999999999e-227 < l

   1. Initial program 62.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 59.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 59.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 59.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 59.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 59.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. frac-times 62.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-/r* 62.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. metadata-eval 62.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. pow1 62.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. div-inv 62.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. *-commutative 62.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. associate-*r* 62.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot \frac{1}{2}\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. div-inv 62.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \color{blue}{\frac{M}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. associate-*r/ 62.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{\frac{M}{2}}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      14. associate-/l/ 62.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      15. *-un-lft-identity 62.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{\color{blue}{1 \cdot M}}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      16. *-commutative 62.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      17. times-frac 62.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      18. metadata-eval 62.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 62.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 50.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.2 \cdot 10^{-227}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 26.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ d \cdot \sqrt{\frac{1}{h \cdot \ell}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (* d (sqrt (/ 1.0 (* h l)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return d * sqrt((1.0 / (h * l)));
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = d * sqrt((1.0d0 / (h * l)))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return d * Math.sqrt((1.0 / (h * l)));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return d * math.sqrt((1.0 / (h * l)))

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(d * sqrt(Float64(1.0 / Float64(h * l))))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = d * sqrt((1.0 / (h * l)));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.7%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 62.8%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 62.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

   2. pow2 62.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

   3. sqrt-prod 62.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

   4. sqrt-pow1 63.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   5. frac-times 66.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. associate-/r* 66.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   7. metadata-eval 66.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   8. pow1 66.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   9. div-inv 66.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   10. *-commutative 66.6%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   11. associate-*r* 66.6%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot \frac{1}{2}\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   12. div-inv 66.6%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \color{blue}{\frac{M}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   13. associate-*r/ 64.3%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{\frac{M}{2}}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   14. associate-/l/ 64.3%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   15. *-un-lft-identity 64.3%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{\color{blue}{1 \cdot M}}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   16. *-commutative 64.3%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   17. times-frac 64.3%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   18. metadata-eval 64.3%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

6. Applied egg-rr 64.3%

   \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

7. Taylor expanded in d around inf 32.5%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
8. Add Preprocessing

Alternative 24: 26.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ d \cdot {\left(h \cdot \ell\right)}^{-0.5} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (* d (pow (* h l) -0.5)))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return d * pow((h * l), -0.5);
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = d * ((h * l) ** (-0.5d0))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return d * Math.pow((h * l), -0.5);
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return d * math.pow((h * l), -0.5)

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(d * (Float64(h * l) ^ -0.5))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = d * ((h * l) ^ -0.5);
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.7%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 62.8%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 62.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

   2. pow2 62.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

   3. sqrt-prod 62.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

   4. sqrt-pow1 63.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   5. frac-times 66.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. associate-/r* 66.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   7. metadata-eval 66.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   8. pow1 66.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   9. div-inv 66.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   10. *-commutative 66.6%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   11. associate-*r* 66.6%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot \frac{1}{2}\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   12. div-inv 66.6%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \color{blue}{\frac{M}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   13. associate-*r/ 64.3%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{\frac{M}{2}}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   14. associate-/l/ 64.3%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   15. *-un-lft-identity 64.3%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{\color{blue}{1 \cdot M}}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   16. *-commutative 64.3%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   17. times-frac 64.3%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   18. metadata-eval 64.3%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

6. Applied egg-rr 64.3%

   \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

7. Taylor expanded in d around inf 32.5%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
8. Step-by-step derivation

   1. pow1 32.5%

      \[\leadsto \color{blue}{{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}^{1}} \]

   2. pow1/2 32.5%

      \[\leadsto {\left(d \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right)}^{1} \]

   3. inv-pow 32.5%

      \[\leadsto {\left(d \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right)}^{1} \]

   4. *-commutative 32.5%

      \[\leadsto {\left(d \cdot {\left({\color{blue}{\left(\ell \cdot h\right)}}^{-1}\right)}^{0.5}\right)}^{1} \]

   5. pow-pow 31.7%

      \[\leadsto {\left(d \cdot \color{blue}{{\left(\ell \cdot h\right)}^{\left(-1 \cdot 0.5\right)}}\right)}^{1} \]

   6. *-commutative 31.7%

      \[\leadsto {\left(d \cdot {\color{blue}{\left(h \cdot \ell\right)}}^{\left(-1 \cdot 0.5\right)}\right)}^{1} \]

   7. metadata-eval 31.7%

      \[\leadsto {\left(d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right)}^{1} \]

9. Applied egg-rr 31.7%

   \[\leadsto \color{blue}{{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)}^{1}} \]
10. Step-by-step derivation

    1. unpow1 31.7%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

11. Simplified 31.7%

    \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024148 
(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)))))