Henrywood and Agarwal, Equation (12)

Percentage Accurate: 67.0% → 83.7%

Time: 1.1min

Alternatives: 30

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.0% 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: 83.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := 1 - h \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D\_m}{d}\right)}^{2}}{\ell}\right)\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;h \leq -1.2 \cdot 10^{+65}:\\ \;\;\;\;\frac{t\_1}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot t\_0\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{-\ell}}{t\_1}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(h \cdot -0.5\right) \cdot {\left(D\_m \cdot \frac{M}{d \cdot 2}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t\_0 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \end{array} \]

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 (- 1.0 (* h (* 0.125 (/ (pow (* M (/ D_m d)) 2.0) l)))))
        (t_1 (sqrt (- d))))
   (if (<= h -1.2e+65)
     (* (/ t_1 (sqrt (- h))) (* (sqrt (/ d l)) t_0))
     (if (<= h -5e-310)
       (*
        (/ 1.0 (/ (sqrt (- l)) t_1))
        (*
         (sqrt (/ d h))
         (+ 1.0 (/ (* (* h -0.5) (pow (* D_m (/ M (* d 2.0))) 2.0)) l))))
       (* (/ (sqrt d) (sqrt h)) (* t_0 (/ (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 = 1.0 - (h * (0.125 * (pow((M * (D_m / d)), 2.0) / l)));
	double t_1 = sqrt(-d);
	double tmp;
	if (h <= -1.2e+65) {
		tmp = (t_1 / sqrt(-h)) * (sqrt((d / l)) * t_0);
	} else if (h <= -5e-310) {
		tmp = (1.0 / (sqrt(-l) / t_1)) * (sqrt((d / h)) * (1.0 + (((h * -0.5) * pow((D_m * (M / (d * 2.0))), 2.0)) / l)));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (t_0 * (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 = 1.0d0 - (h * (0.125d0 * (((m * (d_m / d)) ** 2.0d0) / l)))
    t_1 = sqrt(-d)
    if (h <= (-1.2d+65)) then
        tmp = (t_1 / sqrt(-h)) * (sqrt((d / l)) * t_0)
    else if (h <= (-5d-310)) then
        tmp = (1.0d0 / (sqrt(-l) / t_1)) * (sqrt((d / h)) * (1.0d0 + (((h * (-0.5d0)) * ((d_m * (m / (d * 2.0d0))) ** 2.0d0)) / l)))
    else
        tmp = (sqrt(d) / sqrt(h)) * (t_0 * (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 = 1.0 - (h * (0.125 * (Math.pow((M * (D_m / d)), 2.0) / l)));
	double t_1 = Math.sqrt(-d);
	double tmp;
	if (h <= -1.2e+65) {
		tmp = (t_1 / Math.sqrt(-h)) * (Math.sqrt((d / l)) * t_0);
	} else if (h <= -5e-310) {
		tmp = (1.0 / (Math.sqrt(-l) / t_1)) * (Math.sqrt((d / h)) * (1.0 + (((h * -0.5) * Math.pow((D_m * (M / (d * 2.0))), 2.0)) / l)));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * (t_0 * (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 = 1.0 - (h * (0.125 * (math.pow((M * (D_m / d)), 2.0) / l)))
	t_1 = math.sqrt(-d)
	tmp = 0
	if h <= -1.2e+65:
		tmp = (t_1 / math.sqrt(-h)) * (math.sqrt((d / l)) * t_0)
	elif h <= -5e-310:
		tmp = (1.0 / (math.sqrt(-l) / t_1)) * (math.sqrt((d / h)) * (1.0 + (((h * -0.5) * math.pow((D_m * (M / (d * 2.0))), 2.0)) / l)))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * (t_0 * (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 = 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 (h <= -1.2e+65)
		tmp = Float64(Float64(t_1 / sqrt(Float64(-h))) * Float64(sqrt(Float64(d / l)) * t_0));
	elseif (h <= -5e-310)
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(-l)) / t_1)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(Float64(h * -0.5) * (Float64(D_m * Float64(M / Float64(d * 2.0))) ^ 2.0)) / l))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(t_0 * 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 = 1.0 - (h * (0.125 * (((M * (D_m / d)) ^ 2.0) / l)));
	t_1 = sqrt(-d);
	tmp = 0.0;
	if (h <= -1.2e+65)
		tmp = (t_1 / sqrt(-h)) * (sqrt((d / l)) * t_0);
	elseif (h <= -5e-310)
		tmp = (1.0 / (sqrt(-l) / t_1)) * (sqrt((d / h)) * (1.0 + (((h * -0.5) * ((D_m * (M / (d * 2.0))) ^ 2.0)) / l)));
	else
		tmp = (sqrt(d) / sqrt(h)) * (t_0 * (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[(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[h, -1.2e+65], 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[h, -5e-310], N[(N[(1.0 / N[(N[Sqrt[(-l)], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(N[(h * -0.5), $MachinePrecision] * N[Power[N[(D$95$m * N[(M / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.2000000000000001e65

   1. Initial program 55.0%

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

   3. Simplified 55.1%

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

   5. Taylor expanded in h around -inf 31.7%

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

      1. associate-*r* 31.7%

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

      2. neg-mul-1 31.7%

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

      3. sub-neg 31.7%

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

      4. distribute-lft-in 31.7%

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

   7. Simplified 60.6%

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

      1. frac-2neg 60.6%

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

      2. sqrt-div 81.6%

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

   9. Applied egg-rr 81.6%

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

   if -1.2000000000000001e65 < h < -4.999999999999985e-310

   1. Initial program 73.0%

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

   3. Simplified 71.6%

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

      1. associate-*l/ 71.7%

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

      2. *-commutative 71.7%

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

      3. associate-/l/ 71.7%

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

   6. Applied egg-rr 71.7%

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

      1. associate-*r* 71.7%

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

      2. *-commutative 71.7%

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

   8. Simplified 71.7%

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

      1. clear-num 71.7%

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

      2. sqrt-div 72.9%

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

      3. metadata-eval 72.9%

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

   10. Applied egg-rr 72.9%

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

       1. frac-2neg 72.9%

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

       2. sqrt-div 84.4%

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

   12. Applied egg-rr 84.4%

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

   if -4.999999999999985e-310 < h

   1. Initial program 72.6%

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

   3. Simplified 71.0%

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

   5. Taylor expanded in h around -inf 48.6%

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

      1. associate-*r* 48.6%

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

      2. neg-mul-1 48.6%

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

      3. sub-neg 48.6%

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

      4. distribute-lft-in 48.6%

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

   7. Simplified 73.5%

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

      1. sqrt-div 82.0%

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

   9. Applied egg-rr 82.0%

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

       1. sqrt-div 87.8%

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

   11. Applied egg-rr 87.8%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.2 \cdot 10^{+65}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{-\ell}}{\sqrt{-d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(h \cdot -0.5\right) \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\left(1 - h \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := 1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D\_m}{d \cdot 2}\right)}^{2}\right)\\ t_1 := {\left(\frac{d}{h}\right)}^{0.5}\\ \mathbf{if}\;\left(t\_1 \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot t\_0 \leq 5 \cdot 10^{+266}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot {\left(d \cdot \frac{1}{\ell}\right)}^{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \end{array} \]

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 (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* M D_m) (* d 2.0)) 2.0)))))
        (t_1 (pow (/ d h) 0.5)))
   (if (<= (* (* t_1 (pow (/ d l) 0.5)) t_0) 5e+266)
     (* t_0 (* t_1 (pow (* d (/ 1.0 l)) 0.5)))
     (fabs (/ d (sqrt (* h 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 = 1.0 - ((h / l) * (0.5 * pow(((M * D_m) / (d * 2.0)), 2.0)));
	double t_1 = pow((d / h), 0.5);
	double tmp;
	if (((t_1 * pow((d / l), 0.5)) * t_0) <= 5e+266) {
		tmp = t_0 * (t_1 * pow((d * (1.0 / l)), 0.5));
	} else {
		tmp = fabs((d / sqrt((h * 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 = 1.0d0 - ((h / l) * (0.5d0 * (((m * d_m) / (d * 2.0d0)) ** 2.0d0)))
    t_1 = (d / h) ** 0.5d0
    if (((t_1 * ((d / l) ** 0.5d0)) * t_0) <= 5d+266) then
        tmp = t_0 * (t_1 * ((d * (1.0d0 / l)) ** 0.5d0))
    else
        tmp = abs((d / sqrt((h * 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 = 1.0 - ((h / l) * (0.5 * Math.pow(((M * D_m) / (d * 2.0)), 2.0)));
	double t_1 = Math.pow((d / h), 0.5);
	double tmp;
	if (((t_1 * Math.pow((d / l), 0.5)) * t_0) <= 5e+266) {
		tmp = t_0 * (t_1 * Math.pow((d * (1.0 / l)), 0.5));
	} else {
		tmp = Math.abs((d / Math.sqrt((h * 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 = 1.0 - ((h / l) * (0.5 * math.pow(((M * D_m) / (d * 2.0)), 2.0)))
	t_1 = math.pow((d / h), 0.5)
	tmp = 0
	if ((t_1 * math.pow((d / l), 0.5)) * t_0) <= 5e+266:
		tmp = t_0 * (t_1 * math.pow((d * (1.0 / l)), 0.5))
	else:
		tmp = math.fabs((d / math.sqrt((h * 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(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(M * D_m) / Float64(d * 2.0)) ^ 2.0))))
	t_1 = Float64(d / h) ^ 0.5
	tmp = 0.0
	if (Float64(Float64(t_1 * (Float64(d / l) ^ 0.5)) * t_0) <= 5e+266)
		tmp = Float64(t_0 * Float64(t_1 * (Float64(d * Float64(1.0 / l)) ^ 0.5)));
	else
		tmp = abs(Float64(d / sqrt(Float64(h * 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 = 1.0 - ((h / l) * (0.5 * (((M * D_m) / (d * 2.0)) ^ 2.0)));
	t_1 = (d / h) ^ 0.5;
	tmp = 0.0;
	if (((t_1 * ((d / l) ^ 0.5)) * t_0) <= 5e+266)
		tmp = t_0 * (t_1 * ((d * (1.0 / l)) ^ 0.5));
	else
		tmp = abs((d / sqrt((h * 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[(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]}, Block[{t$95$1 = N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], 5e+266], N[(t$95$0 * N[(t$95$1 * N[Power[N[(d * N[(1.0 / l), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 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)))) < 4.9999999999999999e266

   1. Initial program 87.0%

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

      1. div-inv 87.0%

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

   4. Applied egg-rr 87.0%

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

   if 4.9999999999999999e266 < (*.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 22.4%

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

   3. Simplified 22.4%

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

   5. Taylor expanded in d around inf 32.9%

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

      1. *-un-lft-identity 32.9%

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

      2. pow1/2 32.9%

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

      3. inv-pow 32.9%

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

      4. pow-pow 32.9%

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

      5. metadata-eval 32.9%

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

   7. Applied egg-rr 32.9%

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

      1. *-lft-identity 32.9%

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

   9. Simplified 32.9%

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

       1. metadata-eval 32.9%

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

       2. metadata-eval 32.9%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\left(-1 \cdot \color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}\right)} \]

       3. pow-pow 32.9%

          \[\leadsto d \cdot \color{blue}{{\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       4. inv-pow 32.9%

          \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

       5. pow-pow 28.0%

          \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       6. add-sqr-sqrt 27.6%

          \[\leadsto \color{blue}{\sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}}} \]

       7. sqrt-unprod 33.7%

          \[\leadsto \color{blue}{\sqrt{\left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right) \cdot \left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       8. swap-sqr 32.5%

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot d\right) \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       9. unpow2 32.5%

          \[\leadsto \sqrt{\color{blue}{{d}^{2}} \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       10. pow-pow 32.6%

           \[\leadsto \sqrt{{d}^{2} \cdot \left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       11. pow-pow 34.3%

           \[\leadsto \sqrt{{d}^{2} \cdot \left({\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}\right)} \]

       12. pow-sqr 34.3%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \left(1.5 \cdot 0.3333333333333333\right)\right)}}} \]

       13. metadata-eval 34.3%

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

       14. metadata-eval 34.3%

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

       15. pow1 34.3%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \]

       16. div-inv 34.3%

           \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}} \]

       17. associate-/r* 29.0%

           \[\leadsto \sqrt{\color{blue}{\frac{\frac{{d}^{2}}{h}}{\ell}}} \]

       18. add-sqr-sqrt 29.0%

           \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{\frac{{d}^{2}}{h}}{\ell}} \cdot \sqrt{\frac{\frac{{d}^{2}}{h}}{\ell}}}} \]

   11. Applied egg-rr 54.9%

       \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq 5 \cdot 10^{+266}:\\ \;\;\;\;\left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(d \cdot \frac{1}{\ell}\right)}^{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D\_m}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+266}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \end{array} \]

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
         (*
          (* (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)))))))
   (if (<= t_0 5e+266) t_0 (fabs (/ d (sqrt (* h 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 / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((M * D_m) / (d * 2.0)), 2.0))));
	double tmp;
	if (t_0 <= 5e+266) {
		tmp = t_0;
	} else {
		tmp = fabs((d / sqrt((h * 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 = (((d / h) ** 0.5d0) * ((d / l) ** 0.5d0)) * (1.0d0 - ((h / l) * (0.5d0 * (((m * d_m) / (d * 2.0d0)) ** 2.0d0))))
    if (t_0 <= 5d+266) then
        tmp = t_0
    else
        tmp = abs((d / sqrt((h * 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 / 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 tmp;
	if (t_0 <= 5e+266) {
		tmp = t_0;
	} else {
		tmp = Math.abs((d / Math.sqrt((h * 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 / 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))))
	tmp = 0
	if t_0 <= 5e+266:
		tmp = t_0
	else:
		tmp = math.fabs((d / math.sqrt((h * 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(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)))))
	tmp = 0.0
	if (t_0 <= 5e+266)
		tmp = t_0;
	else
		tmp = abs(Float64(d / sqrt(Float64(h * 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 / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((h / l) * (0.5 * (((M * D_m) / (d * 2.0)) ^ 2.0))));
	tmp = 0.0;
	if (t_0 <= 5e+266)
		tmp = t_0;
	else
		tmp = abs((d / sqrt((h * 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[(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]}, If[LessEqual[t$95$0, 5e+266], t$95$0, N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 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)))) < 4.9999999999999999e266

   1. Initial program 87.0%

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

   if 4.9999999999999999e266 < (*.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 22.4%

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

   3. Simplified 22.4%

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

   5. Taylor expanded in d around inf 32.9%

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

      1. *-un-lft-identity 32.9%

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

      2. pow1/2 32.9%

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

      3. inv-pow 32.9%

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

      4. pow-pow 32.9%

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

      5. metadata-eval 32.9%

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

   7. Applied egg-rr 32.9%

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

      1. *-lft-identity 32.9%

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

   9. Simplified 32.9%

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

       1. metadata-eval 32.9%

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

       2. metadata-eval 32.9%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\left(-1 \cdot \color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}\right)} \]

       3. pow-pow 32.9%

          \[\leadsto d \cdot \color{blue}{{\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       4. inv-pow 32.9%

          \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

       5. pow-pow 28.0%

          \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       6. add-sqr-sqrt 27.6%

          \[\leadsto \color{blue}{\sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}}} \]

       7. sqrt-unprod 33.7%

          \[\leadsto \color{blue}{\sqrt{\left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right) \cdot \left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       8. swap-sqr 32.5%

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot d\right) \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       9. unpow2 32.5%

          \[\leadsto \sqrt{\color{blue}{{d}^{2}} \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       10. pow-pow 32.6%

           \[\leadsto \sqrt{{d}^{2} \cdot \left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       11. pow-pow 34.3%

           \[\leadsto \sqrt{{d}^{2} \cdot \left({\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}\right)} \]

       12. pow-sqr 34.3%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \left(1.5 \cdot 0.3333333333333333\right)\right)}}} \]

       13. metadata-eval 34.3%

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

       14. metadata-eval 34.3%

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

       15. pow1 34.3%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \]

       16. div-inv 34.3%

           \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}} \]

       17. associate-/r* 29.0%

           \[\leadsto \sqrt{\color{blue}{\frac{\frac{{d}^{2}}{h}}{\ell}}} \]

       18. add-sqr-sqrt 29.0%

           \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{\frac{{d}^{2}}{h}}{\ell}} \cdot \sqrt{\frac{\frac{{d}^{2}}{h}}{\ell}}}} \]

   11. Applied egg-rr 54.9%

       \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq 5 \cdot 10^{+266}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{-d}\\ t_2 := \frac{t\_1}{\sqrt{-h}}\\ t_3 := M \cdot \frac{D\_m}{d}\\ \mathbf{if}\;h \leq -4.6 \cdot 10^{+67}:\\ \;\;\;\;t\_2 \cdot \left(t\_0 \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{{t\_3}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;h \leq -1.06 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{t\_1}{\sqrt{-\ell}} \cdot \left(1 + h \cdot \left(-0.125 \cdot \frac{{\left(\frac{M \cdot D\_m}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_0 \cdot \left(t\_2 \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D\_m \cdot \frac{\frac{M}{2}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t\_0 \cdot \left(1 - h \cdot \left(0.125 \cdot \left(t\_3 \cdot \frac{t\_3}{\ell}\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 (sqrt (/ d l)))
        (t_1 (sqrt (- d)))
        (t_2 (/ t_1 (sqrt (- h))))
        (t_3 (* M (/ D_m d))))
   (if (<= h -4.6e+67)
     (* t_2 (* t_0 (- 1.0 (* h (* 0.125 (/ (pow t_3 2.0) l))))))
     (if (<= h -1.06e-162)
       (*
        (sqrt (/ d h))
        (*
         (/ t_1 (sqrt (- l)))
         (+ 1.0 (* h (* -0.125 (/ (pow (/ (* M D_m) d) 2.0) l))))))
       (if (<= h -5e-310)
         (*
          t_0
          (*
           t_2
           (+ 1.0 (* (/ h l) (* -0.5 (pow (* D_m (/ (/ M 2.0) d)) 2.0))))))
         (*
          (/ (sqrt d) (sqrt h))
          (* t_0 (- 1.0 (* h (* 0.125 (* t_3 (/ t_3 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 / l));
	double t_1 = sqrt(-d);
	double t_2 = t_1 / sqrt(-h);
	double t_3 = M * (D_m / d);
	double tmp;
	if (h <= -4.6e+67) {
		tmp = t_2 * (t_0 * (1.0 - (h * (0.125 * (pow(t_3, 2.0) / l)))));
	} else if (h <= -1.06e-162) {
		tmp = sqrt((d / h)) * ((t_1 / sqrt(-l)) * (1.0 + (h * (-0.125 * (pow(((M * D_m) / d), 2.0) / l)))));
	} else if (h <= -5e-310) {
		tmp = t_0 * (t_2 * (1.0 + ((h / l) * (-0.5 * pow((D_m * ((M / 2.0) / d)), 2.0)))));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (t_0 * (1.0 - (h * (0.125 * (t_3 * (t_3 / 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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = sqrt(-d)
    t_2 = t_1 / sqrt(-h)
    t_3 = m * (d_m / d)
    if (h <= (-4.6d+67)) then
        tmp = t_2 * (t_0 * (1.0d0 - (h * (0.125d0 * ((t_3 ** 2.0d0) / l)))))
    else if (h <= (-1.06d-162)) then
        tmp = sqrt((d / h)) * ((t_1 / sqrt(-l)) * (1.0d0 + (h * ((-0.125d0) * ((((m * d_m) / d) ** 2.0d0) / l)))))
    else if (h <= (-5d-310)) then
        tmp = t_0 * (t_2 * (1.0d0 + ((h / l) * ((-0.5d0) * ((d_m * ((m / 2.0d0) / d)) ** 2.0d0)))))
    else
        tmp = (sqrt(d) / sqrt(h)) * (t_0 * (1.0d0 - (h * (0.125d0 * (t_3 * (t_3 / 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 / l));
	double t_1 = Math.sqrt(-d);
	double t_2 = t_1 / Math.sqrt(-h);
	double t_3 = M * (D_m / d);
	double tmp;
	if (h <= -4.6e+67) {
		tmp = t_2 * (t_0 * (1.0 - (h * (0.125 * (Math.pow(t_3, 2.0) / l)))));
	} else if (h <= -1.06e-162) {
		tmp = Math.sqrt((d / h)) * ((t_1 / Math.sqrt(-l)) * (1.0 + (h * (-0.125 * (Math.pow(((M * D_m) / d), 2.0) / l)))));
	} else if (h <= -5e-310) {
		tmp = t_0 * (t_2 * (1.0 + ((h / l) * (-0.5 * Math.pow((D_m * ((M / 2.0) / d)), 2.0)))));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * (t_0 * (1.0 - (h * (0.125 * (t_3 * (t_3 / 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 / l))
	t_1 = math.sqrt(-d)
	t_2 = t_1 / math.sqrt(-h)
	t_3 = M * (D_m / d)
	tmp = 0
	if h <= -4.6e+67:
		tmp = t_2 * (t_0 * (1.0 - (h * (0.125 * (math.pow(t_3, 2.0) / l)))))
	elif h <= -1.06e-162:
		tmp = math.sqrt((d / h)) * ((t_1 / math.sqrt(-l)) * (1.0 + (h * (-0.125 * (math.pow(((M * D_m) / d), 2.0) / l)))))
	elif h <= -5e-310:
		tmp = t_0 * (t_2 * (1.0 + ((h / l) * (-0.5 * math.pow((D_m * ((M / 2.0) / d)), 2.0)))))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * (t_0 * (1.0 - (h * (0.125 * (t_3 * (t_3 / 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 / l))
	t_1 = sqrt(Float64(-d))
	t_2 = Float64(t_1 / sqrt(Float64(-h)))
	t_3 = Float64(M * Float64(D_m / d))
	tmp = 0.0
	if (h <= -4.6e+67)
		tmp = Float64(t_2 * Float64(t_0 * Float64(1.0 - Float64(h * Float64(0.125 * Float64((t_3 ^ 2.0) / l))))));
	elseif (h <= -1.06e-162)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(t_1 / sqrt(Float64(-l))) * Float64(1.0 + Float64(h * Float64(-0.125 * Float64((Float64(Float64(M * D_m) / d) ^ 2.0) / l))))));
	elseif (h <= -5e-310)
		tmp = Float64(t_0 * Float64(t_2 * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(D_m * Float64(Float64(M / 2.0) / d)) ^ 2.0))))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(t_0 * Float64(1.0 - Float64(h * Float64(0.125 * Float64(t_3 * Float64(t_3 / 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 / l));
	t_1 = sqrt(-d);
	t_2 = t_1 / sqrt(-h);
	t_3 = M * (D_m / d);
	tmp = 0.0;
	if (h <= -4.6e+67)
		tmp = t_2 * (t_0 * (1.0 - (h * (0.125 * ((t_3 ^ 2.0) / l)))));
	elseif (h <= -1.06e-162)
		tmp = sqrt((d / h)) * ((t_1 / sqrt(-l)) * (1.0 + (h * (-0.125 * ((((M * D_m) / d) ^ 2.0) / l)))));
	elseif (h <= -5e-310)
		tmp = t_0 * (t_2 * (1.0 + ((h / l) * (-0.5 * ((D_m * ((M / 2.0) / d)) ^ 2.0)))));
	else
		tmp = (sqrt(d) / sqrt(h)) * (t_0 * (1.0 - (h * (0.125 * (t_3 * (t_3 / 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 / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(M * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -4.6e+67], N[(t$95$2 * N[(t$95$0 * N[(1.0 - N[(h * N[(0.125 * N[(N[Power[t$95$3, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1.06e-162], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(h * N[(-0.125 * N[(N[Power[N[(N[(M * D$95$m), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-310], N[(t$95$0 * N[(t$95$2 * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D$95$m * N[(N[(M / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(1.0 - N[(h * N[(0.125 * N[(t$95$3 * N[(t$95$3 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -4.5999999999999997e67

