Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.0% → 80.9%

Time: 23.2s

Alternatives: 15

Speedup: 0.8×

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: 80.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := 1 - h \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D\_m}{d}\right)}^{2}}{\ell}\right)\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -2.6 \cdot 10^{-234}:\\ \;\;\;\;\frac{t\_1}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot t\_0\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t\_0 \cdot \frac{t\_1}{\sqrt{-\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D\_m \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (* h (* 0.125 (/ (pow (* M (/ D_m d)) 2.0) l)))))
        (t_1 (sqrt (- d))))
   (if (<= l -2.6e-234)
     (* (/ t_1 (sqrt (- h))) (* (sqrt (/ d l)) t_0))
     (if (<= l -5e-310)
       (* (sqrt (/ d h)) (* t_0 (/ t_1 (sqrt (- l)))))
       (*
        (/ (sqrt d) (sqrt l))
        (*
         (/ (sqrt d) (sqrt h))
         (+ 1.0 (* (/ h l) (* (pow (* D_m (/ (/ M 2.0) d)) 2.0) -0.5)))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 1.0 - (h * (0.125 * (pow((M * (D_m / d)), 2.0) / l)));
	double t_1 = sqrt(-d);
	double tmp;
	if (l <= -2.6e-234) {
		tmp = (t_1 / sqrt(-h)) * (sqrt((d / l)) * t_0);
	} else if (l <= -5e-310) {
		tmp = sqrt((d / h)) * (t_0 * (t_1 / sqrt(-l)));
	} else {
		tmp = (sqrt(d) / sqrt(l)) * ((sqrt(d) / sqrt(h)) * (1.0 + ((h / l) * (pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (h * (0.125d0 * (((m * (d_m / d)) ** 2.0d0) / l)))
    t_1 = sqrt(-d)
    if (l <= (-2.6d-234)) then
        tmp = (t_1 / sqrt(-h)) * (sqrt((d / l)) * t_0)
    else if (l <= (-5d-310)) then
        tmp = sqrt((d / h)) * (t_0 * (t_1 / sqrt(-l)))
    else
        tmp = (sqrt(d) / sqrt(l)) * ((sqrt(d) / sqrt(h)) * (1.0d0 + ((h / l) * (((d_m * ((m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 1.0 - (h * (0.125 * (Math.pow((M * (D_m / d)), 2.0) / l)));
	double t_1 = Math.sqrt(-d);
	double tmp;
	if (l <= -2.6e-234) {
		tmp = (t_1 / Math.sqrt(-h)) * (Math.sqrt((d / l)) * t_0);
	} else if (l <= -5e-310) {
		tmp = Math.sqrt((d / h)) * (t_0 * (t_1 / Math.sqrt(-l)));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(l)) * ((Math.sqrt(d) / Math.sqrt(h)) * (1.0 + ((h / l) * (Math.pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = 1.0 - (h * (0.125 * (math.pow((M * (D_m / d)), 2.0) / l)))
	t_1 = math.sqrt(-d)
	tmp = 0
	if l <= -2.6e-234:
		tmp = (t_1 / math.sqrt(-h)) * (math.sqrt((d / l)) * t_0)
	elif l <= -5e-310:
		tmp = math.sqrt((d / h)) * (t_0 * (t_1 / math.sqrt(-l)))
	else:
		tmp = (math.sqrt(d) / math.sqrt(l)) * ((math.sqrt(d) / math.sqrt(h)) * (1.0 + ((h / l) * (math.pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(1.0 - Float64(h * Float64(0.125 * Float64((Float64(M * Float64(D_m / d)) ^ 2.0) / l))))
	t_1 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -2.6e-234)
		tmp = Float64(Float64(t_1 / sqrt(Float64(-h))) * Float64(sqrt(Float64(d / l)) * t_0));
	elseif (l <= -5e-310)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(t_0 * Float64(t_1 / sqrt(Float64(-l)))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(l)) * Float64(Float64(sqrt(d) / sqrt(h)) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D_m * Float64(Float64(M / 2.0) / d)) ^ 2.0) * -0.5)))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = 1.0 - (h * (0.125 * (((M * (D_m / d)) ^ 2.0) / l)));
	t_1 = sqrt(-d);
	tmp = 0.0;
	if (l <= -2.6e-234)
		tmp = (t_1 / sqrt(-h)) * (sqrt((d / l)) * t_0);
	elseif (l <= -5e-310)
		tmp = sqrt((d / h)) * (t_0 * (t_1 / sqrt(-l)));
	else
		tmp = (sqrt(d) / sqrt(l)) * ((sqrt(d) / sqrt(h)) * (1.0 + ((h / l) * (((D_m * ((M / 2.0) / d)) ^ 2.0) * -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(1.0 - N[(h * N[(0.125 * N[(N[Power[N[(M * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -2.6e-234], N[(N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D$95$m * N[(N[(M / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.59999999999999989e-234

   1. Initial program 68.3%

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

   3. Simplified 67.4%

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

   5. Taylor expanded in h around -inf 42.6%

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

      1. associate-*r* 42.6%

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

      2. neg-mul-1 42.6%

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

      3. sub-neg 42.6%

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

      4. distribute-lft-in 42.6%

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

   7. Simplified 70.3%

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

      1. frac-2neg 70.3%

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

      2. sqrt-div 82.2%

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

   9. Applied egg-rr 82.2%

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

   if -2.59999999999999989e-234 < l < -4.999999999999985e-310

   1. Initial program 67.7%

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

   3. Simplified 61.0%

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

   5. Taylor expanded in h around -inf 41.5%

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

      1. associate-*r* 41.5%

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

      2. neg-mul-1 41.5%

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

      3. sub-neg 41.5%

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

      4. distribute-lft-in 41.5%

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

   7. Simplified 62.0%

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

      1. frac-2neg 62.0%

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

      2. sqrt-div 93.1%

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

   9. Applied egg-rr 93.1%

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

   if -4.999999999999985e-310 < l

   1. Initial program 60.7%

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

   3. Simplified 61.8%

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

      1. sqrt-div 71.3%

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

   6. Applied egg-rr 71.3%

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

      1. sqrt-div 84.4%

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

   8. Applied egg-rr 84.4%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.6 \cdot 10^{-234}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 - h \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right) \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 70.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D\_m}{d \cdot 2}\right)}^{2}\right)\right)\\ t_1 := {\left(D\_m \cdot \frac{M}{d}\right)}^{2}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{+201}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{t\_1}{\ell}\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;d \cdot \left(\frac{h + \sqrt{\frac{h}{\ell}}}{h} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{t\_1 \cdot \left(\frac{h}{\ell} \cdot -0.125\right)}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* M D_m) (* d 2.0)) 2.0))))))
        (t_1 (pow (* D_m (/ M d)) 2.0)))
   (if (<= t_0 4e+201)
     (* (sqrt (/ d h)) (* (sqrt (/ d l)) (- 1.0 (* h (* 0.125 (/ t_1 l))))))
     (if (<= t_0 INFINITY)
       (* d (+ (/ (+ h (sqrt (/ h l))) h) -1.0))
       (* d (/ (* t_1 (* (/ h l) -0.125)) (sqrt (* l h))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((M * D_m) / (d * 2.0)), 2.0))));
	double t_1 = pow((D_m * (M / d)), 2.0);
	double tmp;
	if (t_0 <= 4e+201) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (h * (0.125 * (t_1 / l)))));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = d * (((h + sqrt((h / l))) / h) + -1.0);
	} else {
		tmp = d * ((t_1 * ((h / l) * -0.125)) / sqrt((l * h)));
	}
	return tmp;
}

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * Math.pow(((M * D_m) / (d * 2.0)), 2.0))));
	double t_1 = Math.pow((D_m * (M / d)), 2.0);
	double tmp;
	if (t_0 <= 4e+201) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - (h * (0.125 * (t_1 / l)))));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = d * (((h + Math.sqrt((h / l))) / h) + -1.0);
	} else {
		tmp = d * ((t_1 * ((h / l) * -0.125)) / Math.sqrt((l * h)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = (math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * math.pow(((M * D_m) / (d * 2.0)), 2.0))))
	t_1 = math.pow((D_m * (M / d)), 2.0)
	tmp = 0
	if t_0 <= 4e+201:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - (h * (0.125 * (t_1 / l)))))
	elif t_0 <= math.inf:
		tmp = d * (((h + math.sqrt((h / l))) / h) + -1.0)
	else:
		tmp = d * ((t_1 * ((h / l) * -0.125)) / math.sqrt((l * h)))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(M * D_m) / Float64(d * 2.0)) ^ 2.0)))))
	t_1 = Float64(D_m * Float64(M / d)) ^ 2.0
	tmp = 0.0
	if (t_0 <= 4e+201)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(h * Float64(0.125 * Float64(t_1 / l))))));
	elseif (t_0 <= Inf)
		tmp = Float64(d * Float64(Float64(Float64(h + sqrt(Float64(h / l))) / h) + -1.0));
	else
		tmp = Float64(d * Float64(Float64(t_1 * Float64(Float64(h / l) * -0.125)) / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = (((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((h / l) * (0.5 * (((M * D_m) / (d * 2.0)) ^ 2.0))));
	t_1 = (D_m * (M / d)) ^ 2.0;
	tmp = 0.0;
	if (t_0 <= 4e+201)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (h * (0.125 * (t_1 / l)))));
	elseif (t_0 <= Inf)
		tmp = d * (((h + sqrt((h / l))) / h) + -1.0);
	else
		tmp = d * ((t_1 * ((h / l) * -0.125)) / sqrt((l * h)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(D$95$m * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$0, 4e+201], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(h * N[(0.125 * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(d * N[(N[(N[(h + N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(t$95$1 * N[(N[(h / l), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.00000000000000015e201

   1. Initial program 86.7%

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

   3. Simplified 86.5%

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

   5. Taylor expanded in h around -inf 54.1%

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

      1. associate-*r* 54.1%

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

      2. neg-mul-1 54.1%

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

      3. sub-neg 54.1%

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

      4. distribute-lft-in 54.1%

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

   7. Simplified 87.1%

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

   8. Taylor expanded in M around 0 86.7%

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

      1. associate-/l* 87.1%

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

   10. Simplified 87.1%

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

   if 4.00000000000000015e201 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 64.8%

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

   3. Simplified 64.8%

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

   5. Taylor expanded in d around inf 57.1%

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

      1. expm1-log1p-u 55.8%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

      2. expm1-undefine 51.9%

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

      3. inv-pow 51.9%

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

      4. sqrt-pow1 51.9%

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

      5. metadata-eval 51.9%

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

   7. Applied egg-rr 51.9%

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

   8. Taylor expanded in h around 0 86.0%

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

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in M around inf 5.3%

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

      1. associate-*r* 5.4%

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

      2. times-frac 0.0%

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

      3. *-commutative 0.0%

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

      4. associate-/l* 0.0%

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

      5. unpow2 0.0%

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

      6. unpow2 0.0%

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

      7. unpow2 0.0%

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

      8. times-frac 0.0%

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

      9. swap-sqr 0.0%

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

      10. unpow2 0.0%

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

   7. Simplified 0.0%

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

      1. pow1 0.0%

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

      2. associate-*r* 0.0%

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

      3. pow1/2 0.0%

         \[\leadsto {\left(\left(\color{blue}{{\left(\frac{d}{h}\right)}^{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

