Henrywood and Agarwal, Equation (12)

Percentage Accurate: 65.7% → 84.0%

Time: 26.4s

Alternatives: 22

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: 65.7% 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: 84.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := \frac{t\_0}{\sqrt{-\ell}}\\ \mathbf{if}\;\ell \leq -2.8 \cdot 10^{-99}:\\ \;\;\;\;t\_1 \cdot \left(\frac{t\_0}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D\_m \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t\_1 \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot \left(0.25 \cdot {\left(\frac{M\_m}{\frac{d}{D\_m}}\right)}^{2}\right)\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (- d))) (t_1 (/ t_0 (sqrt (- l)))))
   (if (<= l -2.8e-99)
     (*
      t_1
      (*
       (/ t_0 (sqrt (- h)))
       (+ 1.0 (* (/ h l) (* (pow (* D_m (/ (/ M_m 2.0) d)) 2.0) -0.5)))))
     (if (<= l -2e-310)
       (*
        t_1
        (*
         (sqrt (/ d h))
         (+ 1.0 (/ (* h (* -0.5 (* 0.25 (pow (/ M_m (/ d D_m)) 2.0)))) l))))
       (*
        (/ d (* (sqrt l) (sqrt h)))
        (- 1.0 (* 0.5 (* h (* (pow (* D_m (/ M_m d)) 2.0) (/ 0.25 l))))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt(-d);
	double t_1 = t_0 / sqrt(-l);
	double tmp;
	if (l <= -2.8e-99) {
		tmp = t_1 * ((t_0 / sqrt(-h)) * (1.0 + ((h / l) * (pow((D_m * ((M_m / 2.0) / d)), 2.0) * -0.5))));
	} else if (l <= -2e-310) {
		tmp = t_1 * (sqrt((d / h)) * (1.0 + ((h * (-0.5 * (0.25 * pow((M_m / (d / D_m)), 2.0)))) / l)));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (0.5 * (h * (pow((D_m * (M_m / d)), 2.0) * (0.25 / l)))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(-d)
    t_1 = t_0 / sqrt(-l)
    if (l <= (-2.8d-99)) then
        tmp = t_1 * ((t_0 / sqrt(-h)) * (1.0d0 + ((h / l) * (((d_m * ((m_m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))))
    else if (l <= (-2d-310)) then
        tmp = t_1 * (sqrt((d / h)) * (1.0d0 + ((h * ((-0.5d0) * (0.25d0 * ((m_m / (d / d_m)) ** 2.0d0)))) / l)))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - (0.5d0 * (h * (((d_m * (m_m / d)) ** 2.0d0) * (0.25d0 / l)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt(-d);
	double t_1 = t_0 / Math.sqrt(-l);
	double tmp;
	if (l <= -2.8e-99) {
		tmp = t_1 * ((t_0 / Math.sqrt(-h)) * (1.0 + ((h / l) * (Math.pow((D_m * ((M_m / 2.0) / d)), 2.0) * -0.5))));
	} else if (l <= -2e-310) {
		tmp = t_1 * (Math.sqrt((d / h)) * (1.0 + ((h * (-0.5 * (0.25 * Math.pow((M_m / (d / D_m)), 2.0)))) / l)));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - (0.5 * (h * (Math.pow((D_m * (M_m / d)), 2.0) * (0.25 / l)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt(-d)
	t_1 = t_0 / math.sqrt(-l)
	tmp = 0
	if l <= -2.8e-99:
		tmp = t_1 * ((t_0 / math.sqrt(-h)) * (1.0 + ((h / l) * (math.pow((D_m * ((M_m / 2.0) / d)), 2.0) * -0.5))))
	elif l <= -2e-310:
		tmp = t_1 * (math.sqrt((d / h)) * (1.0 + ((h * (-0.5 * (0.25 * math.pow((M_m / (d / D_m)), 2.0)))) / l)))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - (0.5 * (h * (math.pow((D_m * (M_m / d)), 2.0) * (0.25 / l)))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(-d))
	t_1 = Float64(t_0 / sqrt(Float64(-l)))
	tmp = 0.0
	if (l <= -2.8e-99)
		tmp = Float64(t_1 * Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D_m * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5)))));
	elseif (l <= -2e-310)
		tmp = Float64(t_1 * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h * Float64(-0.5 * Float64(0.25 * (Float64(M_m / Float64(d / D_m)) ^ 2.0)))) / l))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - Float64(0.5 * Float64(h * Float64((Float64(D_m * Float64(M_m / d)) ^ 2.0) * Float64(0.25 / l))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt(-d);
	t_1 = t_0 / sqrt(-l);
	tmp = 0.0;
	if (l <= -2.8e-99)
		tmp = t_1 * ((t_0 / sqrt(-h)) * (1.0 + ((h / l) * (((D_m * ((M_m / 2.0) / d)) ^ 2.0) * -0.5))));
	elseif (l <= -2e-310)
		tmp = t_1 * (sqrt((d / h)) * (1.0 + ((h * (-0.5 * (0.25 * ((M_m / (d / D_m)) ^ 2.0)))) / l)));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (0.5 * (h * (((D_m * (M_m / d)) ^ 2.0) * (0.25 / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.8e-99], N[(t$95$1 * N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D$95$m * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(t$95$1 * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h * N[(-0.5 * N[(0.25 * N[Power[N[(M$95$m / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.25 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.8000000000000001e-99

   1. Initial program 61.7%

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

   3. Simplified 61.7%

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

      1. frac-2neg 61.7%

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

      2. sqrt-div 67.0%

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

   6. Applied egg-rr 67.0%

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

      1. frac-2neg 67.0%

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

      2. sqrt-div 78.9%

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

   8. Applied egg-rr 78.9%

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

   if -2.8000000000000001e-99 < l < -1.999999999999994e-310

   1. Initial program 76.4%

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

   3. Simplified 76.3%

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

      1. frac-2neg 76.3%

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

      2. sqrt-div 80.3%

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

   6. Applied egg-rr 80.3%

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

      1. associate-*l/ 91.1%

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

   8. Applied egg-rr 93.3%

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

   if -1.999999999999994e-310 < l

   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 64.7%

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

   5. Taylor expanded in M around 0 41.6%

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

      1. associate-*r* 45.7%

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

      2. times-frac 43.8%

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

      3. associate-/l* 43.5%

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

      4. *-commutative 43.5%

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

      5. unpow2 43.5%

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

      6. unpow2 43.5%

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

      7. times-frac 56.0%

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

      8. unpow2 56.0%

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

      9. swap-sqr 67.8%

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

      10. associate-/r/ 65.5%

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

      11. associate-/r/ 65.5%

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

      12. unpow2 65.5%

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

      13. associate-*l* 65.5%

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

      14. *-commutative 65.5%

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

      15. associate-*l/ 66.9%

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

      16. associate-/l* 68.6%

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

      17. *-commutative 68.6%

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

   7. Simplified 71.0%

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

      1. *-commutative 71.0%

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

      2. sqrt-div 77.9%

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

      3. sqrt-div 85.5%

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

      4. frac-times 85.5%

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

      5. add-sqr-sqrt 85.6%

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

   9. Applied egg-rr 85.6%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 79.3% accurate, 0.8× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= h -2e-310)
   (*
    (/ (sqrt (- d)) (sqrt (- l)))
    (*
     (sqrt (/ d h))
     (+ 1.0 (/ (* h (* -0.5 (* 0.25 (pow (/ M_m (/ d D_m)) 2.0)))) l))))
   (*
    (/ d (* (sqrt l) (sqrt h)))
    (- 1.0 (* 0.5 (* h (* (pow (* D_m (/ M_m d)) 2.0) (/ 0.25 l))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= -2e-310) {
		tmp = (sqrt(-d) / sqrt(-l)) * (sqrt((d / h)) * (1.0 + ((h * (-0.5 * (0.25 * pow((M_m / (d / D_m)), 2.0)))) / l)));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (0.5 * (h * (pow((D_m * (M_m / d)), 2.0) * (0.25 / l)))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (h <= (-2d-310)) then
        tmp = (sqrt(-d) / sqrt(-l)) * (sqrt((d / h)) * (1.0d0 + ((h * ((-0.5d0) * (0.25d0 * ((m_m / (d / d_m)) ** 2.0d0)))) / l)))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - (0.5d0 * (h * (((d_m * (m_m / d)) ** 2.0d0) * (0.25d0 / l)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= -2e-310) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-l)) * (Math.sqrt((d / h)) * (1.0 + ((h * (-0.5 * (0.25 * Math.pow((M_m / (d / D_m)), 2.0)))) / l)));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - (0.5 * (h * (Math.pow((D_m * (M_m / d)), 2.0) * (0.25 / l)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if h <= -2e-310:
		tmp = (math.sqrt(-d) / math.sqrt(-l)) * (math.sqrt((d / h)) * (1.0 + ((h * (-0.5 * (0.25 * math.pow((M_m / (d / D_m)), 2.0)))) / l)))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - (0.5 * (h * (math.pow((D_m * (M_m / d)), 2.0) * (0.25 / l)))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (h <= -2e-310)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h * Float64(-0.5 * Float64(0.25 * (Float64(M_m / Float64(d / D_m)) ^ 2.0)))) / l))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - Float64(0.5 * Float64(h * Float64((Float64(D_m * Float64(M_m / d)) ^ 2.0) * Float64(0.25 / l))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (h <= -2e-310)
		tmp = (sqrt(-d) / sqrt(-l)) * (sqrt((d / h)) * (1.0 + ((h * (-0.5 * (0.25 * ((M_m / (d / D_m)) ^ 2.0)))) / l)));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (0.5 * (h * (((D_m * (M_m / d)) ^ 2.0) * (0.25 / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[h, -2e-310], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h * N[(-0.5 * N[(0.25 * N[Power[N[(M$95$m / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.25 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -1.999999999999994e-310

   1. Initial program 66.7%

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

   3. Simplified 66.7%

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

      1. frac-2neg 66.7%

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

      2. sqrt-div 71.5%

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

   6. Applied egg-rr 71.5%

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

      1. associate-*l/ 75.4%

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

   8. Applied egg-rr 76.2%

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

   if -1.999999999999994e-310 < h

   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 64.7%

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

   5. Taylor expanded in M around 0 41.6%

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

      1. associate-*r* 45.7%

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

      2. times-frac 43.8%

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

      3. associate-/l* 43.5%

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

      4. *-commutative 43.5%

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

      5. unpow2 43.5%

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

      6. unpow2 43.5%

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

      7. times-frac 56.0%

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

      8. unpow2 56.0%

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

      9. swap-sqr 67.8%

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

      10. associate-/r/ 65.5%

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

      11. associate-/r/ 65.5%

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

      12. unpow2 65.5%

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

      13. associate-*l* 65.5%

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

      14. *-commutative 65.5%

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

      15. associate-*l/ 66.9%

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

      16. associate-/l* 68.6%

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

      17. *-commutative 68.6%

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

   7. Simplified 71.0%

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

      1. *-commutative 71.0%

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

      2. sqrt-div 77.9%

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

      3. sqrt-div 85.5%

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

      4. frac-times 85.5%

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

      5. add-sqr-sqrt 85.6%

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

   9. Applied egg-rr 85.6%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 80.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(h \cdot \left({\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\\ \mathbf{if}\;\ell \leq -7.6 \cdot 10^{+124}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D\_m \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(t\_0 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - t\_0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (* h (* (pow (* D_m (/ M_m d)) 2.0) (/ 0.25 l))))))
   (if (<= l -7.6e+124)
     (*
      (/ (sqrt (- d)) (sqrt (- l)))
      (*
       (+ 1.0 (* (/ h l) (* (pow (* D_m (/ (/ M_m 2.0) d)) 2.0) -0.5)))
       (sqrt (/ d h))))
     (if (<= l -2e-310)
       (* (* d (sqrt (/ 1.0 (* l h)))) (+ t_0 -1.0))
       (* (/ d (* (sqrt l) (sqrt h))) (- 1.0 t_0))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 0.5 * (h * (pow((D_m * (M_m / d)), 2.0) * (0.25 / l)));
	double tmp;
	if (l <= -7.6e+124) {
		tmp = (sqrt(-d) / sqrt(-l)) * ((1.0 + ((h / l) * (pow((D_m * ((M_m / 2.0) / d)), 2.0) * -0.5))) * sqrt((d / h)));
	} else if (l <= -2e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (t_0 + -1.0);
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - t_0);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (h * (((d_m * (m_m / d)) ** 2.0d0) * (0.25d0 / l)))
    if (l <= (-7.6d+124)) then
        tmp = (sqrt(-d) / sqrt(-l)) * ((1.0d0 + ((h / l) * (((d_m * ((m_m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))) * sqrt((d / h)))
    else if (l <= (-2d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * (t_0 + (-1.0d0))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - t_0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 0.5 * (h * (Math.pow((D_m * (M_m / d)), 2.0) * (0.25 / l)));
	double tmp;
	if (l <= -7.6e+124) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-l)) * ((1.0 + ((h / l) * (Math.pow((D_m * ((M_m / 2.0) / d)), 2.0) * -0.5))) * Math.sqrt((d / h)));
	} else if (l <= -2e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (t_0 + -1.0);
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - t_0);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = 0.5 * (h * (math.pow((D_m * (M_m / d)), 2.0) * (0.25 / l)))
	tmp = 0
	if l <= -7.6e+124:
		tmp = (math.sqrt(-d) / math.sqrt(-l)) * ((1.0 + ((h / l) * (math.pow((D_m * ((M_m / 2.0) / d)), 2.0) * -0.5))) * math.sqrt((d / h)))
	elif l <= -2e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (t_0 + -1.0)
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - t_0)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(0.5 * Float64(h * Float64((Float64(D_m * Float64(M_m / d)) ^ 2.0) * Float64(0.25 / l))))
	tmp = 0.0
	if (l <= -7.6e+124)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D_m * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5))) * sqrt(Float64(d / h))));
	elseif (l <= -2e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(t_0 + -1.0));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - t_0));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = 0.5 * (h * (((D_m * (M_m / d)) ^ 2.0) * (0.25 / l)));
	tmp = 0.0;
	if (l <= -7.6e+124)
		tmp = (sqrt(-d) / sqrt(-l)) * ((1.0 + ((h / l) * (((D_m * ((M_m / 2.0) / d)) ^ 2.0) * -0.5))) * sqrt((d / h)));
	elseif (l <= -2e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * (t_0 + -1.0);
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - t_0);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(0.5 * N[(h * N[(N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.25 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -7.6e+124], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D$95$m * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -7.5999999999999997e124