   1. Initial program 55.0%

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

   3. Simplified 55.1%

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

   5. Taylor expanded in h around -inf 31.7%

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

      1. associate-*r* 31.7%

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

      2. neg-mul-1 31.7%

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

      3. sub-neg 31.7%

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

      4. distribute-lft-in 31.7%

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

   7. Simplified 60.6%

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

      1. frac-2neg 60.6%

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

      2. sqrt-div 81.6%

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

   9. Applied egg-rr 81.6%

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

   if -4.5999999999999997e67 < h < -1.06000000000000003e-162

   1. Initial program 80.0%

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

   3. Simplified 80.0%

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

   5. Taylor expanded in h around -inf 50.2%

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

      1. associate-*r* 50.2%

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

      2. neg-mul-1 50.2%

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

      3. sub-neg 50.2%

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

      4. distribute-lft-in 50.2%

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

   7. Simplified 80.0%

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

      1. pow1 80.0%

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

      2. associate-*r/ 80.0%

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

   9. Applied egg-rr 80.0%

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

       1. unpow1 80.0%

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

       2. distribute-lft-neg-in 80.0%

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

       3. distribute-rgt-neg-in 80.0%

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

       4. associate-/l* 80.0%

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

       5. distribute-lft-neg-in 80.0%

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

       6. metadata-eval 80.0%

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

       7. associate-*r/ 80.0%

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

       8. *-commutative 80.0%

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

   11. Simplified 80.0%

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

       1. frac-2neg 80.0%

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

       2. sqrt-div 95.5%

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

   13. Applied egg-rr 95.5%

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

   if -1.06000000000000003e-162 < h < -4.999999999999985e-310

   1. Initial program 60.3%

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

   3. Simplified 56.3%

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

      1. frac-2neg 48.5%

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

      2. sqrt-div 71.6%

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

   6. Applied egg-rr 79.5%

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

   if -4.999999999999985e-310 < h

   1. Initial program 72.6%

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

   3. Simplified 71.0%

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

   5. Taylor expanded in h around -inf 48.6%

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

      1. associate-*r* 48.6%

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

      2. neg-mul-1 48.6%

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

      3. sub-neg 48.6%

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

      4. distribute-lft-in 48.6%

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

   7. Simplified 73.5%

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

      1. sqrt-div 82.0%

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

   9. Applied egg-rr 82.0%

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

       1. unpow2 82.0%

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

       2. associate-/l* 83.5%

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

   11. Applied egg-rr 83.5%

       \[\leadsto \frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-h\right) \cdot \left(0.125 \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \frac{D}{d}}{\ell}\right)}\right) + 1\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -4.6 \cdot 10^{+67}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;h \leq -1.06 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 + h \cdot \left(-0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \frac{D}{d}}{\ell}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{-d}\\ t_2 := M \cdot \frac{D\_m}{d}\\ \mathbf{if}\;h \leq -1.02 \cdot 10^{+67}:\\ \;\;\;\;\frac{t\_1}{\sqrt{-h}} \cdot \left(t\_0 \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{{t\_2}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{-\ell}}{t\_1}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(h \cdot -0.5\right) \cdot {\left(D\_m \cdot \frac{M}{d \cdot 2}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t\_0 \cdot \left(1 - h \cdot \left(0.125 \cdot \left(t\_2 \cdot \frac{t\_2}{\ell}\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 (sqrt (/ d l))) (t_1 (sqrt (- d))) (t_2 (* M (/ D_m d))))
   (if (<= h -1.02e+67)
     (*
      (/ t_1 (sqrt (- h)))
      (* t_0 (- 1.0 (* h (* 0.125 (/ (pow t_2 2.0) l))))))
     (if (<= h -5e-310)
       (*
        (/ 1.0 (/ (sqrt (- l)) t_1))
        (*
         (sqrt (/ d h))
         (+ 1.0 (/ (* (* h -0.5) (pow (* D_m (/ M (* d 2.0))) 2.0)) l))))
       (*
        (/ (sqrt d) (sqrt h))
        (* t_0 (- 1.0 (* h (* 0.125 (* t_2 (/ t_2 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 / l));
	double t_1 = sqrt(-d);
	double t_2 = M * (D_m / d);
	double tmp;
	if (h <= -1.02e+67) {
		tmp = (t_1 / sqrt(-h)) * (t_0 * (1.0 - (h * (0.125 * (pow(t_2, 2.0) / l)))));
	} else if (h <= -5e-310) {
		tmp = (1.0 / (sqrt(-l) / t_1)) * (sqrt((d / h)) * (1.0 + (((h * -0.5) * pow((D_m * (M / (d * 2.0))), 2.0)) / l)));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (t_0 * (1.0 - (h * (0.125 * (t_2 * (t_2 / 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) :: t_2
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = sqrt(-d)
    t_2 = m * (d_m / d)
    if (h <= (-1.02d+67)) then
        tmp = (t_1 / sqrt(-h)) * (t_0 * (1.0d0 - (h * (0.125d0 * ((t_2 ** 2.0d0) / l)))))
    else if (h <= (-5d-310)) then
        tmp = (1.0d0 / (sqrt(-l) / t_1)) * (sqrt((d / h)) * (1.0d0 + (((h * (-0.5d0)) * ((d_m * (m / (d * 2.0d0))) ** 2.0d0)) / l)))
    else
        tmp = (sqrt(d) / sqrt(h)) * (t_0 * (1.0d0 - (h * (0.125d0 * (t_2 * (t_2 / 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 / l));
	double t_1 = Math.sqrt(-d);
	double t_2 = M * (D_m / d);
	double tmp;
	if (h <= -1.02e+67) {
		tmp = (t_1 / Math.sqrt(-h)) * (t_0 * (1.0 - (h * (0.125 * (Math.pow(t_2, 2.0) / l)))));
	} else if (h <= -5e-310) {
		tmp = (1.0 / (Math.sqrt(-l) / t_1)) * (Math.sqrt((d / h)) * (1.0 + (((h * -0.5) * Math.pow((D_m * (M / (d * 2.0))), 2.0)) / l)));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * (t_0 * (1.0 - (h * (0.125 * (t_2 * (t_2 / 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 / l))
	t_1 = math.sqrt(-d)
	t_2 = M * (D_m / d)
	tmp = 0
	if h <= -1.02e+67:
		tmp = (t_1 / math.sqrt(-h)) * (t_0 * (1.0 - (h * (0.125 * (math.pow(t_2, 2.0) / l)))))
	elif h <= -5e-310:
		tmp = (1.0 / (math.sqrt(-l) / t_1)) * (math.sqrt((d / h)) * (1.0 + (((h * -0.5) * math.pow((D_m * (M / (d * 2.0))), 2.0)) / l)))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * (t_0 * (1.0 - (h * (0.125 * (t_2 * (t_2 / 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 / l))
	t_1 = sqrt(Float64(-d))
	t_2 = Float64(M * Float64(D_m / d))
	tmp = 0.0
	if (h <= -1.02e+67)
		tmp = Float64(Float64(t_1 / sqrt(Float64(-h))) * Float64(t_0 * Float64(1.0 - Float64(h * Float64(0.125 * Float64((t_2 ^ 2.0) / l))))));
	elseif (h <= -5e-310)
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(-l)) / t_1)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(Float64(h * -0.5) * (Float64(D_m * Float64(M / Float64(d * 2.0))) ^ 2.0)) / l))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(t_0 * Float64(1.0 - Float64(h * Float64(0.125 * Float64(t_2 * Float64(t_2 / 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 / l));
	t_1 = sqrt(-d);
	t_2 = M * (D_m / d);
	tmp = 0.0;
	if (h <= -1.02e+67)
		tmp = (t_1 / sqrt(-h)) * (t_0 * (1.0 - (h * (0.125 * ((t_2 ^ 2.0) / l)))));
	elseif (h <= -5e-310)
		tmp = (1.0 / (sqrt(-l) / t_1)) * (sqrt((d / h)) * (1.0 + (((h * -0.5) * ((D_m * (M / (d * 2.0))) ^ 2.0)) / l)));
	else
		tmp = (sqrt(d) / sqrt(h)) * (t_0 * (1.0 - (h * (0.125 * (t_2 * (t_2 / 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 / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$2 = N[(M * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -1.02e+67], N[(N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(1.0 - N[(h * N[(0.125 * N[(N[Power[t$95$2, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-310], N[(N[(1.0 / N[(N[Sqrt[(-l)], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(N[(h * -0.5), $MachinePrecision] * N[Power[N[(D$95$m * N[(M / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(1.0 - N[(h * N[(0.125 * N[(t$95$2 * N[(t$95$2 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.02000000000000002e67

   1. Initial program 55.0%

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

   3. Simplified 55.1%

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

   5. Taylor expanded in h around -inf 31.7%

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

      1. associate-*r* 31.7%

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

      2. neg-mul-1 31.7%

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

      3. sub-neg 31.7%

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

      4. distribute-lft-in 31.7%

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

   7. Simplified 60.6%

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

      1. frac-2neg 60.6%

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

      2. sqrt-div 81.6%

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

   9. Applied egg-rr 81.6%

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

   if -1.02000000000000002e67 < h < -4.999999999999985e-310

   1. Initial program 73.0%

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

   3. Simplified 71.6%

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

      1. associate-*l/ 71.7%

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

      2. *-commutative 71.7%

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

      3. associate-/l/ 71.7%

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

   6. Applied egg-rr 71.7%

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

      1. associate-*r* 71.7%

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

      2. *-commutative 71.7%

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

   8. Simplified 71.7%

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

      1. clear-num 71.7%

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

      2. sqrt-div 72.9%

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

      3. metadata-eval 72.9%

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

   10. Applied egg-rr 72.9%

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

       1. frac-2neg 72.9%

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

       2. sqrt-div 84.4%

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

   12. Applied egg-rr 84.4%

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

   if -4.999999999999985e-310 < h

   1. Initial program 72.6%

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

   3. Simplified 71.0%

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

   5. Taylor expanded in h around -inf 48.6%

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

      1. associate-*r* 48.6%

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

      2. neg-mul-1 48.6%

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

      3. sub-neg 48.6%

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

      4. distribute-lft-in 48.6%

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

   7. Simplified 73.5%

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

      1. sqrt-div 82.0%

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

   9. Applied egg-rr 82.0%

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

       1. unpow2 82.0%

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

       2. associate-/l* 83.5%

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

   11. Applied egg-rr 83.5%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.02 \cdot 10^{+67}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{-\ell}}{\sqrt{-d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(h \cdot -0.5\right) \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \frac{D}{d}}{\ell}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := 1 + h \cdot \left(-0.125 \cdot \frac{{\left(\frac{M \cdot D\_m}{d}\right)}^{2}}{\ell}\right)\\ t_3 := M \cdot \frac{D\_m}{d}\\ \mathbf{if}\;\ell \leq -2.2 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{t\_0}{\sqrt{-\ell}} \cdot t\_2\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{t\_0}{\sqrt{-h}} \cdot \left(t\_1 \cdot t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t\_1 \cdot \left(1 - h \cdot \left(0.125 \cdot \left(t\_3 \cdot \frac{t\_3}{\ell}\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 (sqrt (- d)))
        (t_1 (sqrt (/ d l)))
        (t_2 (+ 1.0 (* h (* -0.125 (/ (pow (/ (* M D_m) d) 2.0) l)))))
        (t_3 (* M (/ D_m d))))
   (if (<= l -2.2e+111)
     (* (sqrt (/ d h)) (* (/ t_0 (sqrt (- l))) t_2))
     (if (<= l -5e-310)
       (* (/ t_0 (sqrt (- h))) (* t_1 t_2))
       (*
        (/ (sqrt d) (sqrt h))
        (* t_1 (- 1.0 (* h (* 0.125 (* t_3 (/ t_3 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);
	double t_1 = sqrt((d / l));
	double t_2 = 1.0 + (h * (-0.125 * (pow(((M * D_m) / d), 2.0) / l)));
	double t_3 = M * (D_m / d);
	double tmp;
	if (l <= -2.2e+111) {
		tmp = sqrt((d / h)) * ((t_0 / sqrt(-l)) * t_2);
	} else if (l <= -5e-310) {
		tmp = (t_0 / sqrt(-h)) * (t_1 * t_2);
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (t_1 * (1.0 - (h * (0.125 * (t_3 * (t_3 / 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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sqrt(-d)
    t_1 = sqrt((d / l))
    t_2 = 1.0d0 + (h * ((-0.125d0) * ((((m * d_m) / d) ** 2.0d0) / l)))
    t_3 = m * (d_m / d)
    if (l <= (-2.2d+111)) then
        tmp = sqrt((d / h)) * ((t_0 / sqrt(-l)) * t_2)
    else if (l <= (-5d-310)) then
        tmp = (t_0 / sqrt(-h)) * (t_1 * t_2)
    else
        tmp = (sqrt(d) / sqrt(h)) * (t_1 * (1.0d0 - (h * (0.125d0 * (t_3 * (t_3 / 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);
	double t_1 = Math.sqrt((d / l));
	double t_2 = 1.0 + (h * (-0.125 * (Math.pow(((M * D_m) / d), 2.0) / l)));
	double t_3 = M * (D_m / d);
	double tmp;
	if (l <= -2.2e+111) {
		tmp = Math.sqrt((d / h)) * ((t_0 / Math.sqrt(-l)) * t_2);
	} else if (l <= -5e-310) {
		tmp = (t_0 / Math.sqrt(-h)) * (t_1 * t_2);
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * (t_1 * (1.0 - (h * (0.125 * (t_3 * (t_3 / 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)
	t_1 = math.sqrt((d / l))
	t_2 = 1.0 + (h * (-0.125 * (math.pow(((M * D_m) / d), 2.0) / l)))
	t_3 = M * (D_m / d)
	tmp = 0
	if l <= -2.2e+111:
		tmp = math.sqrt((d / h)) * ((t_0 / math.sqrt(-l)) * t_2)
	elif l <= -5e-310:
		tmp = (t_0 / math.sqrt(-h)) * (t_1 * t_2)
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * (t_1 * (1.0 - (h * (0.125 * (t_3 * (t_3 / 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))
	t_1 = sqrt(Float64(d / l))
	t_2 = Float64(1.0 + Float64(h * Float64(-0.125 * Float64((Float64(Float64(M * D_m) / d) ^ 2.0) / l))))
	t_3 = Float64(M * Float64(D_m / d))
	tmp = 0.0
	if (l <= -2.2e+111)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(t_0 / sqrt(Float64(-l))) * t_2));
	elseif (l <= -5e-310)
		tmp = Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(t_1 * t_2));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(t_1 * Float64(1.0 - Float64(h * Float64(0.125 * Float64(t_3 * Float64(t_3 / 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);
	t_1 = sqrt((d / l));
	t_2 = 1.0 + (h * (-0.125 * ((((M * D_m) / d) ^ 2.0) / l)));
	t_3 = M * (D_m / d);
	tmp = 0.0;
	if (l <= -2.2e+111)
		tmp = sqrt((d / h)) * ((t_0 / sqrt(-l)) * t_2);
	elseif (l <= -5e-310)
		tmp = (t_0 / sqrt(-h)) * (t_1 * t_2);
	else
		tmp = (sqrt(d) / sqrt(h)) * (t_1 * (1.0 - (h * (0.125 * (t_3 * (t_3 / 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[(-d)], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(h * N[(-0.125 * N[(N[Power[N[(N[(M * D$95$m), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(M * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.2e+111], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(1.0 - N[(h * N[(0.125 * N[(t$95$3 * N[(t$95$3 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.19999999999999999e111

   1. Initial program 57.8%

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

   3. Simplified 57.9%

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

   5. Taylor expanded in h around -inf 26.5%

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

      1. associate-*r* 26.5%

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

      2. neg-mul-1 26.5%

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

      3. sub-neg 26.5%

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

      4. distribute-lft-in 26.5%

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

   7. Simplified 57.9%

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

      1. pow1 57.9%

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

      2. associate-*r/ 57.9%

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

   9. Applied egg-rr 57.9%

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

       1. unpow1 57.9%

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

       2. distribute-lft-neg-in 57.9%

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

       3. distribute-rgt-neg-in 57.9%

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

       4. associate-/l* 57.9%

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

       5. distribute-lft-neg-in 57.9%

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

       6. metadata-eval 57.9%

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

       7. associate-*r/ 57.9%

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

       8. *-commutative 57.9%

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

   11. Simplified 57.9%

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

       1. frac-2neg 57.9%

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

       2. sqrt-div 76.4%

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

   13. Applied egg-rr 76.4%

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

   if -2.19999999999999999e111 < l < -4.999999999999985e-310

   1. Initial program 67.8%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in h around -inf 41.7%

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

      1. associate-*r* 41.7%

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

      2. neg-mul-1 41.7%

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

      3. sub-neg 41.7%

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

      4. distribute-lft-in 41.7%

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

   7. Simplified 68.2%

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

      1. pow1 68.2%

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

      2. associate-*r/ 68.2%

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

   9. Applied egg-rr 68.2%

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

       1. unpow1 68.2%

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

       2. distribute-lft-neg-in 68.2%

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

       3. distribute-rgt-neg-in 68.2%

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

       4. associate-/l* 68.2%

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

       5. distribute-lft-neg-in 68.2%

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

       6. metadata-eval 68.2%

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

       7. associate-*r/ 69.2%

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

       8. *-commutative 69.2%

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

   11. Simplified 69.2%

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

       1. frac-2neg 68.2%

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

       2. sqrt-div 85.7%

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

   13. Applied egg-rr 86.0%

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

   if -4.999999999999985e-310 < l

   1. Initial program 72.6%

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

   3. Simplified 71.0%

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

   5. Taylor expanded in h around -inf 48.6%

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

      1. associate-*r* 48.6%

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

      2. neg-mul-1 48.6%

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

      3. sub-neg 48.6%

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

      4. distribute-lft-in 48.6%

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

   7. Simplified 73.5%

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

      1. sqrt-div 82.0%

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

   9. Applied egg-rr 82.0%

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

       1. unpow2 82.0%

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

       2. associate-/l* 83.5%

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

   11. Applied egg-rr 83.5%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.2 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 + h \cdot \left(-0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(-0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \frac{D}{d}}{\ell}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := M \cdot \frac{D\_m}{d}\\ \mathbf{if}\;\ell \leq -2.4 \cdot 10^{-301}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 + h \cdot \left(-0.125 \cdot \frac{{\left(\frac{M \cdot D\_m}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \left(t\_0 \cdot \frac{t\_0}{\ell}\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 (* M (/ D_m d))))
   (if (<= l -2.4e-301)
     (*
      (sqrt (/ d h))
      (*
       (/ (sqrt (- d)) (sqrt (- l)))
       (+ 1.0 (* h (* -0.125 (/ (pow (/ (* M D_m) d) 2.0) l))))))
     (*
      (/ (sqrt d) (sqrt h))
      (* (sqrt (/ d l)) (- 1.0 (* h (* 0.125 (* t_0 (/ t_0 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 = M * (D_m / d);
	double tmp;
	if (l <= -2.4e-301) {
		tmp = sqrt((d / h)) * ((sqrt(-d) / sqrt(-l)) * (1.0 + (h * (-0.125 * (pow(((M * D_m) / d), 2.0) / l)))));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * (1.0 - (h * (0.125 * (t_0 * (t_0 / 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 = m * (d_m / d)
    if (l <= (-2.4d-301)) then
        tmp = sqrt((d / h)) * ((sqrt(-d) / sqrt(-l)) * (1.0d0 + (h * ((-0.125d0) * ((((m * d_m) / d) ** 2.0d0) / l)))))
    else
        tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * (1.0d0 - (h * (0.125d0 * (t_0 * (t_0 / 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 = M * (D_m / d);
	double tmp;
	if (l <= -2.4e-301) {
		tmp = Math.sqrt((d / h)) * ((Math.sqrt(-d) / Math.sqrt(-l)) * (1.0 + (h * (-0.125 * (Math.pow(((M * D_m) / d), 2.0) / l)))));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * (Math.sqrt((d / l)) * (1.0 - (h * (0.125 * (t_0 * (t_0 / 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 = M * (D_m / d)
	tmp = 0
	if l <= -2.4e-301:
		tmp = math.sqrt((d / h)) * ((math.sqrt(-d) / math.sqrt(-l)) * (1.0 + (h * (-0.125 * (math.pow(((M * D_m) / d), 2.0) / l)))))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * (math.sqrt((d / l)) * (1.0 - (h * (0.125 * (t_0 * (t_0 / 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(M * Float64(D_m / d))
	tmp = 0.0
	if (l <= -2.4e-301)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * Float64(1.0 + Float64(h * Float64(-0.125 * Float64((Float64(Float64(M * D_m) / d) ^ 2.0) / l))))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(h * Float64(0.125 * Float64(t_0 * Float64(t_0 / 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 = M * (D_m / d);
	tmp = 0.0;
	if (l <= -2.4e-301)
		tmp = sqrt((d / h)) * ((sqrt(-d) / sqrt(-l)) * (1.0 + (h * (-0.125 * ((((M * D_m) / d) ^ 2.0) / l)))));
	else
		tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * (1.0 - (h * (0.125 * (t_0 * (t_0 / 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[(M * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.4e-301], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(h * N[(-0.125 * N[(N[Power[N[(N[(M * D$95$m), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(h * N[(0.125 * N[(t$95$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -2.39999999999999991e-301

   1. Initial program 65.4%

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

   3. Simplified 64.6%

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

   5. Taylor expanded in h around -inf 37.5%

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

      1. associate-*r* 37.5%

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

      2. neg-mul-1 37.5%

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

      3. sub-neg 37.5%

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

      4. distribute-lft-in 37.5%

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

   7. Simplified 65.6%

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

      1. pow1 65.6%

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

      2. associate-*r/ 65.6%

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

   9. Applied egg-rr 65.6%

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

       1. unpow1 65.6%

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

       2. distribute-lft-neg-in 65.6%

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

       3. distribute-rgt-neg-in 65.6%

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

       4. associate-/l* 65.6%

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

       5. distribute-lft-neg-in 65.6%

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

       6. metadata-eval 65.6%

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

       7. associate-*r/ 66.4%

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

       8. *-commutative 66.4%

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

   11. Simplified 66.4%

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

       1. frac-2neg 66.4%

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

       2. sqrt-div 75.0%

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

   13. Applied egg-rr 75.0%

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

   if -2.39999999999999991e-301 < l

   1. Initial program 72.1%

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

   3. Simplified 70.5%

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

   5. Taylor expanded in h around -inf 48.2%

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

      1. associate-*r* 48.2%

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

      2. neg-mul-1 48.2%

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

      3. sub-neg 48.2%

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

      4. distribute-lft-in 48.2%

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

   7. Simplified 72.9%

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

      1. sqrt-div 81.4%

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

   9. Applied egg-rr 81.4%

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

       1. unpow2 81.4%

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

       2. associate-/l* 82.9%

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

   11. Applied egg-rr 82.9%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{-301}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 + h \cdot \left(-0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \frac{D}{d}}{\ell}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M}{2}\right)}^{2}\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := M \cdot \frac{D\_m}{d}\\ \mathbf{if}\;d \leq -1.3 \cdot 10^{-128}:\\ \;\;\;\;\left(t\_1 \cdot \left(1 + \frac{\left(h \cdot -0.5\right) \cdot {\left(D\_m \cdot \frac{M}{d \cdot 2}\right)}^{2}}{\ell}\right)\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{-275}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + t\_0\right)\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{+33}:\\ \;\;\;\;t\_1 \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \left(t\_2 \cdot \frac{t\_2}{\ell}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - t\_0\right) \cdot \left({h}^{-0.5} \cdot \left(d \cdot {\ell}^{-0.5}\right)\right)\\ \end{array} \end{array} \]

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 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M 2.0)) 2.0))))
        (t_1 (sqrt (/ d h)))
        (t_2 (* M (/ D_m d))))
   (if (<= d -1.3e-128)
     (*
      (* t_1 (+ 1.0 (/ (* (* h -0.5) (pow (* D_m (/ M (* d 2.0))) 2.0)) l)))
      (/ 1.0 (sqrt (/ l d))))
     (if (<= d 2.5e-275)
       (* (* d (sqrt (/ 1.0 (* h l)))) (+ -1.0 t_0))
       (if (<= d 1.4e+33)
         (*
          t_1
          (* (/ (sqrt d) (sqrt l)) (- 1.0 (* h (* 0.125 (* t_2 (/ t_2 l)))))))
         (* (- 1.0 t_0) (* (pow h -0.5) (* d (pow l -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 = 0.5 * ((h / l) * pow(((D_m / d) * (M / 2.0)), 2.0));
	double t_1 = sqrt((d / h));
	double t_2 = M * (D_m / d);
	double tmp;
	if (d <= -1.3e-128) {
		tmp = (t_1 * (1.0 + (((h * -0.5) * pow((D_m * (M / (d * 2.0))), 2.0)) / l))) * (1.0 / sqrt((l / d)));
	} else if (d <= 2.5e-275) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + t_0);
	} else if (d <= 1.4e+33) {
		tmp = t_1 * ((sqrt(d) / sqrt(l)) * (1.0 - (h * (0.125 * (t_2 * (t_2 / l))))));
	} else {
		tmp = (1.0 - t_0) * (pow(h, -0.5) * (d * pow(l, -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) :: t_2
    real(8) :: tmp
    t_0 = 0.5d0 * ((h / l) * (((d_m / d) * (m / 2.0d0)) ** 2.0d0))
    t_1 = sqrt((d / h))
    t_2 = m * (d_m / d)
    if (d <= (-1.3d-128)) then
        tmp = (t_1 * (1.0d0 + (((h * (-0.5d0)) * ((d_m * (m / (d * 2.0d0))) ** 2.0d0)) / l))) * (1.0d0 / sqrt((l / d)))
    else if (d <= 2.5d-275) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) + t_0)
    else if (d <= 1.4d+33) then
        tmp = t_1 * ((sqrt(d) / sqrt(l)) * (1.0d0 - (h * (0.125d0 * (t_2 * (t_2 / l))))))
    else
        tmp = (1.0d0 - t_0) * ((h ** (-0.5d0)) * (d * (l ** (-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 = 0.5 * ((h / l) * Math.pow(((D_m / d) * (M / 2.0)), 2.0));
	double t_1 = Math.sqrt((d / h));
	double t_2 = M * (D_m / d);
	double tmp;
	if (d <= -1.3e-128) {
		tmp = (t_1 * (1.0 + (((h * -0.5) * Math.pow((D_m * (M / (d * 2.0))), 2.0)) / l))) * (1.0 / Math.sqrt((l / d)));
	} else if (d <= 2.5e-275) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 + t_0);
	} else if (d <= 1.4e+33) {
		tmp = t_1 * ((Math.sqrt(d) / Math.sqrt(l)) * (1.0 - (h * (0.125 * (t_2 * (t_2 / l))))));
	} else {
		tmp = (1.0 - t_0) * (Math.pow(h, -0.5) * (d * Math.pow(l, -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 = 0.5 * ((h / l) * math.pow(((D_m / d) * (M / 2.0)), 2.0))
	t_1 = math.sqrt((d / h))
	t_2 = M * (D_m / d)
	tmp = 0
	if d <= -1.3e-128:
		tmp = (t_1 * (1.0 + (((h * -0.5) * math.pow((D_m * (M / (d * 2.0))), 2.0)) / l))) * (1.0 / math.sqrt((l / d)))
	elif d <= 2.5e-275:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 + t_0)
	elif d <= 1.4e+33:
		tmp = t_1 * ((math.sqrt(d) / math.sqrt(l)) * (1.0 - (h * (0.125 * (t_2 * (t_2 / l))))))
	else:
		tmp = (1.0 - t_0) * (math.pow(h, -0.5) * (d * math.pow(l, -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(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M / 2.0)) ^ 2.0)))
	t_1 = sqrt(Float64(d / h))
	t_2 = Float64(M * Float64(D_m / d))
	tmp = 0.0
	if (d <= -1.3e-128)
		tmp = Float64(Float64(t_1 * Float64(1.0 + Float64(Float64(Float64(h * -0.5) * (Float64(D_m * Float64(M / Float64(d * 2.0))) ^ 2.0)) / l))) * Float64(1.0 / sqrt(Float64(l / d))));
	elseif (d <= 2.5e-275)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 + t_0));
	elseif (d <= 1.4e+33)
		tmp = Float64(t_1 * Float64(Float64(sqrt(d) / sqrt(l)) * Float64(1.0 - Float64(h * Float64(0.125 * Float64(t_2 * Float64(t_2 / l)))))));
	else
		tmp = Float64(Float64(1.0 - t_0) * Float64((h ^ -0.5) * Float64(d * (l ^ -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 = 0.5 * ((h / l) * (((D_m / d) * (M / 2.0)) ^ 2.0));
	t_1 = sqrt((d / h));
	t_2 = M * (D_m / d);
	tmp = 0.0;
	if (d <= -1.3e-128)
		tmp = (t_1 * (1.0 + (((h * -0.5) * ((D_m * (M / (d * 2.0))) ^ 2.0)) / l))) * (1.0 / sqrt((l / d)));
	elseif (d <= 2.5e-275)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + t_0);
	elseif (d <= 1.4e+33)
		tmp = t_1 * ((sqrt(d) / sqrt(l)) * (1.0 - (h * (0.125 * (t_2 * (t_2 / l))))));
	else
		tmp = (1.0 - t_0) * ((h ^ -0.5) * (d * (l ^ -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[(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]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(M * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.3e-128], N[(N[(t$95$1 * N[(1.0 + N[(N[(N[(h * -0.5), $MachinePrecision] * N[Power[N[(D$95$m * N[(M / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.5e-275], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.4e+33], N[(t$95$1 * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(h * N[(0.125 * N[(t$95$2 * N[(t$95$2 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] * N[(N[Power[h, -0.5], $MachinePrecision] * N[(d * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.2999999999999999e-128