      4. pow1/2 0.0%

         \[\leadsto {\left(\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}}\right) \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

      5. pow-prod-down 0.0%

         \[\leadsto {\left(\color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

   9. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{{\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 0.0%

          \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

       2. unpow1/2 0.0%

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

   11. Simplified 0.0%

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

       1. sqrt-prod 0.0%

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

       2. frac-2neg 0.0%

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

       3. sqrt-undiv 0.2%

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

       4. associate-*r* 0.2%

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

       5. sqrt-div 0.0%

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

       6. associate-*l/ 0.0%

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

       7. frac-times 0.0%

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

   13. Applied egg-rr 19.7%

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

       1. associate-*r* 19.7%

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

       2. rem-square-sqrt 19.7%

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

       3. associate-/l* 19.7%

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

       4. *-commutative 19.7%

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

       5. associate-*r/ 19.7%

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

       6. *-commutative 19.7%

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

       7. associate-*r/ 19.7%

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

       8. associate-*l* 19.7%

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

   15. Simplified 19.7%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq 4 \cdot 10^{+201}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq \infty:\\ \;\;\;\;d \cdot \left(\frac{h + \sqrt{\frac{h}{\ell}}}{h} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.125\right)}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := 1 - h \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D\_m}{d}\right)}^{2}}{\ell}\right)\\ \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(t\_0 \cdot t\_1\right)\\ \mathbf{elif}\;h \leq 5.3 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t\_1 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1 (- 1.0 (* h (* 0.125 (/ (pow (* M (/ D_m d)) 2.0) l))))))
   (if (<= h -5e-310)
     (* (/ (sqrt (- d)) (sqrt (- h))) (* t_0 t_1))
     (if (<= h 5.3e+73)
       (* (sqrt (/ d h)) (* t_1 (/ (sqrt d) (sqrt l))))
       (*
        (* t_0 (/ (sqrt d) (sqrt h)))
        (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M 2.0)) 2.0)))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt((d / l));
	double t_1 = 1.0 - (h * (0.125 * (pow((M * (D_m / d)), 2.0) / l)));
	double tmp;
	if (h <= -5e-310) {
		tmp = (sqrt(-d) / sqrt(-h)) * (t_0 * t_1);
	} else if (h <= 5.3e+73) {
		tmp = sqrt((d / h)) * (t_1 * (sqrt(d) / sqrt(l)));
	} else {
		tmp = (t_0 * (sqrt(d) / sqrt(h))) * (1.0 - (0.5 * ((h / l) * pow(((D_m / d) * (M / 2.0)), 2.0))));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = 1.0d0 - (h * (0.125d0 * (((m * (d_m / d)) ** 2.0d0) / l)))
    if (h <= (-5d-310)) then
        tmp = (sqrt(-d) / sqrt(-h)) * (t_0 * t_1)
    else if (h <= 5.3d+73) then
        tmp = sqrt((d / h)) * (t_1 * (sqrt(d) / sqrt(l)))
    else
        tmp = (t_0 * (sqrt(d) / sqrt(h))) * (1.0d0 - (0.5d0 * ((h / l) * (((d_m / d) * (m / 2.0d0)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = 1.0 - (h * (0.125 * (Math.pow((M * (D_m / d)), 2.0) / l)));
	double tmp;
	if (h <= -5e-310) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * (t_0 * t_1);
	} else if (h <= 5.3e+73) {
		tmp = Math.sqrt((d / h)) * (t_1 * (Math.sqrt(d) / Math.sqrt(l)));
	} else {
		tmp = (t_0 * (Math.sqrt(d) / Math.sqrt(h))) * (1.0 - (0.5 * ((h / l) * Math.pow(((D_m / d) * (M / 2.0)), 2.0))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.sqrt((d / l))
	t_1 = 1.0 - (h * (0.125 * (math.pow((M * (D_m / d)), 2.0) / l)))
	tmp = 0
	if h <= -5e-310:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * (t_0 * t_1)
	elif h <= 5.3e+73:
		tmp = math.sqrt((d / h)) * (t_1 * (math.sqrt(d) / math.sqrt(l)))
	else:
		tmp = (t_0 * (math.sqrt(d) / math.sqrt(h))) * (1.0 - (0.5 * ((h / l) * math.pow(((D_m / d) * (M / 2.0)), 2.0))))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(1.0 - Float64(h * Float64(0.125 * Float64((Float64(M * Float64(D_m / d)) ^ 2.0) / l))))
	tmp = 0.0
	if (h <= -5e-310)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(t_0 * t_1));
	elseif (h <= 5.3e+73)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(t_1 * Float64(sqrt(d) / sqrt(l))));
	else
		tmp = Float64(Float64(t_0 * Float64(sqrt(d) / sqrt(h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M / 2.0)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = sqrt((d / l));
	t_1 = 1.0 - (h * (0.125 * (((M * (D_m / d)) ^ 2.0) / l)));
	tmp = 0.0;
	if (h <= -5e-310)
		tmp = (sqrt(-d) / sqrt(-h)) * (t_0 * t_1);
	elseif (h <= 5.3e+73)
		tmp = sqrt((d / h)) * (t_1 * (sqrt(d) / sqrt(l)));
	else
		tmp = (t_0 * (sqrt(d) / sqrt(h))) * (1.0 - (0.5 * ((h / l) * (((D_m / d) * (M / 2.0)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(h * N[(0.125 * N[(N[Power[N[(M * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -5e-310], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 5.3e+73], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -4.999999999999985e-310

   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 66.6%

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

   5. Taylor expanded in h around -inf 42.5%

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

      1. associate-*r* 42.5%

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

      2. neg-mul-1 42.5%

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

      3. sub-neg 42.5%

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

      4. distribute-lft-in 42.5%

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

   7. Simplified 69.3%

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

      1. frac-2neg 69.3%

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

      2. sqrt-div 79.7%

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

   9. Applied egg-rr 79.7%

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

   if -4.999999999999985e-310 < h < 5.29999999999999996e73

   1. Initial program 67.3%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in h around -inf 51.7%

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

      1. associate-*r* 51.7%

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

      2. neg-mul-1 51.7%

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

      3. sub-neg 51.7%

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

      4. distribute-lft-in 51.7%

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

   7. Simplified 70.5%

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

      1. sqrt-div 81.9%

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

   9. Applied egg-rr 84.3%

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

   if 5.29999999999999996e73 < h

   1. Initial program 48.6%

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

   3. Simplified 50.4%

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

      1. sqrt-div 77.6%

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

   6. Applied egg-rr 67.5%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;h \leq 5.3 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 - h \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := 1 - h \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D\_m}{d}\right)}^{2}}{\ell}\right)\\ \mathbf{if}\;d \leq -4.8 \cdot 10^{-213}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left|\left(\frac{h}{\ell} \cdot {\left(D\_m \cdot \frac{M}{d}\right)}^{2}\right) \cdot \frac{-0.125}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (- 1.0 (* h (* 0.125 (/ (pow (* M (/ D_m d)) 2.0) l))))))
   (if (<= d -4.8e-213)
     (* t_0 (* t_1 (/ (sqrt (- d)) (sqrt (- l)))))
     (if (<= d -5e-310)
       (*
        d
        (fabs
         (* (* (/ h l) (pow (* D_m (/ M d)) 2.0)) (/ -0.125 (sqrt (* l h))))))
       (* t_0 (* t_1 (/ (sqrt d) (sqrt l))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt((d / h));
	double t_1 = 1.0 - (h * (0.125 * (pow((M * (D_m / d)), 2.0) / l)));
	double tmp;
	if (d <= -4.8e-213) {
		tmp = t_0 * (t_1 * (sqrt(-d) / sqrt(-l)));
	} else if (d <= -5e-310) {
		tmp = d * fabs((((h / l) * pow((D_m * (M / d)), 2.0)) * (-0.125 / sqrt((l * h)))));
	} else {
		tmp = t_0 * (t_1 * (sqrt(d) / sqrt(l)));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((d / h))
    t_1 = 1.0d0 - (h * (0.125d0 * (((m * (d_m / d)) ** 2.0d0) / l)))
    if (d <= (-4.8d-213)) then
        tmp = t_0 * (t_1 * (sqrt(-d) / sqrt(-l)))
    else if (d <= (-5d-310)) then
        tmp = d * abs((((h / l) * ((d_m * (m / d)) ** 2.0d0)) * ((-0.125d0) / sqrt((l * h)))))
    else
        tmp = t_0 * (t_1 * (sqrt(d) / sqrt(l)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.sqrt((d / h));
	double t_1 = 1.0 - (h * (0.125 * (Math.pow((M * (D_m / d)), 2.0) / l)));
	double tmp;
	if (d <= -4.8e-213) {
		tmp = t_0 * (t_1 * (Math.sqrt(-d) / Math.sqrt(-l)));
	} else if (d <= -5e-310) {
		tmp = d * Math.abs((((h / l) * Math.pow((D_m * (M / d)), 2.0)) * (-0.125 / Math.sqrt((l * h)))));
	} else {
		tmp = t_0 * (t_1 * (Math.sqrt(d) / Math.sqrt(l)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.sqrt((d / h))
	t_1 = 1.0 - (h * (0.125 * (math.pow((M * (D_m / d)), 2.0) / l)))
	tmp = 0
	if d <= -4.8e-213:
		tmp = t_0 * (t_1 * (math.sqrt(-d) / math.sqrt(-l)))
	elif d <= -5e-310:
		tmp = d * math.fabs((((h / l) * math.pow((D_m * (M / d)), 2.0)) * (-0.125 / math.sqrt((l * h)))))
	else:
		tmp = t_0 * (t_1 * (math.sqrt(d) / math.sqrt(l)))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(d / h))
	t_1 = Float64(1.0 - Float64(h * Float64(0.125 * Float64((Float64(M * Float64(D_m / d)) ^ 2.0) / l))))
	tmp = 0.0
	if (d <= -4.8e-213)
		tmp = Float64(t_0 * Float64(t_1 * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l)))));
	elseif (d <= -5e-310)
		tmp = Float64(d * abs(Float64(Float64(Float64(h / l) * (Float64(D_m * Float64(M / d)) ^ 2.0)) * Float64(-0.125 / sqrt(Float64(l * h))))));
	else
		tmp = Float64(t_0 * Float64(t_1 * Float64(sqrt(d) / sqrt(l))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = sqrt((d / h));
	t_1 = 1.0 - (h * (0.125 * (((M * (D_m / d)) ^ 2.0) / l)));
	tmp = 0.0;
	if (d <= -4.8e-213)
		tmp = t_0 * (t_1 * (sqrt(-d) / sqrt(-l)));
	elseif (d <= -5e-310)
		tmp = d * abs((((h / l) * ((D_m * (M / d)) ^ 2.0)) * (-0.125 / sqrt((l * h)))));
	else
		tmp = t_0 * (t_1 * (sqrt(d) / sqrt(l)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(h * N[(0.125 * N[(N[Power[N[(M * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -4.8e-213], N[(t$95$0 * N[(t$95$1 * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(d * N[Abs[N[(N[(N[(h / l), $MachinePrecision] * N[Power[N[(D$95$m * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.125 / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(t$95$1 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.79999999999999991e-213