   1. Initial program 46.9%

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

   3. Simplified 46.9%

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

      1. frac-2neg 46.9%

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

      2. sqrt-div 57.0%

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

   6. Applied egg-rr 57.0%

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

   if -7.5999999999999997e124 < l < -1.999999999999994e-310

   1. Initial program 74.2%

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

   3. Simplified 75.2%

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

   5. Taylor expanded in M around 0 47.6%

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

      1. associate-*r* 48.5%

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

      2. times-frac 49.2%

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

      3. associate-/l* 47.4%

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

      4. *-commutative 47.4%

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

      5. unpow2 47.4%

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

      6. unpow2 47.4%

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

      7. times-frac 59.5%

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

      8. unpow2 59.5%

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

      9. swap-sqr 74.2%

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

      10. associate-/r/ 74.2%

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

      11. associate-/r/ 75.2%

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

      12. unpow2 75.2%

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

      13. associate-*l* 75.2%

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

      14. *-commutative 75.2%

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

      15. associate-*l/ 77.5%

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

      16. associate-/l* 76.5%

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

      17. *-commutative 76.5%

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

   7. Simplified 75.5%

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

      1. expm1-log1p-u 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)\right)}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]

      2. expm1-undefine 57.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)} - 1\right)}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]

   9. Applied egg-rr 57.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)} - 1\right)}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]
   10. Step-by-step derivation

       1. expm1-define 73.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)\right)}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]

   11. Simplified 73.9%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)\right)}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]

   12. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. *-commutative 0.0%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 82.8%

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

       5. neg-mul-1 82.8%

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

   14. Simplified 82.8%

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

   if -1.999999999999994e-310 < l

   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 64.7%

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

   5. Taylor expanded in M around 0 41.6%

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

      1. associate-*r* 45.7%

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

      2. times-frac 43.8%

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

      3. associate-/l* 43.5%

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

      4. *-commutative 43.5%

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

      5. unpow2 43.5%

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

      6. unpow2 43.5%

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

      7. times-frac 56.0%

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

      8. unpow2 56.0%

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

      9. swap-sqr 67.8%

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

      10. associate-/r/ 65.5%

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

      11. associate-/r/ 65.5%

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

      12. unpow2 65.5%

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

      13. associate-*l* 65.5%

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

      14. *-commutative 65.5%

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

      15. associate-*l/ 66.9%

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

      16. associate-/l* 68.6%

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

      17. *-commutative 68.6%

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

   7. Simplified 71.0%

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

      1. *-commutative 71.0%

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

      2. sqrt-div 77.9%

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

      3. sqrt-div 85.5%

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

      4. frac-times 85.5%

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

      5. add-sqr-sqrt 85.6%

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

   9. Applied egg-rr 85.6%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.6 \cdot 10^{+124}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(h \cdot \left({\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\\ \mathbf{if}\;d \leq -1.85 \cdot 10^{-19}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D\_m \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(t\_0 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - t\_0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (* h (* (pow (* D_m (/ M_m d)) 2.0) (/ 0.25 l))))))
   (if (<= d -1.85e-19)
     (*
      (*
       (/ (sqrt (- d)) (sqrt (- h)))
       (+ 1.0 (* (/ h l) (* (pow (* D_m (/ (/ M_m 2.0) d)) 2.0) -0.5))))
      (sqrt (/ d l)))
     (if (<= d -4e-310)
       (* (* d (sqrt (/ 1.0 (* l h)))) (+ t_0 -1.0))
       (* (/ d (* (sqrt l) (sqrt h))) (- 1.0 t_0))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 0.5 * (h * (pow((D_m * (M_m / d)), 2.0) * (0.25 / l)));
	double tmp;
	if (d <= -1.85e-19) {
		tmp = ((sqrt(-d) / sqrt(-h)) * (1.0 + ((h / l) * (pow((D_m * ((M_m / 2.0) / d)), 2.0) * -0.5)))) * sqrt((d / l));
	} else if (d <= -4e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (t_0 + -1.0);
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - t_0);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (h * (((d_m * (m_m / d)) ** 2.0d0) * (0.25d0 / l)))
    if (d <= (-1.85d-19)) then
        tmp = ((sqrt(-d) / sqrt(-h)) * (1.0d0 + ((h / l) * (((d_m * ((m_m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0))))) * sqrt((d / l))
    else if (d <= (-4d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * (t_0 + (-1.0d0))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - t_0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 0.5 * (h * (Math.pow((D_m * (M_m / d)), 2.0) * (0.25 / l)));
	double tmp;
	if (d <= -1.85e-19) {
		tmp = ((Math.sqrt(-d) / Math.sqrt(-h)) * (1.0 + ((h / l) * (Math.pow((D_m * ((M_m / 2.0) / d)), 2.0) * -0.5)))) * Math.sqrt((d / l));
	} else if (d <= -4e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (t_0 + -1.0);
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - t_0);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = 0.5 * (h * (math.pow((D_m * (M_m / d)), 2.0) * (0.25 / l)))
	tmp = 0
	if d <= -1.85e-19:
		tmp = ((math.sqrt(-d) / math.sqrt(-h)) * (1.0 + ((h / l) * (math.pow((D_m * ((M_m / 2.0) / d)), 2.0) * -0.5)))) * math.sqrt((d / l))
	elif d <= -4e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (t_0 + -1.0)
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - t_0)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(0.5 * Float64(h * Float64((Float64(D_m * Float64(M_m / d)) ^ 2.0) * Float64(0.25 / l))))
	tmp = 0.0
	if (d <= -1.85e-19)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D_m * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5)))) * sqrt(Float64(d / l)));
	elseif (d <= -4e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(t_0 + -1.0));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - t_0));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = 0.5 * (h * (((D_m * (M_m / d)) ^ 2.0) * (0.25 / l)));
	tmp = 0.0;
	if (d <= -1.85e-19)
		tmp = ((sqrt(-d) / sqrt(-h)) * (1.0 + ((h / l) * (((D_m * ((M_m / 2.0) / d)) ^ 2.0) * -0.5)))) * sqrt((d / l));
	elseif (d <= -4e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * (t_0 + -1.0);
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - t_0);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(0.5 * N[(h * N[(N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.25 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.85e-19], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D$95$m * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.85000000000000003e-19

   1. Initial program 83.2%

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

   3. Simplified 81.8%

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

      1. frac-2neg 84.2%

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

      2. sqrt-div 89.4%

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

   6. Applied egg-rr 86.9%

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

   if -1.85000000000000003e-19 < d < -3.999999999999988e-310

   1. Initial program 49.0%

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

   3. Simplified 50.5%

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

   5. Taylor expanded in M around 0 30.0%

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

      1. associate-*r* 30.0%

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

      2. times-frac 32.7%

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

      3. associate-/l* 31.3%

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

      4. *-commutative 31.3%

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

      5. unpow2 31.3%

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

      6. unpow2 31.3%

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

      7. times-frac 39.0%

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

      8. unpow2 39.0%

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

      9. swap-sqr 50.5%

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

      10. associate-/r/ 50.5%

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

      11. associate-/r/ 50.5%

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

      12. unpow2 50.5%

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

      13. associate-*l* 50.5%

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

      14. *-commutative 50.5%

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

      15. associate-*l/ 52.2%

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

      16. associate-/l* 50.8%

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

      17. *-commutative 50.8%

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

   7. Simplified 50.8%

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

      1. expm1-log1p-u 50.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)\right)}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]

      2. expm1-undefine 28.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)} - 1\right)}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]

   9. Applied egg-rr 28.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)} - 1\right)}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]
   10. Step-by-step derivation

       1. expm1-define 50.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)\right)}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]

   11. Simplified 50.3%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)\right)}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]

   12. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. *-commutative 0.0%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 63.6%

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

       5. neg-mul-1 63.6%

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

   14. Simplified 63.6%

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

   if -3.999999999999988e-310 < d

   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 64.7%

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

   5. Taylor expanded in M around 0 41.6%

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

      1. associate-*r* 45.7%

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

      2. times-frac 43.8%

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

      3. associate-/l* 43.5%

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

      4. *-commutative 43.5%

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

      5. unpow2 43.5%

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

      6. unpow2 43.5%

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

      7. times-frac 56.0%

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

      8. unpow2 56.0%

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

      9. swap-sqr 67.8%

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

      10. associate-/r/ 65.5%

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

      11. associate-/r/ 65.5%

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

      12. unpow2 65.5%

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

      13. associate-*l* 65.5%

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

      14. *-commutative 65.5%

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

      15. associate-*l/ 66.9%

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

      16. associate-/l* 68.6%

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

      17. *-commutative 68.6%

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

   7. Simplified 71.0%

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

      1. *-commutative 71.0%

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

      2. sqrt-div 77.9%

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

      3. sqrt-div 85.5%

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

      4. frac-times 85.5%

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

      5. add-sqr-sqrt 85.6%

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

   9. Applied egg-rr 85.6%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.85 \cdot 10^{-19}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{+194}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left(h \cdot \left({\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+122}:\\ \;\;\;\;d \cdot \left({\left(\ell \cdot h\right)}^{-0.5} \cdot \mathsf{fma}\left(\frac{0.25}{\ell}, \left(h \cdot -0.5\right) \cdot {\left(\frac{D\_m \cdot M\_m}{d}\right)}^{2}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -1.55e+194)
   (* (/ (sqrt (- d)) (sqrt (- l))) (sqrt (/ d h)))
   (if (<= l -2e-310)
     (*
      (* d (sqrt (/ 1.0 (* l h))))
      (+ (* 0.5 (* h (* (pow (* D_m (/ M_m d)) 2.0) (/ 0.25 l)))) -1.0))
     (if (<= l 3.2e+122)
       (*
        d
        (*
         (pow (* l h) -0.5)
         (fma (/ 0.25 l) (* (* h -0.5) (pow (/ (* D_m M_m) d) 2.0)) 1.0)))
       (* d (* (pow l -0.5) (pow h -0.5)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1.55e+194) {
		tmp = (sqrt(-d) / sqrt(-l)) * sqrt((d / h));
	} else if (l <= -2e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * ((0.5 * (h * (pow((D_m * (M_m / d)), 2.0) * (0.25 / l)))) + -1.0);
	} else if (l <= 3.2e+122) {
		tmp = d * (pow((l * h), -0.5) * fma((0.25 / l), ((h * -0.5) * pow(((D_m * M_m) / d), 2.0)), 1.0));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -1.55e+194)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * sqrt(Float64(d / h)));
	elseif (l <= -2e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(Float64(0.5 * Float64(h * Float64((Float64(D_m * Float64(M_m / d)) ^ 2.0) * Float64(0.25 / l)))) + -1.0));
	elseif (l <= 3.2e+122)
		tmp = Float64(d * Float64((Float64(l * h) ^ -0.5) * fma(Float64(0.25 / l), Float64(Float64(h * -0.5) * (Float64(Float64(D_m * M_m) / d) ^ 2.0)), 1.0)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -1.55e+194], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(h * N[(N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.25 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.2e+122], N[(d * N[(N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision] * N[(N[(0.25 / l), $MachinePrecision] * N[(N[(h * -0.5), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.55e194

   1. Initial program 37.5%

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

   3. Simplified 37.5%

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

      1. frac-2neg 37.5%

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

      2. sqrt-div 54.4%

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

   6. Applied egg-rr 54.4%

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

   7. Taylor expanded in d around inf 59.6%

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

   if -1.55e194 < l < -1.999999999999994e-310

   1. Initial program 72.4%

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

   3. Simplified 73.3%

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

   5. Taylor expanded in M around 0 45.0%

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

      1. associate-*r* 45.8%

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

      2. times-frac 46.4%

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

      3. associate-/l* 44.8%

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

      4. *-commutative 44.8%

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

      5. unpow2 44.8%

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

      6. unpow2 44.8%

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

      7. times-frac 57.0%

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

      8. unpow2 57.0%

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

      9. swap-sqr 72.4%

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

      10. associate-/r/ 72.4%

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

      11. associate-/r/ 73.3%

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

      12. unpow2 73.3%

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

      13. associate-*l* 73.3%

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

      14. *-commutative 73.3%

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

      15. associate-*l/ 74.5%

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

      16. associate-/l* 74.5%

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

      17. *-commutative 74.5%

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

   7. Simplified 73.7%

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

      1. expm1-log1p-u 72.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)\right)}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]

      2. expm1-undefine 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)} - 1\right)}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]

   9. Applied egg-rr 55.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)} - 1\right)}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]
   10. Step-by-step derivation

       1. expm1-define 72.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)\right)}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]

   11. Simplified 72.1%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)\right)}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]

   12. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. *-commutative 0.0%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 77.4%