   1. Initial program 76.7%

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

   3. Simplified 76.8%

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

      1. associate-*l/ 80.4%

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

      2. *-commutative 80.4%

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

      3. associate-/l/ 80.4%

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

   6. Applied egg-rr 80.4%

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

      1. associate-*r* 80.4%

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

      2. *-commutative 80.4%

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

   8. Simplified 80.4%

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

      1. clear-num 80.4%

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

      2. sqrt-div 81.4%

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

      3. metadata-eval 81.4%

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

   10. Applied egg-rr 81.4%

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

   if -1.2999999999999999e-128 < d < 2.49999999999999992e-275

   1. Initial program 39.1%

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

   3. Simplified 39.1%

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

      1. add-sqr-sqrt 39.0%

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

      2. pow2 39.0%

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

      3. sqrt-unprod 30.6%

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

      4. pow1/2 30.6%

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

      5. sqrt-pow1 30.6%

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

      6. metadata-eval 30.6%

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

   6. Applied egg-rr 30.6%

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

   7. Taylor expanded in d around -inf 57.9%

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

   if 2.49999999999999992e-275 < d < 1.4e33

   1. Initial program 74.5%

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

   3. Simplified 71.8%

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

   5. Taylor expanded in h around -inf 59.2%

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

      1. associate-*r* 59.2%

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

      2. neg-mul-1 59.2%

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

      3. sub-neg 59.2%

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

      4. distribute-lft-in 59.2%

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

   7. Simplified 74.7%

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

      1. sqrt-div 82.6%

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

   9. Applied egg-rr 80.0%

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

       1. unpow2 77.3%

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

       2. associate-/l* 78.7%

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

   11. Applied egg-rr 82.7%

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

   if 1.4e33 < d

   1. Initial program 75.0%

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

   3. Simplified 74.7%

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

      1. add-sqr-sqrt 74.6%

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

      2. pow2 74.6%

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

      3. sqrt-unprod 71.1%

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

      4. pow1/2 71.1%

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

      5. sqrt-pow1 71.1%

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

      6. metadata-eval 71.1%

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

   6. Applied egg-rr 71.1%

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

      1. *-un-lft-identity 71.1%

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

      2. pow-pow 71.2%

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

      3. metadata-eval 71.2%

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

      4. pow1/2 71.2%

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

      5. frac-times 58.1%

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

      6. pow2 58.1%

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

   8. Applied egg-rr 58.1%

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

      1. *-lft-identity 58.1%

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

      2. associate-/r* 60.5%

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

   10. Simplified 60.5%

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

       1. associate-/r* 58.1%

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

       2. div-inv 58.1%

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

       3. unpow2 58.1%

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

       4. add-cbrt-cube 45.4%

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

       5. pow1/3 44.7%

          \[\leadsto \sqrt{\left(d \cdot d\right) \cdot \color{blue}{{\left(\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}\right)}^{0.3333333333333333}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       6. pow3 44.7%

          \[\leadsto \sqrt{\left(d \cdot d\right) \cdot {\color{blue}{\left({\left(\frac{1}{h \cdot \ell}\right)}^{3}\right)}}^{0.3333333333333333}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       7. metadata-eval 44.7%

          \[\leadsto \sqrt{\left(d \cdot d\right) \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{\left(1.5 + 1.5\right)}}\right)}^{0.3333333333333333}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       8. pow-prod-up 44.7%

          \[\leadsto \sqrt{\left(d \cdot d\right) \cdot {\color{blue}{\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}}^{0.3333333333333333}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       9. pow-prod-down 51.7%

          \[\leadsto \sqrt{\left(d \cdot d\right) \cdot \color{blue}{\left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       10. swap-sqr 52.0%

           \[\leadsto \sqrt{\color{blue}{\left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right) \cdot \left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       11. sqrt-unprod 61.6%

           \[\leadsto \color{blue}{\left(\sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       12. add-sqr-sqrt 61.6%

           \[\leadsto \color{blue}{\left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       13. *-commutative 61.6%

           \[\leadsto \color{blue}{\left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot d\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   12. Applied egg-rr 90.5%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.3 \cdot 10^{-128}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(h \cdot -0.5\right) \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}}{\ell}\right)\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{-275}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \frac{D}{d}}{\ell}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \left({h}^{-0.5} \cdot \left(d \cdot {\ell}^{-0.5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M}{d}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+122}:\\ \;\;\;\;t\_1 \cdot \left(t\_2 \cdot \left(1 + h \cdot \left(-0.125 \cdot \left(t\_0 \cdot \frac{t\_0}{\ell}\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -4.6 \cdot 10^{-81}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{+177}:\\ \;\;\;\;t\_1 \cdot \left(t\_2 \cdot \left(1 - \frac{h \cdot \left(0.125 \cdot {\left(\frac{M \cdot D\_m}{d}\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{1}{\sqrt{\ell}}\\ \end{array} \end{array} \]

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 (* D_m (/ M d))) (t_1 (sqrt (/ d h))) (t_2 (sqrt (/ d l))))
   (if (<= l -1.9e+122)
     (* t_1 (* t_2 (+ 1.0 (* h (* -0.125 (* t_0 (/ t_0 l)))))))
     (if (<= l -4.6e-81)
       (*
        (* d (sqrt (/ 1.0 (* h l))))
        (+ -1.0 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M 2.0)) 2.0)))))
       (if (<= l 7.5e+177)
         (*
          t_1
          (* t_2 (- 1.0 (/ (* h (* 0.125 (pow (/ (* M D_m) d) 2.0))) l))))
         (* (/ d (sqrt h)) (/ 1.0 (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 = D_m * (M / d);
	double t_1 = sqrt((d / h));
	double t_2 = sqrt((d / l));
	double tmp;
	if (l <= -1.9e+122) {
		tmp = t_1 * (t_2 * (1.0 + (h * (-0.125 * (t_0 * (t_0 / l))))));
	} else if (l <= -4.6e-81) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + (0.5 * ((h / l) * pow(((D_m / d) * (M / 2.0)), 2.0))));
	} else if (l <= 7.5e+177) {
		tmp = t_1 * (t_2 * (1.0 - ((h * (0.125 * pow(((M * D_m) / d), 2.0))) / l)));
	} else {
		tmp = (d / sqrt(h)) * (1.0 / 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) :: t_2
    real(8) :: tmp
    t_0 = d_m * (m / d)
    t_1 = sqrt((d / h))
    t_2 = sqrt((d / l))
    if (l <= (-1.9d+122)) then
        tmp = t_1 * (t_2 * (1.0d0 + (h * ((-0.125d0) * (t_0 * (t_0 / l))))))
    else if (l <= (-4.6d-81)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) + (0.5d0 * ((h / l) * (((d_m / d) * (m / 2.0d0)) ** 2.0d0))))
    else if (l <= 7.5d+177) then
        tmp = t_1 * (t_2 * (1.0d0 - ((h * (0.125d0 * (((m * d_m) / d) ** 2.0d0))) / l)))
    else
        tmp = (d / sqrt(h)) * (1.0d0 / 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 = D_m * (M / d);
	double t_1 = Math.sqrt((d / h));
	double t_2 = Math.sqrt((d / l));
	double tmp;
	if (l <= -1.9e+122) {
		tmp = t_1 * (t_2 * (1.0 + (h * (-0.125 * (t_0 * (t_0 / l))))));
	} else if (l <= -4.6e-81) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 + (0.5 * ((h / l) * Math.pow(((D_m / d) * (M / 2.0)), 2.0))));
	} else if (l <= 7.5e+177) {
		tmp = t_1 * (t_2 * (1.0 - ((h * (0.125 * Math.pow(((M * D_m) / d), 2.0))) / l)));
	} else {
		tmp = (d / Math.sqrt(h)) * (1.0 / 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 = D_m * (M / d)
	t_1 = math.sqrt((d / h))
	t_2 = math.sqrt((d / l))
	tmp = 0
	if l <= -1.9e+122:
		tmp = t_1 * (t_2 * (1.0 + (h * (-0.125 * (t_0 * (t_0 / l))))))
	elif l <= -4.6e-81:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 + (0.5 * ((h / l) * math.pow(((D_m / d) * (M / 2.0)), 2.0))))
	elif l <= 7.5e+177:
		tmp = t_1 * (t_2 * (1.0 - ((h * (0.125 * math.pow(((M * D_m) / d), 2.0))) / l)))
	else:
		tmp = (d / math.sqrt(h)) * (1.0 / 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))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -1.9e+122)
		tmp = Float64(t_1 * Float64(t_2 * Float64(1.0 + Float64(h * Float64(-0.125 * Float64(t_0 * Float64(t_0 / l)))))));
	elseif (l <= -4.6e-81)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 + Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M / 2.0)) ^ 2.0)))));
	elseif (l <= 7.5e+177)
		tmp = Float64(t_1 * Float64(t_2 * Float64(1.0 - Float64(Float64(h * Float64(0.125 * (Float64(Float64(M * D_m) / d) ^ 2.0))) / l))));
	else
		tmp = Float64(Float64(d / sqrt(h)) * Float64(1.0 / 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);
	t_1 = sqrt((d / h));
	t_2 = sqrt((d / l));
	tmp = 0.0;
	if (l <= -1.9e+122)
		tmp = t_1 * (t_2 * (1.0 + (h * (-0.125 * (t_0 * (t_0 / l))))));
	elseif (l <= -4.6e-81)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + (0.5 * ((h / l) * (((D_m / d) * (M / 2.0)) ^ 2.0))));
	elseif (l <= 7.5e+177)
		tmp = t_1 * (t_2 * (1.0 - ((h * (0.125 * (((M * D_m) / d) ^ 2.0))) / l)));
	else
		tmp = (d / sqrt(h)) * (1.0 / 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[(D$95$m * N[(M / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.9e+122], N[(t$95$1 * N[(t$95$2 * N[(1.0 + N[(h * N[(-0.125 * N[(t$95$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -4.6e-81], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.5e+177], N[(t$95$1 * N[(t$95$2 * N[(1.0 - N[(N[(h * N[(0.125 * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.8999999999999999e122

   1. Initial program 59.4%

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

   3. Simplified 59.5%

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

   5. Taylor expanded in h around -inf 27.2%

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

      1. associate-*r* 27.2%

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

      2. neg-mul-1 27.2%

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

      3. sub-neg 27.2%

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

      4. distribute-lft-in 27.2%

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

   7. Simplified 59.5%

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

      1. pow1 59.5%

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

      2. associate-*r/ 59.5%

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

   9. Applied egg-rr 59.5%

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

       1. unpow1 59.5%

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

       2. distribute-lft-neg-in 59.5%

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

       3. distribute-rgt-neg-in 59.5%

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

       4. associate-/l* 59.5%

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

       5. distribute-lft-neg-in 59.5%

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

       6. metadata-eval 59.5%

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

       7. associate-*r/ 59.5%

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

       8. *-commutative 59.5%

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

   11. Simplified 59.5%

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

       1. unpow2 59.5%

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

       2. associate-/l* 62.1%

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

       3. associate-/l* 62.1%

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

       4. associate-/l* 62.1%

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

   13. Applied egg-rr 62.1%

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

   if -1.8999999999999999e122 < l < -4.59999999999999982e-81

   1. Initial program 58.8%

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

   3. Simplified 56.7%

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

      1. add-sqr-sqrt 56.5%

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

      2. pow2 56.5%

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

      3. sqrt-unprod 38.4%

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

      4. pow1/2 38.4%

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

      5. sqrt-pow1 38.4%

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

      6. metadata-eval 38.4%

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

   6. Applied egg-rr 38.4%

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

   7. Taylor expanded in d around -inf 79.0%

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

   if -4.59999999999999982e-81 < l < 7.50000000000000039e177

   1. Initial program 77.5%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in h around -inf 51.6%

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

      1. associate-*r* 51.6%

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

      2. neg-mul-1 51.6%

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

      3. sub-neg 51.6%

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

      4. distribute-lft-in 51.6%

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

   7. Simplified 79.9%

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

      1. pow1 79.9%

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

      2. associate-*r/ 79.9%

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

   9. Applied egg-rr 79.9%

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

       1. unpow1 79.9%

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

       2. associate-*r/ 81.9%

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

       3. associate-*r/ 82.0%

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

       4. *-commutative 82.0%

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

   11. Simplified 82.0%

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

   if 7.50000000000000039e177 < l

   1. Initial program 57.0%

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

   3. Simplified 54.1%

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

   5. Taylor expanded in d around inf 59.9%

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

      1. *-un-lft-identity 59.9%

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

      2. pow1/2 59.9%

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

      3. inv-pow 59.9%

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

      4. pow-pow 59.8%

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

      5. metadata-eval 59.8%

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

   7. Applied egg-rr 59.8%

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

      1. *-lft-identity 59.8%

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

   9. Simplified 59.8%

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

       1. metadata-eval 59.8%

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

       2. metadata-eval 59.8%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\left(-1 \cdot \color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}\right)} \]

       3. pow-pow 59.9%

          \[\leadsto d \cdot \color{blue}{{\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       4. inv-pow 59.9%

          \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

       5. pow-pow 52.0%

          \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       6. add-sqr-sqrt 52.0%

          \[\leadsto \color{blue}{\sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}}} \]

       7. sqrt-unprod 41.8%

          \[\leadsto \color{blue}{\sqrt{\left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right) \cdot \left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       8. swap-sqr 41.5%

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot d\right) \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       9. unpow2 41.5%

          \[\leadsto \sqrt{\color{blue}{{d}^{2}} \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       10. pow-pow 41.9%

           \[\leadsto \sqrt{{d}^{2} \cdot \left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       11. pow-pow 45.9%

           \[\leadsto \sqrt{{d}^{2} \cdot \left({\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}\right)} \]

       12. pow-sqr 45.9%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \left(1.5 \cdot 0.3333333333333333\right)\right)}}} \]

       13. metadata-eval 45.9%

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

       14. metadata-eval 45.9%

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

       15. pow1 45.9%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \]

       16. div-inv 45.9%

           \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}} \]

       17. associate-/r* 43.4%

           \[\leadsto \sqrt{\color{blue}{\frac{\frac{{d}^{2}}{h}}{\ell}}} \]

   11. Applied egg-rr 70.8%

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

       1. associate-/r/ 71.0%

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

   13. Simplified 71.0%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+122}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(-0.125 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \frac{D \cdot \frac{M}{d}}{\ell}\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -4.6 \cdot 10^{-81}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{+177}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{h \cdot \left(0.125 \cdot {\left(\frac{M \cdot D}{d}\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{1}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M}{d}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+129}:\\ \;\;\;\;t\_1 \cdot \left(t\_2 \cdot \left(1 + h \cdot \left(-0.125 \cdot \left(t\_0 \cdot \frac{t\_0}{\ell}\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1.3 \cdot 10^{-81}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+178}:\\ \;\;\;\;t\_1 \cdot \left(t\_2 \cdot \left(1 + \frac{\left(h \cdot -0.125\right) \cdot {t\_0}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{1}{\sqrt{\ell}}\\ \end{array} \end{array} \]

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 (* D_m (/ M d))) (t_1 (sqrt (/ d h))) (t_2 (sqrt (/ d l))))
   (if (<= l -2.5e+129)
     (* t_1 (* t_2 (+ 1.0 (* h (* -0.125 (* t_0 (/ t_0 l)))))))
     (if (<= l -1.3e-81)
       (*
        (* d (sqrt (/ 1.0 (* h l))))
        (+ -1.0 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M 2.0)) 2.0)))))
       (if (<= l 1.2e+178)
         (* t_1 (* t_2 (+ 1.0 (/ (* (* h -0.125) (pow t_0 2.0)) l))))
         (* (/ d (sqrt h)) (/ 1.0 (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 = D_m * (M / d);
	double t_1 = sqrt((d / h));
	double t_2 = sqrt((d / l));
	double tmp;
	if (l <= -2.5e+129) {
		tmp = t_1 * (t_2 * (1.0 + (h * (-0.125 * (t_0 * (t_0 / l))))));
	} else if (l <= -1.3e-81) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + (0.5 * ((h / l) * pow(((D_m / d) * (M / 2.0)), 2.0))));
	} else if (l <= 1.2e+178) {
		tmp = t_1 * (t_2 * (1.0 + (((h * -0.125) * pow(t_0, 2.0)) / l)));
	} else {
		tmp = (d / sqrt(h)) * (1.0 / 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) :: t_2
    real(8) :: tmp
    t_0 = d_m * (m / d)
    t_1 = sqrt((d / h))
    t_2 = sqrt((d / l))
    if (l <= (-2.5d+129)) then
        tmp = t_1 * (t_2 * (1.0d0 + (h * ((-0.125d0) * (t_0 * (t_0 / l))))))
    else if (l <= (-1.3d-81)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) + (0.5d0 * ((h / l) * (((d_m / d) * (m / 2.0d0)) ** 2.0d0))))
    else if (l <= 1.2d+178) then
        tmp = t_1 * (t_2 * (1.0d0 + (((h * (-0.125d0)) * (t_0 ** 2.0d0)) / l)))
    else
        tmp = (d / sqrt(h)) * (1.0d0 / 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 = D_m * (M / d);
	double t_1 = Math.sqrt((d / h));
	double t_2 = Math.sqrt((d / l));
	double tmp;
	if (l <= -2.5e+129) {
		tmp = t_1 * (t_2 * (1.0 + (h * (-0.125 * (t_0 * (t_0 / l))))));
	} else if (l <= -1.3e-81) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 + (0.5 * ((h / l) * Math.pow(((D_m / d) * (M / 2.0)), 2.0))));
	} else if (l <= 1.2e+178) {
		tmp = t_1 * (t_2 * (1.0 + (((h * -0.125) * Math.pow(t_0, 2.0)) / l)));
	} else {
		tmp = (d / Math.sqrt(h)) * (1.0 / 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 = D_m * (M / d)
	t_1 = math.sqrt((d / h))
	t_2 = math.sqrt((d / l))
	tmp = 0
	if l <= -2.5e+129:
		tmp = t_1 * (t_2 * (1.0 + (h * (-0.125 * (t_0 * (t_0 / l))))))
	elif l <= -1.3e-81:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 + (0.5 * ((h / l) * math.pow(((D_m / d) * (M / 2.0)), 2.0))))
	elif l <= 1.2e+178:
		tmp = t_1 * (t_2 * (1.0 + (((h * -0.125) * math.pow(t_0, 2.0)) / l)))
	else:
		tmp = (d / math.sqrt(h)) * (1.0 / 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))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -2.5e+129)
		tmp = Float64(t_1 * Float64(t_2 * Float64(1.0 + Float64(h * Float64(-0.125 * Float64(t_0 * Float64(t_0 / l)))))));
	elseif (l <= -1.3e-81)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 + Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M / 2.0)) ^ 2.0)))));
	elseif (l <= 1.2e+178)
		tmp = Float64(t_1 * Float64(t_2 * Float64(1.0 + Float64(Float64(Float64(h * -0.125) * (t_0 ^ 2.0)) / l))));
	else
		tmp = Float64(Float64(d / sqrt(h)) * Float64(1.0 / 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);
	t_1 = sqrt((d / h));
	t_2 = sqrt((d / l));
	tmp = 0.0;
	if (l <= -2.5e+129)
		tmp = t_1 * (t_2 * (1.0 + (h * (-0.125 * (t_0 * (t_0 / l))))));
	elseif (l <= -1.3e-81)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + (0.5 * ((h / l) * (((D_m / d) * (M / 2.0)) ^ 2.0))));
	elseif (l <= 1.2e+178)
		tmp = t_1 * (t_2 * (1.0 + (((h * -0.125) * (t_0 ^ 2.0)) / l)));
	else
		tmp = (d / sqrt(h)) * (1.0 / 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[(D$95$m * N[(M / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -2.5e+129], N[(t$95$1 * N[(t$95$2 * N[(1.0 + N[(h * N[(-0.125 * N[(t$95$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1.3e-81], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.2e+178], N[(t$95$1 * N[(t$95$2 * N[(1.0 + N[(N[(N[(h * -0.125), $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.5000000000000001e129

   1. Initial program 59.4%

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

   3. Simplified 59.5%

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

   5. Taylor expanded in h around -inf 27.2%

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

      1. associate-*r* 27.2%

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

      2. neg-mul-1 27.2%

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

      3. sub-neg 27.2%

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

      4. distribute-lft-in 27.2%

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

   7. Simplified 59.5%

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

      1. pow1 59.5%

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

      2. associate-*r/ 59.5%

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

   9. Applied egg-rr 59.5%

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

       1. unpow1 59.5%

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

       2. distribute-lft-neg-in 59.5%

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

       3. distribute-rgt-neg-in 59.5%

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

       4. associate-/l* 59.5%

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

       5. distribute-lft-neg-in 59.5%

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

       6. metadata-eval 59.5%

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

       7. associate-*r/ 59.5%

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

       8. *-commutative 59.5%

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

   11. Simplified 59.5%

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

       1. unpow2 59.5%

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

       2. associate-/l* 62.1%

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

       3. associate-/l* 62.1%

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

       4. associate-/l* 62.1%

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

   13. Applied egg-rr 62.1%

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

   if -2.5000000000000001e129 < l < -1.2999999999999999e-81

   1. Initial program 58.8%

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

   3. Simplified 56.7%

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

      1. add-sqr-sqrt 56.5%

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

      2. pow2 56.5%

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

      3. sqrt-unprod 38.4%

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

      4. pow1/2 38.4%

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

      5. sqrt-pow1 38.4%

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

      6. metadata-eval 38.4%

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

   6. Applied egg-rr 38.4%

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

   7. Taylor expanded in d around -inf 79.0%

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

   if -1.2999999999999999e-81 < l < 1.2e178

   1. Initial program 77.5%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in h around -inf 51.6%

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

      1. associate-*r* 51.6%

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

      2. neg-mul-1 51.6%

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

      3. sub-neg 51.6%

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

      4. distribute-lft-in 51.6%

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

   7. Simplified 79.9%

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

      1. pow1 79.9%

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

      2. associate-*r/ 79.9%

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

   9. Applied egg-rr 79.9%

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

       1. unpow1 79.9%

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

       2. distribute-lft-neg-in 79.9%

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

       3. distribute-rgt-neg-in 79.9%

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

       4. associate-/l* 79.9%

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

       5. distribute-lft-neg-in 79.9%

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

       6. metadata-eval 79.9%

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

       7. associate-*r/ 80.7%

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

       8. *-commutative 80.7%

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

   11. Simplified 80.7%

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

       1. associate-*r* 80.7%

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

       2. associate-*r/ 82.0%

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

       3. associate-/l* 81.3%

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

   13. Applied egg-rr 81.3%

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

   if 1.2e178 < l

   1. Initial program 57.0%

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

   3. Simplified 54.1%

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

   5. Taylor expanded in d around inf 59.9%

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

      1. *-un-lft-identity 59.9%

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

      2. pow1/2 59.9%

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

      3. inv-pow 59.9%

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

      4. pow-pow 59.8%

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

      5. metadata-eval 59.8%

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

   7. Applied egg-rr 59.8%

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

      1. *-lft-identity 59.8%

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

   9. Simplified 59.8%

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

       1. metadata-eval 59.8%

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

       2. metadata-eval 59.8%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\left(-1 \cdot \color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}\right)} \]