   1. Initial program 76.1%

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

   3. Simplified 74.2%

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

   5. Taylor expanded in h around -inf 48.9%

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

      1. associate-*r* 48.9%

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

      2. neg-mul-1 48.9%

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

      3. sub-neg 48.9%

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

      4. distribute-lft-in 48.9%

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

   7. Simplified 76.4%

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

      1. frac-2neg 76.4%

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

      2. sqrt-div 83.7%

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

   9. Applied egg-rr 83.7%

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

   if -4.79999999999999991e-213 < d < -4.999999999999985e-310

   1. Initial program 22.7%

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

   3. Simplified 22.7%

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

   5. Taylor expanded in M around inf 0.3%

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

      1. associate-*r* 5.9%

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

      2. times-frac 5.9%

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

      3. *-commutative 5.9%

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

      4. associate-/l* 6.0%

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

      5. unpow2 6.0%

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

      6. unpow2 6.0%

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

      7. unpow2 6.0%

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

      8. times-frac 22.3%

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

      9. swap-sqr 22.7%

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

      10. unpow2 22.7%

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

   7. Simplified 22.7%

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

      1. pow1 22.7%

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

      2. associate-*r* 22.7%

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

      3. pow1/2 22.7%

         \[\leadsto {\left(\left(\color{blue}{{\left(\frac{d}{h}\right)}^{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

      4. pow1/2 22.7%

         \[\leadsto {\left(\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}}\right) \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

      5. pow-prod-down 6.5%

         \[\leadsto {\left(\color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

   9. Applied egg-rr 6.5%

      \[\leadsto \color{blue}{{\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 6.5%

          \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

       2. unpow1/2 6.5%

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

   11. Simplified 6.5%

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

       1. add-cube-cbrt 6.5%

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

       2. pow3 6.5%

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

       3. frac-times 6.5%

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

       4. sqrt-div 6.5%

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

       5. sqrt-unprod 0.0%

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

       6. add-sqr-sqrt 6.6%

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

   13. Applied egg-rr 6.6%

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

       1. rem-cube-cbrt 6.6%

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

       2. add-sqr-sqrt 6.6%

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

       3. sqrt-unprod 6.6%

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

       4. pow2 6.6%

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

       5. *-commutative 6.6%

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

   15. Applied egg-rr 6.6%

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

       1. unpow2 6.6%

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

       2. rem-sqrt-square 6.6%

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

       3. associate-*r/ 6.6%

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

       4. associate-*l/ 6.6%

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

       5. *-commutative 6.6%

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

       6. fabs-mul 6.6%

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

       7. rem-square-sqrt 0.0%

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

       8. fabs-sqr 0.0%

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

       9. rem-square-sqrt 62.4%

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

       10. *-commutative 62.4%

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

       11. associate-/l* 62.4%

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

   17. Simplified 57.1%

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

   if -4.999999999999985e-310 < d

   1. Initial program 60.7%

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

   3. Simplified 61.8%

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

   5. Taylor expanded in h around -inf 43.8%

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

      1. associate-*r* 43.8%

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

      2. neg-mul-1 43.8%

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

      3. sub-neg 43.8%

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

      4. distribute-lft-in 43.8%

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

   7. Simplified 64.9%

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

      1. sqrt-div 71.3%

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

   9. Applied egg-rr 74.5%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \left(\left(-h\right) \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) + 1\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.8 \cdot 10^{-213}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 - h \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right) \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left|\left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right) \cdot \frac{-0.125}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 - h \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := 1 - h \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D\_m}{d}\right)}^{2}}{\ell}\right)\\ \mathbf{if}\;d \leq -2.9 \cdot 10^{-197}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot t\_1\right) \cdot t\_0\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left|\left(\frac{h}{\ell} \cdot {\left(D\_m \cdot \frac{M}{d}\right)}^{2}\right) \cdot \frac{-0.125}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (- 1.0 (* h (* 0.125 (/ (pow (* M (/ D_m d)) 2.0) l))))))
   (if (<= d -2.9e-197)
     (* (* (sqrt (/ d l)) t_1) t_0)
     (if (<= d -5e-310)
       (*
        d
        (fabs
         (* (* (/ h l) (pow (* D_m (/ M d)) 2.0)) (/ -0.125 (sqrt (* l h))))))
       (* t_0 (* t_1 (/ (sqrt d) (sqrt l))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt((d / h));
	double t_1 = 1.0 - (h * (0.125 * (pow((M * (D_m / d)), 2.0) / l)));
	double tmp;
	if (d <= -2.9e-197) {
		tmp = (sqrt((d / l)) * t_1) * t_0;
	} else if (d <= -5e-310) {
		tmp = d * fabs((((h / l) * pow((D_m * (M / d)), 2.0)) * (-0.125 / sqrt((l * h)))));
	} else {
		tmp = t_0 * (t_1 * (sqrt(d) / sqrt(l)));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((d / h))
    t_1 = 1.0d0 - (h * (0.125d0 * (((m * (d_m / d)) ** 2.0d0) / l)))
    if (d <= (-2.9d-197)) then
        tmp = (sqrt((d / l)) * t_1) * t_0
    else if (d <= (-5d-310)) then
        tmp = d * abs((((h / l) * ((d_m * (m / d)) ** 2.0d0)) * ((-0.125d0) / sqrt((l * h)))))
    else
        tmp = t_0 * (t_1 * (sqrt(d) / sqrt(l)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.sqrt((d / h));
	double t_1 = 1.0 - (h * (0.125 * (Math.pow((M * (D_m / d)), 2.0) / l)));
	double tmp;
	if (d <= -2.9e-197) {
		tmp = (Math.sqrt((d / l)) * t_1) * t_0;
	} else if (d <= -5e-310) {
		tmp = d * Math.abs((((h / l) * Math.pow((D_m * (M / d)), 2.0)) * (-0.125 / Math.sqrt((l * h)))));
	} else {
		tmp = t_0 * (t_1 * (Math.sqrt(d) / Math.sqrt(l)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.sqrt((d / h))
	t_1 = 1.0 - (h * (0.125 * (math.pow((M * (D_m / d)), 2.0) / l)))
	tmp = 0
	if d <= -2.9e-197:
		tmp = (math.sqrt((d / l)) * t_1) * t_0
	elif d <= -5e-310:
		tmp = d * math.fabs((((h / l) * math.pow((D_m * (M / d)), 2.0)) * (-0.125 / math.sqrt((l * h)))))
	else:
		tmp = t_0 * (t_1 * (math.sqrt(d) / math.sqrt(l)))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(d / h))
	t_1 = Float64(1.0 - Float64(h * Float64(0.125 * Float64((Float64(M * Float64(D_m / d)) ^ 2.0) / l))))
	tmp = 0.0
	if (d <= -2.9e-197)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * t_1) * t_0);
	elseif (d <= -5e-310)
		tmp = Float64(d * abs(Float64(Float64(Float64(h / l) * (Float64(D_m * Float64(M / d)) ^ 2.0)) * Float64(-0.125 / sqrt(Float64(l * h))))));
	else
		tmp = Float64(t_0 * Float64(t_1 * Float64(sqrt(d) / sqrt(l))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = sqrt((d / h));
	t_1 = 1.0 - (h * (0.125 * (((M * (D_m / d)) ^ 2.0) / l)));
	tmp = 0.0;
	if (d <= -2.9e-197)
		tmp = (sqrt((d / l)) * t_1) * t_0;
	elseif (d <= -5e-310)
		tmp = d * abs((((h / l) * ((D_m * (M / d)) ^ 2.0)) * (-0.125 / sqrt((l * h)))));
	else
		tmp = t_0 * (t_1 * (sqrt(d) / sqrt(l)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(h * N[(0.125 * N[(N[Power[N[(M * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.9e-197], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[d, -5e-310], N[(d * N[Abs[N[(N[(N[(h / l), $MachinePrecision] * N[Power[N[(D$95$m * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.125 / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(t$95$1 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.90000000000000023e-197

   1. Initial program 77.9%

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

   3. Simplified 75.9%

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

   5. Taylor expanded in h around -inf 50.3%

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

      1. associate-*r* 50.3%

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

      2. neg-mul-1 50.3%

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

      3. sub-neg 50.3%

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

      4. distribute-lft-in 50.3%

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

   7. Simplified 78.2%

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

   if -2.90000000000000023e-197 < d < -4.999999999999985e-310

   1. Initial program 27.0%

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

   3. Simplified 26.9%

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

   5. Taylor expanded in M around inf 4.6%

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

      1. associate-*r* 8.9%

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

      2. times-frac 8.9%

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

      3. *-commutative 8.9%

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

      4. associate-/l* 9.0%

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

      5. unpow2 9.0%

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

      6. unpow2 9.0%

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

      7. unpow2 9.0%

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

      8. times-frac 22.6%

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

      9. swap-sqr 22.9%

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

      10. unpow2 22.9%

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

   7. Simplified 22.9%

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

      1. pow1 22.9%

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

      2. associate-*r* 22.9%

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

      3. pow1/2 22.9%

         \[\leadsto {\left(\left(\color{blue}{{\left(\frac{d}{h}\right)}^{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

      4. pow1/2 22.9%

         \[\leadsto {\left(\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}}\right) \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

      5. pow-prod-down 5.9%

         \[\leadsto {\left(\color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

   9. Applied egg-rr 5.9%

      \[\leadsto \color{blue}{{\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 5.9%