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

       5. neg-mul-1 77.4%

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

   14. Simplified 77.4%

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

   if -1.999999999999994e-310 < l < 3.20000000000000012e122

   1. Initial program 74.6%

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

   3. Simplified 72.4%

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

      1. sqrt-unprod 64.3%

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

      2. pow1/2 64.3%

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

      3. frac-times 51.4%

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

      4. pow2 51.4%

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

   6. Applied egg-rr 51.4%

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

      1. unpow1/2 51.4%

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

   8. Simplified 51.4%

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

   9. Taylor expanded in d around 0 76.2%

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

       1. unpow-1 76.2%

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

       2. metadata-eval 76.2%

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

       3. pow-sqr 76.2%

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

       4. rem-sqrt-square 76.4%

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

       5. rem-square-sqrt 76.3%

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

       6. fabs-sqr 76.3%

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

       7. rem-square-sqrt 76.4%

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

   11. Simplified 76.4%

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

       1. pow1 76.4%

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

       2. cancel-sign-sub-inv 76.4%

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

       3. metadata-eval 76.4%

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

       4. div-inv 76.4%

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

       5. metadata-eval 76.4%

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

   13. Applied egg-rr 76.4%

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

   15. Simplified 87.2%

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

   if 3.20000000000000012e122 < l

   1. Initial program 53.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in d around inf 48.6%

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

      1. *-un-lft-identity 48.6%

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

      2. pow1/2 48.6%

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

      3. inv-pow 48.6%

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

      4. pow-pow 48.6%

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

      5. metadata-eval 48.6%

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

   7. Applied egg-rr 48.6%

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

      1. *-lft-identity 48.6%

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

   9. Simplified 48.6%

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

       1. *-commutative 48.6%

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

       2. unpow-prod-down 75.0%

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

   11. Applied egg-rr 75.0%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{+194}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+122}:\\ \;\;\;\;d \cdot \left({\left(\ell \cdot h\right)}^{-0.5} \cdot \mathsf{fma}\left(\frac{0.25}{\ell}, \left(h \cdot -0.5\right) \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 1 - 0.5 \cdot \left(h \cdot \left({\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\\ \mathbf{if}\;h \leq -2.55 \cdot 10^{+125}:\\ \;\;\;\;t\_0 \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (* 0.5 (* h (* (pow (* D_m (/ M_m d)) 2.0) (/ 0.25 l)))))))
   (if (<= h -2.55e+125)
     (* t_0 (* (sqrt (/ d h)) (sqrt (/ d l))))
     (if (<= h -2e-310)
       (*
        (* d (sqrt (/ 1.0 (* l h))))
        (+ -1.0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0)))))
       (* (/ d (* (sqrt l) (sqrt h))) t_0)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 1.0 - (0.5 * (h * (pow((D_m * (M_m / d)), 2.0) * (0.25 / l))));
	double tmp;
	if (h <= -2.55e+125) {
		tmp = t_0 * (sqrt((d / h)) * sqrt((d / l)));
	} else if (h <= -2e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0))));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * t_0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (0.5d0 * (h * (((d_m * (m_m / d)) ** 2.0d0) * (0.25d0 / l))))
    if (h <= (-2.55d+125)) then
        tmp = t_0 * (sqrt((d / h)) * sqrt((d / l)))
    else if (h <= (-2d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((-1.0d0) + (0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_m / d)) ** 2.0d0))))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 1.0 - (0.5 * (h * (Math.pow((D_m * (M_m / d)), 2.0) * (0.25 / l))));
	double tmp;
	if (h <= -2.55e+125) {
		tmp = t_0 * (Math.sqrt((d / h)) * Math.sqrt((d / l)));
	} else if (h <= -2e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D_m / d)), 2.0))));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = 1.0 - (0.5 * (h * (math.pow((D_m * (M_m / d)), 2.0) * (0.25 / l))))
	tmp = 0
	if h <= -2.55e+125:
		tmp = t_0 * (math.sqrt((d / h)) * math.sqrt((d / l)))
	elif h <= -2e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D_m / d)), 2.0))))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * t_0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(1.0 - Float64(0.5 * Float64(h * Float64((Float64(D_m * Float64(M_m / d)) ^ 2.0) * Float64(0.25 / l)))))
	tmp = 0.0
	if (h <= -2.55e+125)
		tmp = Float64(t_0 * Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))));
	elseif (h <= -2e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0)))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * t_0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = 1.0 - (0.5 * (h * (((D_m * (M_m / d)) ^ 2.0) * (0.25 / l))));
	tmp = 0.0;
	if (h <= -2.55e+125)
		tmp = t_0 * (sqrt((d / h)) * sqrt((d / l)));
	elseif (h <= -2e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h / l) * (((M_m / 2.0) * (D_m / d)) ^ 2.0))));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(1.0 - N[(0.5 * N[(h * N[(N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.25 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -2.55e+125], N[(t$95$0 * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -2e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -2.5499999999999999e125

   1. Initial program 59.6%

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

   3. Simplified 59.6%

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

   5. Taylor expanded in M around 0 31.5%

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

      1. associate-*r* 33.8%

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

      2. times-frac 31.1%

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

      3. associate-/l* 28.7%

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

      4. *-commutative 28.7%

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

      5. unpow2 28.7%

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

      6. unpow2 28.7%

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

      7. times-frac 42.8%

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

      8. unpow2 42.8%

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

      9. swap-sqr 59.6%

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

      10. associate-/r/ 59.6%

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

      11. associate-/r/ 59.6%

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

      12. unpow2 59.6%

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

      13. associate-*l* 59.6%

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

      14. *-commutative 59.6%

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

      15. associate-*l/ 62.7%

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

      16. associate-/l* 65.0%

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

      17. *-commutative 65.0%

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

   7. Simplified 65.0%

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

   if -2.5499999999999999e125 < h < -1.999999999999994e-310

   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 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. sqrt-unprod 58.3%

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

      2. pow1/2 58.3%

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

      3. frac-times 51.0%

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

      4. pow2 51.0%

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

   6. Applied egg-rr 51.0%

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

      1. unpow1/2 51.0%

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

   8. Simplified 51.0%

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

   9. Taylor expanded in d around -inf 79.3%

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

   if -1.999999999999994e-310 < h

   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 64.7%

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

   5. Taylor expanded in M around 0 41.6%

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

      1. associate-*r* 45.7%

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

      2. times-frac 43.8%

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

      3. associate-/l* 43.5%

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

      4. *-commutative 43.5%

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

      5. unpow2 43.5%

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

      6. unpow2 43.5%

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

      7. times-frac 56.0%

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

      8. unpow2 56.0%

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

      9. swap-sqr 67.8%

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

      10. associate-/r/ 65.5%

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

      11. associate-/r/ 65.5%

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

      12. unpow2 65.5%

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

      13. associate-*l* 65.5%

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

      14. *-commutative 65.5%

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

      15. associate-*l/ 66.9%

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

      16. associate-/l* 68.6%

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

      17. *-commutative 68.6%

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

   7. Simplified 71.0%

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

      1. *-commutative 71.0%

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

      2. sqrt-div 77.9%

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

      3. sqrt-div 85.5%

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

      4. frac-times 85.5%

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

      5. add-sqr-sqrt 85.6%

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

   9. Applied egg-rr 85.6%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.55 \cdot 10^{+125}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(h \cdot \left({\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\\ \mathbf{if}\;\ell \leq -6.5 \cdot 10^{+179}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(t\_0 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - t\_0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (* h (* (pow (* D_m (/ M_m d)) 2.0) (/ 0.25 l))))))
   (if (<= l -6.5e+179)
     (* (/ (sqrt (- d)) (sqrt (- l))) (sqrt (/ d h)))
     (if (<= l -2e-310)
       (* (* d (sqrt (/ 1.0 (* l h)))) (+ t_0 -1.0))
       (* (/ d (* (sqrt l) (sqrt h))) (- 1.0 t_0))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 0.5 * (h * (pow((D_m * (M_m / d)), 2.0) * (0.25 / l)));
	double tmp;
	if (l <= -6.5e+179) {
		tmp = (sqrt(-d) / sqrt(-l)) * sqrt((d / h));
	} else if (l <= -2e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (t_0 + -1.0);
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - t_0);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (h * (((d_m * (m_m / d)) ** 2.0d0) * (0.25d0 / l)))
    if (l <= (-6.5d+179)) then
        tmp = (sqrt(-d) / sqrt(-l)) * sqrt((d / h))
    else if (l <= (-2d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * (t_0 + (-1.0d0))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - t_0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 0.5 * (h * (Math.pow((D_m * (M_m / d)), 2.0) * (0.25 / l)));
	double tmp;
	if (l <= -6.5e+179) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-l)) * Math.sqrt((d / h));
	} else if (l <= -2e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (t_0 + -1.0);
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - t_0);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = 0.5 * (h * (math.pow((D_m * (M_m / d)), 2.0) * (0.25 / l)))
	tmp = 0
	if l <= -6.5e+179:
		tmp = (math.sqrt(-d) / math.sqrt(-l)) * math.sqrt((d / h))
	elif l <= -2e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (t_0 + -1.0)
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - t_0)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(0.5 * Float64(h * Float64((Float64(D_m * Float64(M_m / d)) ^ 2.0) * Float64(0.25 / l))))
	tmp = 0.0
	if (l <= -6.5e+179)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * sqrt(Float64(d / h)));
	elseif (l <= -2e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(t_0 + -1.0));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - t_0));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = 0.5 * (h * (((D_m * (M_m / d)) ^ 2.0) * (0.25 / l)));
	tmp = 0.0;
	if (l <= -6.5e+179)
		tmp = (sqrt(-d) / sqrt(-l)) * sqrt((d / h));
	elseif (l <= -2e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * (t_0 + -1.0);
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - t_0);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(0.5 * N[(h * N[(N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.25 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -6.5e+179], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.50000000000000052e179

   1. Initial program 37.5%

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

   3. Simplified 37.5%

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

      1. frac-2neg 37.5%

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

      2. sqrt-div 54.4%

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

   6. Applied egg-rr 54.4%

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

   7. Taylor expanded in d around inf 59.6%

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

   if -6.50000000000000052e179 < l < -1.999999999999994e-310

   1. Initial program 72.4%

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

   3. Simplified 73.3%

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

   5. Taylor expanded in M around 0 45.0%

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

      1. associate-*r* 45.8%

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

      2. times-frac 46.4%

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

      3. associate-/l* 44.8%

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

      4. *-commutative 44.8%

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

      5. unpow2 44.8%

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

      6. unpow2 44.8%

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

      7. times-frac 57.0%

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

      8. unpow2 57.0%

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

      9. swap-sqr 72.4%

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

      10. associate-/r/ 72.4%

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

      11. associate-/r/ 73.3%

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

      12. unpow2 73.3%

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

      13. associate-*l* 73.3%

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

      14. *-commutative 73.3%

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

      15. associate-*l/ 74.5%

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

      16. associate-/l* 74.5%

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

      17. *-commutative 74.5%

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

   7. Simplified 73.7%

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

      1. expm1-log1p-u 72.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)\right)}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]

      2. expm1-undefine 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)} - 1\right)}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]

   9. Applied egg-rr 55.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)} - 1\right)}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]
   10. Step-by-step derivation

       1. expm1-define 72.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)\right)}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]

   11. Simplified 72.1%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)\right)}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]

   12. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. *-commutative 0.0%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 77.4%

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

       5. neg-mul-1 77.4%

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

   14. Simplified 77.4%

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

   if -1.999999999999994e-310 < l

   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 64.7%

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

   5. Taylor expanded in M around 0 41.6%

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

      1. associate-*r* 45.7%

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

      2. times-frac 43.8%

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

      3. associate-/l* 43.5%

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

      4. *-commutative 43.5%

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

      5. unpow2 43.5%

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

      6. unpow2 43.5%

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

      7. times-frac 56.0%

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

      8. unpow2 56.0%

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

      9. swap-sqr 67.8%

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

      10. associate-/r/ 65.5%

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

      11. associate-/r/ 65.5%

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

      12. unpow2 65.5%

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

      13. associate-*l* 65.5%

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

      14. *-commutative 65.5%

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

      15. associate-*l/ 66.9%

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

      16. associate-/l* 68.6%

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

      17. *-commutative 68.6%

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

   7. Simplified 71.0%

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

      1. *-commutative 71.0%

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

      2. sqrt-div 77.9%

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

      3. sqrt-div 85.5%

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

      4. frac-times 85.5%

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

      5. add-sqr-sqrt 85.6%

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

   9. Applied egg-rr 85.6%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{+179}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(h \cdot \left({\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\\ \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+181}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(t\_0 + -1\right)\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+153}:\\ \;\;\;\;\left(1 - t\_0\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (* h (* (pow (* D_m (/ M_m d)) 2.0) (/ 0.25 l))))))
   (if (<= l -2.5e+181)
     (* (/ (sqrt (- d)) (sqrt (- l))) (sqrt (/ d h)))
     (if (<= l -2e-310)
       (* (* d (sqrt (/ 1.0 (* l h)))) (+ t_0 -1.0))
       (if (<= l 9e+153)
         (* (- 1.0 t_0) (* d (pow (* l h) -0.5)))
         (* d (* (pow l -0.5) (pow h -0.5))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 0.5 * (h * (pow((D_m * (M_m / d)), 2.0) * (0.25 / l)));
	double tmp;
	if (l <= -2.5e+181) {
		tmp = (sqrt(-d) / sqrt(-l)) * sqrt((d / h));
	} else if (l <= -2e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (t_0 + -1.0);
	} else if (l <= 9e+153) {
		tmp = (1.0 - t_0) * (d * pow((l * h), -0.5));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (h * (((d_m * (m_m / d)) ** 2.0d0) * (0.25d0 / l)))
    if (l <= (-2.5d+181)) then
        tmp = (sqrt(-d) / sqrt(-l)) * sqrt((d / h))
    else if (l <= (-2d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * (t_0 + (-1.0d0))
    else if (l <= 9d+153) then
        tmp = (1.0d0 - t_0) * (d * ((l * h) ** (-0.5d0)))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 0.5 * (h * (Math.pow((D_m * (M_m / d)), 2.0) * (0.25 / l)));
	double tmp;
	if (l <= -2.5e+181) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-l)) * Math.sqrt((d / h));
	} else if (l <= -2e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (t_0 + -1.0);
	} else if (l <= 9e+153) {
		tmp = (1.0 - t_0) * (d * Math.pow((l * h), -0.5));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = 0.5 * (h * (math.pow((D_m * (M_m / d)), 2.0) * (0.25 / l)))
	tmp = 0
	if l <= -2.5e+181:
		tmp = (math.sqrt(-d) / math.sqrt(-l)) * math.sqrt((d / h))
	elif l <= -2e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (t_0 + -1.0)
	elif l <= 9e+153:
		tmp = (1.0 - t_0) * (d * math.pow((l * h), -0.5))
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(0.5 * Float64(h * Float64((Float64(D_m * Float64(M_m / d)) ^ 2.0) * Float64(0.25 / l))))
	tmp = 0.0
	if (l <= -2.5e+181)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * sqrt(Float64(d / h)));
	elseif (l <= -2e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(t_0 + -1.0));
	elseif (l <= 9e+153)
		tmp = Float64(Float64(1.0 - t_0) * Float64(d * (Float64(l * h) ^ -0.5)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = 0.5 * (h * (((D_m * (M_m / d)) ^ 2.0) * (0.25 / l)));
	tmp = 0.0;
	if (l <= -2.5e+181)
		tmp = (sqrt(-d) / sqrt(-l)) * sqrt((d / h));
	elseif (l <= -2e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * (t_0 + -1.0);
	elseif (l <= 9e+153)
		tmp = (1.0 - t_0) * (d * ((l * h) ^ -0.5));
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(0.5 * N[(h * N[(N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.25 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.5e+181], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 9e+153], N[(N[(1.0 - t$95$0), $MachinePrecision] * N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.5000000000000002e181

   1. Initial program 37.5%

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

   3. Simplified 37.5%

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

      1. frac-2neg 37.5%

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

      2. sqrt-div 54.4%

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

   6. Applied egg-rr 54.4%

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

   7. Taylor expanded in d around inf 59.6%

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

   if -2.5000000000000002e181 < l < -1.999999999999994e-310

   1. Initial program 72.4%

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

   3. Simplified 73.3%

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

   5. Taylor expanded in M around 0 45.0%

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

      1. associate-*r* 45.8%

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

      2. times-frac 46.4%

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

      3. associate-/l* 44.8%

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

      4. *-commutative 44.8%

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

      5. unpow2 44.8%

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

      6. unpow2 44.8%

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

      7. times-frac 57.0%

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

      8. unpow2 57.0%

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

      9. swap-sqr 72.4%

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

      10. associate-/r/ 72.4%

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

      11. associate-/r/ 73.3%

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

      12. unpow2 73.3%

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

      13. associate-*l* 73.3%

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

      14. *-commutative 73.3%

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

      15. associate-*l/ 74.5%

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

      16. associate-/l* 74.5%

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

      17. *-commutative 74.5%

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

   7. Simplified 73.7%

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

      1. expm1-log1p-u 72.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)\right)}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]