       3. pow-pow 59.9%

          \[\leadsto d \cdot \color{blue}{{\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       4. inv-pow 59.9%

          \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

       5. pow-pow 52.0%

          \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       6. add-sqr-sqrt 52.0%

          \[\leadsto \color{blue}{\sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}}} \]

       7. sqrt-unprod 41.8%

          \[\leadsto \color{blue}{\sqrt{\left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right) \cdot \left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       8. swap-sqr 41.5%

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot d\right) \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       9. unpow2 41.5%

          \[\leadsto \sqrt{\color{blue}{{d}^{2}} \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       10. pow-pow 41.9%

           \[\leadsto \sqrt{{d}^{2} \cdot \left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       11. pow-pow 45.9%

           \[\leadsto \sqrt{{d}^{2} \cdot \left({\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}\right)} \]

       12. pow-sqr 45.9%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \left(1.5 \cdot 0.3333333333333333\right)\right)}}} \]

       13. metadata-eval 45.9%

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

       14. metadata-eval 45.9%

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

       15. pow1 45.9%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \]

       16. div-inv 45.9%

           \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}} \]

       17. associate-/r* 43.4%

           \[\leadsto \sqrt{\color{blue}{\frac{\frac{{d}^{2}}{h}}{\ell}}} \]

   11. Applied egg-rr 70.8%

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

       1. associate-/r/ 71.0%

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

   13. Simplified 71.0%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+129}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(-0.125 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \frac{D \cdot \frac{M}{d}}{\ell}\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1.3 \cdot 10^{-81}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+178}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{\left(h \cdot -0.125\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{1}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := M \cdot \frac{D\_m}{d}\\ \mathbf{if}\;d \leq -5.8 \cdot 10^{-126}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(h \cdot -0.5\right) \cdot {\left(D\_m \cdot \frac{M}{d \cdot 2}\right)}^{2}}{\ell}\right)\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \left(t\_0 \cdot \frac{t\_0}{\ell}\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 (* M (/ D_m d))))
   (if (<= d -5.8e-126)
     (*
      (*
       (sqrt (/ d h))
       (+ 1.0 (/ (* (* h -0.5) (pow (* D_m (/ M (* d 2.0))) 2.0)) l)))
      (/ 1.0 (sqrt (/ l d))))
     (if (<= d -5e-311)
       (*
        (* d (sqrt (/ 1.0 (* h l))))
        (+ -1.0 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M 2.0)) 2.0)))))
       (*
        (/ (sqrt d) (sqrt h))
        (* (sqrt (/ d l)) (- 1.0 (* h (* 0.125 (* t_0 (/ t_0 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 = M * (D_m / d);
	double tmp;
	if (d <= -5.8e-126) {
		tmp = (sqrt((d / h)) * (1.0 + (((h * -0.5) * pow((D_m * (M / (d * 2.0))), 2.0)) / l))) * (1.0 / sqrt((l / d)));
	} else if (d <= -5e-311) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + (0.5 * ((h / l) * pow(((D_m / d) * (M / 2.0)), 2.0))));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * (1.0 - (h * (0.125 * (t_0 * (t_0 / 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 = m * (d_m / d)
    if (d <= (-5.8d-126)) then
        tmp = (sqrt((d / h)) * (1.0d0 + (((h * (-0.5d0)) * ((d_m * (m / (d * 2.0d0))) ** 2.0d0)) / l))) * (1.0d0 / sqrt((l / d)))
    else if (d <= (-5d-311)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) + (0.5d0 * ((h / l) * (((d_m / d) * (m / 2.0d0)) ** 2.0d0))))
    else
        tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * (1.0d0 - (h * (0.125d0 * (t_0 * (t_0 / 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 = M * (D_m / d);
	double tmp;
	if (d <= -5.8e-126) {
		tmp = (Math.sqrt((d / h)) * (1.0 + (((h * -0.5) * Math.pow((D_m * (M / (d * 2.0))), 2.0)) / l))) * (1.0 / Math.sqrt((l / d)));
	} else if (d <= -5e-311) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 + (0.5 * ((h / l) * Math.pow(((D_m / d) * (M / 2.0)), 2.0))));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * (Math.sqrt((d / l)) * (1.0 - (h * (0.125 * (t_0 * (t_0 / 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 = M * (D_m / d)
	tmp = 0
	if d <= -5.8e-126:
		tmp = (math.sqrt((d / h)) * (1.0 + (((h * -0.5) * math.pow((D_m * (M / (d * 2.0))), 2.0)) / l))) * (1.0 / math.sqrt((l / d)))
	elif d <= -5e-311:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 + (0.5 * ((h / l) * math.pow(((D_m / d) * (M / 2.0)), 2.0))))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * (math.sqrt((d / l)) * (1.0 - (h * (0.125 * (t_0 * (t_0 / 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(M * Float64(D_m / d))
	tmp = 0.0
	if (d <= -5.8e-126)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(Float64(h * -0.5) * (Float64(D_m * Float64(M / Float64(d * 2.0))) ^ 2.0)) / l))) * Float64(1.0 / sqrt(Float64(l / d))));
	elseif (d <= -5e-311)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 + Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M / 2.0)) ^ 2.0)))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(h * Float64(0.125 * Float64(t_0 * Float64(t_0 / 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 = M * (D_m / d);
	tmp = 0.0;
	if (d <= -5.8e-126)
		tmp = (sqrt((d / h)) * (1.0 + (((h * -0.5) * ((D_m * (M / (d * 2.0))) ^ 2.0)) / l))) * (1.0 / sqrt((l / d)));
	elseif (d <= -5e-311)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + (0.5 * ((h / l) * (((D_m / d) * (M / 2.0)) ^ 2.0))));
	else
		tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * (1.0 - (h * (0.125 * (t_0 * (t_0 / 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[(M * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5.8e-126], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(N[(h * -0.5), $MachinePrecision] * N[Power[N[(D$95$m * N[(M / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-311], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(h * N[(0.125 * N[(t$95$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.79999999999999975e-126

   1. Initial program 76.7%

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

   3. Simplified 76.8%

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

      1. associate-*l/ 80.4%

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

      2. *-commutative 80.4%

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

      3. associate-/l/ 80.4%

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

   6. Applied egg-rr 80.4%

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

      1. associate-*r* 80.4%

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

      2. *-commutative 80.4%

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

   8. Simplified 80.4%

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

      1. clear-num 80.4%

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

      2. sqrt-div 81.4%

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

      3. metadata-eval 81.4%

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

   10. Applied egg-rr 81.4%

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

   if -5.79999999999999975e-126 < d < -5.00000000000023e-311

   1. Initial program 41.3%

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

   3. Simplified 41.4%

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

      1. add-sqr-sqrt 41.2%

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

      2. pow2 41.2%

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

      3. sqrt-unprod 31.8%

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

      4. pow1/2 31.8%

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

      5. sqrt-pow1 31.8%

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

      6. metadata-eval 31.8%

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

   6. Applied egg-rr 31.8%

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

   7. Taylor expanded in d around -inf 62.4%

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

   if -5.00000000000023e-311 < d

   1. Initial program 72.6%

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

   3. Simplified 71.0%

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

   5. Taylor expanded in h around -inf 48.6%

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

      1. associate-*r* 48.6%

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

      2. neg-mul-1 48.6%

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

      3. sub-neg 48.6%

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

      4. distribute-lft-in 48.6%

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

   7. Simplified 73.5%

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

      1. sqrt-div 82.0%

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

   9. Applied egg-rr 82.0%

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

       1. unpow2 82.0%

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

       2. associate-/l* 83.5%

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

   11. Applied egg-rr 83.5%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.8 \cdot 10^{-126}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(h \cdot -0.5\right) \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}}{\ell}\right)\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \frac{D}{d}}{\ell}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M}{2}\right)}^{2}\right)\\ \mathbf{if}\;d \leq -6 \cdot 10^{-126}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(h \cdot -0.5\right) \cdot {\left(D\_m \cdot \frac{M}{d \cdot 2}\right)}^{2}}{\ell}\right)\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - t\_0\right) \cdot \left({h}^{-0.5} \cdot \left(d \cdot {\ell}^{-0.5}\right)\right)\\ \end{array} \end{array} \]

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 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M 2.0)) 2.0)))))
   (if (<= d -6e-126)
     (*
      (*
       (sqrt (/ d h))
       (+ 1.0 (/ (* (* h -0.5) (pow (* D_m (/ M (* d 2.0))) 2.0)) l)))
      (/ 1.0 (sqrt (/ l d))))
     (if (<= d -5e-311)
       (* (* d (sqrt (/ 1.0 (* h l)))) (+ -1.0 t_0))
       (* (- 1.0 t_0) (* (pow h -0.5) (* d (pow l -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 = 0.5 * ((h / l) * pow(((D_m / d) * (M / 2.0)), 2.0));
	double tmp;
	if (d <= -6e-126) {
		tmp = (sqrt((d / h)) * (1.0 + (((h * -0.5) * pow((D_m * (M / (d * 2.0))), 2.0)) / l))) * (1.0 / sqrt((l / d)));
	} else if (d <= -5e-311) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + t_0);
	} else {
		tmp = (1.0 - t_0) * (pow(h, -0.5) * (d * pow(l, -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) :: tmp
    t_0 = 0.5d0 * ((h / l) * (((d_m / d) * (m / 2.0d0)) ** 2.0d0))
    if (d <= (-6d-126)) then
        tmp = (sqrt((d / h)) * (1.0d0 + (((h * (-0.5d0)) * ((d_m * (m / (d * 2.0d0))) ** 2.0d0)) / l))) * (1.0d0 / sqrt((l / d)))
    else if (d <= (-5d-311)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) + t_0)
    else
        tmp = (1.0d0 - t_0) * ((h ** (-0.5d0)) * (d * (l ** (-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 = 0.5 * ((h / l) * Math.pow(((D_m / d) * (M / 2.0)), 2.0));
	double tmp;
	if (d <= -6e-126) {
		tmp = (Math.sqrt((d / h)) * (1.0 + (((h * -0.5) * Math.pow((D_m * (M / (d * 2.0))), 2.0)) / l))) * (1.0 / Math.sqrt((l / d)));
	} else if (d <= -5e-311) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 + t_0);
	} else {
		tmp = (1.0 - t_0) * (Math.pow(h, -0.5) * (d * Math.pow(l, -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 = 0.5 * ((h / l) * math.pow(((D_m / d) * (M / 2.0)), 2.0))
	tmp = 0
	if d <= -6e-126:
		tmp = (math.sqrt((d / h)) * (1.0 + (((h * -0.5) * math.pow((D_m * (M / (d * 2.0))), 2.0)) / l))) * (1.0 / math.sqrt((l / d)))
	elif d <= -5e-311:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 + t_0)
	else:
		tmp = (1.0 - t_0) * (math.pow(h, -0.5) * (d * math.pow(l, -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(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M / 2.0)) ^ 2.0)))
	tmp = 0.0
	if (d <= -6e-126)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(Float64(h * -0.5) * (Float64(D_m * Float64(M / Float64(d * 2.0))) ^ 2.0)) / l))) * Float64(1.0 / sqrt(Float64(l / d))));
	elseif (d <= -5e-311)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 + t_0));
	else
		tmp = Float64(Float64(1.0 - t_0) * Float64((h ^ -0.5) * Float64(d * (l ^ -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 = 0.5 * ((h / l) * (((D_m / d) * (M / 2.0)) ^ 2.0));
	tmp = 0.0;
	if (d <= -6e-126)
		tmp = (sqrt((d / h)) * (1.0 + (((h * -0.5) * ((D_m * (M / (d * 2.0))) ^ 2.0)) / l))) * (1.0 / sqrt((l / d)));
	elseif (d <= -5e-311)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + t_0);
	else
		tmp = (1.0 - t_0) * ((h ^ -0.5) * (d * (l ^ -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[(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]}, If[LessEqual[d, -6e-126], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(N[(h * -0.5), $MachinePrecision] * N[Power[N[(D$95$m * N[(M / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-311], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] * N[(N[Power[h, -0.5], $MachinePrecision] * N[(d * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -6.0000000000000003e-126

   1. Initial program 76.7%

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

   3. Simplified 76.8%

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

      1. associate-*l/ 80.4%

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

      2. *-commutative 80.4%

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

      3. associate-/l/ 80.4%

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

   6. Applied egg-rr 80.4%

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

      1. associate-*r* 80.4%

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

      2. *-commutative 80.4%

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

   8. Simplified 80.4%

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

      1. clear-num 80.4%

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

      2. sqrt-div 81.4%

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

      3. metadata-eval 81.4%

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

   10. Applied egg-rr 81.4%

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

   if -6.0000000000000003e-126 < d < -5.00000000000023e-311

   1. Initial program 41.3%

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

   3. Simplified 41.4%

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

      1. add-sqr-sqrt 41.2%

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

      2. pow2 41.2%

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

      3. sqrt-unprod 31.8%

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

      4. pow1/2 31.8%

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

      5. sqrt-pow1 31.8%

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

      6. metadata-eval 31.8%

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

   6. Applied egg-rr 31.8%

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

   7. Taylor expanded in d around -inf 62.4%

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

   if -5.00000000000023e-311 < d

   1. Initial program 72.6%

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

   3. Simplified 71.0%

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

      1. add-sqr-sqrt 70.9%

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

      2. pow2 70.9%

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

      3. sqrt-unprod 64.3%

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

      4. pow1/2 64.3%

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

      5. sqrt-pow1 64.3%

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

      6. metadata-eval 64.3%

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

   6. Applied egg-rr 64.3%

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

      1. *-un-lft-identity 64.3%

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

      2. pow-pow 64.4%

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

      3. metadata-eval 64.4%

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

      4. pow1/2 64.4%

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

      5. frac-times 53.1%

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

      6. pow2 53.1%

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

   8. Applied egg-rr 53.1%

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

      1. *-lft-identity 53.1%

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

      2. associate-/r* 53.9%

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

   10. Simplified 53.9%

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

       1. associate-/r* 53.1%

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

       2. div-inv 53.0%

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

       3. unpow2 53.0%

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

       4. add-cbrt-cube 41.2%

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

       5. pow1/3 40.6%

          \[\leadsto \sqrt{\left(d \cdot d\right) \cdot \color{blue}{{\left(\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}\right)}^{0.3333333333333333}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       6. pow3 40.6%

          \[\leadsto \sqrt{\left(d \cdot d\right) \cdot {\color{blue}{\left({\left(\frac{1}{h \cdot \ell}\right)}^{3}\right)}}^{0.3333333333333333}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       7. metadata-eval 40.6%

          \[\leadsto \sqrt{\left(d \cdot d\right) \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{\left(1.5 + 1.5\right)}}\right)}^{0.3333333333333333}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       8. pow-prod-up 40.6%

          \[\leadsto \sqrt{\left(d \cdot d\right) \cdot {\color{blue}{\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}}^{0.3333333333333333}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       9. pow-prod-down 48.8%

          \[\leadsto \sqrt{\left(d \cdot d\right) \cdot \color{blue}{\left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       10. swap-sqr 54.3%

           \[\leadsto \sqrt{\color{blue}{\left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right) \cdot \left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       11. sqrt-unprod 62.5%

           \[\leadsto \color{blue}{\left(\sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       12. add-sqr-sqrt 62.5%

           \[\leadsto \color{blue}{\left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       13. *-commutative 62.5%

           \[\leadsto \color{blue}{\left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot d\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   12. Applied egg-rr 79.7%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6 \cdot 10^{-126}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(h \cdot -0.5\right) \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}}{\ell}\right)\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \left({h}^{-0.5} \cdot \left(d \cdot {\ell}^{-0.5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M}{d}\\ t_1 := 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M}{2}\right)}^{2}\right)\\ \mathbf{if}\;d \leq -9 \cdot 10^{-127}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \left(-0.125 \cdot \left(t\_0 \cdot \left(t\_0 \cdot \frac{-1}{\ell}\right)\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - t\_1\right) \cdot \left({h}^{-0.5} \cdot \left(d \cdot {\ell}^{-0.5}\right)\right)\\ \end{array} \end{array} \]

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 (* D_m (/ M d)))
        (t_1 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M 2.0)) 2.0)))))
   (if (<= d -9e-127)
     (*
      (sqrt (/ d h))
      (* (sqrt (/ d l)) (- 1.0 (* h (* -0.125 (* t_0 (* t_0 (/ -1.0 l))))))))
     (if (<= d -5e-311)
       (* (* d (sqrt (/ 1.0 (* h l)))) (+ -1.0 t_1))
       (* (- 1.0 t_1) (* (pow h -0.5) (* d (pow l -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 = D_m * (M / d);
	double t_1 = 0.5 * ((h / l) * pow(((D_m / d) * (M / 2.0)), 2.0));
	double tmp;
	if (d <= -9e-127) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (h * (-0.125 * (t_0 * (t_0 * (-1.0 / l)))))));
	} else if (d <= -5e-311) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + t_1);
	} else {
		tmp = (1.0 - t_1) * (pow(h, -0.5) * (d * pow(l, -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 = d_m * (m / d)
    t_1 = 0.5d0 * ((h / l) * (((d_m / d) * (m / 2.0d0)) ** 2.0d0))
    if (d <= (-9d-127)) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 - (h * ((-0.125d0) * (t_0 * (t_0 * ((-1.0d0) / l)))))))
    else if (d <= (-5d-311)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) + t_1)
    else
        tmp = (1.0d0 - t_1) * ((h ** (-0.5d0)) * (d * (l ** (-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 = D_m * (M / d);
	double t_1 = 0.5 * ((h / l) * Math.pow(((D_m / d) * (M / 2.0)), 2.0));
	double tmp;
	if (d <= -9e-127) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - (h * (-0.125 * (t_0 * (t_0 * (-1.0 / l)))))));
	} else if (d <= -5e-311) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 + t_1);
	} else {
		tmp = (1.0 - t_1) * (Math.pow(h, -0.5) * (d * Math.pow(l, -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 = D_m * (M / d)
	t_1 = 0.5 * ((h / l) * math.pow(((D_m / d) * (M / 2.0)), 2.0))
	tmp = 0
	if d <= -9e-127:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - (h * (-0.125 * (t_0 * (t_0 * (-1.0 / l)))))))
	elif d <= -5e-311:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 + t_1)
	else:
		tmp = (1.0 - t_1) * (math.pow(h, -0.5) * (d * math.pow(l, -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(D_m * Float64(M / d))
	t_1 = Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M / 2.0)) ^ 2.0)))
	tmp = 0.0
	if (d <= -9e-127)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(h * Float64(-0.125 * Float64(t_0 * Float64(t_0 * Float64(-1.0 / l))))))));
	elseif (d <= -5e-311)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 + t_1));
	else
		tmp = Float64(Float64(1.0 - t_1) * Float64((h ^ -0.5) * Float64(d * (l ^ -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 = D_m * (M / d);
	t_1 = 0.5 * ((h / l) * (((D_m / d) * (M / 2.0)) ^ 2.0));
	tmp = 0.0;
	if (d <= -9e-127)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (h * (-0.125 * (t_0 * (t_0 * (-1.0 / l)))))));
	elseif (d <= -5e-311)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + t_1);
	else
		tmp = (1.0 - t_1) * ((h ^ -0.5) * (d * (l ^ -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[(D$95$m * N[(M / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[d, -9e-127], 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$0 * N[(t$95$0 * N[(-1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-311], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$1), $MachinePrecision] * N[(N[Power[h, -0.5], $MachinePrecision] * N[(d * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -8.9999999999999998e-127

   1. Initial program 76.7%

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

   3. Simplified 75.6%

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

   5. Taylor expanded in h around -inf 48.7%

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

      1. associate-*r* 48.7%

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

      2. neg-mul-1 48.7%

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

      3. sub-neg 48.7%

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

      4. distribute-lft-in 48.7%

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

   7. Simplified 78.2%

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

      1. pow1 78.2%

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

      2. associate-*r/ 78.2%

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

   9. Applied egg-rr 78.2%

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

       1. unpow1 78.2%

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

       2. distribute-lft-neg-in 78.2%

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

       3. distribute-rgt-neg-in 78.2%

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

       4. associate-/l* 78.2%

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

       5. distribute-lft-neg-in 78.2%

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

       6. metadata-eval 78.2%

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

       7. associate-*r/ 79.3%

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

       8. *-commutative 79.3%

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

   11. Simplified 79.3%

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

       1. div-inv 79.3%

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

       2. unpow2 79.3%

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

       3. associate-*l* 80.5%

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

       4. associate-/l* 80.6%

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

       5. associate-/l* 80.5%

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

   13. Applied egg-rr 80.5%

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

   if -8.9999999999999998e-127 < d < -5.00000000000023e-311

   1. Initial program 41.3%

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

   3. Simplified 41.4%

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

      1. add-sqr-sqrt 41.2%

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

      2. pow2 41.2%

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

      3. sqrt-unprod 31.8%

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

      4. pow1/2 31.8%

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

      5. sqrt-pow1 31.8%

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

      6. metadata-eval 31.8%

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

   6. Applied egg-rr 31.8%

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

   7. Taylor expanded in d around -inf 62.4%

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

   if -5.00000000000023e-311 < d

   1. Initial program 72.6%

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

   3. Simplified 71.0%

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

      1. add-sqr-sqrt 70.9%

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

      2. pow2 70.9%

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

      3. sqrt-unprod 64.3%

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

      4. pow1/2 64.3%

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

      5. sqrt-pow1 64.3%

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

      6. metadata-eval 64.3%

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

   6. Applied egg-rr 64.3%

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

      1. *-un-lft-identity 64.3%

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

      2. pow-pow 64.4%

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

      3. metadata-eval 64.4%

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

      4. pow1/2 64.4%

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

      5. frac-times 53.1%

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

      6. pow2 53.1%

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

   8. Applied egg-rr 53.1%

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

      1. *-lft-identity 53.1%

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

      2. associate-/r* 53.9%

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

   10. Simplified 53.9%

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

       1. associate-/r* 53.1%

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

       2. div-inv 53.0%

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

       3. unpow2 53.0%

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

       4. add-cbrt-cube 41.2%

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

       5. pow1/3 40.6%

          \[\leadsto \sqrt{\left(d \cdot d\right) \cdot \color{blue}{{\left(\left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right) \cdot \frac{1}{h \cdot \ell}\right)}^{0.3333333333333333}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       6. pow3 40.6%

          \[\leadsto \sqrt{\left(d \cdot d\right) \cdot {\color{blue}{\left({\left(\frac{1}{h \cdot \ell}\right)}^{3}\right)}}^{0.3333333333333333}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       7. metadata-eval 40.6%

          \[\leadsto \sqrt{\left(d \cdot d\right) \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{\left(1.5 + 1.5\right)}}\right)}^{0.3333333333333333}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       8. pow-prod-up 40.6%

          \[\leadsto \sqrt{\left(d \cdot d\right) \cdot {\color{blue}{\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}}^{0.3333333333333333}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       9. pow-prod-down 48.8%

          \[\leadsto \sqrt{\left(d \cdot d\right) \cdot \color{blue}{\left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       10. swap-sqr 54.3%

           \[\leadsto \sqrt{\color{blue}{\left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right) \cdot \left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       11. sqrt-unprod 62.5%

           \[\leadsto \color{blue}{\left(\sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       12. add-sqr-sqrt 62.5%

           \[\leadsto \color{blue}{\left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       13. *-commutative 62.5%

           \[\leadsto \color{blue}{\left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot d\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   12. Applied egg-rr 79.7%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9 \cdot 10^{-127}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \left(-0.125 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \frac{-1}{\ell}\right)\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \left({h}^{-0.5} \cdot \left(d \cdot {\ell}^{-0.5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M}{d}\\ t_1 := 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M}{2}\right)}^{2}\right)\\ \mathbf{if}\;d \leq -1.12 \cdot 10^{-125}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \left(-0.125 \cdot \left(t\_0 \cdot \left(t\_0 \cdot \frac{-1}{\ell}\right)\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - t\_1\right) \cdot \left({\ell}^{-0.5} \cdot \frac{d}{\sqrt{h}}\right)\\ \end{array} \end{array} \]