          \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

       2. unpow1/2 5.9%

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

   11. Simplified 5.9%

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

       1. add-cube-cbrt 5.9%

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

       2. pow3 5.9%

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

       3. frac-times 5.9%

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

       4. sqrt-div 5.9%

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

       5. sqrt-unprod 0.0%

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

       6. add-sqr-sqrt 5.9%

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

   13. Applied egg-rr 5.9%

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

       1. rem-cube-cbrt 5.9%

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

       2. add-sqr-sqrt 5.9%

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

       3. sqrt-unprod 6.0%

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

       4. pow2 6.0%

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

       5. *-commutative 6.0%

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

   15. Applied egg-rr 6.0%

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

       1. unpow2 6.0%

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

       2. rem-sqrt-square 5.9%

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

       3. associate-*r/ 6.0%

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

       4. associate-*l/ 6.0%

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

       5. *-commutative 6.0%

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

       6. fabs-mul 6.0%

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

       7. rem-square-sqrt 0.0%

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

       8. fabs-sqr 0.0%

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

       9. rem-square-sqrt 58.3%

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

       10. *-commutative 58.3%

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

       11. associate-/l* 58.3%

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

   17. Simplified 54.1%

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

   if -4.999999999999985e-310 < d

   1. Initial program 60.7%

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

   3. Simplified 61.8%

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

   5. Taylor expanded in h around -inf 43.8%

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

      1. associate-*r* 43.8%

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

      2. neg-mul-1 43.8%

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

      3. sub-neg 43.8%

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

      4. distribute-lft-in 43.8%

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

   7. Simplified 64.9%

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

      1. sqrt-div 71.3%

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

   9. Applied egg-rr 74.5%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \left(\left(-h\right) \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) + 1\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.9 \cdot 10^{-197}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left|\left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right) \cdot \frac{-0.125}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 - h \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := {\left(D\_m \cdot \frac{M}{d}\right)}^{2}\\ t_1 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;d \leq -3.1 \cdot 10^{-181}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left|\left(\frac{h}{\ell} \cdot t\_0\right) \cdot \frac{-0.125}{t\_1}\right|\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{-132}:\\ \;\;\;\;d \cdot \frac{t\_0 \cdot \left(\frac{h}{\ell} \cdot -0.125\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (pow (* D_m (/ M d)) 2.0)) (t_1 (sqrt (* l h))))
   (if (<= d -3.1e-181)
     (*
      (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M 2.0)) 2.0))))
      (sqrt (* (/ d l) (/ d h))))
     (if (<= d -5e-310)
       (* d (fabs (* (* (/ h l) t_0) (/ -0.125 t_1))))
       (if (<= d 1.1e-132)
         (* d (/ (* t_0 (* (/ h l) -0.125)) t_1))
         (* d (/ (sqrt (/ 1.0 h)) (sqrt l))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = pow((D_m * (M / d)), 2.0);
	double t_1 = sqrt((l * h));
	double tmp;
	if (d <= -3.1e-181) {
		tmp = (1.0 - (0.5 * ((h / l) * pow(((D_m / d) * (M / 2.0)), 2.0)))) * sqrt(((d / l) * (d / h)));
	} else if (d <= -5e-310) {
		tmp = d * fabs((((h / l) * t_0) * (-0.125 / t_1)));
	} else if (d <= 1.1e-132) {
		tmp = d * ((t_0 * ((h / l) * -0.125)) / t_1);
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (d_m * (m / d)) ** 2.0d0
    t_1 = sqrt((l * h))
    if (d <= (-3.1d-181)) then
        tmp = (1.0d0 - (0.5d0 * ((h / l) * (((d_m / d) * (m / 2.0d0)) ** 2.0d0)))) * sqrt(((d / l) * (d / h)))
    else if (d <= (-5d-310)) then
        tmp = d * abs((((h / l) * t_0) * ((-0.125d0) / t_1)))
    else if (d <= 1.1d-132) then
        tmp = d * ((t_0 * ((h / l) * (-0.125d0))) / t_1)
    else
        tmp = d * (sqrt((1.0d0 / h)) / sqrt(l))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.pow((D_m * (M / d)), 2.0);
	double t_1 = Math.sqrt((l * h));
	double tmp;
	if (d <= -3.1e-181) {
		tmp = (1.0 - (0.5 * ((h / l) * Math.pow(((D_m / d) * (M / 2.0)), 2.0)))) * Math.sqrt(((d / l) * (d / h)));
	} else if (d <= -5e-310) {
		tmp = d * Math.abs((((h / l) * t_0) * (-0.125 / t_1)));
	} else if (d <= 1.1e-132) {
		tmp = d * ((t_0 * ((h / l) * -0.125)) / t_1);
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.pow((D_m * (M / d)), 2.0)
	t_1 = math.sqrt((l * h))
	tmp = 0
	if d <= -3.1e-181:
		tmp = (1.0 - (0.5 * ((h / l) * math.pow(((D_m / d) * (M / 2.0)), 2.0)))) * math.sqrt(((d / l) * (d / h)))
	elif d <= -5e-310:
		tmp = d * math.fabs((((h / l) * t_0) * (-0.125 / t_1)))
	elif d <= 1.1e-132:
		tmp = d * ((t_0 * ((h / l) * -0.125)) / t_1)
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(D_m * Float64(M / d)) ^ 2.0
	t_1 = sqrt(Float64(l * h))
	tmp = 0.0
	if (d <= -3.1e-181)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M / 2.0)) ^ 2.0)))) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	elseif (d <= -5e-310)
		tmp = Float64(d * abs(Float64(Float64(Float64(h / l) * t_0) * Float64(-0.125 / t_1))));
	elseif (d <= 1.1e-132)
		tmp = Float64(d * Float64(Float64(t_0 * Float64(Float64(h / l) * -0.125)) / t_1));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = (D_m * (M / d)) ^ 2.0;
	t_1 = sqrt((l * h));
	tmp = 0.0;
	if (d <= -3.1e-181)
		tmp = (1.0 - (0.5 * ((h / l) * (((D_m / d) * (M / 2.0)) ^ 2.0)))) * sqrt(((d / l) * (d / h)));
	elseif (d <= -5e-310)
		tmp = d * abs((((h / l) * t_0) * (-0.125 / t_1)));
	elseif (d <= 1.1e-132)
		tmp = d * ((t_0 * ((h / l) * -0.125)) / t_1);
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Power[N[(D$95$m * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -3.1e-181], N[(N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(d * N[Abs[N[(N[(N[(h / l), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(-0.125 / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.1e-132], N[(d * N[(N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.10000000000000021e-181

   1. Initial program 78.2%

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

   3. Simplified 77.1%

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

      1. pow1 77.1%

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

      2. sqrt-unprod 69.4%

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

   6. Applied egg-rr 69.4%

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

      1. unpow1 69.4%

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

   8. Simplified 69.4%

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

   if -3.10000000000000021e-181 < d < -4.999999999999985e-310

   1. Initial program 31.5%

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

   3. Simplified 31.5%

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

   5. Taylor expanded in M around inf 11.8%

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

      1. associate-*r* 15.6%

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

      2. times-frac 11.8%

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

      3. *-commutative 11.8%

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

      4. associate-/l* 11.8%

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

      5. unpow2 11.8%

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

      6. unpow2 11.8%

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

      7. unpow2 11.8%

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

      8. times-frac 23.8%

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

      9. swap-sqr 24.2%

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

      10. unpow2 24.2%

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

   7. Simplified 24.2%

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

      1. pow1 24.2%

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

      2. associate-*r* 24.2%

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

      3. pow1/2 24.2%

         \[\leadsto {\left(\left(\color{blue}{{\left(\frac{d}{h}\right)}^{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

      4. pow1/2 24.2%

         \[\leadsto {\left(\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}}\right) \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

      5. pow-prod-down 5.3%

         \[\leadsto {\left(\color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

   9. Applied egg-rr 5.3%

      \[\leadsto \color{blue}{{\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 5.3%

          \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

       2. unpow1/2 5.3%

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

   11. Simplified 5.3%

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

       1. add-cube-cbrt 5.3%

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

       2. pow3 5.3%

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

       3. frac-times 5.4%

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

       4. sqrt-div 5.4%

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

       5. sqrt-unprod 0.0%

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

       6. add-sqr-sqrt 5.5%

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

   13. Applied egg-rr 5.5%

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

       1. rem-cube-cbrt 5.5%

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

       2. add-sqr-sqrt 5.5%

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

       3. sqrt-unprod 5.4%

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

       4. pow2 5.4%

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

       5. *-commutative 5.4%

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

   15. Applied egg-rr 5.4%

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

       1. unpow2 5.4%

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

       2. rem-sqrt-square 5.5%

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

       3. associate-*r/ 5.5%

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

       4. associate-*l/ 5.5%

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

       5. *-commutative 5.5%

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

       6. fabs-mul 5.5%

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

       7. rem-square-sqrt 0.0%

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

       8. fabs-sqr 0.0%

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

       9. rem-square-sqrt 55.5%

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

       10. *-commutative 55.5%

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

       11. associate-/l* 55.5%

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

   17. Simplified 51.9%

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

   if -4.999999999999985e-310 < d < 1.09999999999999995e-132

   1. Initial program 40.5%

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

   3. Simplified 42.0%

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

   5. Taylor expanded in M around inf 20.1%

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

      1. associate-*r* 20.1%

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

      2. times-frac 20.1%

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

      3. *-commutative 20.1%

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

      4. associate-/l* 19.5%

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

      5. unpow2 19.5%

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

      6. unpow2 19.5%

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

      7. unpow2 19.5%

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

      8. times-frac 26.6%

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

      9. swap-sqr 31.8%

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

      10. unpow2 31.8%

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

   7. Simplified 31.8%

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

      1. pow1 31.8%

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

      2. associate-*r* 31.8%

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

      3. pow1/2 31.8%

         \[\leadsto {\left(\left(\color{blue}{{\left(\frac{d}{h}\right)}^{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

      4. pow1/2 31.8%

         \[\leadsto {\left(\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}}\right) \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

      5. pow-prod-down 24.7%

         \[\leadsto {\left(\color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

   9. Applied egg-rr 24.7%

      \[\leadsto \color{blue}{{\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 24.7%