      2. expm1-undefine 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)} - 1\right)}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]

   9. Applied egg-rr 55.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)} - 1\right)}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]
   10. Step-by-step derivation

       1. expm1-define 72.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)\right)}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]

   11. Simplified 72.1%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)\right)}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]

   12. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. *-commutative 0.0%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 77.4%

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

       5. neg-mul-1 77.4%

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

   14. Simplified 77.4%

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

   if -1.999999999999994e-310 < l < 9.0000000000000002e153

   1. Initial program 73.7%

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

   3. Simplified 71.6%

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

      1. sqrt-unprod 62.9%

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

      2. pow1/2 62.9%

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

      3. frac-times 50.5%

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

      4. pow2 50.5%

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

   6. Applied egg-rr 50.5%

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

      1. unpow1/2 50.5%

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

   8. Simplified 50.5%

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

   9. Taylor expanded in d around 0 76.1%

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

       1. unpow-1 76.1%

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

       2. metadata-eval 76.1%

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

       3. pow-sqr 76.2%

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

       4. rem-sqrt-square 76.4%

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

       5. rem-square-sqrt 76.2%

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

       6. fabs-sqr 76.2%

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

       7. rem-square-sqrt 76.4%

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

   11. Simplified 76.4%

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

   12. Taylor expanded in M around 0 58.7%

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

       1. associate-*r* 54.3%

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

       2. times-frac 51.6%

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

       3. associate-/l* 50.5%

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

       4. *-commutative 50.5%

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

       5. unpow2 50.5%

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

       6. unpow2 50.5%

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

       7. times-frac 63.4%

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

       8. unpow2 63.4%

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

       9. swap-sqr 73.8%

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

       10. associate-/r/ 71.6%

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

       11. associate-/r/ 71.6%

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

       12. unpow2 71.6%

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

       13. associate-*l* 71.6%

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

       14. *-commutative 71.6%

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

       15. associate-*l/ 75.9%

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

       16. associate-/l* 75.9%

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

       17. *-commutative 75.9%

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

   14. Simplified 86.2%

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

   if 9.0000000000000002e153 < l

   1. Initial program 53.2%

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

   3. Simplified 47.7%

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

   5. Taylor expanded in d around inf 45.5%

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

      1. *-un-lft-identity 45.5%

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

      2. pow1/2 45.5%

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

      3. inv-pow 45.5%

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

      4. pow-pow 45.5%

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

      5. metadata-eval 45.5%

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

   7. Applied egg-rr 45.5%

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

      1. *-lft-identity 45.5%

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

   9. Simplified 45.5%

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

       1. *-commutative 45.5%

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

       2. unpow-prod-down 74.9%

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

   11. Applied egg-rr 74.9%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+181}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+153}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.25 \cdot 10^{-174}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -2.45 \cdot 10^{-293}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(\left(\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{d}\right)}^{2}\right) \cdot -0.125\right)\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+126}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \frac{h \cdot {\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -1.25e-174)
   (* d (- (sqrt (/ (/ 1.0 l) h))))
   (if (<= l -2.45e-293)
     (*
      (sqrt (* (/ d h) (/ d l)))
      (* (* (/ h l) (pow (/ (* D_m M_m) d) 2.0)) -0.125))
     (if (<= l 9e+126)
       (*
        (* d (pow (* l h) -0.5))
        (- 1.0 (* 0.125 (/ (* h (pow (* D_m (/ M_m d)) 2.0)) l))))
       (* d (* (pow l -0.5) (pow h -0.5)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1.25e-174) {
		tmp = d * -sqrt(((1.0 / l) / h));
	} else if (l <= -2.45e-293) {
		tmp = sqrt(((d / h) * (d / l))) * (((h / l) * pow(((D_m * M_m) / d), 2.0)) * -0.125);
	} else if (l <= 9e+126) {
		tmp = (d * pow((l * h), -0.5)) * (1.0 - (0.125 * ((h * pow((D_m * (M_m / d)), 2.0)) / l)));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-1.25d-174)) then
        tmp = d * -sqrt(((1.0d0 / l) / h))
    else if (l <= (-2.45d-293)) then
        tmp = sqrt(((d / h) * (d / l))) * (((h / l) * (((d_m * m_m) / d) ** 2.0d0)) * (-0.125d0))
    else if (l <= 9d+126) then
        tmp = (d * ((l * h) ** (-0.5d0))) * (1.0d0 - (0.125d0 * ((h * ((d_m * (m_m / d)) ** 2.0d0)) / l)))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1.25e-174) {
		tmp = d * -Math.sqrt(((1.0 / l) / h));
	} else if (l <= -2.45e-293) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (((h / l) * Math.pow(((D_m * M_m) / d), 2.0)) * -0.125);
	} else if (l <= 9e+126) {
		tmp = (d * Math.pow((l * h), -0.5)) * (1.0 - (0.125 * ((h * Math.pow((D_m * (M_m / d)), 2.0)) / l)));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -1.25e-174:
		tmp = d * -math.sqrt(((1.0 / l) / h))
	elif l <= -2.45e-293:
		tmp = math.sqrt(((d / h) * (d / l))) * (((h / l) * math.pow(((D_m * M_m) / d), 2.0)) * -0.125)
	elif l <= 9e+126:
		tmp = (d * math.pow((l * h), -0.5)) * (1.0 - (0.125 * ((h * math.pow((D_m * (M_m / d)), 2.0)) / l)))
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -1.25e-174)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / l) / h))));
	elseif (l <= -2.45e-293)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / d) ^ 2.0)) * -0.125));
	elseif (l <= 9e+126)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(1.0 - Float64(0.125 * Float64(Float64(h * (Float64(D_m * Float64(M_m / d)) ^ 2.0)) / l))));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -1.25e-174)
		tmp = d * -sqrt(((1.0 / l) / h));
	elseif (l <= -2.45e-293)
		tmp = sqrt(((d / h) * (d / l))) * (((h / l) * (((D_m * M_m) / d) ^ 2.0)) * -0.125);
	elseif (l <= 9e+126)
		tmp = (d * ((l * h) ^ -0.5)) * (1.0 - (0.125 * ((h * ((D_m * (M_m / d)) ^ 2.0)) / l)));
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -1.25e-174], N[(d * (-N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, -2.45e-293], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 9e+126], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.125 * N[(N[(h * N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.2500000000000001e-174

   1. Initial program 63.3%

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

   3. Simplified 63.3%

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

   5. Taylor expanded in d around inf 6.4%

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

   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 42.3%

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

      4. associate-*r* 42.3%

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

      5. *-commutative 42.3%

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

      6. neg-mul-1 42.3%

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

      7. distribute-lft-neg-in 42.3%

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

      8. distribute-rgt-neg-in 42.3%

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

      9. unpow-1 42.3%

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

      10. metadata-eval 42.3%

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

      11. pow-sqr 42.3%

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

      12. rem-sqrt-square 42.3%

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

      13. rem-square-sqrt 42.2%

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

      14. fabs-sqr 42.2%

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

      15. rem-square-sqrt 42.3%

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

   8. Simplified 42.3%

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

   9. Taylor expanded in d around 0 42.3%

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

       1. mul-1-neg 42.3%

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

       2. distribute-rgt-neg-in 42.3%

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

       3. *-commutative 42.3%

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

       4. associate-/r* 43.2%

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

   11. Simplified 43.2%

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

   if -1.2500000000000001e-174 < l < -2.45e-293

   1. Initial program 87.9%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in M around inf 33.4%

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

      1. associate-*r* 33.4%

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

      2. times-frac 33.5%

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

      3. associate-/l* 29.3%

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

      4. *-commutative 29.3%

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

      5. unpow2 29.3%

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

      6. unpow2 29.3%

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

      7. times-frac 41.8%

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

      8. unpow2 41.8%

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

      9. swap-sqr 66.7%

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

      10. associate-/r/ 66.7%

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

      11. associate-/r/ 70.9%

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

      12. unpow2 70.9%

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

      13. associate-/r/ 66.7%

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

      14. *-commutative 66.7%

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

   7. Simplified 66.7%

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

      1. pow1 66.7%

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

      2. associate-*r* 66.8%

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

      3. pow1/2 66.8%

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

      4. pow1/2 66.8%

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

      5. pow-prod-down 60.3%

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

   9. Applied egg-rr 60.3%

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

       1. unpow1 60.3%

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

       2. unpow1/2 60.3%

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

       3. *-commutative 60.3%

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

       4. *-commutative 60.3%

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

       5. associate-*r/ 63.1%

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

   11. Simplified 63.1%

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

   if -2.45e-293 < l < 8.99999999999999947e126

   1. Initial program 72.9%

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

   3. Simplified 70.7%

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

      1. sqrt-unprod 62.8%

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

      2. pow1/2 62.8%

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

      3. frac-times 50.1%

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

      4. pow2 50.1%

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

   6. Applied egg-rr 50.1%

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

      1. unpow1/2 50.1%

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

   8. Simplified 50.1%

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

   9. Taylor expanded in d around 0 74.3%

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

       1. unpow-1 74.3%

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

       2. metadata-eval 74.3%

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

       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 74.6%

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

       5. rem-square-sqrt 74.5%

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

       6. fabs-sqr 74.5%

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

       7. rem-square-sqrt 74.6%

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

   11. Simplified 74.6%

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

   12. Taylor expanded in M around 0 56.5%

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

       1. associate-*r* 59.0%

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

       2. times-frac 53.0%

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

       3. associate-/l* 53.0%

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

       4. unpow2 53.0%

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

       5. unpow2 53.0%

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

       6. unpow2 53.0%

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

       7. times-frac 66.2%

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

       8. swap-sqr 76.9%

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

       9. unpow2 76.9%

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

       10. *-commutative 76.9%

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

       11. associate-*r/ 76.8%

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

   14. Simplified 76.8%

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

       1. associate-*l/ 84.0%

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

       2. associate-*r/ 84.7%

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

   16. Applied egg-rr 84.7%

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

   if 8.99999999999999947e126 < l

   1. Initial program 53.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in d around inf 48.6%

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

      1. *-un-lft-identity 48.6%

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

      2. pow1/2 48.6%

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

      3. inv-pow 48.6%

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

      4. pow-pow 48.6%

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

      5. metadata-eval 48.6%

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

   7. Applied egg-rr 48.6%

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

      1. *-lft-identity 48.6%

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

   9. Simplified 48.6%

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

       1. *-commutative 48.6%

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

       2. unpow-prod-down 75.0%

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

   11. Applied egg-rr 75.0%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.25 \cdot 10^{-174}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -2.45 \cdot 10^{-293}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(\left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right) \cdot -0.125\right)\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+126}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \frac{h \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{d}\right)}^{2}\\ \mathbf{if}\;\ell \leq -6.5 \cdot 10^{-172}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -2.45 \cdot 10^{-293}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(t\_0 \cdot -0.125\right)\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+153}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ h l) (pow (/ (* D_m M_m) d) 2.0))))
   (if (<= l -6.5e-172)
     (* d (- (sqrt (/ (/ 1.0 l) h))))
     (if (<= l -2.45e-293)
       (* (sqrt (* (/ d h) (/ d l))) (* t_0 -0.125))
       (if (<= l 1.65e+153)
         (* (* d (pow (* l h) -0.5)) (- 1.0 (* 0.125 t_0)))
         (* d (* (pow l -0.5) (pow h -0.5))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (h / l) * pow(((D_m * M_m) / d), 2.0);
	double tmp;
	if (l <= -6.5e-172) {
		tmp = d * -sqrt(((1.0 / l) / h));
	} else if (l <= -2.45e-293) {
		tmp = sqrt(((d / h) * (d / l))) * (t_0 * -0.125);
	} else if (l <= 1.65e+153) {
		tmp = (d * pow((l * h), -0.5)) * (1.0 - (0.125 * t_0));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (h / l) * (((d_m * m_m) / d) ** 2.0d0)
    if (l <= (-6.5d-172)) then
        tmp = d * -sqrt(((1.0d0 / l) / h))
    else if (l <= (-2.45d-293)) then
        tmp = sqrt(((d / h) * (d / l))) * (t_0 * (-0.125d0))
    else if (l <= 1.65d+153) then
        tmp = (d * ((l * h) ** (-0.5d0))) * (1.0d0 - (0.125d0 * t_0))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (h / l) * Math.pow(((D_m * M_m) / d), 2.0);
	double tmp;
	if (l <= -6.5e-172) {
		tmp = d * -Math.sqrt(((1.0 / l) / h));
	} else if (l <= -2.45e-293) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (t_0 * -0.125);
	} else if (l <= 1.65e+153) {
		tmp = (d * Math.pow((l * h), -0.5)) * (1.0 - (0.125 * t_0));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (h / l) * math.pow(((D_m * M_m) / d), 2.0)
	tmp = 0
	if l <= -6.5e-172:
		tmp = d * -math.sqrt(((1.0 / l) / h))
	elif l <= -2.45e-293:
		tmp = math.sqrt(((d / h) * (d / l))) * (t_0 * -0.125)
	elif l <= 1.65e+153:
		tmp = (d * math.pow((l * h), -0.5)) * (1.0 - (0.125 * t_0))
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / d) ^ 2.0))
	tmp = 0.0
	if (l <= -6.5e-172)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / l) / h))));
	elseif (l <= -2.45e-293)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(t_0 * -0.125));
	elseif (l <= 1.65e+153)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(1.0 - Float64(0.125 * t_0)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (h / l) * (((D_m * M_m) / d) ^ 2.0);
	tmp = 0.0;
	if (l <= -6.5e-172)
		tmp = d * -sqrt(((1.0 / l) / h));
	elseif (l <= -2.45e-293)
		tmp = sqrt(((d / h) * (d / l))) * (t_0 * -0.125);
	elseif (l <= 1.65e+153)
		tmp = (d * ((l * h) ^ -0.5)) * (1.0 - (0.125 * t_0));
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -6.5e-172], N[(d * (-N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, -2.45e-293], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * -0.125), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.65e+153], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.125 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -6.50000000000000012e-172