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 (* D_m (/ M d)))
        (t_1 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M 2.0)) 2.0)))))
   (if (<= d -1.12e-125)
     (*
      (sqrt (/ d h))
      (* (sqrt (/ d l)) (- 1.0 (* h (* -0.125 (* t_0 (* t_0 (/ -1.0 l))))))))
     (if (<= d -5e-311)
       (* (* d (sqrt (/ 1.0 (* h l)))) (+ -1.0 t_1))
       (* (- 1.0 t_1) (* (pow l -0.5) (/ d (sqrt 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 = D_m * (M / d);
	double t_1 = 0.5 * ((h / l) * pow(((D_m / d) * (M / 2.0)), 2.0));
	double tmp;
	if (d <= -1.12e-125) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (h * (-0.125 * (t_0 * (t_0 * (-1.0 / l)))))));
	} else if (d <= -5e-311) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + t_1);
	} else {
		tmp = (1.0 - t_1) * (pow(l, -0.5) * (d / sqrt(h)));
	}
	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)
    t_1 = 0.5d0 * ((h / l) * (((d_m / d) * (m / 2.0d0)) ** 2.0d0))
    if (d <= (-1.12d-125)) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 - (h * ((-0.125d0) * (t_0 * (t_0 * ((-1.0d0) / l)))))))
    else if (d <= (-5d-311)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) + t_1)
    else
        tmp = (1.0d0 - t_1) * ((l ** (-0.5d0)) * (d / sqrt(h)))
    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 = D_m * (M / d);
	double t_1 = 0.5 * ((h / l) * Math.pow(((D_m / d) * (M / 2.0)), 2.0));
	double tmp;
	if (d <= -1.12e-125) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - (h * (-0.125 * (t_0 * (t_0 * (-1.0 / l)))))));
	} else if (d <= -5e-311) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 + t_1);
	} else {
		tmp = (1.0 - t_1) * (Math.pow(l, -0.5) * (d / Math.sqrt(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 = D_m * (M / d)
	t_1 = 0.5 * ((h / l) * math.pow(((D_m / d) * (M / 2.0)), 2.0))
	tmp = 0
	if d <= -1.12e-125:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - (h * (-0.125 * (t_0 * (t_0 * (-1.0 / l)))))))
	elif d <= -5e-311:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 + t_1)
	else:
		tmp = (1.0 - t_1) * (math.pow(l, -0.5) * (d / math.sqrt(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(D_m * Float64(M / d))
	t_1 = Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M / 2.0)) ^ 2.0)))
	tmp = 0.0
	if (d <= -1.12e-125)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(h * Float64(-0.125 * Float64(t_0 * Float64(t_0 * Float64(-1.0 / l))))))));
	elseif (d <= -5e-311)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 + t_1));
	else
		tmp = Float64(Float64(1.0 - t_1) * Float64((l ^ -0.5) * Float64(d / sqrt(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_m * (M / d);
	t_1 = 0.5 * ((h / l) * (((D_m / d) * (M / 2.0)) ^ 2.0));
	tmp = 0.0;
	if (d <= -1.12e-125)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (h * (-0.125 * (t_0 * (t_0 * (-1.0 / l)))))));
	elseif (d <= -5e-311)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + t_1);
	else
		tmp = (1.0 - t_1) * ((l ^ -0.5) * (d / sqrt(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[(D$95$m * N[(M / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[d, -1.12e-125], 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$0 * N[(t$95$0 * N[(-1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-311], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$1), $MachinePrecision] * N[(N[Power[l, -0.5], $MachinePrecision] * N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.11999999999999997e-125

   1. Initial program 76.7%

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

   3. Simplified 75.6%

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

   5. Taylor expanded in h around -inf 48.7%

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

      1. associate-*r* 48.7%

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

      2. neg-mul-1 48.7%

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

      3. sub-neg 48.7%

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

      4. distribute-lft-in 48.7%

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

   7. Simplified 78.2%

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

      1. pow1 78.2%

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

      2. associate-*r/ 78.2%

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

   9. Applied egg-rr 78.2%

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

       1. unpow1 78.2%

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

       2. distribute-lft-neg-in 78.2%

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

       3. distribute-rgt-neg-in 78.2%

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

       4. associate-/l* 78.2%

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

       5. distribute-lft-neg-in 78.2%

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

       6. metadata-eval 78.2%

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

       7. associate-*r/ 79.3%

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

       8. *-commutative 79.3%

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

   11. Simplified 79.3%

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

       1. div-inv 79.3%

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

       2. unpow2 79.3%

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

       3. associate-*l* 80.5%

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

       4. associate-/l* 80.6%

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

       5. associate-/l* 80.5%

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

   13. Applied egg-rr 80.5%

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

   if -1.11999999999999997e-125 < d < -5.00000000000023e-311

   1. Initial program 41.3%

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

   3. Simplified 41.4%

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

      1. add-sqr-sqrt 41.2%

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

      2. pow2 41.2%

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

      3. sqrt-unprod 31.8%

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

      4. pow1/2 31.8%

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

      5. sqrt-pow1 31.8%

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

      6. metadata-eval 31.8%

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

   6. Applied egg-rr 31.8%

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

   7. Taylor expanded in d around -inf 62.4%

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

   if -5.00000000000023e-311 < d

   1. Initial program 72.6%

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

   3. Simplified 71.0%

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

      1. add-sqr-sqrt 70.9%

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

      2. pow2 70.9%

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

      3. sqrt-unprod 64.3%

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

      4. pow1/2 64.3%

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

      5. sqrt-pow1 64.3%

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

      6. metadata-eval 64.3%

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

   6. Applied egg-rr 64.3%

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

      1. *-un-lft-identity 64.3%

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

      2. pow-pow 64.4%

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

      3. metadata-eval 64.4%

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

      4. pow1/2 64.4%

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

      5. frac-times 53.1%

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

      6. pow2 53.1%

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

   8. Applied egg-rr 53.1%

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

      1. *-lft-identity 53.1%

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

      2. associate-/r* 53.9%

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

   10. Simplified 53.9%

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

       1. div-inv 53.9%

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

       2. sqrt-prod 56.8%

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

       3. sqrt-div 64.9%

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

       4. unpow2 64.9%

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

       5. sqrt-prod 79.4%

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

       6. add-sqr-sqrt 79.7%

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

       7. inv-pow 79.7%

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

       8. sqrt-pow1 79.7%

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

       9. metadata-eval 79.7%

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

   12. Applied egg-rr 79.7%

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

       1. *-commutative 79.7%

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

   14. Simplified 79.7%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.12 \cdot 10^{-125}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \left(-0.125 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \frac{-1}{\ell}\right)\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \left({\ell}^{-0.5} \cdot \frac{d}{\sqrt{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 71.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M}{d}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -2.15 \cdot 10^{+117}:\\ \;\;\;\;t\_1 \cdot \left(t\_2 \cdot \left(1 + h \cdot \left(-0.125 \cdot \left(t\_0 \cdot \frac{t\_0}{\ell}\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1.6 \cdot 10^{-105}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{+178}:\\ \;\;\;\;t\_1 \cdot \left(t\_2 \cdot \left(1 - h \cdot \left(-0.125 \cdot \left(t\_0 \cdot \left(t\_0 \cdot \frac{-1}{\ell}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{1}{\sqrt{\ell}}\\ \end{array} \end{array} \]

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 (* D_m (/ M d))) (t_1 (sqrt (/ d h))) (t_2 (sqrt (/ d l))))
   (if (<= l -2.15e+117)
     (* t_1 (* t_2 (+ 1.0 (* h (* -0.125 (* t_0 (/ t_0 l)))))))
     (if (<= l -1.6e-105)
       (*
        (* d (sqrt (/ 1.0 (* h l))))
        (+ -1.0 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M 2.0)) 2.0)))))
       (if (<= l 2.9e+178)
         (* t_1 (* t_2 (- 1.0 (* h (* -0.125 (* t_0 (* t_0 (/ -1.0 l))))))))
         (* (/ d (sqrt h)) (/ 1.0 (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 = D_m * (M / d);
	double t_1 = sqrt((d / h));
	double t_2 = sqrt((d / l));
	double tmp;
	if (l <= -2.15e+117) {
		tmp = t_1 * (t_2 * (1.0 + (h * (-0.125 * (t_0 * (t_0 / l))))));
	} else if (l <= -1.6e-105) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + (0.5 * ((h / l) * pow(((D_m / d) * (M / 2.0)), 2.0))));
	} else if (l <= 2.9e+178) {
		tmp = t_1 * (t_2 * (1.0 - (h * (-0.125 * (t_0 * (t_0 * (-1.0 / l)))))));
	} else {
		tmp = (d / sqrt(h)) * (1.0 / 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) :: t_2
    real(8) :: tmp
    t_0 = d_m * (m / d)
    t_1 = sqrt((d / h))
    t_2 = sqrt((d / l))
    if (l <= (-2.15d+117)) then
        tmp = t_1 * (t_2 * (1.0d0 + (h * ((-0.125d0) * (t_0 * (t_0 / l))))))
    else if (l <= (-1.6d-105)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) + (0.5d0 * ((h / l) * (((d_m / d) * (m / 2.0d0)) ** 2.0d0))))
    else if (l <= 2.9d+178) then
        tmp = t_1 * (t_2 * (1.0d0 - (h * ((-0.125d0) * (t_0 * (t_0 * ((-1.0d0) / l)))))))
    else
        tmp = (d / sqrt(h)) * (1.0d0 / 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 = D_m * (M / d);
	double t_1 = Math.sqrt((d / h));
	double t_2 = Math.sqrt((d / l));
	double tmp;
	if (l <= -2.15e+117) {
		tmp = t_1 * (t_2 * (1.0 + (h * (-0.125 * (t_0 * (t_0 / l))))));
	} else if (l <= -1.6e-105) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 + (0.5 * ((h / l) * Math.pow(((D_m / d) * (M / 2.0)), 2.0))));
	} else if (l <= 2.9e+178) {
		tmp = t_1 * (t_2 * (1.0 - (h * (-0.125 * (t_0 * (t_0 * (-1.0 / l)))))));
	} else {
		tmp = (d / Math.sqrt(h)) * (1.0 / 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 = D_m * (M / d)
	t_1 = math.sqrt((d / h))
	t_2 = math.sqrt((d / l))
	tmp = 0
	if l <= -2.15e+117:
		tmp = t_1 * (t_2 * (1.0 + (h * (-0.125 * (t_0 * (t_0 / l))))))
	elif l <= -1.6e-105:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 + (0.5 * ((h / l) * math.pow(((D_m / d) * (M / 2.0)), 2.0))))
	elif l <= 2.9e+178:
		tmp = t_1 * (t_2 * (1.0 - (h * (-0.125 * (t_0 * (t_0 * (-1.0 / l)))))))
	else:
		tmp = (d / math.sqrt(h)) * (1.0 / 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))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -2.15e+117)
		tmp = Float64(t_1 * Float64(t_2 * Float64(1.0 + Float64(h * Float64(-0.125 * Float64(t_0 * Float64(t_0 / l)))))));
	elseif (l <= -1.6e-105)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 + Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M / 2.0)) ^ 2.0)))));
	elseif (l <= 2.9e+178)
		tmp = Float64(t_1 * Float64(t_2 * Float64(1.0 - Float64(h * Float64(-0.125 * Float64(t_0 * Float64(t_0 * Float64(-1.0 / l))))))));
	else
		tmp = Float64(Float64(d / sqrt(h)) * Float64(1.0 / 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);
	t_1 = sqrt((d / h));
	t_2 = sqrt((d / l));
	tmp = 0.0;
	if (l <= -2.15e+117)
		tmp = t_1 * (t_2 * (1.0 + (h * (-0.125 * (t_0 * (t_0 / l))))));
	elseif (l <= -1.6e-105)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + (0.5 * ((h / l) * (((D_m / d) * (M / 2.0)) ^ 2.0))));
	elseif (l <= 2.9e+178)
		tmp = t_1 * (t_2 * (1.0 - (h * (-0.125 * (t_0 * (t_0 * (-1.0 / l)))))));
	else
		tmp = (d / sqrt(h)) * (1.0 / 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[(D$95$m * N[(M / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -2.15e+117], N[(t$95$1 * N[(t$95$2 * N[(1.0 + N[(h * N[(-0.125 * N[(t$95$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1.6e-105], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.9e+178], N[(t$95$1 * N[(t$95$2 * N[(1.0 - N[(h * N[(-0.125 * N[(t$95$0 * N[(t$95$0 * N[(-1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.14999999999999999e117

   1. Initial program 59.4%

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

   3. Simplified 59.5%

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

   5. Taylor expanded in h around -inf 27.2%

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

      1. associate-*r* 27.2%

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

      2. neg-mul-1 27.2%

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

      3. sub-neg 27.2%

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

      4. distribute-lft-in 27.2%

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

   7. Simplified 59.5%

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

      1. pow1 59.5%

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

      2. associate-*r/ 59.5%

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

   9. Applied egg-rr 59.5%

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

       1. unpow1 59.5%

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

       2. distribute-lft-neg-in 59.5%

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

       3. distribute-rgt-neg-in 59.5%

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

       4. associate-/l* 59.5%

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

       5. distribute-lft-neg-in 59.5%

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

       6. metadata-eval 59.5%

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

       7. associate-*r/ 59.5%

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

       8. *-commutative 59.5%

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

   11. Simplified 59.5%

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

       1. unpow2 59.5%

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

       2. associate-/l* 62.1%

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

       3. associate-/l* 62.1%

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

       4. associate-/l* 62.1%

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

   13. Applied egg-rr 62.1%

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

   if -2.14999999999999999e117 < l < -1.59999999999999991e-105

   1. Initial program 61.0%

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

   3. Simplified 59.1%

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

      1. add-sqr-sqrt 58.9%

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

      2. pow2 58.9%

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

      3. sqrt-unprod 42.5%

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

      4. pow1/2 42.5%

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

      5. sqrt-pow1 42.5%

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

      6. metadata-eval 42.5%

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

   6. Applied egg-rr 42.5%

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

   7. Taylor expanded in d around -inf 79.2%

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

   if -1.59999999999999991e-105 < l < 2.9e178

   1. Initial program 77.4%

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

   3. Simplified 76.6%

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

   5. Taylor expanded in h around -inf 52.7%

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

      1. associate-*r* 52.7%

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

      2. neg-mul-1 52.7%

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

      3. sub-neg 52.7%

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

      4. distribute-lft-in 52.7%

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

   7. Simplified 79.9%

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

      1. pow1 79.9%

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

      2. associate-*r/ 79.9%

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

   9. Applied egg-rr 79.9%

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

       1. unpow1 79.9%

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

       2. distribute-lft-neg-in 79.9%

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

       3. distribute-rgt-neg-in 79.9%

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

       4. associate-/l* 79.9%

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

       5. distribute-lft-neg-in 79.9%

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

       6. metadata-eval 79.9%

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

       7. associate-*r/ 80.6%

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

       8. *-commutative 80.6%

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

   11. Simplified 80.6%

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

       1. div-inv 80.6%

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

       2. unpow2 80.6%

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

       3. associate-*l* 81.4%

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

       4. associate-/l* 81.4%

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

       5. associate-/l* 81.4%

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

   13. Applied egg-rr 81.4%

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

   if 2.9e178 < l

   1. Initial program 57.0%

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

   3. Simplified 54.1%

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

   5. Taylor expanded in d around inf 59.9%

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

      1. *-un-lft-identity 59.9%

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

      2. pow1/2 59.9%

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

      3. inv-pow 59.9%

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

      4. pow-pow 59.8%

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

      5. metadata-eval 59.8%

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

   7. Applied egg-rr 59.8%

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

      1. *-lft-identity 59.8%

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

   9. Simplified 59.8%

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

       1. metadata-eval 59.8%

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

       2. metadata-eval 59.8%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\left(-1 \cdot \color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}\right)} \]

       3. pow-pow 59.9%

          \[\leadsto d \cdot \color{blue}{{\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       4. inv-pow 59.9%

          \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

       5. pow-pow 52.0%

          \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       6. add-sqr-sqrt 52.0%

          \[\leadsto \color{blue}{\sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}}} \]

       7. sqrt-unprod 41.8%

          \[\leadsto \color{blue}{\sqrt{\left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right) \cdot \left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       8. swap-sqr 41.5%

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot d\right) \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       9. unpow2 41.5%

          \[\leadsto \sqrt{\color{blue}{{d}^{2}} \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       10. pow-pow 41.9%

           \[\leadsto \sqrt{{d}^{2} \cdot \left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       11. pow-pow 45.9%

           \[\leadsto \sqrt{{d}^{2} \cdot \left({\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}\right)} \]

       12. pow-sqr 45.9%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \left(1.5 \cdot 0.3333333333333333\right)\right)}}} \]

       13. metadata-eval 45.9%

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

       14. metadata-eval 45.9%

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

       15. pow1 45.9%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \]

       16. div-inv 45.9%

           \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}} \]

       17. associate-/r* 43.4%

           \[\leadsto \sqrt{\color{blue}{\frac{\frac{{d}^{2}}{h}}{\ell}}} \]

   11. Applied egg-rr 70.8%

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

       1. associate-/r/ 71.0%

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

   13. Simplified 71.0%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.15 \cdot 10^{+117}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(-0.125 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \frac{D \cdot \frac{M}{d}}{\ell}\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1.6 \cdot 10^{-105}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{+178}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \left(-0.125 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \frac{-1}{\ell}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{1}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 71.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M}{d}\\ t_1 := \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(-0.125 \cdot \left(t\_0 \cdot \frac{t\_0}{\ell}\right)\right)\right)\right)\\ \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+115}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -6 \cdot 10^{-105}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{+176}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{1}{\sqrt{\ell}}\\ \end{array} \end{array} \]

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 (* D_m (/ M d)))
        (t_1
         (*
          (sqrt (/ d h))
          (* (sqrt (/ d l)) (+ 1.0 (* h (* -0.125 (* t_0 (/ t_0 l)))))))))
   (if (<= l -1.4e+115)
     t_1
     (if (<= l -6e-105)
       (*
        (* d (sqrt (/ 1.0 (* h l))))
        (+ -1.0 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M 2.0)) 2.0)))))
       (if (<= l 5.2e+176) t_1 (* (/ d (sqrt h)) (/ 1.0 (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 = D_m * (M / d);
	double t_1 = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + (h * (-0.125 * (t_0 * (t_0 / l))))));
	double tmp;
	if (l <= -1.4e+115) {
		tmp = t_1;
	} else if (l <= -6e-105) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + (0.5 * ((h / l) * pow(((D_m / d) * (M / 2.0)), 2.0))));
	} else if (l <= 5.2e+176) {
		tmp = t_1;
	} else {
		tmp = (d / sqrt(h)) * (1.0 / 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)
    t_1 = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 + (h * ((-0.125d0) * (t_0 * (t_0 / l))))))
    if (l <= (-1.4d+115)) then
        tmp = t_1
    else if (l <= (-6d-105)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) + (0.5d0 * ((h / l) * (((d_m / d) * (m / 2.0d0)) ** 2.0d0))))
    else if (l <= 5.2d+176) then
        tmp = t_1
    else
        tmp = (d / sqrt(h)) * (1.0d0 / 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 = D_m * (M / d);
	double t_1 = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 + (h * (-0.125 * (t_0 * (t_0 / l))))));
	double tmp;
	if (l <= -1.4e+115) {
		tmp = t_1;
	} else if (l <= -6e-105) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 + (0.5 * ((h / l) * Math.pow(((D_m / d) * (M / 2.0)), 2.0))));
	} else if (l <= 5.2e+176) {
		tmp = t_1;
	} else {
		tmp = (d / Math.sqrt(h)) * (1.0 / 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 = D_m * (M / d)
	t_1 = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 + (h * (-0.125 * (t_0 * (t_0 / l))))))
	tmp = 0
	if l <= -1.4e+115:
		tmp = t_1
	elif l <= -6e-105:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 + (0.5 * ((h / l) * math.pow(((D_m / d) * (M / 2.0)), 2.0))))
	elif l <= 5.2e+176:
		tmp = t_1
	else:
		tmp = (d / math.sqrt(h)) * (1.0 / 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))
	t_1 = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(h * Float64(-0.125 * Float64(t_0 * Float64(t_0 / l)))))))
	tmp = 0.0
	if (l <= -1.4e+115)
		tmp = t_1;
	elseif (l <= -6e-105)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 + Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M / 2.0)) ^ 2.0)))));
	elseif (l <= 5.2e+176)
		tmp = t_1;
	else
		tmp = Float64(Float64(d / sqrt(h)) * Float64(1.0 / 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);
	t_1 = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + (h * (-0.125 * (t_0 * (t_0 / l))))));
	tmp = 0.0;
	if (l <= -1.4e+115)
		tmp = t_1;
	elseif (l <= -6e-105)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + (0.5 * ((h / l) * (((D_m / d) * (M / 2.0)) ^ 2.0))));
	elseif (l <= 5.2e+176)
		tmp = t_1;
	else
		tmp = (d / sqrt(h)) * (1.0 / 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[(D$95$m * N[(M / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.4e+115], t$95$1, If[LessEqual[l, -6e-105], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.2e+176], t$95$1, N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.4e115 or -6.0000000000000002e-105 < l < 5.19999999999999981e176

   1. Initial program 73.6%

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

   3. Simplified 73.0%

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

   5. Taylor expanded in h around -inf 47.3%

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

      1. associate-*r* 47.3%

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

      2. neg-mul-1 47.3%

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

      3. sub-neg 47.3%

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

      4. distribute-lft-in 47.3%

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

   7. Simplified 75.5%

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

      1. pow1 75.5%

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

      2. associate-*r/ 75.5%

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

   9. Applied egg-rr 75.5%

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

       1. unpow1 75.5%

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

       2. distribute-lft-neg-in 75.5%

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

       3. distribute-rgt-neg-in 75.5%

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

       4. associate-/l* 75.5%

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

       5. distribute-lft-neg-in 75.5%

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

       6. metadata-eval 75.5%

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

       7. associate-*r/ 76.2%

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

       8. *-commutative 76.2%

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

   11. Simplified 76.2%

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

       1. unpow2 76.2%

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

       2. associate-/l* 77.3%

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

       3. associate-/l* 77.3%

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

       4. associate-/l* 77.3%

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

   13. Applied egg-rr 77.3%

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

   if -1.4e115 < l < -6.0000000000000002e-105

   1. Initial program 61.0%

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

   3. Simplified 59.1%

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

      1. add-sqr-sqrt 58.9%

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

      2. pow2 58.9%

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

      3. sqrt-unprod 42.5%

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

      4. pow1/2 42.5%

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

      5. sqrt-pow1 42.5%

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

      6. metadata-eval 42.5%

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

   6. Applied egg-rr 42.5%

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

   7. Taylor expanded in d around -inf 79.2%

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

   if 5.19999999999999981e176 < l

   1. Initial program 57.0%

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

   3. Simplified 54.1%

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

   5. Taylor expanded in d around inf 59.9%

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

      1. *-un-lft-identity 59.9%

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

      2. pow1/2 59.9%

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

      3. inv-pow 59.9%

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

      4. pow-pow 59.8%

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

      5. metadata-eval 59.8%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 59.8%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 59.8%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 59.8%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. metadata-eval 59.8%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{\left(-1 \cdot 0.5\right)}} \]

       2. metadata-eval 59.8%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\left(-1 \cdot \color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}\right)} \]

       3. pow-pow 59.9%

          \[\leadsto d \cdot \color{blue}{{\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       4. inv-pow 59.9%

          \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

       5. pow-pow 52.0%

          \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       6. add-sqr-sqrt 52.0%

          \[\leadsto \color{blue}{\sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}}} \]

       7. sqrt-unprod 41.8%

          \[\leadsto \color{blue}{\sqrt{\left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right) \cdot \left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       8. swap-sqr 41.5%

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot d\right) \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       9. unpow2 41.5%

          \[\leadsto \sqrt{\color{blue}{{d}^{2}} \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       10. pow-pow 41.9%

           \[\leadsto \sqrt{{d}^{2} \cdot \left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       11. pow-pow 45.9%

           \[\leadsto \sqrt{{d}^{2} \cdot \left({\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}\right)} \]

       12. pow-sqr 45.9%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \left(1.5 \cdot 0.3333333333333333\right)\right)}}} \]

       13. metadata-eval 45.9%

           \[\leadsto \sqrt{{d}^{2} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \color{blue}{0.5}\right)}} \]

       14. metadata-eval 45.9%

           \[\leadsto \sqrt{{d}^{2} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{1}}} \]

       15. pow1 45.9%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \]

       16. div-inv 45.9%

           \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}} \]

       17. associate-/r* 43.4%

           \[\leadsto \sqrt{\color{blue}{\frac{\frac{{d}^{2}}{h}}{\ell}}} \]

   11. Applied egg-rr 70.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\frac{d}{\sqrt{h}}}}} \]
   12. Step-by-step derivation

       1. associate-/r/ 71.0%

          \[\leadsto \color{blue}{\frac{1}{\sqrt{\ell}} \cdot \frac{d}{\sqrt{h}}} \]

   13. Simplified 71.0%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{\ell}} \cdot \frac{d}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+115}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(-0.125 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \frac{D \cdot \frac{M}{d}}{\ell}\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -6 \cdot 10^{-105}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{+176}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(-0.125 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \frac{D \cdot \frac{M}{d}}{\ell}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h}} \cdot \frac{1}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 62.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := 1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M}{2}\right)}^{2}\right)\\ \mathbf{if}\;\ell \leq -2.9 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -2.4 \cdot 10^{-301}:\\ \;\;\;\;t\_0 \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+130}:\\ \;\;\;\;t\_0 \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