          \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

       2. unpow1/2 24.7%

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

   11. Simplified 24.7%

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

       1. sqrt-prod 31.8%

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

       2. frac-2neg 31.8%

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

       3. sqrt-undiv 0.0%

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

       4. associate-*r* 0.0%

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

       5. sqrt-div 0.0%

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

       6. associate-*l/ 0.0%

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

       7. frac-times 0.0%

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

   13. Applied egg-rr 54.9%

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

       1. associate-*r* 54.9%

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

       2. rem-square-sqrt 54.8%

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

       3. associate-/l* 54.8%

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

       4. *-commutative 54.8%

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

       5. associate-*r/ 54.8%

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

       6. *-commutative 54.8%

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

       7. associate-*r/ 54.7%

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

       8. associate-*l* 54.7%

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

   15. Simplified 54.7%

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

   if 1.09999999999999995e-132 < d

   1. Initial program 71.5%

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

   3. Simplified 72.4%

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

   5. Taylor expanded in d around inf 60.9%

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

      1. associate-/r* 60.9%

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

      2. sqrt-div 72.6%

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

   7. Applied egg-rr 72.6%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.1 \cdot 10^{-181}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left|\left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right) \cdot \frac{-0.125}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{-132}:\\ \;\;\;\;d \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.125\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;d \leq -5 \cdot 10^{-156}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{t\_0} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D\_m}{d}\right)}^{2} \cdot \frac{h}{-\ell}\right)\right)\\ \mathbf{elif}\;d \leq 1.28 \cdot 10^{-142}:\\ \;\;\;\;d \cdot \frac{{\left(D\_m \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.125\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* l h))))
   (if (<= d -5e-156)
     (*
      (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M 2.0)) 2.0))))
      (sqrt (* (/ d l) (/ d h))))
     (if (<= d -5e-310)
       (* (/ d t_0) (* -0.125 (* (pow (* M (/ D_m d)) 2.0) (/ h (- l)))))
       (if (<= d 1.28e-142)
         (* d (/ (* (pow (* D_m (/ M d)) 2.0) (* (/ h l) -0.125)) t_0))
         (* d (/ (sqrt (/ 1.0 h)) (sqrt l))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (d <= -5e-156) {
		tmp = (1.0 - (0.5 * ((h / l) * pow(((D_m / d) * (M / 2.0)), 2.0)))) * sqrt(((d / l) * (d / h)));
	} else if (d <= -5e-310) {
		tmp = (d / t_0) * (-0.125 * (pow((M * (D_m / d)), 2.0) * (h / -l)));
	} else if (d <= 1.28e-142) {
		tmp = d * ((pow((D_m * (M / d)), 2.0) * ((h / l) * -0.125)) / t_0);
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((l * h))
    if (d <= (-5d-156)) then
        tmp = (1.0d0 - (0.5d0 * ((h / l) * (((d_m / d) * (m / 2.0d0)) ** 2.0d0)))) * sqrt(((d / l) * (d / h)))
    else if (d <= (-5d-310)) then
        tmp = (d / t_0) * ((-0.125d0) * (((m * (d_m / d)) ** 2.0d0) * (h / -l)))
    else if (d <= 1.28d-142) then
        tmp = d * ((((d_m * (m / d)) ** 2.0d0) * ((h / l) * (-0.125d0))) / t_0)
    else
        tmp = d * (sqrt((1.0d0 / h)) / sqrt(l))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.sqrt((l * h));
	double tmp;
	if (d <= -5e-156) {
		tmp = (1.0 - (0.5 * ((h / l) * Math.pow(((D_m / d) * (M / 2.0)), 2.0)))) * Math.sqrt(((d / l) * (d / h)));
	} else if (d <= -5e-310) {
		tmp = (d / t_0) * (-0.125 * (Math.pow((M * (D_m / d)), 2.0) * (h / -l)));
	} else if (d <= 1.28e-142) {
		tmp = d * ((Math.pow((D_m * (M / d)), 2.0) * ((h / l) * -0.125)) / t_0);
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.sqrt((l * h))
	tmp = 0
	if d <= -5e-156:
		tmp = (1.0 - (0.5 * ((h / l) * math.pow(((D_m / d) * (M / 2.0)), 2.0)))) * math.sqrt(((d / l) * (d / h)))
	elif d <= -5e-310:
		tmp = (d / t_0) * (-0.125 * (math.pow((M * (D_m / d)), 2.0) * (h / -l)))
	elif d <= 1.28e-142:
		tmp = d * ((math.pow((D_m * (M / d)), 2.0) * ((h / l) * -0.125)) / t_0)
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (d <= -5e-156)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M / 2.0)) ^ 2.0)))) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	elseif (d <= -5e-310)
		tmp = Float64(Float64(d / t_0) * Float64(-0.125 * Float64((Float64(M * Float64(D_m / d)) ^ 2.0) * Float64(h / Float64(-l)))));
	elseif (d <= 1.28e-142)
		tmp = Float64(d * Float64(Float64((Float64(D_m * Float64(M / d)) ^ 2.0) * Float64(Float64(h / l) * -0.125)) / t_0));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = sqrt((l * h));
	tmp = 0.0;
	if (d <= -5e-156)
		tmp = (1.0 - (0.5 * ((h / l) * (((D_m / d) * (M / 2.0)) ^ 2.0)))) * sqrt(((d / l) * (d / h)));
	elseif (d <= -5e-310)
		tmp = (d / t_0) * (-0.125 * (((M * (D_m / d)) ^ 2.0) * (h / -l)));
	elseif (d <= 1.28e-142)
		tmp = d * ((((D_m * (M / d)) ^ 2.0) * ((h / l) * -0.125)) / t_0);
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -5e-156], N[(N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(d / t$95$0), $MachinePrecision] * N[(-0.125 * N[(N[Power[N[(M * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / (-l)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.28e-142], N[(d * N[(N[(N[Power[N[(D$95$m * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -5.00000000000000007e-156

   1. Initial program 77.9%

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

   3. Simplified 76.8%

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

      1. pow1 76.8%

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

      2. sqrt-unprod 69.6%

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

   6. Applied egg-rr 69.6%

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

      1. unpow1 69.6%

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

   8. Simplified 69.6%

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

   if -5.00000000000000007e-156 < d < -4.999999999999985e-310

   1. Initial program 41.2%

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

   3. Simplified 41.1%

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

   5. Taylor expanded in M around inf 12.7%

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

      1. associate-*r* 15.8%

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

      2. times-frac 12.7%

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

      3. *-commutative 12.7%

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

      4. associate-/l* 12.7%

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

      5. unpow2 12.7%

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

      6. unpow2 12.7%

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

      7. unpow2 12.7%

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

      8. times-frac 25.8%

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

      9. swap-sqr 32.2%

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

      10. unpow2 32.2%

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

   7. Simplified 32.2%

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

      1. pow1 32.2%

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

      2. associate-*r* 32.2%

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

      3. pow1/2 32.2%

         \[\leadsto {\left(\left(\color{blue}{{\left(\frac{d}{h}\right)}^{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

      4. pow1/2 32.2%

         \[\leadsto {\left(\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}}\right) \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

      5. pow-prod-down 16.9%

         \[\leadsto {\left(\color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

   9. Applied egg-rr 16.9%

      \[\leadsto \color{blue}{{\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 16.9%

          \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

       2. unpow1/2 16.9%

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

   11. Simplified 16.9%

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

       1. add-cube-cbrt 16.8%

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

       2. pow3 16.8%

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

       3. frac-times 4.7%

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

       4. sqrt-div 4.7%

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

       5. sqrt-unprod 0.0%

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

       6. add-sqr-sqrt 4.7%

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

   13. Applied egg-rr 4.7%

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

       1. rem-cube-cbrt 4.7%

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

       2. div-inv 4.7%

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

       3. pow1/2 4.7%

          \[\leadsto \left(d \cdot \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}}\right) \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       4. pow-flip 4.7%

          \[\leadsto \left(d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-0.5\right)}}\right) \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       5. metadata-eval 4.7%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       6. add-sqr-sqrt 3.1%

          \[\leadsto \color{blue}{\left(\sqrt{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{d \cdot {\left(h \cdot \ell\right)}^{-0.5}}\right)} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       7. sqrt-unprod 20.0%

          \[\leadsto \color{blue}{\sqrt{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)}} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       8. sqr-neg 20.0%

          \[\leadsto \sqrt{\color{blue}{\left(-d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(-d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)}} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       9. distribute-rgt-neg-out 20.0%

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \cdot \left(-d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       10. distribute-rgt-neg-out 20.0%

           \[\leadsto \sqrt{\left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right)}} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       11. sqrt-unprod 48.3%

           \[\leadsto \color{blue}{\left(\sqrt{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \cdot \sqrt{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)}\right)} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       12. add-sqr-sqrt 48.3%

           \[\leadsto \color{blue}{\left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       13. distribute-rgt-neg-out 48.3%

           \[\leadsto \color{blue}{\left(-d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       14. neg-mul-1 48.3%

           \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       15. metadata-eval 48.3%

           \[\leadsto \left(-1 \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{\left(-0.5\right)}}\right)\right) \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       16. pow-flip 48.3%

           \[\leadsto \left(-1 \cdot \left(d \cdot \color{blue}{\frac{1}{{\left(h \cdot \ell\right)}^{0.5}}}\right)\right) \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       17. pow1/2 48.3%

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

       18. div-inv 48.3%

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

   15. Applied egg-rr 48.3%

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

   if -4.999999999999985e-310 < d < 1.2799999999999999e-142

   1. Initial program 40.5%

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

   3. Simplified 42.0%

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

   5. Taylor expanded in M around inf 20.1%

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

      1. associate-*r* 20.1%

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

      2. times-frac 20.1%

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

      3. *-commutative 20.1%

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

      4. associate-/l* 19.5%

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

      5. unpow2 19.5%

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

      6. unpow2 19.5%

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

      7. unpow2 19.5%

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

      8. times-frac 26.6%

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

      9. swap-sqr 31.8%

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

      10. unpow2 31.8%

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

   7. Simplified 31.8%

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

      1. pow1 31.8%

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

      2. associate-*r* 31.8%

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

      3. pow1/2 31.8%

         \[\leadsto {\left(\left(\color{blue}{{\left(\frac{d}{h}\right)}^{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

      4. pow1/2 31.8%

         \[\leadsto {\left(\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}}\right) \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

      5. pow-prod-down 24.7%

         \[\leadsto {\left(\color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

   9. Applied egg-rr 24.7%

      \[\leadsto \color{blue}{{\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 24.7%