   1. Initial program 63.3%

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

   3. Simplified 63.3%

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

   5. Taylor expanded in d around inf 6.4%

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

   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 42.3%

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

      4. associate-*r* 42.3%

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

      5. *-commutative 42.3%

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

      6. neg-mul-1 42.3%

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

      7. distribute-lft-neg-in 42.3%

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

      8. distribute-rgt-neg-in 42.3%

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

      9. unpow-1 42.3%

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

      10. metadata-eval 42.3%

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

      11. pow-sqr 42.3%

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

      12. rem-sqrt-square 42.3%

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

      13. rem-square-sqrt 42.2%

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

      14. fabs-sqr 42.2%

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

      15. rem-square-sqrt 42.3%

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

   8. Simplified 42.3%

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

   9. Taylor expanded in d around 0 42.3%

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

       1. mul-1-neg 42.3%

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

       2. distribute-rgt-neg-in 42.3%

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

       3. *-commutative 42.3%

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

       4. associate-/r* 43.2%

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

   11. Simplified 43.2%

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

   if -6.50000000000000012e-172 < l < -2.45e-293

   1. Initial program 87.9%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in M around inf 33.4%

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

      1. associate-*r* 33.4%

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

      2. times-frac 33.5%

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

      3. associate-/l* 29.3%

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

      4. *-commutative 29.3%

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

      5. unpow2 29.3%

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

      6. unpow2 29.3%

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

      7. times-frac 41.8%

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

      8. unpow2 41.8%

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

      9. swap-sqr 66.7%

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

      10. associate-/r/ 66.7%

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

      11. associate-/r/ 70.9%

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

      12. unpow2 70.9%

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

      13. associate-/r/ 66.7%

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

      14. *-commutative 66.7%

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

   7. Simplified 66.7%

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

      1. pow1 66.7%

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

      2. associate-*r* 66.8%

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

      3. pow1/2 66.8%

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

      4. pow1/2 66.8%

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

      5. pow-prod-down 60.3%

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

   9. Applied egg-rr 60.3%

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

       1. unpow1 60.3%

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

       2. unpow1/2 60.3%

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

       3. *-commutative 60.3%

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

       4. *-commutative 60.3%

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

       5. associate-*r/ 63.1%

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

   11. Simplified 63.1%

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

   if -2.45e-293 < l < 1.64999999999999997e153

   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 70.0%

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

      1. sqrt-unprod 61.4%

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

      2. pow1/2 61.4%

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

      3. frac-times 49.3%

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

      4. pow2 49.3%

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

   6. Applied egg-rr 49.3%

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

      1. unpow1/2 49.3%

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

   8. Simplified 49.3%

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

   9. Taylor expanded in d around 0 74.4%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   10. 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 74.6%

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

       5. rem-square-sqrt 74.5%

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

       6. fabs-sqr 74.5%

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

       7. rem-square-sqrt 74.6%

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

   11. Simplified 74.6%

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

   12. Taylor expanded in M around 0 57.4%

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

       1. associate-*r* 59.7%

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

       2. times-frac 54.0%

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

       3. associate-/l* 52.9%

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

       4. unpow2 52.9%

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

       5. unpow2 52.9%

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

       6. unpow2 52.9%

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

       7. times-frac 65.5%

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

       8. swap-sqr 76.8%

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

       9. unpow2 76.8%

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

       10. *-commutative 76.8%

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

       11. associate-*r/ 76.8%

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

   14. Simplified 76.8%

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

   if 1.64999999999999997e153 < l

   1. Initial program 53.2%

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

   3. Simplified 47.7%

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

   5. Taylor expanded in d around inf 45.5%

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

      1. *-un-lft-identity 45.5%

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

      2. pow1/2 45.5%

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

      3. inv-pow 45.5%

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

      4. pow-pow 45.5%

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

      5. metadata-eval 45.5%

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

   7. Applied egg-rr 45.5%

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

      1. *-lft-identity 45.5%

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

   9. Simplified 45.5%

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

       1. *-commutative 45.5%

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

       2. unpow-prod-down 74.9%

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

   11. Applied egg-rr 74.9%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{-172}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -2.45 \cdot 10^{-293}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(\left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right) \cdot -0.125\right)\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+153}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(h \cdot \left({\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(t\_0 + -1\right)\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{+154}:\\ \;\;\;\;\left(1 - t\_0\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (* h (* (pow (* D_m (/ M_m d)) 2.0) (/ 0.25 l))))))
   (if (<= l -2e-310)
     (* (* d (sqrt (/ 1.0 (* l h)))) (+ t_0 -1.0))
     (if (<= l 7.2e+154)
       (* (- 1.0 t_0) (* d (pow (* l h) -0.5)))
       (* d (* (pow l -0.5) (pow h -0.5)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 0.5 * (h * (pow((D_m * (M_m / d)), 2.0) * (0.25 / l)));
	double tmp;
	if (l <= -2e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (t_0 + -1.0);
	} else if (l <= 7.2e+154) {
		tmp = (1.0 - t_0) * (d * pow((l * h), -0.5));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (h * (((d_m * (m_m / d)) ** 2.0d0) * (0.25d0 / l)))
    if (l <= (-2d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * (t_0 + (-1.0d0))
    else if (l <= 7.2d+154) then
        tmp = (1.0d0 - t_0) * (d * ((l * h) ** (-0.5d0)))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 0.5 * (h * (Math.pow((D_m * (M_m / d)), 2.0) * (0.25 / l)));
	double tmp;
	if (l <= -2e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (t_0 + -1.0);
	} else if (l <= 7.2e+154) {
		tmp = (1.0 - t_0) * (d * Math.pow((l * h), -0.5));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = 0.5 * (h * (math.pow((D_m * (M_m / d)), 2.0) * (0.25 / l)))
	tmp = 0
	if l <= -2e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (t_0 + -1.0)
	elif l <= 7.2e+154:
		tmp = (1.0 - t_0) * (d * math.pow((l * h), -0.5))
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(0.5 * Float64(h * Float64((Float64(D_m * Float64(M_m / d)) ^ 2.0) * Float64(0.25 / l))))
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(t_0 + -1.0));
	elseif (l <= 7.2e+154)
		tmp = Float64(Float64(1.0 - t_0) * Float64(d * (Float64(l * h) ^ -0.5)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = 0.5 * (h * (((D_m * (M_m / d)) ^ 2.0) * (0.25 / l)));
	tmp = 0.0;
	if (l <= -2e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * (t_0 + -1.0);
	elseif (l <= 7.2e+154)
		tmp = (1.0 - t_0) * (d * ((l * h) ^ -0.5));
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(0.5 * N[(h * N[(N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.25 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.2e+154], N[(N[(1.0 - t$95$0), $MachinePrecision] * N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 66.7%

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

   3. Simplified 67.4%

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

   5. Taylor expanded in M around 0 40.7%

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

      1. associate-*r* 41.3%

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

      2. times-frac 41.8%

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

      3. associate-/l* 40.5%

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

      4. *-commutative 40.5%

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

      5. unpow2 40.5%

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

      6. unpow2 40.5%

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

      7. times-frac 51.5%

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

      8. unpow2 51.5%

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

      9. swap-sqr 66.7%

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

      10. associate-/r/ 66.7%

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

      11. associate-/r/ 67.4%

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

      12. unpow2 67.4%

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

      13. associate-*l* 67.4%

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

      14. *-commutative 67.4%

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

      15. associate-*l/ 68.5%

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

      16. associate-/l* 68.5%

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

      17. *-commutative 68.5%

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

   7. Simplified 67.8%

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

      1. expm1-log1p-u 66.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)\right)}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]

      2. expm1-undefine 48.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)} - 1\right)}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]

   9. Applied egg-rr 48.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)} - 1\right)}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]
   10. Step-by-step derivation

       1. expm1-define 66.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)\right)}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]

   11. Simplified 66.4%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)\right)}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \]

   12. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. *-commutative 0.0%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 71.0%

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

       5. neg-mul-1 71.0%

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

   14. Simplified 71.0%

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

   if -1.999999999999994e-310 < l < 7.2000000000000001e154

   1. Initial program 73.7%

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

   3. Simplified 71.6%

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

      1. sqrt-unprod 62.9%

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

      2. pow1/2 62.9%

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

      3. frac-times 50.5%

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

      4. pow2 50.5%

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

   6. Applied egg-rr 50.5%

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

      1. unpow1/2 50.5%

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

   8. Simplified 50.5%

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

   9. Taylor expanded in d around 0 76.1%

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

       1. unpow-1 76.1%

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

       2. metadata-eval 76.1%

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

       3. pow-sqr 76.2%

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

       4. rem-sqrt-square 76.4%

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

       5. rem-square-sqrt 76.2%

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

       6. fabs-sqr 76.2%

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

       7. rem-square-sqrt 76.4%

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

   11. Simplified 76.4%

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

   12. Taylor expanded in M around 0 58.7%

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

       1. associate-*r* 54.3%

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

       2. times-frac 51.6%

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

       3. associate-/l* 50.5%

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

       4. *-commutative 50.5%

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

       5. unpow2 50.5%

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

       6. unpow2 50.5%

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

       7. times-frac 63.4%

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

       8. unpow2 63.4%

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

       9. swap-sqr 73.8%

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

       10. associate-/r/ 71.6%

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

       11. associate-/r/ 71.6%

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

       12. unpow2 71.6%

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

       13. associate-*l* 71.6%

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

       14. *-commutative 71.6%

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

       15. associate-*l/ 75.9%

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

       16. associate-/l* 75.9%

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

       17. *-commutative 75.9%

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

   14. Simplified 86.2%

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

   if 7.2000000000000001e154 < l

   1. Initial program 53.2%

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

   3. Simplified 47.7%

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

   5. Taylor expanded in d around inf 45.5%

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

      1. *-un-lft-identity 45.5%

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

      2. pow1/2 45.5%

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

      3. inv-pow 45.5%

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

      4. pow-pow 45.5%

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

      5. metadata-eval 45.5%

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

   7. Applied egg-rr 45.5%

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

      1. *-lft-identity 45.5%

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

   9. Simplified 45.5%

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

       1. *-commutative 45.5%

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

       2. unpow-prod-down 74.9%

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

   11. Applied egg-rr 74.9%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{+154}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.45 \cdot 10^{-293}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+153}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \left({\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -2.45e-293)
   (*
    (/ d (sqrt (* l h)))
    (+ -1.0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0)))))
   (if (<= l 1.2e+153)
     (*
      (- 1.0 (* 0.5 (* h (* (pow (* D_m (/ M_m d)) 2.0) (/ 0.25 l)))))
      (* d (pow (* l h) -0.5)))
     (* d (* (pow l -0.5) (pow h -0.5))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -2.45e-293) {
		tmp = (d / sqrt((l * h))) * (-1.0 + (0.5 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0))));
	} else if (l <= 1.2e+153) {
		tmp = (1.0 - (0.5 * (h * (pow((D_m * (M_m / d)), 2.0) * (0.25 / l))))) * (d * pow((l * h), -0.5));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-2.45d-293)) then
        tmp = (d / sqrt((l * h))) * ((-1.0d0) + (0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_m / d)) ** 2.0d0))))
    else if (l <= 1.2d+153) then
        tmp = (1.0d0 - (0.5d0 * (h * (((d_m * (m_m / d)) ** 2.0d0) * (0.25d0 / l))))) * (d * ((l * h) ** (-0.5d0)))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -2.45e-293) {
		tmp = (d / Math.sqrt((l * h))) * (-1.0 + (0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D_m / d)), 2.0))));
	} else if (l <= 1.2e+153) {
		tmp = (1.0 - (0.5 * (h * (Math.pow((D_m * (M_m / d)), 2.0) * (0.25 / l))))) * (d * Math.pow((l * h), -0.5));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -2.45e-293:
		tmp = (d / math.sqrt((l * h))) * (-1.0 + (0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D_m / d)), 2.0))))
	elif l <= 1.2e+153:
		tmp = (1.0 - (0.5 * (h * (math.pow((D_m * (M_m / d)), 2.0) * (0.25 / l))))) * (d * math.pow((l * h), -0.5))
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -2.45e-293)
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * Float64(-1.0 + Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0)))));
	elseif (l <= 1.2e+153)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(h * Float64((Float64(D_m * Float64(M_m / d)) ^ 2.0) * Float64(0.25 / l))))) * Float64(d * (Float64(l * h) ^ -0.5)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -2.45e-293)
		tmp = (d / sqrt((l * h))) * (-1.0 + (0.5 * ((h / l) * (((M_m / 2.0) * (D_m / d)) ^ 2.0))));
	elseif (l <= 1.2e+153)
		tmp = (1.0 - (0.5 * (h * (((D_m * (M_m / d)) ^ 2.0) * (0.25 / l))))) * (d * ((l * h) ^ -0.5));
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -2.45e-293], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.2e+153], N[(N[(1.0 - N[(0.5 * N[(h * N[(N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.25 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.45e-293

   1. Initial program 67.7%

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

   3. Simplified 68.4%

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

      1. sqrt-unprod 55.3%

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

      2. pow1/2 55.3%

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

      3. frac-times 43.7%

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

      4. pow2 43.7%

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

   6. Applied egg-rr 43.7%

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

      1. unpow1/2 43.7%

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

   8. Simplified 43.7%

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

      1. sqrt-div 48.8%

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

   10. Applied egg-rr 48.8%

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

       1. unpow2 48.8%

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

       2. sqr-neg 48.8%

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

       3. rem-sqrt-square 68.0%

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

       4. rem-square-sqrt 67.9%

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

       5. fabs-sqr 67.9%

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

       6. rem-square-sqrt 68.0%

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

   12. Simplified 68.0%

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

   if -2.45e-293 < l < 1.19999999999999996e153

   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 70.0%

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

      1. sqrt-unprod 61.4%

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

      2. pow1/2 61.4%

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

      3. frac-times 49.3%

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

      4. pow2 49.3%

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

   6. Applied egg-rr 49.3%

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

      1. unpow1/2 49.3%

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

   8. Simplified 49.3%

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

   9. Taylor expanded in d around 0 74.4%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   10. 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 74.6%