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 (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M 2.0)) 2.0))))))
   (if (<= l -2.9e+41)
     (* (sqrt (/ d l)) (sqrt (/ d h)))
     (if (<= l -2.4e-301)
       (* t_0 (sqrt (* d (/ d (* h l)))))
       (if (<= l 8e+130)
         (* t_0 (* d (pow (* h l) -0.5)))
         (/ (/ d (sqrt 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 = 1.0 - (0.5 * ((h / l) * pow(((D_m / d) * (M / 2.0)), 2.0)));
	double tmp;
	if (l <= -2.9e+41) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (l <= -2.4e-301) {
		tmp = t_0 * sqrt((d * (d / (h * l))));
	} else if (l <= 8e+130) {
		tmp = t_0 * (d * pow((h * l), -0.5));
	} else {
		tmp = (d / sqrt(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 = 1.0d0 - (0.5d0 * ((h / l) * (((d_m / d) * (m / 2.0d0)) ** 2.0d0)))
    if (l <= (-2.9d+41)) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else if (l <= (-2.4d-301)) then
        tmp = t_0 * sqrt((d * (d / (h * l))))
    else if (l <= 8d+130) then
        tmp = t_0 * (d * ((h * l) ** (-0.5d0)))
    else
        tmp = (d / sqrt(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 = 1.0 - (0.5 * ((h / l) * Math.pow(((D_m / d) * (M / 2.0)), 2.0)));
	double tmp;
	if (l <= -2.9e+41) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (l <= -2.4e-301) {
		tmp = t_0 * Math.sqrt((d * (d / (h * l))));
	} else if (l <= 8e+130) {
		tmp = t_0 * (d * Math.pow((h * l), -0.5));
	} else {
		tmp = (d / Math.sqrt(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 = 1.0 - (0.5 * ((h / l) * math.pow(((D_m / d) * (M / 2.0)), 2.0)))
	tmp = 0
	if l <= -2.9e+41:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	elif l <= -2.4e-301:
		tmp = t_0 * math.sqrt((d * (d / (h * l))))
	elif l <= 8e+130:
		tmp = t_0 * (d * math.pow((h * l), -0.5))
	else:
		tmp = (d / math.sqrt(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(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M / 2.0)) ^ 2.0))))
	tmp = 0.0
	if (l <= -2.9e+41)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (l <= -2.4e-301)
		tmp = Float64(t_0 * sqrt(Float64(d * Float64(d / Float64(h * l)))));
	elseif (l <= 8e+130)
		tmp = Float64(t_0 * Float64(d * (Float64(h * l) ^ -0.5)));
	else
		tmp = Float64(Float64(d / sqrt(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 = 1.0 - (0.5 * ((h / l) * (((D_m / d) * (M / 2.0)) ^ 2.0)));
	tmp = 0.0;
	if (l <= -2.9e+41)
		tmp = sqrt((d / l)) * sqrt((d / h));
	elseif (l <= -2.4e-301)
		tmp = t_0 * sqrt((d * (d / (h * l))));
	elseif (l <= 8e+130)
		tmp = t_0 * (d * ((h * l) ^ -0.5));
	else
		tmp = (d / sqrt(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[(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]}, If[LessEqual[l, -2.9e+41], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2.4e-301], N[(t$95$0 * N[Sqrt[N[(d * N[(d / N[(h * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8e+130], N[(t$95$0 * N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.89999999999999988e41

   1. Initial program 55.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 55.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 49.0%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

   if -2.89999999999999988e41 < l < -2.39999999999999991e-301

   1. Initial program 72.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 71.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 70.7%

         \[\leadsto \color{blue}{\left(\sqrt{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \cdot \sqrt{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. pow2 70.7%

         \[\leadsto \color{blue}{{\left(\sqrt{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}\right)}^{2}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. sqrt-unprod 61.7%

         \[\leadsto {\left(\sqrt{\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}}}\right)}^{2} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. pow1/2 61.7%

         \[\leadsto {\left(\sqrt{\color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}}}\right)}^{2} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. sqrt-pow1 61.7%

         \[\leadsto {\color{blue}{\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. metadata-eval 61.7%

         \[\leadsto {\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{0.25}}\right)}^{2} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 61.7%

      \[\leadsto \color{blue}{{\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.25}\right)}^{2}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. *-un-lft-identity 61.7%

         \[\leadsto \color{blue}{\left(1 \cdot {\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.25}\right)}^{2}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. pow-pow 61.9%

         \[\leadsto \left(1 \cdot \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(0.25 \cdot 2\right)}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. metadata-eval 61.9%

         \[\leadsto \left(1 \cdot {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. pow1/2 61.9%

         \[\leadsto \left(1 \cdot \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. frac-times 48.8%

         \[\leadsto \left(1 \cdot \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. pow2 48.8%

         \[\leadsto \left(1 \cdot \sqrt{\frac{\color{blue}{{d}^{2}}}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. Applied egg-rr 48.8%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{{d}^{2}}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   9. Step-by-step derivation

      1. *-lft-identity 48.8%

         \[\leadsto \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. associate-/r* 47.3%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{{d}^{2}}{h}}{\ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   10. Simplified 47.3%

       \[\leadsto \color{blue}{\sqrt{\frac{\frac{{d}^{2}}{h}}{\ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   11. Step-by-step derivation

       1. associate-/r* 48.8%

          \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       2. unpow2 48.8%

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       3. associate-/l* 59.5%

          \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{h \cdot \ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       4. *-commutative 59.5%

          \[\leadsto \sqrt{d \cdot \frac{d}{\color{blue}{\ell \cdot h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   12. Applied egg-rr 59.5%

       \[\leadsto \sqrt{\color{blue}{d \cdot \frac{d}{\ell \cdot h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   if -2.39999999999999991e-301 < l < 8.0000000000000005e130

   1. Initial program 76.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 75.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 75.4%

         \[\leadsto \color{blue}{\left(\sqrt{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \cdot \sqrt{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. pow2 75.4%

         \[\leadsto \color{blue}{{\left(\sqrt{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}\right)}^{2}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. sqrt-unprod 68.0%

         \[\leadsto {\left(\sqrt{\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}}}\right)}^{2} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. pow1/2 68.0%

         \[\leadsto {\left(\sqrt{\color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}}}\right)}^{2} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. sqrt-pow1 68.1%

         \[\leadsto {\color{blue}{\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. metadata-eval 68.1%

         \[\leadsto {\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{0.25}}\right)}^{2} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 68.1%

      \[\leadsto \color{blue}{{\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.25}\right)}^{2}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around 0 74.4%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   8. Step-by-step derivation

      1. unpow-1 74.4%

         \[\leadsto \left(d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. metadata-eval 74.4%

         \[\leadsto \left(d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. pow-sqr 74.4%

         \[\leadsto \left(d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. rem-sqrt-square 74.4%

         \[\leadsto \left(d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. rem-square-sqrt 74.3%

         \[\leadsto \left(d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. fabs-sqr 74.3%

         \[\leadsto \left(d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      7. rem-square-sqrt 74.4%

         \[\leadsto \left(d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   9. Simplified 74.4%

      \[\leadsto \color{blue}{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   if 8.0000000000000005e130 < l

   1. Initial program 61.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 59.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 59.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 59.5%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. pow1/2 59.5%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right) \]

      3. inv-pow 59.5%

         \[\leadsto d \cdot \left(1 \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right) \]

      4. pow-pow 59.5%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right) \]

      5. metadata-eval 59.5%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 59.5%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 59.5%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 59.5%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. metadata-eval 59.5%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{\left(-1 \cdot 0.5\right)}} \]

       2. metadata-eval 59.5%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\left(-1 \cdot \color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}\right)} \]

       3. pow-pow 59.5%

          \[\leadsto d \cdot \color{blue}{{\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       4. inv-pow 59.5%

          \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

       5. pow-pow 52.4%

          \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       6. add-sqr-sqrt 52.4%

          \[\leadsto \color{blue}{\sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}}} \]

       7. sqrt-unprod 44.0%

          \[\leadsto \color{blue}{\sqrt{\left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right) \cdot \left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       8. swap-sqr 43.7%

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot d\right) \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       9. unpow2 43.7%

          \[\leadsto \sqrt{\color{blue}{{d}^{2}} \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       10. pow-pow 44.1%

           \[\leadsto \sqrt{{d}^{2} \cdot \left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       11. pow-pow 48.0%

           \[\leadsto \sqrt{{d}^{2} \cdot \left({\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}\right)} \]

       12. pow-sqr 47.9%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \left(1.5 \cdot 0.3333333333333333\right)\right)}}} \]

       13. metadata-eval 47.9%

           \[\leadsto \sqrt{{d}^{2} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \color{blue}{0.5}\right)}} \]

       14. metadata-eval 47.9%

           \[\leadsto \sqrt{{d}^{2} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{1}}} \]

       15. pow1 47.9%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \]

       16. div-inv 48.0%

           \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}} \]

       17. associate-/r* 45.9%

           \[\leadsto \sqrt{\color{blue}{\frac{\frac{{d}^{2}}{h}}{\ell}}} \]

   11. Applied egg-rr 71.0%

       \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.9 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -2.4 \cdot 10^{-301}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+130}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 58.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.9 \cdot 10^{-214}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt{-1 + \mathsf{fma}\left(\ell, h, 1\right)}}\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{+132}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

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 -4.9e-214)
   (* (- d) (sqrt (/ (/ 1.0 l) h)))
   (if (<= l -5e-310)
     (/ d (sqrt (+ -1.0 (fma l h 1.0))))
     (if (<= l 1.25e+132)
       (*
        (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M 2.0)) 2.0))))
        (* d (pow (* h l) -0.5)))
       (/ (/ d (sqrt 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 <= -4.9e-214) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else if (l <= -5e-310) {
		tmp = d / sqrt((-1.0 + fma(l, h, 1.0)));
	} else if (l <= 1.25e+132) {
		tmp = (1.0 - (0.5 * ((h / l) * pow(((D_m / d) * (M / 2.0)), 2.0)))) * (d * pow((h * l), -0.5));
	} else {
		tmp = (d / sqrt(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 (l <= -4.9e-214)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	elseif (l <= -5e-310)
		tmp = Float64(d / sqrt(Float64(-1.0 + fma(l, h, 1.0))));
	elseif (l <= 1.25e+132)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M / 2.0)) ^ 2.0)))) * Float64(d * (Float64(h * l) ^ -0.5)));
	else
		tmp = Float64(Float64(d / sqrt(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[l, -4.9e-214], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(d / N[Sqrt[N[(-1.0 + N[(l * h + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.25e+132], 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[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.89999999999999968e-214

   1. Initial program 61.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 60.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 7.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   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 45.0%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. neg-mul-1 45.0%

         \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. *-commutative 45.0%

         \[\leadsto d \cdot \left(-\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \]

      6. associate-/r* 45.2%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \]

   8. Simplified 45.2%

      \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \]

   if -4.89999999999999968e-214 < l < -4.999999999999985e-310

   1. Initial program 78.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 78.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 27.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 27.9%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. pow1/2 27.9%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right) \]

      3. inv-pow 27.9%

         \[\leadsto d \cdot \left(1 \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right) \]

      4. pow-pow 27.9%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right) \]

      5. metadata-eval 27.9%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 27.9%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 27.9%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 27.9%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. metadata-eval 27.9%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{\left(-1 \cdot 0.5\right)}} \]

       2. metadata-eval 27.9%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\left(-1 \cdot \color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}\right)} \]

       3. pow-pow 27.9%

          \[\leadsto d \cdot \color{blue}{{\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       4. inv-pow 27.9%

          \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

       5. pow-pow 48.9%

          \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       6. add-sqr-sqrt 0.0%

          \[\leadsto \color{blue}{\sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}}} \]

       7. sqrt-unprod 9.4%

          \[\leadsto \color{blue}{\sqrt{\left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right) \cdot \left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       8. swap-sqr 9.4%

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot d\right) \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       9. unpow2 9.4%

          \[\leadsto \sqrt{\color{blue}{{d}^{2}} \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       10. pow-pow 9.4%

           \[\leadsto \sqrt{{d}^{2} \cdot \left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       11. pow-pow 9.6%

           \[\leadsto \sqrt{{d}^{2} \cdot \left({\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}\right)} \]

       12. pow-sqr 9.6%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \left(1.5 \cdot 0.3333333333333333\right)\right)}}} \]

       13. metadata-eval 9.6%

           \[\leadsto \sqrt{{d}^{2} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \color{blue}{0.5}\right)}} \]

       14. metadata-eval 9.6%

           \[\leadsto \sqrt{{d}^{2} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{1}}} \]

       15. pow1 9.6%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \]

       16. div-inv 9.6%

           \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}} \]

       17. sqrt-div 13.7%

           \[\leadsto \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}}} \]

   11. Applied egg-rr 27.9%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   12. Step-by-step derivation

       1. expm1-log1p-u 27.9%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot h\right)\right)}}} \]

       2. expm1-undefine 70.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{e^{\mathsf{log1p}\left(\ell \cdot h\right)} - 1}}} \]

   13. Applied egg-rr 70.0%

       \[\leadsto \frac{d}{\sqrt{\color{blue}{e^{\mathsf{log1p}\left(\ell \cdot h\right)} - 1}}} \]
   14. Step-by-step derivation

       1. sub-neg 70.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{e^{\mathsf{log1p}\left(\ell \cdot h\right)} + \left(-1\right)}}} \]

       2. metadata-eval 70.0%

          \[\leadsto \frac{d}{\sqrt{e^{\mathsf{log1p}\left(\ell \cdot h\right)} + \color{blue}{-1}}} \]

       3. +-commutative 70.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{-1 + e^{\mathsf{log1p}\left(\ell \cdot h\right)}}}} \]

       4. log1p-undefine 70.0%

          \[\leadsto \frac{d}{\sqrt{-1 + e^{\color{blue}{\log \left(1 + \ell \cdot h\right)}}}} \]

       5. rem-exp-log 70.0%

          \[\leadsto \frac{d}{\sqrt{-1 + \color{blue}{\left(1 + \ell \cdot h\right)}}} \]

       6. +-commutative 70.0%

          \[\leadsto \frac{d}{\sqrt{-1 + \color{blue}{\left(\ell \cdot h + 1\right)}}} \]

       7. fma-define 70.0%

          \[\leadsto \frac{d}{\sqrt{-1 + \color{blue}{\mathsf{fma}\left(\ell, h, 1\right)}}} \]

   15. Simplified 70.0%

       \[\leadsto \frac{d}{\sqrt{\color{blue}{-1 + \mathsf{fma}\left(\ell, h, 1\right)}}} \]

   if -4.999999999999985e-310 < l < 1.25e132

   1. Initial program 77.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 76.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 76.2%

         \[\leadsto \color{blue}{\left(\sqrt{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \cdot \sqrt{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. pow2 76.2%

         \[\leadsto \color{blue}{{\left(\sqrt{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}\right)}^{2}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. sqrt-unprod 68.8%

         \[\leadsto {\left(\sqrt{\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}}}\right)}^{2} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. pow1/2 68.8%

         \[\leadsto {\left(\sqrt{\color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}}}\right)}^{2} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. sqrt-pow1 68.8%

         \[\leadsto {\color{blue}{\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. metadata-eval 68.8%

         \[\leadsto {\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{0.25}}\right)}^{2} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 68.8%

      \[\leadsto \color{blue}{{\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.25}\right)}^{2}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around 0 75.2%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   8. Step-by-step derivation

      1. unpow-1 75.2%

         \[\leadsto \left(d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. metadata-eval 75.2%

         \[\leadsto \left(d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. pow-sqr 75.3%

         \[\leadsto \left(d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. rem-sqrt-square 75.3%

         \[\leadsto \left(d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. rem-square-sqrt 75.1%

         \[\leadsto \left(d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. fabs-sqr 75.1%

         \[\leadsto \left(d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      7. rem-square-sqrt 75.3%

         \[\leadsto \left(d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   9. Simplified 75.3%

      \[\leadsto \color{blue}{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   if 1.25e132 < l

   1. Initial program 61.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 59.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 59.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 59.5%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. pow1/2 59.5%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right) \]

      3. inv-pow 59.5%

         \[\leadsto d \cdot \left(1 \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right) \]

      4. pow-pow 59.5%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right) \]

      5. metadata-eval 59.5%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 59.5%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 59.5%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 59.5%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. metadata-eval 59.5%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{\left(-1 \cdot 0.5\right)}} \]

       2. metadata-eval 59.5%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\left(-1 \cdot \color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}\right)} \]

       3. pow-pow 59.5%

          \[\leadsto d \cdot \color{blue}{{\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       4. inv-pow 59.5%

          \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

       5. pow-pow 52.4%

          \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       6. add-sqr-sqrt 52.4%

          \[\leadsto \color{blue}{\sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}}} \]

       7. sqrt-unprod 44.0%

          \[\leadsto \color{blue}{\sqrt{\left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right) \cdot \left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       8. swap-sqr 43.7%

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot d\right) \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       9. unpow2 43.7%

          \[\leadsto \sqrt{\color{blue}{{d}^{2}} \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       10. pow-pow 44.1%

           \[\leadsto \sqrt{{d}^{2} \cdot \left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       11. pow-pow 48.0%

           \[\leadsto \sqrt{{d}^{2} \cdot \left({\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}\right)} \]

       12. pow-sqr 47.9%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \left(1.5 \cdot 0.3333333333333333\right)\right)}}} \]

       13. metadata-eval 47.9%

           \[\leadsto \sqrt{{d}^{2} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \color{blue}{0.5}\right)}} \]

       14. metadata-eval 47.9%

           \[\leadsto \sqrt{{d}^{2} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{1}}} \]

       15. pow1 47.9%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \]

       16. div-inv 48.0%

           \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}} \]

       17. associate-/r* 45.9%

           \[\leadsto \sqrt{\color{blue}{\frac{\frac{{d}^{2}}{h}}{\ell}}} \]

   11. Applied egg-rr 71.0%

       \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.9 \cdot 10^{-214}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt{-1 + \mathsf{fma}\left(\ell, h, 1\right)}}\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{+132}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 68.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M}{2}\right)}^{2}\right)\\ t_1 := 1 - t\_0\\ \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + t\_0\right)\\ \mathbf{elif}\;h \leq 1.6 \cdot 10^{+155}:\\ \;\;\;\;t\_1 \cdot \left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \end{array} \end{array} \]

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 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M 2.0)) 2.0))))
        (t_1 (- 1.0 t_0)))
   (if (<= h -5e-310)
     (* (* d (sqrt (/ 1.0 (* h l)))) (+ -1.0 t_0))
     (if (<= h 1.6e+155)
       (* t_1 (* d (sqrt (/ (/ 1.0 l) h))))
       (* t_1 (sqrt (* (/ d l) (/ d 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 = 0.5 * ((h / l) * pow(((D_m / d) * (M / 2.0)), 2.0));
	double t_1 = 1.0 - t_0;
	double tmp;
	if (h <= -5e-310) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + t_0);
	} else if (h <= 1.6e+155) {
		tmp = t_1 * (d * sqrt(((1.0 / l) / h)));
	} else {
		tmp = t_1 * sqrt(((d / l) * (d / h)));
	}
	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 = 0.5d0 * ((h / l) * (((d_m / d) * (m / 2.0d0)) ** 2.0d0))
    t_1 = 1.0d0 - t_0
    if (h <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((-1.0d0) + t_0)
    else if (h <= 1.6d+155) then
        tmp = t_1 * (d * sqrt(((1.0d0 / l) / h)))
    else
        tmp = t_1 * sqrt(((d / l) * (d / h)))
    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 = 0.5 * ((h / l) * Math.pow(((D_m / d) * (M / 2.0)), 2.0));
	double t_1 = 1.0 - t_0;
	double tmp;
	if (h <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * (-1.0 + t_0);
	} else if (h <= 1.6e+155) {
		tmp = t_1 * (d * Math.sqrt(((1.0 / l) / h)));
	} else {
		tmp = t_1 * Math.sqrt(((d / l) * (d / 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 = 0.5 * ((h / l) * math.pow(((D_m / d) * (M / 2.0)), 2.0))
	t_1 = 1.0 - t_0
	tmp = 0
	if h <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * (-1.0 + t_0)
	elif h <= 1.6e+155:
		tmp = t_1 * (d * math.sqrt(((1.0 / l) / h)))
	else:
		tmp = t_1 * math.sqrt(((d / l) * (d / 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(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M / 2.0)) ^ 2.0)))
	t_1 = Float64(1.0 - t_0)
	tmp = 0.0
	if (h <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(-1.0 + t_0));
	elseif (h <= 1.6e+155)
		tmp = Float64(t_1 * Float64(d * sqrt(Float64(Float64(1.0 / l) / h))));
	else
		tmp = Float64(t_1 * sqrt(Float64(Float64(d / l) * Float64(d / 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 = 0.5 * ((h / l) * (((D_m / d) * (M / 2.0)) ^ 2.0));
	t_1 = 1.0 - t_0;
	tmp = 0.0;
	if (h <= -5e-310)
		tmp = (d * sqrt((1.0 / (h * l)))) * (-1.0 + t_0);
	elseif (h <= 1.6e+155)
		tmp = t_1 * (d * sqrt(((1.0 / l) / h)));
	else
		tmp = t_1 * sqrt(((d / l) * (d / 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[(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]}, Block[{t$95$1 = N[(1.0 - t$95$0), $MachinePrecision]}, If[LessEqual[h, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 1.6e+155], N[(t$95$1 * N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -4.999999999999985e-310

   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 64.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 63.9%

         \[\leadsto \color{blue}{\left(\sqrt{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \cdot \sqrt{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. pow2 63.9%

         \[\leadsto \color{blue}{{\left(\sqrt{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}\right)}^{2}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. sqrt-unprod 53.3%

         \[\leadsto {\left(\sqrt{\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}}}\right)}^{2} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. pow1/2 53.3%

         \[\leadsto {\left(\sqrt{\color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}}}\right)}^{2} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. sqrt-pow1 53.3%

         \[\leadsto {\color{blue}{\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. metadata-eval 53.3%

         \[\leadsto {\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{0.25}}\right)}^{2} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 53.3%

      \[\leadsto \color{blue}{{\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.25}\right)}^{2}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around -inf 67.9%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   if -4.999999999999985e-310 < h < 1.60000000000000006e155

   1. Initial program 73.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 72.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 72.5%

         \[\leadsto \color{blue}{\left(\sqrt{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \cdot \sqrt{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. pow2 72.5%

         \[\leadsto \color{blue}{{\left(\sqrt{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}\right)}^{2}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. sqrt-unprod 65.3%

         \[\leadsto {\left(\sqrt{\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}}}\right)}^{2} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. pow1/2 65.3%

         \[\leadsto {\left(\sqrt{\color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}}}\right)}^{2} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. sqrt-pow1 65.4%

         \[\leadsto {\color{blue}{\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. metadata-eval 65.4%

         \[\leadsto {\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{0.25}}\right)}^{2} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 65.4%

      \[\leadsto \color{blue}{{\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.25}\right)}^{2}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around 0 77.9%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   8. Step-by-step derivation

      1. unpow-1 77.9%

         \[\leadsto \left(d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. metadata-eval 77.9%

         \[\leadsto \left(d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. pow-sqr 78.0%

         \[\leadsto \left(d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. rem-sqrt-square 78.0%

         \[\leadsto \left(d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. rem-square-sqrt 77.8%

         \[\leadsto \left(d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. fabs-sqr 77.8%

         \[\leadsto \left(d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      7. fabs-sqr 77.8%

         \[\leadsto \left(d \cdot \color{blue}{\left|\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right|}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. rem-square-sqrt 78.0%

         \[\leadsto \left(d \cdot \left|\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right|\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      9. rem-sqrt-square 78.0%

         \[\leadsto \left(d \cdot \color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      10. pow-sqr 77.9%

          \[\leadsto \left(d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(2 \cdot -0.5\right)}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      11. metadata-eval 77.9%

          \[\leadsto \left(d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{-1}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      12. unpow-1 77.9%

          \[\leadsto \left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      13. *-commutative 77.9%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      14. associate-/r* 78.9%

          \[\leadsto \left(d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   9. Simplified 78.9%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   if 1.60000000000000006e155 < h

   1. Initial program 68.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 64.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 64.4%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}^{1}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-unprod 59.9%

         \[\leadsto {\color{blue}{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}}^{1} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 59.9%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}^{1}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. unpow1 59.9%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. Simplified 59.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq 1.6 \cdot 10^{+155}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 50.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.75 \cdot 10^{-151}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \end{array} \end{array} \]

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 (<= M 1.75e-151)
   (fabs (/ d (sqrt (* h l))))
   (*
    (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M 2.0)) 2.0))))
    (sqrt (* (/ d l) (/ d 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 tmp;
	if (M <= 1.75e-151) {
		tmp = fabs((d / sqrt((h * l))));
	} else {
		tmp = (1.0 - (0.5 * ((h / l) * pow(((D_m / d) * (M / 2.0)), 2.0)))) * sqrt(((d / l) * (d / h)));
	}
	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 (m <= 1.75d-151) then
        tmp = abs((d / sqrt((h * l))))
    else
        tmp = (1.0d0 - (0.5d0 * ((h / l) * (((d_m / d) * (m / 2.0d0)) ** 2.0d0)))) * sqrt(((d / l) * (d / h)))
    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 (M <= 1.75e-151) {
		tmp = Math.abs((d / Math.sqrt((h * l))));
	} else {
		tmp = (1.0 - (0.5 * ((h / l) * Math.pow(((D_m / d) * (M / 2.0)), 2.0)))) * Math.sqrt(((d / l) * (d / 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):
	tmp = 0
	if M <= 1.75e-151:
		tmp = math.fabs((d / math.sqrt((h * l))))
	else:
		tmp = (1.0 - (0.5 * ((h / l) * math.pow(((D_m / d) * (M / 2.0)), 2.0)))) * math.sqrt(((d / l) * (d / 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)
	tmp = 0.0
	if (M <= 1.75e-151)
		tmp = abs(Float64(d / sqrt(Float64(h * l))));
	else
		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))));
	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 (M <= 1.75e-151)
		tmp = abs((d / sqrt((h * l))));
	else
		tmp = (1.0 - (0.5 * ((h / l) * (((D_m / d) * (M / 2.0)) ^ 2.0)))) * sqrt(((d / l) * (d / 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_] := If[LessEqual[M, 1.75e-151], N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if M < 1.74999999999999998e-151