          \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

       2. unpow1/2 24.7%

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

   11. Simplified 24.7%

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

       1. sqrt-prod 31.8%

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

       2. frac-2neg 31.8%

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

       3. sqrt-undiv 0.0%

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

       4. associate-*r* 0.0%

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

       5. sqrt-div 0.0%

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

       6. associate-*l/ 0.0%

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

       7. frac-times 0.0%

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

   13. Applied egg-rr 54.9%

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

       1. associate-*r* 54.9%

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

       2. rem-square-sqrt 54.8%

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

       3. associate-/l* 54.8%

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

       4. *-commutative 54.8%

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

       5. associate-*r/ 54.8%

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

       6. *-commutative 54.8%

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

       7. associate-*r/ 54.7%

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

       8. associate-*l* 54.7%

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

   15. Simplified 54.7%

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

   if 1.2799999999999999e-142 < d

   1. Initial program 71.5%

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

   3. Simplified 72.4%

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

   5. Taylor expanded in d around inf 60.9%

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

      1. associate-/r* 60.9%

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

      2. sqrt-div 72.6%

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

   7. Applied egg-rr 72.6%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-156}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{-\ell}\right)\right)\\ \mathbf{elif}\;d \leq 1.28 \cdot 10^{-142}:\\ \;\;\;\;d \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.125\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;d \leq -1.46 \cdot 10^{-61}:\\ \;\;\;\;\frac{d}{-t\_0}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{t\_0} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D\_m}{d}\right)}^{2} \cdot \frac{h}{-\ell}\right)\right)\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{-139}:\\ \;\;\;\;d \cdot \frac{{\left(D\_m \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.125\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* l h))))
   (if (<= d -1.46e-61)
     (/ d (- t_0))
     (if (<= d -5e-310)
       (* (/ d t_0) (* -0.125 (* (pow (* M (/ D_m d)) 2.0) (/ h (- l)))))
       (if (<= d 1.85e-139)
         (* d (/ (* (pow (* D_m (/ M d)) 2.0) (* (/ h l) -0.125)) t_0))
         (* d (/ (sqrt (/ 1.0 h)) (sqrt l))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (d <= -1.46e-61) {
		tmp = d / -t_0;
	} else if (d <= -5e-310) {
		tmp = (d / t_0) * (-0.125 * (pow((M * (D_m / d)), 2.0) * (h / -l)));
	} else if (d <= 1.85e-139) {
		tmp = d * ((pow((D_m * (M / d)), 2.0) * ((h / l) * -0.125)) / t_0);
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((l * h))
    if (d <= (-1.46d-61)) then
        tmp = d / -t_0
    else if (d <= (-5d-310)) then
        tmp = (d / t_0) * ((-0.125d0) * (((m * (d_m / d)) ** 2.0d0) * (h / -l)))
    else if (d <= 1.85d-139) then
        tmp = d * ((((d_m * (m / d)) ** 2.0d0) * ((h / l) * (-0.125d0))) / t_0)
    else
        tmp = d * (sqrt((1.0d0 / h)) / sqrt(l))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.sqrt((l * h));
	double tmp;
	if (d <= -1.46e-61) {
		tmp = d / -t_0;
	} else if (d <= -5e-310) {
		tmp = (d / t_0) * (-0.125 * (Math.pow((M * (D_m / d)), 2.0) * (h / -l)));
	} else if (d <= 1.85e-139) {
		tmp = d * ((Math.pow((D_m * (M / d)), 2.0) * ((h / l) * -0.125)) / t_0);
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.sqrt((l * h))
	tmp = 0
	if d <= -1.46e-61:
		tmp = d / -t_0
	elif d <= -5e-310:
		tmp = (d / t_0) * (-0.125 * (math.pow((M * (D_m / d)), 2.0) * (h / -l)))
	elif d <= 1.85e-139:
		tmp = d * ((math.pow((D_m * (M / d)), 2.0) * ((h / l) * -0.125)) / t_0)
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (d <= -1.46e-61)
		tmp = Float64(d / Float64(-t_0));
	elseif (d <= -5e-310)
		tmp = Float64(Float64(d / t_0) * Float64(-0.125 * Float64((Float64(M * Float64(D_m / d)) ^ 2.0) * Float64(h / Float64(-l)))));
	elseif (d <= 1.85e-139)
		tmp = Float64(d * Float64(Float64((Float64(D_m * Float64(M / d)) ^ 2.0) * Float64(Float64(h / l) * -0.125)) / t_0));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = sqrt((l * h));
	tmp = 0.0;
	if (d <= -1.46e-61)
		tmp = d / -t_0;
	elseif (d <= -5e-310)
		tmp = (d / t_0) * (-0.125 * (((M * (D_m / d)) ^ 2.0) * (h / -l)));
	elseif (d <= 1.85e-139)
		tmp = d * ((((D_m * (M / d)) ^ 2.0) * ((h / l) * -0.125)) / t_0);
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.46e-61], N[(d / (-t$95$0)), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(d / t$95$0), $MachinePrecision] * N[(-0.125 * N[(N[Power[N[(M * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / (-l)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.85e-139], N[(d * N[(N[(N[Power[N[(D$95$m * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.46e-61

   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 76.7%

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

   5. Taylor expanded in d around inf 6.5%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 54.1%

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

      4. unpow-1 54.1%

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

      5. metadata-eval 54.1%

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

      6. pow-sqr 54.1%

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

      7. rem-sqrt-square 54.1%

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

      8. rem-square-sqrt 53.7%

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

      9. fabs-sqr 53.7%

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

      10. rem-square-sqrt 54.1%

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

      11. mul-1-neg 54.1%

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

   8. Simplified 54.1%

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

   9. Taylor expanded in d around 0 54.1%

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

       1. mul-1-neg 54.1%

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

       2. distribute-rgt-neg-in 54.1%

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

       3. unpow1/2 54.1%

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

       4. rem-exp-log 51.1%

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

       5. exp-neg 51.1%

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

       6. exp-prod 51.1%

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

       7. distribute-lft-neg-out 51.1%

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

       8. rec-exp 51.1%

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

       9. exp-to-pow 54.1%

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

       10. unpow1/2 54.1%

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

       11. unpow-1 54.1%

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

       12. metadata-eval 54.1%

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

       13. pow-sqr 53.7%

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

       14. pow-sqr 54.1%

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

       15. metadata-eval 54.1%

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

       16. unpow-1 54.1%

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

       17. distribute-frac-neg2 54.1%

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

   11. Simplified 54.2%

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

   if -1.46e-61 < d < -4.999999999999985e-310

   1. Initial program 52.3%

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

   3. Simplified 52.2%

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

   5. Taylor expanded in M around inf 22.3%

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

      1. associate-*r* 24.3%

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

      2. times-frac 20.5%

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

      3. *-commutative 20.5%

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

      4. associate-/l* 20.5%

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

      5. unpow2 20.5%

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

      6. unpow2 20.5%

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

      7. unpow2 20.5%

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

      8. times-frac 28.8%

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

      9. swap-sqr 34.9%

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

      10. unpow2 34.9%

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

   7. Simplified 34.9%

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

      1. pow1 34.9%

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

      2. associate-*r* 34.9%

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

      3. pow1/2 34.9%

         \[\leadsto {\left(\left(\color{blue}{{\left(\frac{d}{h}\right)}^{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

      4. pow1/2 34.9%

         \[\leadsto {\left(\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}}\right) \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

      5. pow-prod-down 23.2%

         \[\leadsto {\left(\color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

   9. Applied egg-rr 23.2%

      \[\leadsto \color{blue}{{\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 23.2%

          \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

       2. unpow1/2 23.2%

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

   11. Simplified 23.2%

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

       1. add-cube-cbrt 23.1%

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

       2. pow3 23.1%

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

       3. frac-times 15.4%

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

       4. sqrt-div 19.3%

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

       5. sqrt-unprod 0.0%

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

       6. add-sqr-sqrt 3.6%

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

   13. Applied egg-rr 3.6%

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

       1. rem-cube-cbrt 3.6%

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

       2. div-inv 3.6%

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

       3. pow1/2 3.6%

          \[\leadsto \left(d \cdot \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}}\right) \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       4. pow-flip 3.6%

          \[\leadsto \left(d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-0.5\right)}}\right) \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       5. metadata-eval 3.6%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       6. add-sqr-sqrt 2.0%

          \[\leadsto \color{blue}{\left(\sqrt{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{d \cdot {\left(h \cdot \ell\right)}^{-0.5}}\right)} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       7. sqrt-unprod 25.2%

          \[\leadsto \color{blue}{\sqrt{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)}} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       8. sqr-neg 25.2%

          \[\leadsto \sqrt{\color{blue}{\left(-d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(-d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)}} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       9. distribute-rgt-neg-out 25.2%

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \cdot \left(-d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       10. distribute-rgt-neg-out 25.2%

           \[\leadsto \sqrt{\left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right) \cdot \color{blue}{\left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right)}} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       11. sqrt-unprod 47.2%

           \[\leadsto \color{blue}{\left(\sqrt{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \cdot \sqrt{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)}\right)} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       12. add-sqr-sqrt 47.2%

           \[\leadsto \color{blue}{\left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       13. distribute-rgt-neg-out 47.2%

           \[\leadsto \color{blue}{\left(-d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       14. neg-mul-1 47.2%

           \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       15. metadata-eval 47.2%

           \[\leadsto \left(-1 \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{\left(-0.5\right)}}\right)\right) \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       16. pow-flip 47.2%

           \[\leadsto \left(-1 \cdot \left(d \cdot \color{blue}{\frac{1}{{\left(h \cdot \ell\right)}^{0.5}}}\right)\right) \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       17. pow1/2 47.2%

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

       18. div-inv 47.2%

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

   15. Applied egg-rr 47.2%

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

   if -4.999999999999985e-310 < d < 1.85000000000000003e-139

   1. Initial program 40.5%

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

   3. Simplified 42.0%

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

   5. Taylor expanded in M around inf 20.1%

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

      1. associate-*r* 20.1%

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

      2. times-frac 20.1%

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

      3. *-commutative 20.1%

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

      4. associate-/l* 19.5%

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

      5. unpow2 19.5%

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

      6. unpow2 19.5%

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

      7. unpow2 19.5%

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

      8. times-frac 26.6%

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

      9. swap-sqr 31.8%

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

      10. unpow2 31.8%

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

   7. Simplified 31.8%

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

      1. pow1 31.8%

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

      2. associate-*r* 31.8%

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

      3. pow1/2 31.8%

         \[\leadsto {\left(\left(\color{blue}{{\left(\frac{d}{h}\right)}^{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

      4. pow1/2 31.8%

         \[\leadsto {\left(\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}}\right) \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

      5. pow-prod-down 24.7%

         \[\leadsto {\left(\color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

   9. Applied egg-rr 24.7%

      \[\leadsto \color{blue}{{\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 24.7%

          \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

       2. unpow1/2 24.7%

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

   11. Simplified 24.7%

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

       1. sqrt-prod 31.8%

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

       2. frac-2neg 31.8%

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

       3. sqrt-undiv 0.0%

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

       4. associate-*r* 0.0%

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

       5. sqrt-div 0.0%

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

       6. associate-*l/ 0.0%

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

       7. frac-times 0.0%

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

   13. Applied egg-rr 54.9%

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

       1. associate-*r* 54.9%

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

       2. rem-square-sqrt 54.8%

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

       3. associate-/l* 54.8%

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

       4. *-commutative 54.8%

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

       5. associate-*r/ 54.8%

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

       6. *-commutative 54.8%

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

       7. associate-*r/ 54.7%

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

       8. associate-*l* 54.7%

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

   15. Simplified 54.7%

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

   if 1.85000000000000003e-139 < d

   1. Initial program 71.5%

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

   3. Simplified 72.4%

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

   5. Taylor expanded in d around inf 60.9%

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

      1. associate-/r* 60.9%

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

      2. sqrt-div 72.6%

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

   7. Applied egg-rr 72.6%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.46 \cdot 10^{-61}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{-\ell}\right)\right)\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{-139}:\\ \;\;\;\;d \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.125\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -6.6 \cdot 10^{-131}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}^{-0.5}\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{-137}:\\ \;\;\;\;d \cdot \frac{{\left(D\_m \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.125\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= d -6.6e-131)
   (* (sqrt (/ d l)) (sqrt (/ d h)))
   (if (<= d -5e-310)
     (* d (pow (+ -1.0 (fma h l 1.0)) -0.5))
     (if (<= d 1.5e-137)
       (*
        d
        (/ (* (pow (* D_m (/ M d)) 2.0) (* (/ h l) -0.125)) (sqrt (* l h))))
       (* d (/ (sqrt (/ 1.0 h)) (sqrt l)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= -6.6e-131) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (d <= -5e-310) {
		tmp = d * pow((-1.0 + fma(h, l, 1.0)), -0.5);
	} else if (d <= 1.5e-137) {
		tmp = d * ((pow((D_m * (M / d)), 2.0) * ((h / l) * -0.125)) / sqrt((l * h)));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (d <= -6.6e-131)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (d <= -5e-310)
		tmp = Float64(d * (Float64(-1.0 + fma(h, l, 1.0)) ^ -0.5));
	elseif (d <= 1.5e-137)
		tmp = Float64(d * Float64(Float64((Float64(D_m * Float64(M / d)) ^ 2.0) * Float64(Float64(h / l) * -0.125)) / sqrt(Float64(l * h))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[d, -6.6e-131], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(d * N[Power[N[(-1.0 + N[(h * l + 1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.5e-137], N[(d * N[(N[(N[Power[N[(D$95$m * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -6.6000000000000004e-131