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

       5. rem-square-sqrt 74.5%

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

       6. fabs-sqr 74.5%

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

       7. rem-square-sqrt 74.6%

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

   11. Simplified 74.6%

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

   12. Taylor expanded in M around 0 57.4%

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

       1. associate-*r* 53.0%

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

       2. times-frac 50.4%

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

       3. associate-/l* 49.3%

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

       4. *-commutative 49.3%

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

       5. unpow2 49.3%

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

       6. unpow2 49.3%

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

       7. times-frac 61.9%

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

       8. unpow2 61.9%

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

       9. swap-sqr 72.1%

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

       10. associate-/r/ 70.0%

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

       11. associate-/r/ 69.9%

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

       12. unpow2 69.9%

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

       13. associate-*l* 69.9%

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

       14. *-commutative 69.9%

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

       15. associate-*l/ 75.3%

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

       16. associate-/l* 75.3%

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

       17. *-commutative 75.3%

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

   14. Simplified 84.3%

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

   if 1.19999999999999996e153 < l

   1. Initial program 53.2%

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

   3. Simplified 47.7%

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

   5. Taylor expanded in d around inf 45.5%

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

      1. *-un-lft-identity 45.5%

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

      2. pow1/2 45.5%

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

      3. inv-pow 45.5%

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

      4. pow-pow 45.5%

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

      5. metadata-eval 45.5%

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

   7. Applied egg-rr 45.5%

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

      1. *-lft-identity 45.5%

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

   9. Simplified 45.5%

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

       1. *-commutative 45.5%

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

       2. unpow-prod-down 74.9%

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

   11. Applied egg-rr 74.9%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.45 \cdot 10^{-293}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+153}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 72.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.45 \cdot 10^{-293}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{+155}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \left({\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -2.45e-293)
   (*
    (* d (sqrt (/ 1.0 (* l h))))
    (+ -1.0 (* 0.125 (* (/ h l) (pow (/ (* D_m M_m) d) 2.0)))))
   (if (<= l 2.1e+155)
     (*
      (- 1.0 (* 0.5 (* h (* (pow (* D_m (/ M_m d)) 2.0) (/ 0.25 l)))))
      (* d (pow (* l h) -0.5)))
     (* d (* (pow l -0.5) (pow h -0.5))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -2.45e-293) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.125 * ((h / l) * pow(((D_m * M_m) / d), 2.0))));
	} else if (l <= 2.1e+155) {
		tmp = (1.0 - (0.5 * (h * (pow((D_m * (M_m / d)), 2.0) * (0.25 / l))))) * (d * pow((l * h), -0.5));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-2.45d-293)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((-1.0d0) + (0.125d0 * ((h / l) * (((d_m * m_m) / d) ** 2.0d0))))
    else if (l <= 2.1d+155) then
        tmp = (1.0d0 - (0.5d0 * (h * (((d_m * (m_m / d)) ** 2.0d0) * (0.25d0 / l))))) * (d * ((l * h) ** (-0.5d0)))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -2.45e-293) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (-1.0 + (0.125 * ((h / l) * Math.pow(((D_m * M_m) / d), 2.0))));
	} else if (l <= 2.1e+155) {
		tmp = (1.0 - (0.5 * (h * (Math.pow((D_m * (M_m / d)), 2.0) * (0.25 / l))))) * (d * Math.pow((l * h), -0.5));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -2.45e-293:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (-1.0 + (0.125 * ((h / l) * math.pow(((D_m * M_m) / d), 2.0))))
	elif l <= 2.1e+155:
		tmp = (1.0 - (0.5 * (h * (math.pow((D_m * (M_m / d)), 2.0) * (0.25 / l))))) * (d * math.pow((l * h), -0.5))
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -2.45e-293)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + Float64(0.125 * Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / d) ^ 2.0)))));
	elseif (l <= 2.1e+155)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(h * Float64((Float64(D_m * Float64(M_m / d)) ^ 2.0) * Float64(0.25 / l))))) * Float64(d * (Float64(l * h) ^ -0.5)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -2.45e-293)
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.125 * ((h / l) * (((D_m * M_m) / d) ^ 2.0))));
	elseif (l <= 2.1e+155)
		tmp = (1.0 - (0.5 * (h * (((D_m * (M_m / d)) ^ 2.0) * (0.25 / l))))) * (d * ((l * h) ^ -0.5));
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -2.45e-293], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(0.125 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.1e+155], N[(N[(1.0 - N[(0.5 * N[(h * N[(N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.25 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.45e-293

   1. Initial program 67.7%

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

   3. Simplified 68.4%

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

      1. sqrt-unprod 55.3%

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

      2. pow1/2 55.3%

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

      3. frac-times 43.7%

         \[\leadsto {\color{blue}{\left(\frac{d \cdot d}{h \cdot \ell}\right)}}^{0.5} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. pow2 43.7%

         \[\leadsto {\left(\frac{\color{blue}{{d}^{2}}}{h \cdot \ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 43.7%

      \[\leadsto \color{blue}{{\left(\frac{{d}^{2}}{h \cdot \ell}\right)}^{0.5}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. unpow1/2 43.7%

         \[\leadsto \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. Simplified 43.7%

      \[\leadsto \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   9. Taylor expanded in d around 0 3.1%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   10. Step-by-step derivation

       1. unpow-1 3.1%

          \[\leadsto \left(d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       2. metadata-eval 3.1%

          \[\leadsto \left(d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       3. pow-sqr 3.1%

          \[\leadsto \left(d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       4. rem-sqrt-square 3.1%

          \[\leadsto \left(d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       5. rem-square-sqrt 3.1%

          \[\leadsto \left(d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       6. fabs-sqr 3.1%

          \[\leadsto \left(d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       7. rem-square-sqrt 3.1%

          \[\leadsto \left(d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   11. Simplified 3.1%

       \[\leadsto \color{blue}{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   12. Taylor expanded in M around 0 0.7%

       \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   13. Step-by-step derivation

       1. associate-*r* 0.8%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}\right) \]

       2. times-frac 0.7%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right)}\right) \]

       3. associate-/l* 0.6%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \left(\color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right)\right) \]

       4. unpow2 0.6%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot \frac{{M}^{2}}{{d}^{2}}\right) \cdot \frac{h}{\ell}\right)\right) \]

       5. unpow2 0.6%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot \frac{\color{blue}{M \cdot M}}{{d}^{2}}\right) \cdot \frac{h}{\ell}\right)\right) \]

       6. unpow2 0.6%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{\color{blue}{d \cdot d}}\right) \cdot \frac{h}{\ell}\right)\right) \]

       7. times-frac 2.1%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{M}{d}\right)}\right) \cdot \frac{h}{\ell}\right)\right) \]

       8. swap-sqr 3.8%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \left(\color{blue}{\left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)} \cdot \frac{h}{\ell}\right)\right) \]

       9. unpow2 3.8%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \left(\color{blue}{{\left(D \cdot \frac{M}{d}\right)}^{2}} \cdot \frac{h}{\ell}\right)\right) \]

       10. *-commutative 3.8%

           \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}\right) \]

       11. associate-*r/ 3.1%

           \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot {\color{blue}{\left(\frac{D \cdot M}{d}\right)}}^{2}\right)\right) \]

   14. Simplified 3.1%

       \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)}\right) \]

   15. Taylor expanded in h around -inf 0.0%

       \[\leadsto \left(d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \]
   16. Step-by-step derivation

       1. *-commutative 0.0%

          \[\leadsto \left(d \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \]

       2. unpow2 0.0%

          \[\leadsto \left(d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \]

       3. rem-square-sqrt 67.3%

          \[\leadsto \left(d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \]

       4. mul-1-neg 67.3%

          \[\leadsto \left(d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \]

   17. Simplified 67.3%

       \[\leadsto \left(d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \]

   if -2.45e-293 < l < 2.1e155

   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 70.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sqrt-unprod 61.4%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. pow1/2 61.4%

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. frac-times 49.3%

         \[\leadsto {\color{blue}{\left(\frac{d \cdot d}{h \cdot \ell}\right)}}^{0.5} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. pow2 49.3%

         \[\leadsto {\left(\frac{\color{blue}{{d}^{2}}}{h \cdot \ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 49.3%

      \[\leadsto \color{blue}{{\left(\frac{{d}^{2}}{h \cdot \ell}\right)}^{0.5}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. unpow1/2 49.3%

         \[\leadsto \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. Simplified 49.3%

      \[\leadsto \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   9. Taylor expanded in d around 0 74.4%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   10. 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 74.6%

          \[\leadsto \left(d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       5. rem-square-sqrt 74.5%

          \[\leadsto \left(d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       6. fabs-sqr 74.5%

          \[\leadsto \left(d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       7. rem-square-sqrt 74.6%

          \[\leadsto \left(d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   11. Simplified 74.6%

       \[\leadsto \color{blue}{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   12. Taylor expanded in M around 0 57.4%

       \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   13. Step-by-step derivation

       1. associate-*r* 53.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}\right)\right) \]

       2. times-frac 50.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right)}\right)\right) \]

       3. associate-/l* 49.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right)\right)\right) \]

       4. *-commutative 49.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\left(\frac{{M}^{2}}{{d}^{2}} \cdot {D}^{2}\right)} \cdot \frac{h}{\ell}\right)\right)\right) \]

       5. unpow2 49.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\frac{\color{blue}{M \cdot M}}{{d}^{2}} \cdot {D}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

       6. unpow2 49.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\frac{M \cdot M}{\color{blue}{d \cdot d}} \cdot {D}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

       7. times-frac 61.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\color{blue}{\left(\frac{M}{d} \cdot \frac{M}{d}\right)} \cdot {D}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

       8. unpow2 61.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\left(\frac{M}{d} \cdot \frac{M}{d}\right) \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

       9. swap-sqr 72.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right)} \cdot \frac{h}{\ell}\right)\right)\right) \]

       10. associate-/r/ 70.0%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{D}}} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

       11. associate-/r/ 69.9%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\left(\frac{M}{\frac{d}{D}} \cdot \color{blue}{\frac{M}{\frac{d}{D}}}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

       12. unpow2 69.9%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2}} \cdot \frac{h}{\ell}\right)\right)\right) \]

       13. associate-*l* 69.9%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right) \]

       14. *-commutative 69.9%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)\right)}\right) \]

       15. associate-*l/ 75.3%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{h \cdot \left(0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)}{\ell}}\right) \]

       16. associate-/l* 75.3%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{0.25 \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}{\ell}\right)}\right) \]

       17. *-commutative 75.3%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2} \cdot 0.25}}{\ell}\right)\right) \]

   14. Simplified 84.3%

       \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)}\right) \]

   if 2.1e155 < l

   1. Initial program 53.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 47.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 45.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 45.5%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. pow1/2 45.5%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right) \]

      3. inv-pow 45.5%

         \[\leadsto d \cdot \left(1 \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right) \]

      4. pow-pow 45.5%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right) \]

      5. metadata-eval 45.5%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 45.5%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 45.5%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 45.5%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. *-commutative 45.5%

          \[\leadsto d \cdot {\color{blue}{\left(\ell \cdot h\right)}}^{-0.5} \]

       2. unpow-prod-down 74.9%

          \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]

   11. Applied egg-rr 74.9%

       \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.45 \cdot 10^{-293}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{+155}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{0.25}{\ell}\right)\right)\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 72.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.45 \cdot 10^{-293}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 5.7 \cdot 10^{+122}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \frac{h \cdot {\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -2.45e-293)
   (*
    (* d (sqrt (/ 1.0 (* l h))))
    (+ -1.0 (* 0.125 (* (/ h l) (pow (/ (* D_m M_m) d) 2.0)))))
   (if (<= l 5.7e+122)
     (*
      (* d (pow (* l h) -0.5))
      (- 1.0 (* 0.125 (/ (* h (pow (* D_m (/ M_m d)) 2.0)) l))))
     (* d (* (pow l -0.5) (pow h -0.5))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -2.45e-293) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.125 * ((h / l) * pow(((D_m * M_m) / d), 2.0))));
	} else if (l <= 5.7e+122) {
		tmp = (d * pow((l * h), -0.5)) * (1.0 - (0.125 * ((h * pow((D_m * (M_m / d)), 2.0)) / l)));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-2.45d-293)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((-1.0d0) + (0.125d0 * ((h / l) * (((d_m * m_m) / d) ** 2.0d0))))
    else if (l <= 5.7d+122) then
        tmp = (d * ((l * h) ** (-0.5d0))) * (1.0d0 - (0.125d0 * ((h * ((d_m * (m_m / d)) ** 2.0d0)) / l)))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -2.45e-293) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (-1.0 + (0.125 * ((h / l) * Math.pow(((D_m * M_m) / d), 2.0))));
	} else if (l <= 5.7e+122) {
		tmp = (d * Math.pow((l * h), -0.5)) * (1.0 - (0.125 * ((h * Math.pow((D_m * (M_m / d)), 2.0)) / l)));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -2.45e-293:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (-1.0 + (0.125 * ((h / l) * math.pow(((D_m * M_m) / d), 2.0))))
	elif l <= 5.7e+122:
		tmp = (d * math.pow((l * h), -0.5)) * (1.0 - (0.125 * ((h * math.pow((D_m * (M_m / d)), 2.0)) / l)))
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -2.45e-293)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + Float64(0.125 * Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / d) ^ 2.0)))));
	elseif (l <= 5.7e+122)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(1.0 - Float64(0.125 * Float64(Float64(h * (Float64(D_m * Float64(M_m / d)) ^ 2.0)) / l))));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -2.45e-293)
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.125 * ((h / l) * (((D_m * M_m) / d) ^ 2.0))));
	elseif (l <= 5.7e+122)
		tmp = (d * ((l * h) ^ -0.5)) * (1.0 - (0.125 * ((h * ((D_m * (M_m / d)) ^ 2.0)) / l)));
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -2.45e-293], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(0.125 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.7e+122], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.125 * N[(N[(h * N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.45e-293

   1. Initial program 67.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sqrt-unprod 55.3%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. pow1/2 55.3%

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. frac-times 43.7%

         \[\leadsto {\color{blue}{\left(\frac{d \cdot d}{h \cdot \ell}\right)}}^{0.5} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. pow2 43.7%

         \[\leadsto {\left(\frac{\color{blue}{{d}^{2}}}{h \cdot \ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 43.7%

      \[\leadsto \color{blue}{{\left(\frac{{d}^{2}}{h \cdot \ell}\right)}^{0.5}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. unpow1/2 43.7%

         \[\leadsto \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. Simplified 43.7%

      \[\leadsto \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   9. Taylor expanded in d around 0 3.1%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   10. Step-by-step derivation

       1. unpow-1 3.1%

          \[\leadsto \left(d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       2. metadata-eval 3.1%

          \[\leadsto \left(d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       3. pow-sqr 3.1%

          \[\leadsto \left(d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       4. rem-sqrt-square 3.1%

          \[\leadsto \left(d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       5. rem-square-sqrt 3.1%

          \[\leadsto \left(d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       6. fabs-sqr 3.1%

          \[\leadsto \left(d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       7. rem-square-sqrt 3.1%

          \[\leadsto \left(d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   11. Simplified 3.1%

       \[\leadsto \color{blue}{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   12. Taylor expanded in M around 0 0.7%

       \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   13. Step-by-step derivation

       1. associate-*r* 0.8%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}\right) \]

       2. times-frac 0.7%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right)}\right) \]

       3. associate-/l* 0.6%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \left(\color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right)\right) \]

       4. unpow2 0.6%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot \frac{{M}^{2}}{{d}^{2}}\right) \cdot \frac{h}{\ell}\right)\right) \]

       5. unpow2 0.6%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot \frac{\color{blue}{M \cdot M}}{{d}^{2}}\right) \cdot \frac{h}{\ell}\right)\right) \]