   1. Initial program 68.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 66.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 33.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 33.2%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. pow1/2 33.2%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right) \]

      3. inv-pow 33.2%

         \[\leadsto d \cdot \left(1 \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right) \]

      4. pow-pow 33.2%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right) \]

      5. metadata-eval 33.2%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 33.2%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 33.2%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 33.2%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. metadata-eval 33.2%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{\left(-1 \cdot 0.5\right)}} \]

       2. metadata-eval 33.2%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\left(-1 \cdot \color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}\right)} \]

       3. pow-pow 33.2%

          \[\leadsto d \cdot \color{blue}{{\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       4. inv-pow 33.2%

          \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

       5. pow-pow 32.2%

          \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       6. add-sqr-sqrt 26.0%

          \[\leadsto \color{blue}{\sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}}} \]

       7. sqrt-unprod 29.4%

          \[\leadsto \color{blue}{\sqrt{\left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right) \cdot \left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       8. swap-sqr 27.2%

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot d\right) \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       9. unpow2 27.2%

          \[\leadsto \sqrt{\color{blue}{{d}^{2}} \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       10. pow-pow 27.4%

           \[\leadsto \sqrt{{d}^{2} \cdot \left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       11. pow-pow 30.6%

           \[\leadsto \sqrt{{d}^{2} \cdot \left({\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}\right)} \]

       12. pow-sqr 30.6%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \left(1.5 \cdot 0.3333333333333333\right)\right)}}} \]

       13. metadata-eval 30.6%

           \[\leadsto \sqrt{{d}^{2} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \color{blue}{0.5}\right)}} \]

       14. metadata-eval 30.6%

           \[\leadsto \sqrt{{d}^{2} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{1}}} \]

       15. pow1 30.6%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \]

       16. div-inv 30.6%

           \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}} \]

       17. associate-/r* 31.8%

           \[\leadsto \sqrt{\color{blue}{\frac{\frac{{d}^{2}}{h}}{\ell}}} \]

       18. add-sqr-sqrt 31.8%

           \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{\frac{{d}^{2}}{h}}{\ell}} \cdot \sqrt{\frac{\frac{{d}^{2}}{h}}{\ell}}}} \]

   11. Applied egg-rr 47.9%

       \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \]

   if 1.74999999999999998e-151 < M

   1. Initial program 69.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 69.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 69.2%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}^{1}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-unprod 62.6%

         \[\leadsto {\color{blue}{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}}^{1} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 62.6%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}^{1}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. unpow1 62.6%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. Simplified 62.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.75 \cdot 10^{-151}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 44.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.9 \cdot 10^{-214}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -6.6 \cdot 10^{-301}:\\ \;\;\;\;\frac{d}{\sqrt{-1 + \mathsf{fma}\left(\ell, h, 1\right)}}\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{-168}:\\ \;\;\;\;\frac{d}{-\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

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 -5.9e-214)
   (* (- d) (sqrt (/ (/ 1.0 l) h)))
   (if (<= l -6.6e-301)
     (/ d (sqrt (+ -1.0 (fma l h 1.0))))
     (if (<= l 7.5e-168)
       (/ d (- (sqrt (* h l))))
       (/ (/ d (sqrt 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 <= -5.9e-214) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else if (l <= -6.6e-301) {
		tmp = d / sqrt((-1.0 + fma(l, h, 1.0)));
	} else if (l <= 7.5e-168) {
		tmp = d / -sqrt((h * l));
	} else {
		tmp = (d / sqrt(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 (l <= -5.9e-214)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	elseif (l <= -6.6e-301)
		tmp = Float64(d / sqrt(Float64(-1.0 + fma(l, h, 1.0))));
	elseif (l <= 7.5e-168)
		tmp = Float64(d / Float64(-sqrt(Float64(h * l))));
	else
		tmp = Float64(Float64(d / sqrt(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[l, -5.9e-214], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -6.6e-301], N[(d / N[Sqrt[N[(-1.0 + N[(l * h + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.5e-168], N[(d / (-N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -5.8999999999999998e-214

   1. Initial program 61.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 60.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 7.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   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 45.0%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. neg-mul-1 45.0%

         \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. *-commutative 45.0%

         \[\leadsto d \cdot \left(-\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \]

      6. associate-/r* 45.2%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \]

   8. Simplified 45.2%

      \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \]

   if -5.8999999999999998e-214 < l < -6.6000000000000001e-301

   1. Initial program 80.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 80.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 30.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 30.4%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. pow1/2 30.4%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right) \]

      3. inv-pow 30.4%

         \[\leadsto d \cdot \left(1 \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right) \]

      4. pow-pow 30.4%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right) \]

      5. metadata-eval 30.4%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 30.4%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 30.4%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 30.4%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. metadata-eval 30.4%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{\left(-1 \cdot 0.5\right)}} \]

       2. metadata-eval 30.4%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\left(-1 \cdot \color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}\right)} \]

       3. pow-pow 30.4%

          \[\leadsto d \cdot \color{blue}{{\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       4. inv-pow 30.4%

          \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

       5. pow-pow 53.5%

          \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       6. add-sqr-sqrt 0.0%

          \[\leadsto \color{blue}{\sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}}} \]

       7. sqrt-unprod 5.5%

          \[\leadsto \color{blue}{\sqrt{\left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right) \cdot \left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       8. swap-sqr 5.5%

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot d\right) \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       9. unpow2 5.5%

          \[\leadsto \sqrt{\color{blue}{{d}^{2}} \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       10. pow-pow 5.5%

           \[\leadsto \sqrt{{d}^{2} \cdot \left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       11. pow-pow 5.7%

           \[\leadsto \sqrt{{d}^{2} \cdot \left({\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}\right)} \]

       12. pow-sqr 5.7%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \left(1.5 \cdot 0.3333333333333333\right)\right)}}} \]

       13. metadata-eval 5.7%

           \[\leadsto \sqrt{{d}^{2} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \color{blue}{0.5}\right)}} \]

       14. metadata-eval 5.7%

           \[\leadsto \sqrt{{d}^{2} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{1}}} \]

       15. pow1 5.7%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \]

       16. div-inv 5.7%

           \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}} \]

       17. sqrt-div 10.2%

           \[\leadsto \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}}} \]

   11. Applied egg-rr 30.4%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   12. Step-by-step derivation

       1. expm1-log1p-u 30.4%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot h\right)\right)}}} \]

       2. expm1-undefine 71.9%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{e^{\mathsf{log1p}\left(\ell \cdot h\right)} - 1}}} \]

   13. Applied egg-rr 71.9%

       \[\leadsto \frac{d}{\sqrt{\color{blue}{e^{\mathsf{log1p}\left(\ell \cdot h\right)} - 1}}} \]
   14. Step-by-step derivation

       1. sub-neg 71.9%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{e^{\mathsf{log1p}\left(\ell \cdot h\right)} + \left(-1\right)}}} \]

       2. metadata-eval 71.9%

          \[\leadsto \frac{d}{\sqrt{e^{\mathsf{log1p}\left(\ell \cdot h\right)} + \color{blue}{-1}}} \]

       3. +-commutative 71.9%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{-1 + e^{\mathsf{log1p}\left(\ell \cdot h\right)}}}} \]

       4. log1p-undefine 71.9%

          \[\leadsto \frac{d}{\sqrt{-1 + e^{\color{blue}{\log \left(1 + \ell \cdot h\right)}}}} \]

       5. rem-exp-log 71.9%

          \[\leadsto \frac{d}{\sqrt{-1 + \color{blue}{\left(1 + \ell \cdot h\right)}}} \]

       6. +-commutative 71.9%

          \[\leadsto \frac{d}{\sqrt{-1 + \color{blue}{\left(\ell \cdot h + 1\right)}}} \]

       7. fma-define 71.9%

          \[\leadsto \frac{d}{\sqrt{-1 + \color{blue}{\mathsf{fma}\left(\ell, h, 1\right)}}} \]

   15. Simplified 71.9%

       \[\leadsto \frac{d}{\sqrt{\color{blue}{-1 + \mathsf{fma}\left(\ell, h, 1\right)}}} \]

   if -6.6000000000000001e-301 < l < 7.4999999999999995e-168

   1. Initial program 79.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 79.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 14.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 14.1%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. pow1/2 14.1%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right) \]

      3. inv-pow 14.1%

         \[\leadsto d \cdot \left(1 \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right) \]

      4. pow-pow 14.1%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right) \]

      5. metadata-eval 14.1%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 14.1%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 14.1%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 14.1%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. metadata-eval 14.1%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{\left(-1 \cdot 0.5\right)}} \]

       2. metadata-eval 14.1%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\left(-1 \cdot \color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}\right)} \]

       3. pow-pow 14.1%

          \[\leadsto d \cdot \color{blue}{{\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       4. inv-pow 14.1%

          \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

       5. pow-pow 7.3%

          \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       6. add-sqr-sqrt 7.2%

          \[\leadsto \color{blue}{\sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}}} \]

       7. sqrt-unprod 7.6%

          \[\leadsto \color{blue}{\sqrt{\left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right) \cdot \left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       8. swap-sqr 7.6%

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot d\right) \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       9. unpow2 7.6%

          \[\leadsto \sqrt{\color{blue}{{d}^{2}} \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       10. pow-pow 7.6%

           \[\leadsto \sqrt{{d}^{2} \cdot \left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       11. pow-pow 14.3%

           \[\leadsto \sqrt{{d}^{2} \cdot \left({\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}\right)} \]

       12. pow-sqr 14.3%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \left(1.5 \cdot 0.3333333333333333\right)\right)}}} \]

       13. metadata-eval 14.3%

           \[\leadsto \sqrt{{d}^{2} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \color{blue}{0.5}\right)}} \]

       14. metadata-eval 14.3%

           \[\leadsto \sqrt{{d}^{2} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{1}}} \]

       15. pow1 14.3%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \]

       16. div-inv 14.4%

           \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}} \]

       17. sqrt-div 14.4%

           \[\leadsto \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}}} \]

   11. Applied egg-rr 14.2%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   12. Step-by-step derivation

       1. sqrt-prod 14.2%

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

       2. add-sqr-sqrt 14.1%

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{\ell} \cdot \sqrt{h}} \]

       3. times-frac 14.1%

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}} \]

       4. sqrt-div 10.9%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \frac{\sqrt{d}}{\sqrt{h}} \]

       5. sqrt-div 14.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

       6. add-sqr-sqrt 14.5%

          \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{\ell}}}} \cdot \sqrt{\frac{d}{h}} \]

       7. sqr-neg 14.5%

          \[\leadsto \sqrt{\color{blue}{\left(-\sqrt{\frac{d}{\ell}}\right) \cdot \left(-\sqrt{\frac{d}{\ell}}\right)}} \cdot \sqrt{\frac{d}{h}} \]

       8. sqrt-unprod 0.0%

          \[\leadsto \color{blue}{\left(\sqrt{-\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-\sqrt{\frac{d}{\ell}}}\right)} \cdot \sqrt{\frac{d}{h}} \]

       9. add-sqr-sqrt 13.3%

          \[\leadsto \color{blue}{\left(-\sqrt{\frac{d}{\ell}}\right)} \cdot \sqrt{\frac{d}{h}} \]

       10. distribute-lft-neg-out 13.3%

           \[\leadsto \color{blue}{-\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}} \]

       11. sqrt-div 9.6%

           \[\leadsto -\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \sqrt{\frac{d}{h}} \]

       12. sqrt-div 9.7%

           \[\leadsto -\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \]

       13. times-frac 9.7%

           \[\leadsto -\color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}}} \]

       14. add-sqr-sqrt 9.7%

           \[\leadsto -\frac{\color{blue}{d}}{\sqrt{\ell} \cdot \sqrt{h}} \]

       15. sqrt-prod 33.0%

           \[\leadsto -\frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

       16. neg-sub0 33.0%

           \[\leadsto \color{blue}{0 - \frac{d}{\sqrt{\ell \cdot h}}} \]

   13. Applied egg-rr 33.0%

       \[\leadsto \color{blue}{0 - \frac{d}{\sqrt{\ell \cdot h}}} \]
   14. Step-by-step derivation

       1. neg-sub0 33.0%

          \[\leadsto \color{blue}{-\frac{d}{\sqrt{\ell \cdot h}}} \]

       2. distribute-frac-neg2 33.0%

          \[\leadsto \color{blue}{\frac{d}{-\sqrt{\ell \cdot h}}} \]

   15. Simplified 33.0%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{\ell \cdot h}}} \]

   if 7.4999999999999995e-168 < l

   1. Initial program 70.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 52.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 52.7%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. pow1/2 52.7%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right) \]

      3. inv-pow 52.7%

         \[\leadsto d \cdot \left(1 \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right) \]

      4. pow-pow 52.7%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right) \]

      5. metadata-eval 52.7%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 52.7%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 52.7%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 52.7%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. metadata-eval 52.7%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{\left(-1 \cdot 0.5\right)}} \]

       2. metadata-eval 52.7%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\left(-1 \cdot \color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}\right)} \]

       3. pow-pow 52.7%

          \[\leadsto d \cdot \color{blue}{{\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       4. inv-pow 52.7%

          \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

       5. pow-pow 46.7%

          \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       6. add-sqr-sqrt 46.7%

          \[\leadsto \color{blue}{\sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}}} \]

       7. sqrt-unprod 40.2%

          \[\leadsto \color{blue}{\sqrt{\left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right) \cdot \left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       8. swap-sqr 39.2%

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot d\right) \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       9. unpow2 39.2%

          \[\leadsto \sqrt{\color{blue}{{d}^{2}} \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       10. pow-pow 39.6%

           \[\leadsto \sqrt{{d}^{2} \cdot \left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       11. pow-pow 41.9%

           \[\leadsto \sqrt{{d}^{2} \cdot \left({\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}\right)} \]

       12. pow-sqr 41.9%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \left(1.5 \cdot 0.3333333333333333\right)\right)}}} \]

       13. metadata-eval 41.9%

           \[\leadsto \sqrt{{d}^{2} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \color{blue}{0.5}\right)}} \]

       14. metadata-eval 41.9%

           \[\leadsto \sqrt{{d}^{2} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{1}}} \]

       15. pow1 41.9%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \]

       16. div-inv 42.0%

           \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}} \]

       17. associate-/r* 42.1%

           \[\leadsto \sqrt{\color{blue}{\frac{\frac{{d}^{2}}{h}}{\ell}}} \]

   11. Applied egg-rr 60.9%

       \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.9 \cdot 10^{-214}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -6.6 \cdot 10^{-301}:\\ \;\;\;\;\frac{d}{\sqrt{-1 + \mathsf{fma}\left(\ell, h, 1\right)}}\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{-168}:\\ \;\;\;\;\frac{d}{-\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 43.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{-214}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -8.5 \cdot 10^{-301}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(h \cdot \ell\right)}^{-1.5}}\\ \mathbf{elif}\;\ell \leq 7.1 \cdot 10^{-168}:\\ \;\;\;\;\frac{d}{-\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

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 -1.4e-214)
   (* (- d) (sqrt (/ (/ 1.0 l) h)))
   (if (<= l -8.5e-301)
     (* d (cbrt (pow (* h l) -1.5)))
     (if (<= l 7.1e-168)
       (/ d (- (sqrt (* h l))))
       (/ (/ d (sqrt 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 <= -1.4e-214) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else if (l <= -8.5e-301) {
		tmp = d * cbrt(pow((h * l), -1.5));
	} else if (l <= 7.1e-168) {
		tmp = d / -sqrt((h * l));
	} else {
		tmp = (d / sqrt(h)) / sqrt(l);
	}
	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 tmp;
	if (l <= -1.4e-214) {
		tmp = -d * Math.sqrt(((1.0 / l) / h));
	} else if (l <= -8.5e-301) {
		tmp = d * Math.cbrt(Math.pow((h * l), -1.5));
	} else if (l <= 7.1e-168) {
		tmp = d / -Math.sqrt((h * l));
	} else {
		tmp = (d / Math.sqrt(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 <= -1.4e-214)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	elseif (l <= -8.5e-301)
		tmp = Float64(d * cbrt((Float64(h * l) ^ -1.5)));
	elseif (l <= 7.1e-168)
		tmp = Float64(d / Float64(-sqrt(Float64(h * l))));
	else
		tmp = Float64(Float64(d / sqrt(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[l, -1.4e-214], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -8.5e-301], N[(d * N[Power[N[Power[N[(h * l), $MachinePrecision], -1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.1e-168], N[(d / (-N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.4000000000000001e-214

   1. Initial program 61.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 60.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 7.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   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 45.0%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. neg-mul-1 45.0%

         \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. *-commutative 45.0%

         \[\leadsto d \cdot \left(-\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \]

      6. associate-/r* 45.2%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \]

   8. Simplified 45.2%

      \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \]

   if -1.4000000000000001e-214 < l < -8.50000000000000046e-301

   1. Initial program 80.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 80.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 30.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 30.4%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. pow1/2 30.4%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right) \]

      3. inv-pow 30.4%

         \[\leadsto d \cdot \left(1 \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right) \]

      4. pow-pow 30.4%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right) \]

      5. metadata-eval 30.4%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 30.4%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 30.4%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 30.4%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. metadata-eval 30.4%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{\left(-1 \cdot 0.5\right)}} \]

       2. metadata-eval 30.4%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\left(-1 \cdot \color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}\right)} \]

       3. pow-pow 30.4%

          \[\leadsto d \cdot \color{blue}{{\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       4. inv-pow 30.4%

          \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

       5. pow-pow 53.5%

          \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       6. unpow1/3 53.5%

          \[\leadsto d \cdot \color{blue}{\sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{1.5}}} \]

       7. inv-pow 53.5%

          \[\leadsto d \cdot \sqrt[3]{{\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{1.5}} \]

       8. pow-pow 53.5%

          \[\leadsto d \cdot \sqrt[3]{\color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 1.5\right)}}} \]

       9. *-commutative 53.5%

          \[\leadsto d \cdot \sqrt[3]{{\color{blue}{\left(\ell \cdot h\right)}}^{\left(-1 \cdot 1.5\right)}} \]

       10. metadata-eval 53.5%

           \[\leadsto d \cdot \sqrt[3]{{\left(\ell \cdot h\right)}^{\color{blue}{-1.5}}} \]

   11. Applied egg-rr 53.5%

       \[\leadsto d \cdot \color{blue}{\sqrt[3]{{\left(\ell \cdot h\right)}^{-1.5}}} \]

   if -8.50000000000000046e-301 < l < 7.10000000000000019e-168

   1. Initial program 79.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 79.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 14.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 14.1%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. pow1/2 14.1%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right) \]

      3. inv-pow 14.1%

         \[\leadsto d \cdot \left(1 \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right) \]

      4. pow-pow 14.1%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right) \]

      5. metadata-eval 14.1%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 14.1%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 14.1%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 14.1%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. metadata-eval 14.1%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{\left(-1 \cdot 0.5\right)}} \]

       2. metadata-eval 14.1%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\left(-1 \cdot \color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}\right)} \]

       3. pow-pow 14.1%

          \[\leadsto d \cdot \color{blue}{{\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       4. inv-pow 14.1%

          \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

       5. pow-pow 7.3%

          \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       6. add-sqr-sqrt 7.2%

          \[\leadsto \color{blue}{\sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}}} \]

       7. sqrt-unprod 7.6%

          \[\leadsto \color{blue}{\sqrt{\left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right) \cdot \left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       8. swap-sqr 7.6%

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot d\right) \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       9. unpow2 7.6%

          \[\leadsto \sqrt{\color{blue}{{d}^{2}} \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       10. pow-pow 7.6%

           \[\leadsto \sqrt{{d}^{2} \cdot \left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       11. pow-pow 14.3%

           \[\leadsto \sqrt{{d}^{2} \cdot \left({\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}\right)} \]

       12. pow-sqr 14.3%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \left(1.5 \cdot 0.3333333333333333\right)\right)}}} \]

       13. metadata-eval 14.3%

           \[\leadsto \sqrt{{d}^{2} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \color{blue}{0.5}\right)}} \]

       14. metadata-eval 14.3%

           \[\leadsto \sqrt{{d}^{2} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{1}}} \]

       15. pow1 14.3%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \]

       16. div-inv 14.4%

           \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}} \]

       17. sqrt-div 14.4%

           \[\leadsto \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}}} \]

   11. Applied egg-rr 14.2%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   12. Step-by-step derivation

       1. sqrt-prod 14.2%

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

       2. add-sqr-sqrt 14.1%

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{\ell} \cdot \sqrt{h}} \]

       3. times-frac 14.1%

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}} \]

       4. sqrt-div 10.9%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \frac{\sqrt{d}}{\sqrt{h}} \]

       5. sqrt-div 14.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

       6. add-sqr-sqrt 14.5%

          \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{\ell}}}} \cdot \sqrt{\frac{d}{h}} \]

       7. sqr-neg 14.5%

          \[\leadsto \sqrt{\color{blue}{\left(-\sqrt{\frac{d}{\ell}}\right) \cdot \left(-\sqrt{\frac{d}{\ell}}\right)}} \cdot \sqrt{\frac{d}{h}} \]

       8. sqrt-unprod 0.0%

          \[\leadsto \color{blue}{\left(\sqrt{-\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-\sqrt{\frac{d}{\ell}}}\right)} \cdot \sqrt{\frac{d}{h}} \]

       9. add-sqr-sqrt 13.3%

          \[\leadsto \color{blue}{\left(-\sqrt{\frac{d}{\ell}}\right)} \cdot \sqrt{\frac{d}{h}} \]

       10. distribute-lft-neg-out 13.3%

           \[\leadsto \color{blue}{-\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}} \]

       11. sqrt-div 9.6%

           \[\leadsto -\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \sqrt{\frac{d}{h}} \]

       12. sqrt-div 9.7%

           \[\leadsto -\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \]

       13. times-frac 9.7%

           \[\leadsto -\color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}}} \]

       14. add-sqr-sqrt 9.7%

           \[\leadsto -\frac{\color{blue}{d}}{\sqrt{\ell} \cdot \sqrt{h}} \]

       15. sqrt-prod 33.0%

           \[\leadsto -\frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

       16. neg-sub0 33.0%

           \[\leadsto \color{blue}{0 - \frac{d}{\sqrt{\ell \cdot h}}} \]

   13. Applied egg-rr 33.0%

       \[\leadsto \color{blue}{0 - \frac{d}{\sqrt{\ell \cdot h}}} \]
   14. Step-by-step derivation

       1. neg-sub0 33.0%

          \[\leadsto \color{blue}{-\frac{d}{\sqrt{\ell \cdot h}}} \]

       2. distribute-frac-neg2 33.0%

          \[\leadsto \color{blue}{\frac{d}{-\sqrt{\ell \cdot h}}} \]

   15. Simplified 33.0%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{\ell \cdot h}}} \]

   if 7.10000000000000019e-168 < l

   1. Initial program 70.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 52.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 52.7%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. pow1/2 52.7%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right) \]

      3. inv-pow 52.7%

         \[\leadsto d \cdot \left(1 \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right) \]

      4. pow-pow 52.7%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right) \]

      5. metadata-eval 52.7%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 52.7%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 52.7%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 52.7%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. metadata-eval 52.7%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{\left(-1 \cdot 0.5\right)}} \]

       2. metadata-eval 52.7%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\left(-1 \cdot \color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}\right)} \]

       3. pow-pow 52.7%

          \[\leadsto d \cdot \color{blue}{{\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       4. inv-pow 52.7%

          \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

       5. pow-pow 46.7%

          \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       6. add-sqr-sqrt 46.7%

          \[\leadsto \color{blue}{\sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}}} \]

       7. sqrt-unprod 40.2%

          \[\leadsto \color{blue}{\sqrt{\left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right) \cdot \left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       8. swap-sqr 39.2%

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot d\right) \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       9. unpow2 39.2%

          \[\leadsto \sqrt{\color{blue}{{d}^{2}} \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       10. pow-pow 39.6%

           \[\leadsto \sqrt{{d}^{2} \cdot \left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       11. pow-pow 41.9%

           \[\leadsto \sqrt{{d}^{2} \cdot \left({\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}\right)} \]

       12. pow-sqr 41.9%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \left(1.5 \cdot 0.3333333333333333\right)\right)}}} \]

       13. metadata-eval 41.9%

           \[\leadsto \sqrt{{d}^{2} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \color{blue}{0.5}\right)}} \]

       14. metadata-eval 41.9%

           \[\leadsto \sqrt{{d}^{2} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{1}}} \]