   1. Initial program 78.7%

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

   3. Simplified 76.4%

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

   5. Taylor expanded in d around inf 51.3%

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

   if -6.6000000000000004e-131 < d < -4.999999999999985e-310

   1. Initial program 45.2%

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

   3. Simplified 45.1%

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

   5. Taylor expanded in d around inf 8.6%

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

      1. pow1 8.6%

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

      2. inv-pow 8.6%

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

      3. sqrt-pow1 8.6%

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

      4. metadata-eval 8.6%

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

   7. Applied egg-rr 8.6%

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

      1. unpow1 8.6%

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

   9. Simplified 8.6%

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

       1. expm1-log1p-u 8.6%

          \[\leadsto d \cdot {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(h \cdot \ell\right)\right)\right)}}^{-0.5} \]

       2. expm1-undefine 31.3%

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

   11. Applied egg-rr 31.3%

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

       1. sub-neg 31.3%

          \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \left(-1\right)\right)}}^{-0.5} \]

       2. metadata-eval 31.3%

          \[\leadsto d \cdot {\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \color{blue}{-1}\right)}^{-0.5} \]

       3. +-commutative 31.3%

          \[\leadsto d \cdot {\color{blue}{\left(-1 + e^{\mathsf{log1p}\left(h \cdot \ell\right)}\right)}}^{-0.5} \]

       4. log1p-undefine 31.3%

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

       5. rem-exp-log 31.3%

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

       6. +-commutative 31.3%

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

       7. fma-define 31.3%

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

   13. Simplified 31.3%

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

   if -4.999999999999985e-310 < d < 1.4999999999999999e-137

   1. Initial program 40.5%

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

   3. Simplified 42.0%

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

   5. Taylor expanded in M around inf 20.1%

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

      1. associate-*r* 20.1%

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

      2. times-frac 20.1%

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

      3. *-commutative 20.1%

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

      4. associate-/l* 19.5%

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

      5. unpow2 19.5%

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

      6. unpow2 19.5%

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

      7. unpow2 19.5%

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

      8. times-frac 26.6%

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

      9. swap-sqr 31.8%

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

      10. unpow2 31.8%

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

   7. Simplified 31.8%

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

      1. pow1 31.8%

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

      2. associate-*r* 31.8%

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

      3. pow1/2 31.8%

         \[\leadsto {\left(\left(\color{blue}{{\left(\frac{d}{h}\right)}^{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

      4. pow1/2 31.8%

         \[\leadsto {\left(\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}}\right) \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

      5. pow-prod-down 24.7%

         \[\leadsto {\left(\color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

   9. Applied egg-rr 24.7%

      \[\leadsto \color{blue}{{\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 24.7%

          \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

       2. unpow1/2 24.7%

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

   11. Simplified 24.7%

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

       1. sqrt-prod 31.8%

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

       2. frac-2neg 31.8%

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

       3. sqrt-undiv 0.0%

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

       4. associate-*r* 0.0%

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

       5. sqrt-div 0.0%

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

       6. associate-*l/ 0.0%

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

       7. frac-times 0.0%

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

   13. Applied egg-rr 54.9%

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

       1. associate-*r* 54.9%

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

       2. rem-square-sqrt 54.8%

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

       3. associate-/l* 54.8%

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

       4. *-commutative 54.8%

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

       5. associate-*r/ 54.8%

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

       6. *-commutative 54.8%

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

       7. associate-*r/ 54.7%

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

       8. associate-*l* 54.7%

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

   15. Simplified 54.7%

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

   if 1.4999999999999999e-137 < d

   1. Initial program 71.5%

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

   3. Simplified 72.4%

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

   5. Taylor expanded in d around inf 60.9%

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

      1. associate-/r* 60.9%

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

      2. sqrt-div 72.6%

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

   7. Applied egg-rr 72.6%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.6 \cdot 10^{-131}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}^{-0.5}\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{-137}:\\ \;\;\;\;d \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.125\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{-130}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 1.46 \cdot 10^{-307}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}^{-0.5}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-132}:\\ \;\;\;\;-0.125 \cdot \left(\frac{h}{\ell} \cdot \left({\left(D\_m \cdot \frac{M}{d}\right)}^{2} \cdot \frac{d}{\sqrt{\ell \cdot h}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= d -7.2e-130)
   (* (sqrt (/ d l)) (sqrt (/ d h)))
   (if (<= d 1.46e-307)
     (* d (pow (+ -1.0 (fma h l 1.0)) -0.5))
     (if (<= d 2.7e-132)
       (*
        -0.125
        (* (/ h l) (* (pow (* D_m (/ M d)) 2.0) (/ d (sqrt (* l h))))))
       (* d (/ (sqrt (/ 1.0 h)) (sqrt l)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= -7.2e-130) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (d <= 1.46e-307) {
		tmp = d * pow((-1.0 + fma(h, l, 1.0)), -0.5);
	} else if (d <= 2.7e-132) {
		tmp = -0.125 * ((h / l) * (pow((D_m * (M / d)), 2.0) * (d / sqrt((l * h)))));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (d <= -7.2e-130)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (d <= 1.46e-307)
		tmp = Float64(d * (Float64(-1.0 + fma(h, l, 1.0)) ^ -0.5));
	elseif (d <= 2.7e-132)
		tmp = Float64(-0.125 * Float64(Float64(h / l) * Float64((Float64(D_m * Float64(M / d)) ^ 2.0) * Float64(d / sqrt(Float64(l * h))))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[d, -7.2e-130], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.46e-307], N[(d * N[Power[N[(-1.0 + N[(h * l + 1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.7e-132], N[(-0.125 * N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D$95$m * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -7.2000000000000003e-130

   1. Initial program 78.7%

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

   3. Simplified 76.4%

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

   5. Taylor expanded in d around inf 51.3%

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

   if -7.2000000000000003e-130 < d < 1.4600000000000001e-307

   1. Initial program 45.2%

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

   3. Simplified 45.1%

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

   5. Taylor expanded in d around inf 8.6%

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

      1. pow1 8.6%

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

      2. inv-pow 8.6%

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

      3. sqrt-pow1 8.6%

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

      4. metadata-eval 8.6%

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

   7. Applied egg-rr 8.6%

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

      1. unpow1 8.6%

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

   9. Simplified 8.6%

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

       1. expm1-log1p-u 8.6%

          \[\leadsto d \cdot {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(h \cdot \ell\right)\right)\right)}}^{-0.5} \]

       2. expm1-undefine 31.3%

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

   11. Applied egg-rr 31.3%

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

       1. sub-neg 31.3%

          \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \left(-1\right)\right)}}^{-0.5} \]

       2. metadata-eval 31.3%

          \[\leadsto d \cdot {\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \color{blue}{-1}\right)}^{-0.5} \]

       3. +-commutative 31.3%

          \[\leadsto d \cdot {\color{blue}{\left(-1 + e^{\mathsf{log1p}\left(h \cdot \ell\right)}\right)}}^{-0.5} \]

       4. log1p-undefine 31.3%

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

       5. rem-exp-log 31.3%

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

       6. +-commutative 31.3%

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

       7. fma-define 31.3%

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

   13. Simplified 31.3%

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

   if 1.4600000000000001e-307 < d < 2.6999999999999999e-132

   1. Initial program 40.5%

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

   3. Simplified 42.0%

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

   5. Taylor expanded in M around inf 20.1%

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

      1. associate-*r* 20.1%

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

      2. times-frac 20.1%

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

      3. *-commutative 20.1%

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

      4. associate-/l* 19.5%

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

      5. unpow2 19.5%

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

      6. unpow2 19.5%

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

      7. unpow2 19.5%

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

      8. times-frac 26.6%

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

      9. swap-sqr 31.8%

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

      10. unpow2 31.8%

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

   7. Simplified 31.8%

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

      1. pow1 31.8%

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

      2. associate-*r* 31.8%

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

      3. pow1/2 31.8%

         \[\leadsto {\left(\left(\color{blue}{{\left(\frac{d}{h}\right)}^{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

      4. pow1/2 31.8%

         \[\leadsto {\left(\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}}\right) \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

      5. pow-prod-down 24.7%

         \[\leadsto {\left(\color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

   9. Applied egg-rr 24.7%

      \[\leadsto \color{blue}{{\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 24.7%

          \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

       2. unpow1/2 24.7%

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

   11. Simplified 24.7%

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

       1. add-cube-cbrt 24.7%

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

       2. pow3 24.7%

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

       3. frac-times 3.6%

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

       4. sqrt-div 10.0%

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

       5. sqrt-unprod 46.3%

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

       6. add-sqr-sqrt 46.3%

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

   13. Applied egg-rr 46.3%

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

       1. rem-cube-cbrt 46.3%

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

       2. *-commutative 46.3%

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

       3. clear-num 46.3%

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

       4. un-div-inv 46.3%

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

   15. Applied egg-rr 46.3%

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

       1. associate-/r/ 54.8%

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

       2. associate-*l/ 54.8%

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

       3. associate-*r/ 46.3%

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

       4. associate-*l* 46.3%

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

       5. *-commutative 46.3%

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

       6. associate-*l* 48.3%

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

       7. associate-*r/ 48.3%

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

       8. *-commutative 48.3%

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

       9. associate-/l* 48.1%

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

   17. Simplified 48.1%

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

   if 2.6999999999999999e-132 < d

   1. Initial program 71.5%

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

   3. Simplified 72.4%

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

   5. Taylor expanded in d around inf 60.9%

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

      1. associate-/r* 60.9%

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

      2. sqrt-div 72.6%

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

   7. Applied egg-rr 72.6%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{-130}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 1.46 \cdot 10^{-307}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}^{-0.5}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-132}:\\ \;\;\;\;-0.125 \cdot \left(\frac{h}{\ell} \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{d}{\sqrt{\ell \cdot h}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -6 \cdot 10^{-128}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= d -6e-128)
   (* (sqrt (/ d l)) (sqrt (/ d h)))
   (if (<= d -5e-310)
     (* d (pow (+ -1.0 (fma h l 1.0)) -0.5))
     (* d (/ (sqrt (/ 1.0 h)) (sqrt l))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= -6e-128) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (d <= -5e-310) {
		tmp = d * pow((-1.0 + fma(h, l, 1.0)), -0.5);
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (d <= -6e-128)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (d <= -5e-310)
		tmp = Float64(d * (Float64(-1.0 + fma(h, l, 1.0)) ^ -0.5));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[d, -6e-128], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(d * N[Power[N[(-1.0 + N[(h * l + 1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.99999999999999956e-128