       6. unpow2 0.6%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{\color{blue}{d \cdot d}}\right) \cdot \frac{h}{\ell}\right)\right) \]

       7. times-frac 2.1%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{M}{d}\right)}\right) \cdot \frac{h}{\ell}\right)\right) \]

       8. swap-sqr 3.8%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \left(\color{blue}{\left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)} \cdot \frac{h}{\ell}\right)\right) \]

       9. unpow2 3.8%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \left(\color{blue}{{\left(D \cdot \frac{M}{d}\right)}^{2}} \cdot \frac{h}{\ell}\right)\right) \]

       10. *-commutative 3.8%

           \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}\right) \]

       11. associate-*r/ 3.1%

           \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot {\color{blue}{\left(\frac{D \cdot M}{d}\right)}}^{2}\right)\right) \]

   14. Simplified 3.1%

       \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)}\right) \]

   15. Taylor expanded in h around -inf 0.0%

       \[\leadsto \left(d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \]
   16. Step-by-step derivation

       1. *-commutative 0.0%

          \[\leadsto \left(d \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \]

       2. unpow2 0.0%

          \[\leadsto \left(d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \]

       3. rem-square-sqrt 67.3%

          \[\leadsto \left(d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \]

       4. mul-1-neg 67.3%

          \[\leadsto \left(d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \]

   17. Simplified 67.3%

       \[\leadsto \left(d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \]

   if -2.45e-293 < l < 5.70000000000000006e122

   1. Initial program 72.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 70.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sqrt-unprod 62.8%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. pow1/2 62.8%

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. frac-times 50.1%

         \[\leadsto {\color{blue}{\left(\frac{d \cdot d}{h \cdot \ell}\right)}}^{0.5} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. pow2 50.1%

         \[\leadsto {\left(\frac{\color{blue}{{d}^{2}}}{h \cdot \ell}\right)}^{0.5} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 50.1%

      \[\leadsto \color{blue}{{\left(\frac{{d}^{2}}{h \cdot \ell}\right)}^{0.5}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. unpow1/2 50.1%

         \[\leadsto \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. Simplified 50.1%

      \[\leadsto \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   9. Taylor expanded in d around 0 74.3%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   10. Step-by-step derivation

       1. unpow-1 74.3%

          \[\leadsto \left(d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       2. metadata-eval 74.3%

          \[\leadsto \left(d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       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 74.6%

          \[\leadsto \left(d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       5. rem-square-sqrt 74.5%

          \[\leadsto \left(d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       6. fabs-sqr 74.5%

          \[\leadsto \left(d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       7. rem-square-sqrt 74.6%

          \[\leadsto \left(d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   11. Simplified 74.6%

       \[\leadsto \color{blue}{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   12. Taylor expanded in M around 0 56.5%

       \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   13. Step-by-step derivation

       1. associate-*r* 59.0%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}\right) \]

       2. times-frac 53.0%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right)}\right) \]

       3. associate-/l* 53.0%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \left(\color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right)\right) \]

       4. unpow2 53.0%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot \frac{{M}^{2}}{{d}^{2}}\right) \cdot \frac{h}{\ell}\right)\right) \]

       5. unpow2 53.0%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot \frac{\color{blue}{M \cdot M}}{{d}^{2}}\right) \cdot \frac{h}{\ell}\right)\right) \]

       6. unpow2 53.0%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{\color{blue}{d \cdot d}}\right) \cdot \frac{h}{\ell}\right)\right) \]

       7. times-frac 66.2%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{M}{d}\right)}\right) \cdot \frac{h}{\ell}\right)\right) \]

       8. swap-sqr 76.9%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \left(\color{blue}{\left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)} \cdot \frac{h}{\ell}\right)\right) \]

       9. unpow2 76.9%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \left(\color{blue}{{\left(D \cdot \frac{M}{d}\right)}^{2}} \cdot \frac{h}{\ell}\right)\right) \]

       10. *-commutative 76.9%

           \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}\right) \]

       11. associate-*r/ 76.8%

           \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot {\color{blue}{\left(\frac{D \cdot M}{d}\right)}}^{2}\right)\right) \]

   14. Simplified 76.8%

       \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)}\right) \]
   15. Step-by-step derivation

       1. associate-*l/ 84.0%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{h \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}}{\ell}}\right) \]

       2. associate-*r/ 84.7%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \frac{h \cdot {\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2}}{\ell}\right) \]

   16. Applied egg-rr 84.7%

       \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\frac{h \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}}\right) \]

   if 5.70000000000000006e122 < l

   1. Initial program 53.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 48.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 48.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 48.6%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. pow1/2 48.6%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right) \]

      3. inv-pow 48.6%

         \[\leadsto d \cdot \left(1 \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right) \]

      4. pow-pow 48.6%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right) \]

      5. metadata-eval 48.6%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 48.6%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 48.6%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 48.6%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. *-commutative 48.6%

          \[\leadsto d \cdot {\color{blue}{\left(\ell \cdot h\right)}}^{-0.5} \]

       2. unpow-prod-down 75.0%

          \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]

   11. Applied egg-rr 75.0%

       \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.45 \cdot 10^{-293}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 5.7 \cdot 10^{+122}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - 0.125 \cdot \frac{h \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 51.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 1.04 \cdot 10^{-63}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left({\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.125\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= M_m 1.04e-63)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (*
    (sqrt (* (/ d h) (/ d l)))
    (* (pow (* D_m (/ M_m d)) 2.0) (* (/ h l) -0.125)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (M_m <= 1.04e-63) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = sqrt(((d / h) * (d / l))) * (pow((D_m * (M_m / d)), 2.0) * ((h / l) * -0.125));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (m_m <= 1.04d-63) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else
        tmp = sqrt(((d / h) * (d / l))) * (((d_m * (m_m / d)) ** 2.0d0) * ((h / l) * (-0.125d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (M_m <= 1.04e-63) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else {
		tmp = Math.sqrt(((d / h) * (d / l))) * (Math.pow((D_m * (M_m / d)), 2.0) * ((h / l) * -0.125));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if M_m <= 1.04e-63:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	else:
		tmp = math.sqrt(((d / h) * (d / l))) * (math.pow((D_m * (M_m / d)), 2.0) * ((h / l) * -0.125))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (M_m <= 1.04e-63)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64((Float64(D_m * Float64(M_m / d)) ^ 2.0) * Float64(Float64(h / l) * -0.125)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (M_m <= 1.04e-63)
		tmp = sqrt((d / h)) * sqrt((d / l));
	else
		tmp = sqrt(((d / h) * (d / l))) * (((D_m * (M_m / d)) ^ 2.0) * ((h / l) * -0.125));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[M$95$m, 1.04e-63], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.04e-63

   1. Initial program 65.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 63.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 47.5%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

   if 1.04e-63 < M

   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 73.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in M around inf 27.7%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* 27.8%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}\right)\right) \]

      2. times-frac 29.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right)}\right)\right) \]

      3. associate-/l* 26.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right)\right)\right) \]

      4. *-commutative 26.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\color{blue}{\left(\frac{{M}^{2}}{{d}^{2}} \cdot {D}^{2}\right)} \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow2 26.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\left(\frac{\color{blue}{M \cdot M}}{{d}^{2}} \cdot {D}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. unpow2 26.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\left(\frac{M \cdot M}{\color{blue}{d \cdot d}} \cdot {D}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. times-frac 39.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\left(\color{blue}{\left(\frac{M}{d} \cdot \frac{M}{d}\right)} \cdot {D}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. unpow2 39.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\left(\left(\frac{M}{d} \cdot \frac{M}{d}\right) \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      9. swap-sqr 57.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right)} \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. associate-/r/ 57.7%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{D}}} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. associate-/r/ 57.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\left(\frac{M}{\frac{d}{D}} \cdot \color{blue}{\frac{M}{\frac{d}{D}}}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. unpow2 57.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2}} \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. associate-/r/ 57.7%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left({\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      14. *-commutative 57.7%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left({\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   7. Simplified 57.7%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. associate-*r/ 56.5%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \color{blue}{\frac{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot h}{\ell}}\right)\right) \]

   9. Applied egg-rr 56.5%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \color{blue}{\frac{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot h}{\ell}}\right)\right) \]
   10. Step-by-step derivation

       1. pow1 56.5%

          \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot h}{\ell}\right)\right)\right)}^{1}} \]

       2. associate-*r* 57.8%

          \[\leadsto {\color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot h}{\ell}\right)\right)}}^{1} \]

       3. sqrt-unprod 54.7%

          \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(-0.125 \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot h}{\ell}\right)\right)}^{1} \]

       4. associate-/l* 54.6%

          \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(-0.125 \cdot \color{blue}{\left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)\right)}^{1} \]

       5. associate-*r/ 54.6%

          \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(-0.125 \cdot \left({\color{blue}{\left(\frac{D \cdot M}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

       6. *-commutative 54.6%

          \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(-0.125 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)}\right)\right)}^{1} \]

       7. associate-*r/ 54.6%

          \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\frac{h}{\ell} \cdot {\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2}\right)\right)\right)}^{1} \]

   11. Applied egg-rr 54.6%

       \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)\right)}^{1}} \]
   12. Step-by-step derivation

       1. unpow1 54.6%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)} \]

       2. *-commutative 54.6%

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(-0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right) \]

       3. associate-*r* 54.6%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \color{blue}{\left(\left(-0.125 \cdot \frac{h}{\ell}\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)} \]

   13. Simplified 54.6%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(\left(-0.125 \cdot \frac{h}{\ell}\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.04 \cdot 10^{-63}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.125\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 51.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 5.2 \cdot 10^{-64}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(\left(\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{d}\right)}^{2}\right) \cdot -0.125\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= M_m 5.2e-64)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (*
    (sqrt (* (/ d h) (/ d l)))
    (* (* (/ h l) (pow (/ (* D_m M_m) d) 2.0)) -0.125))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (M_m <= 5.2e-64) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = sqrt(((d / h) * (d / l))) * (((h / l) * pow(((D_m * M_m) / d), 2.0)) * -0.125);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (m_m <= 5.2d-64) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else
        tmp = sqrt(((d / h) * (d / l))) * (((h / l) * (((d_m * m_m) / d) ** 2.0d0)) * (-0.125d0))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (M_m <= 5.2e-64) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else {
		tmp = Math.sqrt(((d / h) * (d / l))) * (((h / l) * Math.pow(((D_m * M_m) / d), 2.0)) * -0.125);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if M_m <= 5.2e-64:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	else:
		tmp = math.sqrt(((d / h) * (d / l))) * (((h / l) * math.pow(((D_m * M_m) / d), 2.0)) * -0.125)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (M_m <= 5.2e-64)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / d) ^ 2.0)) * -0.125));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (M_m <= 5.2e-64)
		tmp = sqrt((d / h)) * sqrt((d / l));
	else
		tmp = sqrt(((d / h) * (d / l))) * (((h / l) * (((D_m * M_m) / d) ^ 2.0)) * -0.125);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[M$95$m, 5.2e-64], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 5.2e-64

   1. Initial program 65.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 63.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 47.5%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

   if 5.2e-64 < M

   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 73.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in M around inf 27.7%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* 27.8%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}\right)\right) \]

      2. times-frac 29.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right)}\right)\right) \]

      3. associate-/l* 26.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right)\right)\right) \]

      4. *-commutative 26.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\color{blue}{\left(\frac{{M}^{2}}{{d}^{2}} \cdot {D}^{2}\right)} \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow2 26.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\left(\frac{\color{blue}{M \cdot M}}{{d}^{2}} \cdot {D}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. unpow2 26.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\left(\frac{M \cdot M}{\color{blue}{d \cdot d}} \cdot {D}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. times-frac 39.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\left(\color{blue}{\left(\frac{M}{d} \cdot \frac{M}{d}\right)} \cdot {D}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. unpow2 39.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\left(\left(\frac{M}{d} \cdot \frac{M}{d}\right) \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      9. swap-sqr 57.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right)} \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. associate-/r/ 57.7%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\left(\color{blue}{\frac{M}{\frac{d}{D}}} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. associate-/r/ 57.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\left(\frac{M}{\frac{d}{D}} \cdot \color{blue}{\frac{M}{\frac{d}{D}}}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. unpow2 57.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\color{blue}{{\left(\frac{M}{\frac{d}{D}}\right)}^{2}} \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. associate-/r/ 57.7%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left({\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      14. *-commutative 57.7%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left({\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   7. Simplified 57.7%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. pow1 57.7%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)}^{1}} \]

      2. associate-*r* 57.7%

         \[\leadsto {\color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}}^{1} \]

      3. pow1/2 57.7%

         \[\leadsto {\left(\left(\color{blue}{{\left(\frac{d}{h}\right)}^{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

      4. pow1/2 57.7%

         \[\leadsto {\left(\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}}\right) \cdot \left(-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

      5. pow-prod-down 54.6%

         \[\leadsto {\left(\color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}} \cdot \left(-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

   9. Applied egg-rr 54.6%

      \[\leadsto \color{blue}{{\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 54.6%

          \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5} \cdot \left(-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

       2. unpow1/2 54.6%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       3. *-commutative 54.6%

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       4. *-commutative 54.6%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(-0.125 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}\right) \]

       5. associate-*r/ 54.6%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(-0.125 \cdot \left(\frac{h}{\ell} \cdot {\color{blue}{\left(\frac{D \cdot M}{d}\right)}}^{2}\right)\right) \]

   11. Simplified 54.6%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(-0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 5.2 \cdot 10^{-64}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(\left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right) \cdot -0.125\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 47.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{-199}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -1.4e-199)
   (* d (- (sqrt (/ (/ 1.0 l) h))))
   (if (<= l -2e-310)
     (* d (cbrt (pow (/ (/ 1.0 h) l) 1.5)))
     (* d (* (pow l -0.5) (pow h -0.5))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1.4e-199) {
		tmp = d * -sqrt(((1.0 / l) / h));
	} else if (l <= -2e-310) {
		tmp = d * cbrt(pow(((1.0 / h) / l), 1.5));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1.4e-199) {
		tmp = d * -Math.sqrt(((1.0 / l) / h));
	} else if (l <= -2e-310) {
		tmp = d * Math.cbrt(Math.pow(((1.0 / h) / l), 1.5));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -1.4e-199)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / l) / h))));
	elseif (l <= -2e-310)
		tmp = Float64(d * cbrt((Float64(Float64(1.0 / h) / l) ^ 1.5)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -1.4e-199], N[(d * (-N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, -2e-310], N[(d * N[Power[N[Power[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.40000000000000009e-199