       15. pow1 41.9%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \]

       16. div-inv 42.0%

           \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}} \]

       17. associate-/r* 42.1%

           \[\leadsto \sqrt{\color{blue}{\frac{\frac{{d}^{2}}{h}}{\ell}}} \]

   11. Applied egg-rr 60.9%

       \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{-214}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -8.5 \cdot 10^{-301}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(h \cdot \ell\right)}^{-1.5}}\\ \mathbf{elif}\;\ell \leq 7.1 \cdot 10^{-168}:\\ \;\;\;\;\frac{d}{-\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 42.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.6 \cdot 10^{-167}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

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 5.6e-167)
   (* (- d) (sqrt (/ (/ 1.0 l) h)))
   (/ (/ d (sqrt 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 <= 5.6e-167) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else {
		tmp = (d / sqrt(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 <= 5.6d-167) then
        tmp = -d * sqrt(((1.0d0 / l) / h))
    else
        tmp = (d / sqrt(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 <= 5.6e-167) {
		tmp = -d * Math.sqrt(((1.0 / l) / h));
	} else {
		tmp = (d / Math.sqrt(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 <= 5.6e-167:
		tmp = -d * math.sqrt(((1.0 / l) / h))
	else:
		tmp = (d / math.sqrt(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 <= 5.6e-167)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	else
		tmp = Float64(Float64(d / sqrt(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 <= 5.6e-167)
		tmp = -d * sqrt(((1.0 / l) / h));
	else
		tmp = (d / sqrt(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, 5.6e-167], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.59999999999999971e-167

   1. Initial program 67.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 12.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   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 37.9%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. neg-mul-1 37.9%

         \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. *-commutative 37.9%

         \[\leadsto d \cdot \left(-\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \]

      6. associate-/r* 38.0%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \]

   8. Simplified 38.0%

      \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \]

   if 5.59999999999999971e-167 < l

   1. Initial program 70.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 52.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 52.7%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. pow1/2 52.7%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right) \]

      3. inv-pow 52.7%

         \[\leadsto d \cdot \left(1 \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right) \]

      4. pow-pow 52.7%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right) \]

      5. metadata-eval 52.7%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 52.7%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 52.7%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 52.7%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. metadata-eval 52.7%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{\left(-1 \cdot 0.5\right)}} \]

       2. metadata-eval 52.7%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\left(-1 \cdot \color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}\right)} \]

       3. pow-pow 52.7%

          \[\leadsto d \cdot \color{blue}{{\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       4. inv-pow 52.7%

          \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

       5. pow-pow 46.7%

          \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       6. add-sqr-sqrt 46.7%

          \[\leadsto \color{blue}{\sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}}} \]

       7. sqrt-unprod 40.2%

          \[\leadsto \color{blue}{\sqrt{\left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right) \cdot \left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       8. swap-sqr 39.2%

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot d\right) \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       9. unpow2 39.2%

          \[\leadsto \sqrt{\color{blue}{{d}^{2}} \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       10. pow-pow 39.6%

           \[\leadsto \sqrt{{d}^{2} \cdot \left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       11. pow-pow 41.9%

           \[\leadsto \sqrt{{d}^{2} \cdot \left({\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}\right)} \]

       12. pow-sqr 41.9%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \left(1.5 \cdot 0.3333333333333333\right)\right)}}} \]

       13. metadata-eval 41.9%

           \[\leadsto \sqrt{{d}^{2} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \color{blue}{0.5}\right)}} \]

       14. metadata-eval 41.9%

           \[\leadsto \sqrt{{d}^{2} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{1}}} \]

       15. pow1 41.9%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \]

       16. div-inv 42.0%

           \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}} \]

       17. associate-/r* 42.1%

           \[\leadsto \sqrt{\color{blue}{\frac{\frac{{d}^{2}}{h}}{\ell}}} \]

   11. Applied egg-rr 60.9%

       \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.6 \cdot 10^{-167}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 42.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 2.5 \cdot 10^{+18}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \end{array} \]

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
 (if (<= M 2.5e+18)
   (fabs (/ d (sqrt (* h l))))
   (* (- d) (sqrt (/ (/ 1.0 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 tmp;
	if (M <= 2.5e+18) {
		tmp = fabs((d / sqrt((h * l))));
	} else {
		tmp = -d * sqrt(((1.0 / l) / h));
	}
	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 (m <= 2.5d+18) then
        tmp = abs((d / sqrt((h * l))))
    else
        tmp = -d * sqrt(((1.0d0 / l) / h))
    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 (M <= 2.5e+18) {
		tmp = Math.abs((d / Math.sqrt((h * l))));
	} else {
		tmp = -d * Math.sqrt(((1.0 / 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):
	tmp = 0
	if M <= 2.5e+18:
		tmp = math.fabs((d / math.sqrt((h * l))))
	else:
		tmp = -d * math.sqrt(((1.0 / 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)
	tmp = 0.0
	if (M <= 2.5e+18)
		tmp = abs(Float64(d / sqrt(Float64(h * l))));
	else
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / 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)
	tmp = 0.0;
	if (M <= 2.5e+18)
		tmp = abs((d / sqrt((h * l))));
	else
		tmp = -d * sqrt(((1.0 / 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_] := If[LessEqual[M, 2.5e+18], N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.5e18

   1. Initial program 68.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 66.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 33.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 33.4%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. pow1/2 33.4%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right) \]

      3. inv-pow 33.4%

         \[\leadsto d \cdot \left(1 \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right) \]

      4. pow-pow 33.5%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right) \]

      5. metadata-eval 33.5%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 33.5%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 33.5%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 33.5%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. metadata-eval 33.5%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{\left(-1 \cdot 0.5\right)}} \]

       2. metadata-eval 33.5%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\left(-1 \cdot \color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}\right)} \]

       3. pow-pow 33.4%

          \[\leadsto d \cdot \color{blue}{{\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       4. inv-pow 33.4%

          \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

       5. pow-pow 31.9%

          \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       6. add-sqr-sqrt 26.1%

          \[\leadsto \color{blue}{\sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}}} \]

       7. sqrt-unprod 29.5%

          \[\leadsto \color{blue}{\sqrt{\left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right) \cdot \left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       8. swap-sqr 27.6%

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot d\right) \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       9. unpow2 27.6%

          \[\leadsto \sqrt{\color{blue}{{d}^{2}} \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       10. pow-pow 27.8%

           \[\leadsto \sqrt{{d}^{2} \cdot \left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       11. pow-pow 31.7%

           \[\leadsto \sqrt{{d}^{2} \cdot \left({\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}\right)} \]

       12. pow-sqr 31.8%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \left(1.5 \cdot 0.3333333333333333\right)\right)}}} \]

       13. metadata-eval 31.8%

           \[\leadsto \sqrt{{d}^{2} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \color{blue}{0.5}\right)}} \]

       14. metadata-eval 31.8%

           \[\leadsto \sqrt{{d}^{2} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{1}}} \]

       15. pow1 31.8%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \]

       16. div-inv 31.8%

           \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}} \]

       17. associate-/r* 33.8%

           \[\leadsto \sqrt{\color{blue}{\frac{\frac{{d}^{2}}{h}}{\ell}}} \]

       18. add-sqr-sqrt 33.8%

           \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{\frac{{d}^{2}}{h}}{\ell}} \cdot \sqrt{\frac{\frac{{d}^{2}}{h}}{\ell}}}} \]

   11. Applied egg-rr 47.9%

       \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \]

   if 2.5e18 < M

   1. Initial program 69.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 70.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 12.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   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 25.0%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. neg-mul-1 25.0%

         \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. *-commutative 25.0%

         \[\leadsto d \cdot \left(-\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \]

      6. associate-/r* 25.0%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \]

   8. Simplified 25.0%

      \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.5 \cdot 10^{+18}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 41.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7.8 \cdot 10^{-168}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell} \cdot \frac{1}{h}}\\ \end{array} \end{array} \]

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
 (if (<= l 7.8e-168)
   (* (- d) (sqrt (/ (/ 1.0 l) h)))
   (* d (sqrt (* (/ 1.0 l) (/ 1.0 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 tmp;
	if (l <= 7.8e-168) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else {
		tmp = d * sqrt(((1.0 / l) * (1.0 / h)));
	}
	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 <= 7.8d-168) then
        tmp = -d * sqrt(((1.0d0 / l) / h))
    else
        tmp = d * sqrt(((1.0d0 / l) * (1.0d0 / h)))
    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 <= 7.8e-168) {
		tmp = -d * Math.sqrt(((1.0 / l) / h));
	} else {
		tmp = d * Math.sqrt(((1.0 / l) * (1.0 / 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):
	tmp = 0
	if l <= 7.8e-168:
		tmp = -d * math.sqrt(((1.0 / l) / h))
	else:
		tmp = d * math.sqrt(((1.0 / l) * (1.0 / 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)
	tmp = 0.0
	if (l <= 7.8e-168)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) * Float64(1.0 / 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)
	tmp = 0.0;
	if (l <= 7.8e-168)
		tmp = -d * sqrt(((1.0 / l) / h));
	else
		tmp = d * sqrt(((1.0 / l) * (1.0 / 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_] := If[LessEqual[l, 7.8e-168], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] * N[(1.0 / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.80000000000000025e-168

   1. Initial program 67.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 12.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   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 37.9%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. neg-mul-1 37.9%

         \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. *-commutative 37.9%

         \[\leadsto d \cdot \left(-\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \]

      6. associate-/r* 38.0%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \]

   8. Simplified 38.0%

      \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \]

   if 7.80000000000000025e-168 < l

   1. Initial program 70.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 52.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. inv-pow 52.7%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. *-commutative 52.7%

         \[\leadsto d \cdot \sqrt{{\color{blue}{\left(\ell \cdot h\right)}}^{-1}} \]

      3. unpow-prod-down 53.9%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\ell}^{-1} \cdot {h}^{-1}}} \]

      4. inv-pow 53.9%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{\ell}} \cdot {h}^{-1}} \]

      5. inv-pow 53.9%

         \[\leadsto d \cdot \sqrt{\frac{1}{\ell} \cdot \color{blue}{\frac{1}{h}}} \]

   7. Applied egg-rr 53.9%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{\ell} \cdot \frac{1}{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.8 \cdot 10^{-168}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell} \cdot \frac{1}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 41.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{if}\;\ell \leq 7.1 \cdot 10^{-168}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot t\_0\\ \end{array} \end{array} \]

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 (/ (/ 1.0 l) h))))
   (if (<= l 7.1e-168) (* (- d) t_0) (* d t_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(((1.0 / l) / h));
	double tmp;
	if (l <= 7.1e-168) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_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) :: tmp
    t_0 = sqrt(((1.0d0 / l) / h))
    if (l <= 7.1d-168) then
        tmp = -d * t_0
    else
        tmp = d * t_0
    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(((1.0 / l) / h));
	double tmp;
	if (l <= 7.1e-168) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_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(((1.0 / l) / h))
	tmp = 0
	if l <= 7.1e-168:
		tmp = -d * t_0
	else:
		tmp = d * t_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(Float64(1.0 / l) / h))
	tmp = 0.0
	if (l <= 7.1e-168)
		tmp = Float64(Float64(-d) * t_0);
	else
		tmp = Float64(d * t_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(((1.0 / l) / h));
	tmp = 0.0;
	if (l <= 7.1e-168)
		tmp = -d * t_0;
	else
		tmp = d * t_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[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, 7.1e-168], N[((-d) * t$95$0), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 7.10000000000000019e-168

   1. Initial program 67.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 12.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   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 37.9%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. neg-mul-1 37.9%

         \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. *-commutative 37.9%

         \[\leadsto d \cdot \left(-\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \]

      6. associate-/r* 38.0%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \]

   8. Simplified 38.0%

      \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \]

   if 7.10000000000000019e-168 < l

   1. Initial program 70.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 52.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 52.7%

         \[\leadsto d \cdot \sqrt{\color{blue}{1 \cdot \frac{1}{h \cdot \ell}}} \]

   7. Applied egg-rr 52.7%

      \[\leadsto d \cdot \sqrt{\color{blue}{1 \cdot \frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-lft-identity 52.7%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      2. *-commutative 52.7%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      3. associate-/r* 53.8%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   9. Simplified 53.8%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.1 \cdot 10^{-168}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 40.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7.2 \cdot 10^{-168}:\\ \;\;\;\;\left(-d\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \end{array} \]

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
 (if (<= l 7.2e-168)
   (* (- d) (pow (* h l) -0.5))
   (* d (sqrt (/ (/ 1.0 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 tmp;
	if (l <= 7.2e-168) {
		tmp = -d * pow((h * l), -0.5);
	} else {
		tmp = d * sqrt(((1.0 / l) / h));
	}
	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 <= 7.2d-168) then
        tmp = -d * ((h * l) ** (-0.5d0))
    else
        tmp = d * sqrt(((1.0d0 / l) / h))
    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 <= 7.2e-168) {
		tmp = -d * Math.pow((h * l), -0.5);
	} else {
		tmp = d * Math.sqrt(((1.0 / 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):
	tmp = 0
	if l <= 7.2e-168:
		tmp = -d * math.pow((h * l), -0.5)
	else:
		tmp = d * math.sqrt(((1.0 / 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)
	tmp = 0.0
	if (l <= 7.2e-168)
		tmp = Float64(Float64(-d) * (Float64(h * l) ^ -0.5));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / 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)
	tmp = 0.0;
	if (l <= 7.2e-168)
		tmp = -d * ((h * l) ^ -0.5);
	else
		tmp = d * sqrt(((1.0 / 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_] := If[LessEqual[l, 7.2e-168], N[((-d) * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.1999999999999998e-168

   1. Initial program 67.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 12.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   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 37.9%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. neg-mul-1 37.9%

         \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. unpow-1 37.9%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      6. metadata-eval 37.9%

         \[\leadsto d \cdot \left(-\sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \]

      7. pow-sqr 37.9%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      8. rem-sqrt-square 37.9%

         \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \]

      9. rem-square-sqrt 37.7%

         \[\leadsto d \cdot \left(-\left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \]

      10. fabs-sqr 37.7%

          \[\leadsto d \cdot \left(-\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      11. rem-square-sqrt 37.9%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   8. Simplified 37.9%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if 7.1999999999999998e-168 < l

   1. Initial program 70.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 52.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 52.7%

         \[\leadsto d \cdot \sqrt{\color{blue}{1 \cdot \frac{1}{h \cdot \ell}}} \]

   7. Applied egg-rr 52.7%

      \[\leadsto d \cdot \sqrt{\color{blue}{1 \cdot \frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-lft-identity 52.7%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      2. *-commutative 52.7%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      3. associate-/r* 53.8%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   9. Simplified 53.8%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.2 \cdot 10^{-168}:\\ \;\;\;\;\left(-d\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 40.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{if}\;\ell \leq 7.8 \cdot 10^{-168}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot t\_0\\ \end{array} \end{array} \]

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 (* h l) -0.5)))
   (if (<= l 7.8e-168) (* (- d) t_0) (* d t_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 = pow((h * l), -0.5);
	double tmp;
	if (l <= 7.8e-168) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_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) :: tmp
    t_0 = (h * l) ** (-0.5d0)
    if (l <= 7.8d-168) then
        tmp = -d * t_0
    else
        tmp = d * t_0
    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((h * l), -0.5);
	double tmp;
	if (l <= 7.8e-168) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_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.pow((h * l), -0.5)
	tmp = 0
	if l <= 7.8e-168:
		tmp = -d * t_0
	else:
		tmp = d * t_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 = Float64(h * l) ^ -0.5
	tmp = 0.0
	if (l <= 7.8e-168)
		tmp = Float64(Float64(-d) * t_0);
	else
		tmp = Float64(d * t_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 = (h * l) ^ -0.5;
	tmp = 0.0;
	if (l <= 7.8e-168)
		tmp = -d * t_0;
	else
		tmp = d * t_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[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[l, 7.8e-168], N[((-d) * t$95$0), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 7.80000000000000025e-168

   1. Initial program 67.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 12.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   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 37.9%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. neg-mul-1 37.9%

         \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. unpow-1 37.9%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      6. metadata-eval 37.9%

         \[\leadsto d \cdot \left(-\sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \]

      7. pow-sqr 37.9%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      8. rem-sqrt-square 37.9%

         \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \]

      9. rem-square-sqrt 37.7%

         \[\leadsto d \cdot \left(-\left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \]

      10. fabs-sqr 37.7%

          \[\leadsto d \cdot \left(-\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      11. rem-square-sqrt 37.9%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   8. Simplified 37.9%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if 7.80000000000000025e-168 < l

   1. Initial program 70.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 52.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 52.7%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. pow1/2 52.7%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right) \]

      3. inv-pow 52.7%

         \[\leadsto d \cdot \left(1 \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right) \]

      4. pow-pow 52.7%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right) \]

      5. metadata-eval 52.7%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 52.7%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 52.7%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 52.7%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.8 \cdot 10^{-168}:\\ \;\;\;\;\left(-d\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 29: 40.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{-167}:\\ \;\;\;\;\frac{d}{-\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \end{array} \]

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 1.15e-167) (/ d (- (sqrt (* h l)))) (* d (pow (* h l) -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 <= 1.15e-167) {
		tmp = d / -sqrt((h * l));
	} else {
		tmp = d * pow((h * l), -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 <= 1.15d-167) then
        tmp = d / -sqrt((h * l))
    else
        tmp = d * ((h * l) ** (-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 <= 1.15e-167) {
		tmp = d / -Math.sqrt((h * l));
	} else {
		tmp = d * Math.pow((h * l), -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 <= 1.15e-167:
		tmp = d / -math.sqrt((h * l))
	else:
		tmp = d * math.pow((h * l), -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 <= 1.15e-167)
		tmp = Float64(d / Float64(-sqrt(Float64(h * l))));
	else
		tmp = Float64(d * (Float64(h * l) ^ -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 <= 1.15e-167)
		tmp = d / -sqrt((h * l));
	else
		tmp = d * ((h * l) ^ -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, 1.15e-167], N[(d / (-N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.1500000000000001e-167

   1. Initial program 67.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 12.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 12.0%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. pow1/2 12.0%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right) \]

      3. inv-pow 12.0%

         \[\leadsto d \cdot \left(1 \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right) \]

      4. pow-pow 12.0%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right) \]

      5. metadata-eval 12.0%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 12.0%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 12.0%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 12.0%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. metadata-eval 12.0%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{\left(-1 \cdot 0.5\right)}} \]

       2. metadata-eval 12.0%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\left(-1 \cdot \color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}\right)} \]

       3. pow-pow 12.0%

          \[\leadsto d \cdot \color{blue}{{\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       4. inv-pow 12.0%

          \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

       5. pow-pow 13.8%

          \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       6. add-sqr-sqrt 4.5%

          \[\leadsto \color{blue}{\sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}}} \]

       7. sqrt-unprod 17.5%

          \[\leadsto \color{blue}{\sqrt{\left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right) \cdot \left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       8. swap-sqr 15.7%

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot d\right) \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

       9. unpow2 15.7%

          \[\leadsto \sqrt{\color{blue}{{d}^{2}} \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       10. pow-pow 15.8%

           \[\leadsto \sqrt{{d}^{2} \cdot \left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

       11. pow-pow 19.5%

           \[\leadsto \sqrt{{d}^{2} \cdot \left({\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}\right)} \]

       12. pow-sqr 19.5%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \left(1.5 \cdot 0.3333333333333333\right)\right)}}} \]

       13. metadata-eval 19.5%

           \[\leadsto \sqrt{{d}^{2} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \color{blue}{0.5}\right)}} \]

       14. metadata-eval 19.5%

           \[\leadsto \sqrt{{d}^{2} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{1}}} \]

       15. pow1 19.5%

           \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \]

       16. div-inv 19.5%

           \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}} \]

       17. sqrt-div 22.6%

           \[\leadsto \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}}} \]

   11. Applied egg-rr 12.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   12. Step-by-step derivation

       1. sqrt-prod 2.7%

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

       2. add-sqr-sqrt 2.7%

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{\ell} \cdot \sqrt{h}} \]

       3. times-frac 2.7%

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}} \]

       4. sqrt-div 2.1%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \frac{\sqrt{d}}{\sqrt{h}} \]

       5. sqrt-div 32.0%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

       6. add-sqr-sqrt 32.0%

          \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{\ell}}}} \cdot \sqrt{\frac{d}{h}} \]

       7. sqr-neg 32.0%

          \[\leadsto \sqrt{\color{blue}{\left(-\sqrt{\frac{d}{\ell}}\right) \cdot \left(-\sqrt{\frac{d}{\ell}}\right)}} \cdot \sqrt{\frac{d}{h}} \]

       8. sqrt-unprod 2.2%

          \[\leadsto \color{blue}{\left(\sqrt{-\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-\sqrt{\frac{d}{\ell}}}\right)} \cdot \sqrt{\frac{d}{h}} \]

       9. add-sqr-sqrt 11.1%

          \[\leadsto \color{blue}{\left(-\sqrt{\frac{d}{\ell}}\right)} \cdot \sqrt{\frac{d}{h}} \]

       10. distribute-lft-neg-out 11.1%

           \[\leadsto \color{blue}{-\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}} \]

       11. sqrt-div 1.8%

           \[\leadsto -\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \sqrt{\frac{d}{h}} \]

       12. sqrt-div 1.8%

           \[\leadsto -\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \]

       13. times-frac 1.8%

           \[\leadsto -\color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}}} \]

       14. add-sqr-sqrt 1.8%

           \[\leadsto -\frac{\color{blue}{d}}{\sqrt{\ell} \cdot \sqrt{h}} \]

       15. sqrt-prod 37.8%

           \[\leadsto -\frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

       16. neg-sub0 37.8%

           \[\leadsto \color{blue}{0 - \frac{d}{\sqrt{\ell \cdot h}}} \]

   13. Applied egg-rr 37.8%

       \[\leadsto \color{blue}{0 - \frac{d}{\sqrt{\ell \cdot h}}} \]
   14. Step-by-step derivation

       1. neg-sub0 37.8%

          \[\leadsto \color{blue}{-\frac{d}{\sqrt{\ell \cdot h}}} \]

       2. distribute-frac-neg2 37.8%

          \[\leadsto \color{blue}{\frac{d}{-\sqrt{\ell \cdot h}}} \]

   15. Simplified 37.8%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{\ell \cdot h}}} \]

   if 1.1500000000000001e-167 < l

   1. Initial program 70.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 52.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 52.7%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. pow1/2 52.7%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right) \]

      3. inv-pow 52.7%

         \[\leadsto d \cdot \left(1 \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right) \]

      4. pow-pow 52.7%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right) \]

      5. metadata-eval 52.7%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 52.7%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 52.7%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 52.7%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{-167}:\\ \;\;\;\;\frac{d}{-\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 30: 26.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \frac{d}{\sqrt{h \cdot \ell}} \end{array} \]

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 (* h 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) {
	return d / sqrt((h * l));
}

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((h * l))
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((h * l));
}

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((h * l))

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(h * l)))
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((h * l));
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[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.8%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 67.6%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in d around inf 28.5%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
6. Step-by-step derivation

   1. *-un-lft-identity 28.5%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   2. pow1/2 28.5%

      \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right) \]

   3. inv-pow 28.5%

      \[\leadsto d \cdot \left(1 \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right) \]

   4. pow-pow 28.5%

      \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right) \]

   5. metadata-eval 28.5%

      \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

7. Applied egg-rr 28.5%

   \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
8. Step-by-step derivation

   1. *-lft-identity 28.5%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

9. Simplified 28.5%

   \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
10. Step-by-step derivation

    1. metadata-eval 28.5%

       \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{\left(-1 \cdot 0.5\right)}} \]

    2. metadata-eval 28.5%

       \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\left(-1 \cdot \color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}\right)} \]

    3. pow-pow 28.5%

       \[\leadsto d \cdot \color{blue}{{\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

    4. inv-pow 28.5%

       \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

    5. pow-pow 27.2%

       \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

    6. add-sqr-sqrt 21.7%

       \[\leadsto \color{blue}{\sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sqrt{d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}}} \]

    7. sqrt-unprod 26.7%

       \[\leadsto \color{blue}{\sqrt{\left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right) \cdot \left(d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

    8. swap-sqr 25.2%

       \[\leadsto \sqrt{\color{blue}{\left(d \cdot d\right) \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

    9. unpow2 25.2%

       \[\leadsto \sqrt{\color{blue}{{d}^{2}} \cdot \left({\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

    10. pow-pow 25.4%

        \[\leadsto \sqrt{{d}^{2} \cdot \left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

    11. pow-pow 28.6%

        \[\leadsto \sqrt{{d}^{2} \cdot \left({\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}\right)} \]

    12. pow-sqr 28.6%

        \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \left(1.5 \cdot 0.3333333333333333\right)\right)}}} \]

    13. metadata-eval 28.6%

        \[\leadsto \sqrt{{d}^{2} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\left(2 \cdot \color{blue}{0.5}\right)}} \]

    14. metadata-eval 28.6%

        \[\leadsto \sqrt{{d}^{2} \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{1}}} \]

    15. pow1 28.6%

        \[\leadsto \sqrt{{d}^{2} \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \]

    16. div-inv 28.6%

        \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}} \]

    17. sqrt-div 31.2%

        \[\leadsto \color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}}} \]

11. Applied egg-rr 28.5%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

12. Final simplification 28.5%

    \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024143 
(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)))))