   1. Initial program 78.7%

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

   3. Simplified 76.4%

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

   5. Taylor expanded in d around inf 51.3%

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

   if -5.99999999999999956e-128 < d < -4.999999999999985e-310

   1. Initial program 45.2%

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

   3. Simplified 45.1%

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

   5. Taylor expanded in d around inf 8.6%

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

      1. pow1 8.6%

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

      2. inv-pow 8.6%

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

      3. sqrt-pow1 8.6%

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

      4. metadata-eval 8.6%

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

   7. Applied egg-rr 8.6%

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

      1. unpow1 8.6%

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

   9. Simplified 8.6%

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

       1. expm1-log1p-u 8.6%

          \[\leadsto d \cdot {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(h \cdot \ell\right)\right)\right)}}^{-0.5} \]

       2. expm1-undefine 31.3%

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

   11. Applied egg-rr 31.3%

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

       1. sub-neg 31.3%

          \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \left(-1\right)\right)}}^{-0.5} \]

       2. metadata-eval 31.3%

          \[\leadsto d \cdot {\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \color{blue}{-1}\right)}^{-0.5} \]

       3. +-commutative 31.3%

          \[\leadsto d \cdot {\color{blue}{\left(-1 + e^{\mathsf{log1p}\left(h \cdot \ell\right)}\right)}}^{-0.5} \]

       4. log1p-undefine 31.3%

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

       5. rem-exp-log 31.3%

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

       6. +-commutative 31.3%

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

       7. fma-define 31.3%

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

   13. Simplified 31.3%

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

   if -4.999999999999985e-310 < d

   1. Initial program 60.7%

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

   3. Simplified 61.8%

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

   5. Taylor expanded in d around inf 48.3%

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

      1. associate-/r* 48.2%

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

      2. sqrt-div 56.1%

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

   7. Applied egg-rr 56.1%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6 \cdot 10^{-128}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.8 \cdot 10^{-300}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= l 2.8e-300)
   (/ d (- (sqrt (* l h))))
   (* d (/ (sqrt (/ 1.0 h)) (sqrt l)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= 2.8e-300) {
		tmp = d / -sqrt((l * h));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: tmp
    if (l <= 2.8d-300) then
        tmp = d / -sqrt((l * h))
    else
        tmp = d * (sqrt((1.0d0 / h)) / sqrt(l))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= 2.8e-300) {
		tmp = d / -Math.sqrt((l * h));
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if l <= 2.8e-300:
		tmp = d / -math.sqrt((l * h))
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (l <= 2.8e-300)
		tmp = Float64(d / Float64(-sqrt(Float64(l * h))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (l <= 2.8e-300)
		tmp = d / -sqrt((l * h));
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[l, 2.8e-300], N[(d / (-N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.79999999999999994e-300

   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 67.1%

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

   5. Taylor expanded in d around inf 6.6%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 40.3%

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

      4. unpow-1 40.3%

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

      5. metadata-eval 40.3%

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

      6. pow-sqr 40.3%

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

      7. rem-sqrt-square 40.3%

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

      8. rem-square-sqrt 40.0%

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

      9. fabs-sqr 40.0%

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

      10. rem-square-sqrt 40.3%

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

      11. mul-1-neg 40.3%

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

   8. Simplified 40.3%

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

   9. Taylor expanded in d around 0 40.3%

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

       1. mul-1-neg 40.3%

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

       2. distribute-rgt-neg-in 40.3%

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

       3. unpow1/2 40.3%

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

       4. rem-exp-log 38.1%

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

       5. exp-neg 38.1%

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

       6. exp-prod 38.1%

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

       7. distribute-lft-neg-out 38.1%

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

       8. rec-exp 38.1%

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

       9. exp-to-pow 40.3%

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

       10. unpow1/2 40.3%

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

       11. unpow-1 40.3%

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

       12. metadata-eval 40.3%

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

       13. pow-sqr 40.0%

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

       14. pow-sqr 40.3%

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

       15. metadata-eval 40.3%

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

       16. unpow-1 40.3%

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

       17. distribute-frac-neg2 40.3%

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

   11. Simplified 40.3%

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

   if 2.79999999999999994e-300 < l

   1. Initial program 60.1%

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

   3. Simplified 61.3%

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

   5. Taylor expanded in d around inf 49.0%

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

      1. associate-/r* 48.9%

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

      2. sqrt-div 56.9%

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

   7. Applied egg-rr 56.9%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.8 \cdot 10^{-300}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 43.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -9.5 \cdot 10^{-294}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= l -9.5e-294) (/ d (- (sqrt (* l h)))) (* d (pow (* l h) -0.5))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= -9.5e-294) {
		tmp = d / -sqrt((l * h));
	} else {
		tmp = d * pow((l * h), -0.5);
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: tmp
    if (l <= (-9.5d-294)) then
        tmp = d / -sqrt((l * h))
    else
        tmp = d * ((l * h) ** (-0.5d0))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= -9.5e-294) {
		tmp = d / -Math.sqrt((l * h));
	} else {
		tmp = d * Math.pow((l * h), -0.5);
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if l <= -9.5e-294:
		tmp = d / -math.sqrt((l * h))
	else:
		tmp = d * math.pow((l * h), -0.5)
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (l <= -9.5e-294)
		tmp = Float64(d / Float64(-sqrt(Float64(l * h))));
	else
		tmp = Float64(d * (Float64(l * h) ^ -0.5));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (l <= -9.5e-294)
		tmp = d / -sqrt((l * h));
	else
		tmp = d * ((l * h) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[l, -9.5e-294], N[(d / (-N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -9.499999999999999e-294

   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 66.6%

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

   5. Taylor expanded in d around inf 4.3%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 41.1%

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

      4. unpow-1 41.1%

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

      5. metadata-eval 41.1%

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

      6. pow-sqr 41.1%

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

      7. rem-sqrt-square 41.1%

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

      8. rem-square-sqrt 40.9%

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

      9. fabs-sqr 40.9%

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

      10. rem-square-sqrt 41.1%

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

      11. mul-1-neg 41.1%

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

   8. Simplified 41.1%

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

   9. Taylor expanded in d around 0 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. distribute-rgt-neg-in 41.1%

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

       3. unpow1/2 41.1%

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

       4. rem-exp-log 38.9%

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

       5. exp-neg 38.9%

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

       6. exp-prod 38.9%

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

       7. distribute-lft-neg-out 38.9%

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

       8. rec-exp 38.9%

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

       9. exp-to-pow 41.1%

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

       10. unpow1/2 41.1%

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

       11. unpow-1 41.1%

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

       12. metadata-eval 41.1%

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

       13. pow-sqr 40.9%

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

       14. pow-sqr 41.1%

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

       15. metadata-eval 41.1%

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

       16. unpow-1 41.1%

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

       17. distribute-frac-neg2 41.1%

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

   11. Simplified 41.2%

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

   if -9.499999999999999e-294 < l

   1. Initial program 61.5%

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

   3. Simplified 61.9%

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

   5. Taylor expanded in d around inf 49.4%

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

      1. pow1 49.4%

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

      2. inv-pow 49.4%

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

      3. sqrt-pow1 49.4%

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

      4. metadata-eval 49.4%

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

   7. Applied egg-rr 49.4%

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

      1. unpow1 49.4%

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

   9. Simplified 49.4%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9.5 \cdot 10^{-294}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 25.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ d \cdot {\left(\ell \cdot h\right)}^{-0.5} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m) :precision binary64 (* d (pow (* l h) -0.5)))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	return d * pow((l * h), -0.5);
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    code = d * ((l * h) ** (-0.5d0))
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	return d * Math.pow((l * h), -0.5);
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	return d * math.pow((l * h), -0.5)

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	return Float64(d * (Float64(l * h) ^ -0.5))
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp = code(d, h, l, M, D_m)
	tmp = d * ((l * h) ^ -0.5);
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.2%

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

3. Simplified 64.1%

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

5. Taylor expanded in d around inf 28.6%

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

   1. pow1 28.6%

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

   2. inv-pow 28.6%

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

   3. sqrt-pow1 28.6%

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

   4. metadata-eval 28.6%

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

7. Applied egg-rr 28.6%

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

   1. unpow1 28.6%

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

9. Simplified 28.6%

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

10. Final simplification 28.6%

    \[\leadsto d \cdot {\left(\ell \cdot h\right)}^{-0.5} \]
11. Add Preprocessing

Alternative 15: 25.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m) :precision binary64 (/ d (sqrt (* l h))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	return d / sqrt((l * h));
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    code = d / sqrt((l * h))
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	return d / Math.sqrt((l * h));
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	return d / math.sqrt((l * h))

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp = code(d, h, l, M, D_m)
	tmp = d / sqrt((l * h));
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.2%

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

3. Simplified 64.1%

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

5. Taylor expanded in d around inf 28.6%

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

6. Taylor expanded in h around -inf 0.0%

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

   1. associate-*l* 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 21.7%

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

   4. unpow-1 21.7%

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

   5. metadata-eval 21.7%

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

   6. pow-sqr 21.7%

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

   7. rem-sqrt-square 21.7%

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

   8. rem-square-sqrt 21.6%

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

   9. fabs-sqr 21.6%

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

   10. rem-square-sqrt 21.7%

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

   11. mul-1-neg 21.7%

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

8. Simplified 21.7%

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

9. Taylor expanded in d around 0 21.7%

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

    1. mul-1-neg 21.7%

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

    2. distribute-lft-neg-in 21.7%

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

    3. unpow1/2 21.7%

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

    4. rem-exp-log 20.7%

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

    5. exp-neg 20.7%

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

    6. exp-prod 20.7%

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

    7. distribute-lft-neg-out 20.7%

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

    8. rec-exp 20.7%

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

    9. exp-to-pow 21.7%

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

    10. metadata-eval 21.7%

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

    11. pow-sqr 21.7%

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

    12. fabs-sqr 21.7%

        \[\leadsto \left(-d\right) \cdot \frac{1}{\color{blue}{\left|{\left(h \cdot \ell\right)}^{0.25} \cdot {\left(h \cdot \ell\right)}^{0.25}\right|}} \]

    13. pow-sqr 21.7%

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

    14. metadata-eval 21.7%

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

    15. unpow1/2 21.7%

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

    16. fabs-neg 21.7%

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

    17. neg-mul-1 21.7%

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

    18. rem-square-sqrt 0.0%

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

    19. unpow1/2 0.0%

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

    20. metadata-eval 0.0%

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

    21. pow-sqr 0.0%

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

    22. unswap-sqr 0.0%

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

11. Simplified 28.6%

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

12. Final simplification 28.6%

    \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024165 
(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)))))