   1. Initial program 63.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 64.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 6.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   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 42.3%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. associate-*r* 42.3%

         \[\leadsto \color{blue}{\left(d \cdot -1\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. *-commutative 42.3%

         \[\leadsto \color{blue}{\left(-1 \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. neg-mul-1 42.3%

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. distribute-lft-neg-in 42.3%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      8. distribute-rgt-neg-in 42.3%

         \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      9. unpow-1 42.3%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      10. metadata-eval 42.3%

          \[\leadsto d \cdot \left(-\sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \]

      11. pow-sqr 42.3%

          \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      12. rem-sqrt-square 42.3%

          \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \]

      13. rem-square-sqrt 42.1%

          \[\leadsto d \cdot \left(-\left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \]

      14. fabs-sqr 42.1%

          \[\leadsto d \cdot \left(-\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      15. rem-square-sqrt 42.3%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   8. Simplified 42.3%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   9. Taylor expanded in d around 0 42.3%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg 42.3%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

       2. distribute-rgt-neg-in 42.3%

          \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

       3. *-commutative 42.3%

          \[\leadsto d \cdot \left(-\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \]

       4. associate-/r* 43.1%

          \[\leadsto d \cdot \left(-\sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \]

   11. Simplified 43.1%

       \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \]

   if -1.40000000000000009e-199 < l < -1.999999999999994e-310

   1. Initial program 85.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 85.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 39.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. add-cbrt-cube 53.5%

         \[\leadsto d \cdot \color{blue}{\sqrt[3]{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}}} \]

      2. add-sqr-sqrt 53.5%

         \[\leadsto d \cdot \sqrt[3]{\color{blue}{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      3. pow1 53.5%

         \[\leadsto d \cdot \sqrt[3]{\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{1}} \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      4. pow1/2 53.5%

         \[\leadsto d \cdot \sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{1} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}} \]

      5. pow-prod-up 53.5%

         \[\leadsto d \cdot \sqrt[3]{\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{\left(1 + 0.5\right)}}} \]

      6. associate-/r* 53.5%

         \[\leadsto d \cdot \sqrt[3]{{\color{blue}{\left(\frac{\frac{1}{h}}{\ell}\right)}}^{\left(1 + 0.5\right)}} \]

      7. metadata-eval 53.5%

         \[\leadsto d \cdot \sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{\color{blue}{1.5}}} \]

   7. Applied egg-rr 53.5%

      \[\leadsto d \cdot \color{blue}{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{1.5}}} \]

   if -1.999999999999994e-310 < l

   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 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 d around inf 43.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 43.3%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. pow1/2 43.3%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right) \]

      3. inv-pow 43.3%

         \[\leadsto d \cdot \left(1 \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right) \]

      4. pow-pow 43.5%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right) \]

      5. metadata-eval 43.5%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 43.5%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 43.5%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 43.5%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. *-commutative 43.5%

          \[\leadsto d \cdot {\color{blue}{\left(\ell \cdot h\right)}}^{-0.5} \]

       2. unpow-prod-down 54.1%

          \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]

   11. Applied egg-rr 54.1%

       \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 46.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.45 \cdot 10^{-184}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d 1.45e-184)
   (* d (- (sqrt (/ (/ 1.0 l) h))))
   (* d (* (pow l -0.5) (pow h -0.5)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 1.45e-184) {
		tmp = d * -sqrt(((1.0 / l) / h));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= 1.45d-184) then
        tmp = d * -sqrt(((1.0d0 / l) / h))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 1.45e-184) {
		tmp = d * -Math.sqrt(((1.0 / l) / h));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= 1.45e-184:
		tmp = d * -math.sqrt(((1.0 / l) / h))
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= 1.45e-184)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / l) / h))));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= 1.45e-184)
		tmp = d * -sqrt(((1.0 / l) / h));
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, 1.45e-184], N[(d * (-N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.45000000000000007e-184

   1. Initial program 63.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 62.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 13.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   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 38.3%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. associate-*r* 38.3%

         \[\leadsto \color{blue}{\left(d \cdot -1\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. *-commutative 38.3%

         \[\leadsto \color{blue}{\left(-1 \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. neg-mul-1 38.3%

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. distribute-lft-neg-in 38.3%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      8. distribute-rgt-neg-in 38.3%

         \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      9. unpow-1 38.3%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      10. metadata-eval 38.3%

          \[\leadsto d \cdot \left(-\sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \]

      11. pow-sqr 38.3%

          \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      12. rem-sqrt-square 38.3%

          \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \]

      13. rem-square-sqrt 38.2%

          \[\leadsto d \cdot \left(-\left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \]

      14. fabs-sqr 38.2%

          \[\leadsto d \cdot \left(-\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      15. rem-square-sqrt 38.3%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   8. Simplified 38.3%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   9. Taylor expanded in d around 0 38.3%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg 38.3%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

       2. distribute-rgt-neg-in 38.3%

          \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

       3. *-commutative 38.3%

          \[\leadsto d \cdot \left(-\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \]

       4. associate-/r* 38.9%

          \[\leadsto d \cdot \left(-\sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \]

   11. Simplified 38.9%

       \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \]

   if 1.45000000000000007e-184 < d

   1. Initial program 72.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 72.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 47.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 47.1%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. pow1/2 47.1%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right) \]

      3. inv-pow 47.1%

         \[\leadsto d \cdot \left(1 \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right) \]

      4. pow-pow 47.3%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right) \]

      5. metadata-eval 47.3%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 47.3%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 47.3%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 47.3%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. *-commutative 47.3%

          \[\leadsto d \cdot {\color{blue}{\left(\ell \cdot h\right)}}^{-0.5} \]

       2. unpow-prod-down 60.2%

          \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]

   11. Applied egg-rr 60.2%

       \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 43.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.2 \cdot 10^{-207}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\frac{\frac{1}{h}}{\ell}\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -2.2e-207)
   (* d (- (sqrt (/ (/ 1.0 l) h))))
   (* d (pow (/ (/ 1.0 h) l) 0.5))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -2.2e-207) {
		tmp = d * -sqrt(((1.0 / l) / h));
	} else {
		tmp = d * pow(((1.0 / h) / l), 0.5);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-2.2d-207)) then
        tmp = d * -sqrt(((1.0d0 / l) / h))
    else
        tmp = d * (((1.0d0 / h) / l) ** 0.5d0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -2.2e-207) {
		tmp = d * -Math.sqrt(((1.0 / l) / h));
	} else {
		tmp = d * Math.pow(((1.0 / h) / l), 0.5);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -2.2e-207:
		tmp = d * -math.sqrt(((1.0 / l) / h))
	else:
		tmp = d * math.pow(((1.0 / h) / l), 0.5)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -2.2e-207)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / l) / h))));
	else
		tmp = Float64(d * (Float64(Float64(1.0 / h) / l) ^ 0.5));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -2.2e-207)
		tmp = d * -sqrt(((1.0 / l) / h));
	else
		tmp = d * (((1.0 / h) / l) ^ 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -2.2e-207], N[(d * (-N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d * N[Power[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.1999999999999999e-207

   1. Initial program 63.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 64.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 6.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   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 42.3%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. associate-*r* 42.3%

         \[\leadsto \color{blue}{\left(d \cdot -1\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. *-commutative 42.3%

         \[\leadsto \color{blue}{\left(-1 \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. neg-mul-1 42.3%

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. distribute-lft-neg-in 42.3%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      8. distribute-rgt-neg-in 42.3%

         \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      9. unpow-1 42.3%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      10. metadata-eval 42.3%

          \[\leadsto d \cdot \left(-\sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \]

      11. pow-sqr 42.3%

          \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      12. rem-sqrt-square 42.3%

          \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \]

      13. rem-square-sqrt 42.1%

          \[\leadsto d \cdot \left(-\left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \]

      14. fabs-sqr 42.1%

          \[\leadsto d \cdot \left(-\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      15. rem-square-sqrt 42.3%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   8. Simplified 42.3%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   9. Taylor expanded in d around 0 42.3%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg 42.3%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

       2. distribute-rgt-neg-in 42.3%

          \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

       3. *-commutative 42.3%

          \[\leadsto d \cdot \left(-\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \]

       4. associate-/r* 43.1%

          \[\leadsto d \cdot \left(-\sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \]

   11. Simplified 43.1%

       \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \]

   if -2.1999999999999999e-207 < l

   1. Initial program 70.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 67.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 42.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. pow1/2 42.8%

         \[\leadsto d \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \]

      2. associate-/r* 43.7%

         \[\leadsto d \cdot {\color{blue}{\left(\frac{\frac{1}{h}}{\ell}\right)}}^{0.5} \]

   7. Applied egg-rr 43.7%

      \[\leadsto d \cdot \color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 42.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -9 \cdot 10^{-211}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -9e-211) (* d (- (sqrt (/ (/ 1.0 l) h)))) (* d (pow (* l h) -0.5))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -9e-211) {
		tmp = d * -sqrt(((1.0 / l) / h));
	} else {
		tmp = d * pow((l * h), -0.5);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-9d-211)) then
        tmp = d * -sqrt(((1.0d0 / l) / h))
    else
        tmp = d * ((l * h) ** (-0.5d0))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -9e-211) {
		tmp = d * -Math.sqrt(((1.0 / l) / h));
	} else {
		tmp = d * Math.pow((l * h), -0.5);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -9e-211:
		tmp = d * -math.sqrt(((1.0 / l) / h))
	else:
		tmp = d * math.pow((l * h), -0.5)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -9e-211)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / l) / h))));
	else
		tmp = Float64(d * (Float64(l * h) ^ -0.5));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -9e-211)
		tmp = d * -sqrt(((1.0 / l) / h));
	else
		tmp = d * ((l * h) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -9e-211], N[(d * (-N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -8.9999999999999997e-211

   1. Initial program 63.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 64.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 6.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   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 42.3%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. associate-*r* 42.3%

         \[\leadsto \color{blue}{\left(d \cdot -1\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. *-commutative 42.3%

         \[\leadsto \color{blue}{\left(-1 \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. neg-mul-1 42.3%

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. distribute-lft-neg-in 42.3%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      8. distribute-rgt-neg-in 42.3%

         \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      9. unpow-1 42.3%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      10. metadata-eval 42.3%

          \[\leadsto d \cdot \left(-\sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \]

      11. pow-sqr 42.3%

          \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      12. rem-sqrt-square 42.3%

          \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \]

      13. rem-square-sqrt 42.1%

          \[\leadsto d \cdot \left(-\left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \]

      14. fabs-sqr 42.1%

          \[\leadsto d \cdot \left(-\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      15. rem-square-sqrt 42.3%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   8. Simplified 42.3%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   9. Taylor expanded in d around 0 42.3%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg 42.3%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

       2. distribute-rgt-neg-in 42.3%

          \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

       3. *-commutative 42.3%

          \[\leadsto d \cdot \left(-\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \]

       4. associate-/r* 43.1%

          \[\leadsto d \cdot \left(-\sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \]

   11. Simplified 43.1%

       \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \]

   if -8.9999999999999997e-211 < l

   1. Initial program 70.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 67.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 42.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 42.8%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. pow1/2 42.8%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right) \]

      3. inv-pow 42.8%

         \[\leadsto d \cdot \left(1 \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right) \]

      4. pow-pow 42.9%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right) \]

      5. metadata-eval 42.9%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 42.9%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 42.9%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 42.9%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9 \cdot 10^{-211}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 42.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{if}\;\ell \leq -6.5 \cdot 10^{-211}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot t\_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (pow (* l h) -0.5)))
   (if (<= l -6.5e-211) (* (- d) t_0) (* d t_0))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = pow((l * h), -0.5);
	double tmp;
	if (l <= -6.5e-211) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (l * h) ** (-0.5d0)
    if (l <= (-6.5d-211)) then
        tmp = -d * t_0
    else
        tmp = d * t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.pow((l * h), -0.5);
	double tmp;
	if (l <= -6.5e-211) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.pow((l * h), -0.5)
	tmp = 0
	if l <= -6.5e-211:
		tmp = -d * t_0
	else:
		tmp = d * t_0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(l * h) ^ -0.5
	tmp = 0.0
	if (l <= -6.5e-211)
		tmp = Float64(Float64(-d) * t_0);
	else
		tmp = Float64(d * t_0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (l * h) ^ -0.5;
	tmp = 0.0;
	if (l <= -6.5e-211)
		tmp = -d * t_0;
	else
		tmp = d * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[l, -6.5e-211], N[((-d) * t$95$0), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -6.4999999999999996e-211

   1. Initial program 63.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 64.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 6.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   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 42.3%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. associate-*r* 42.3%

         \[\leadsto \color{blue}{\left(d \cdot -1\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. *-commutative 42.3%

         \[\leadsto \color{blue}{\left(-1 \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. neg-mul-1 42.3%

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. distribute-lft-neg-in 42.3%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      8. distribute-rgt-neg-in 42.3%

         \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      9. unpow-1 42.3%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      10. metadata-eval 42.3%

          \[\leadsto d \cdot \left(-\sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \]

      11. pow-sqr 42.3%

          \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      12. rem-sqrt-square 42.3%

          \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \]

      13. rem-square-sqrt 42.1%

          \[\leadsto d \cdot \left(-\left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \]

      14. fabs-sqr 42.1%

          \[\leadsto d \cdot \left(-\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      15. rem-square-sqrt 42.3%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   8. Simplified 42.3%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if -6.4999999999999996e-211 < l

   1. Initial program 70.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 67.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 42.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 42.8%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. pow1/2 42.8%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right) \]

      3. inv-pow 42.8%

         \[\leadsto d \cdot \left(1 \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right) \]

      4. pow-pow 42.9%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right) \]

      5. metadata-eval 42.9%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 42.9%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 42.9%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 42.9%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{-211}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 26.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ d \cdot {\left(\ell \cdot h\right)}^{-0.5} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (* d (pow (* l h) -0.5)))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return d * pow((l * h), -0.5);
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = d * ((l * h) ** (-0.5d0))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return d * Math.pow((l * h), -0.5);
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return d * math.pow((l * h), -0.5)

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(d * (Float64(l * h) ^ -0.5))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = d * ((l * h) ^ -0.5);
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.2%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 66.1%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in d around inf 26.5%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
6. Step-by-step derivation

   1. *-un-lft-identity 26.5%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   2. pow1/2 26.5%

      \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right) \]

   3. inv-pow 26.5%

      \[\leadsto d \cdot \left(1 \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right) \]

   4. pow-pow 26.6%

      \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right) \]

   5. metadata-eval 26.6%

      \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

7. Applied egg-rr 26.6%

   \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
8. Step-by-step derivation

   1. *-lft-identity 26.6%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

9. Simplified 26.6%

   \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

10. Final simplification 26.6%

    \[\leadsto d \cdot {\left(\ell \cdot h\right)}^{-0.5} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(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)))))