Henrywood and Agarwal, Equation (12)

Percentage Accurate: 67.5% → 81.6%

Time: 26.7s

Alternatives: 19

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.5% 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: 81.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := 1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{-d}\\ t_3 := 1 + 0.5 \cdot \left(\left(h \cdot {\left(D \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2}\right) \cdot \frac{-1}{\ell}\right)\\ t_4 := \frac{\sqrt{d}}{\sqrt{h}}\\ \mathbf{if}\;\ell \leq -7 \cdot 10^{+97}:\\ \;\;\;\;\frac{t\_2}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot t\_0\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{t\_2}{\sqrt{-h}} \cdot t\_1\right) \cdot t\_3\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-21}:\\ \;\;\;\;t\_3 \cdot \left(t\_1 \cdot t\_4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(t\_0 \cdot t\_4\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (/ h l) (* (pow (* D (/ (/ M_m 2.0) d)) 2.0) -0.5))))
        (t_1 (sqrt (/ d l)))
        (t_2 (sqrt (- d)))
        (t_3
         (+
          1.0
          (* 0.5 (* (* h (pow (* D (* M_m (/ 0.5 d))) 2.0)) (/ -1.0 l)))))
        (t_4 (/ (sqrt d) (sqrt h))))
   (if (<= l -7e+97)
     (* (/ t_2 (sqrt (- l))) (* (sqrt (/ d h)) t_0))
     (if (<= l -5e-310)
       (* (* (/ t_2 (sqrt (- h))) t_1) t_3)
       (if (<= l 1.5e-21)
         (* t_3 (* t_1 t_4))
         (* (/ (sqrt d) (sqrt l)) (* t_0 t_4)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = 1.0 + ((h / l) * (pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5));
	double t_1 = sqrt((d / l));
	double t_2 = sqrt(-d);
	double t_3 = 1.0 + (0.5 * ((h * pow((D * (M_m * (0.5 / d))), 2.0)) * (-1.0 / l)));
	double t_4 = sqrt(d) / sqrt(h);
	double tmp;
	if (l <= -7e+97) {
		tmp = (t_2 / sqrt(-l)) * (sqrt((d / h)) * t_0);
	} else if (l <= -5e-310) {
		tmp = ((t_2 / sqrt(-h)) * t_1) * t_3;
	} else if (l <= 1.5e-21) {
		tmp = t_3 * (t_1 * t_4);
	} else {
		tmp = (sqrt(d) / sqrt(l)) * (t_0 * t_4);
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = 1.0d0 + ((h / l) * (((d_1 * ((m_m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))
    t_1 = sqrt((d / l))
    t_2 = sqrt(-d)
    t_3 = 1.0d0 + (0.5d0 * ((h * ((d_1 * (m_m * (0.5d0 / d))) ** 2.0d0)) * ((-1.0d0) / l)))
    t_4 = sqrt(d) / sqrt(h)
    if (l <= (-7d+97)) then
        tmp = (t_2 / sqrt(-l)) * (sqrt((d / h)) * t_0)
    else if (l <= (-5d-310)) then
        tmp = ((t_2 / sqrt(-h)) * t_1) * t_3
    else if (l <= 1.5d-21) then
        tmp = t_3 * (t_1 * t_4)
    else
        tmp = (sqrt(d) / sqrt(l)) * (t_0 * t_4)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = 1.0 + ((h / l) * (Math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5));
	double t_1 = Math.sqrt((d / l));
	double t_2 = Math.sqrt(-d);
	double t_3 = 1.0 + (0.5 * ((h * Math.pow((D * (M_m * (0.5 / d))), 2.0)) * (-1.0 / l)));
	double t_4 = Math.sqrt(d) / Math.sqrt(h);
	double tmp;
	if (l <= -7e+97) {
		tmp = (t_2 / Math.sqrt(-l)) * (Math.sqrt((d / h)) * t_0);
	} else if (l <= -5e-310) {
		tmp = ((t_2 / Math.sqrt(-h)) * t_1) * t_3;
	} else if (l <= 1.5e-21) {
		tmp = t_3 * (t_1 * t_4);
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(l)) * (t_0 * t_4);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = 1.0 + ((h / l) * (math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))
	t_1 = math.sqrt((d / l))
	t_2 = math.sqrt(-d)
	t_3 = 1.0 + (0.5 * ((h * math.pow((D * (M_m * (0.5 / d))), 2.0)) * (-1.0 / l)))
	t_4 = math.sqrt(d) / math.sqrt(h)
	tmp = 0
	if l <= -7e+97:
		tmp = (t_2 / math.sqrt(-l)) * (math.sqrt((d / h)) * t_0)
	elif l <= -5e-310:
		tmp = ((t_2 / math.sqrt(-h)) * t_1) * t_3
	elif l <= 1.5e-21:
		tmp = t_3 * (t_1 * t_4)
	else:
		tmp = (math.sqrt(d) / math.sqrt(l)) * (t_0 * t_4)
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5)))
	t_1 = sqrt(Float64(d / l))
	t_2 = sqrt(Float64(-d))
	t_3 = Float64(1.0 + Float64(0.5 * Float64(Float64(h * (Float64(D * Float64(M_m * Float64(0.5 / d))) ^ 2.0)) * Float64(-1.0 / l))))
	t_4 = Float64(sqrt(d) / sqrt(h))
	tmp = 0.0
	if (l <= -7e+97)
		tmp = Float64(Float64(t_2 / sqrt(Float64(-l))) * Float64(sqrt(Float64(d / h)) * t_0));
	elseif (l <= -5e-310)
		tmp = Float64(Float64(Float64(t_2 / sqrt(Float64(-h))) * t_1) * t_3);
	elseif (l <= 1.5e-21)
		tmp = Float64(t_3 * Float64(t_1 * t_4));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(l)) * Float64(t_0 * t_4));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = 1.0 + ((h / l) * (((D * ((M_m / 2.0) / d)) ^ 2.0) * -0.5));
	t_1 = sqrt((d / l));
	t_2 = sqrt(-d);
	t_3 = 1.0 + (0.5 * ((h * ((D * (M_m * (0.5 / d))) ^ 2.0)) * (-1.0 / l)));
	t_4 = sqrt(d) / sqrt(h);
	tmp = 0.0;
	if (l <= -7e+97)
		tmp = (t_2 / sqrt(-l)) * (sqrt((d / h)) * t_0);
	elseif (l <= -5e-310)
		tmp = ((t_2 / sqrt(-h)) * t_1) * t_3;
	elseif (l <= 1.5e-21)
		tmp = t_3 * (t_1 * t_4);
	else
		tmp = (sqrt(d) / sqrt(l)) * (t_0 * t_4);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + N[(0.5 * N[(N[(h * N[Power[N[(D * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -7e+97], N[(N[(t$95$2 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(N[(N[(t$95$2 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[l, 1.5e-21], N[(t$95$3 * N[(t$95$1 * t$95$4), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$4), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -7.0000000000000001e97

   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 48.9%

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

      1. frac-2neg 48.9%

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

      2. sqrt-div 63.5%

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

   6. Applied egg-rr 63.5%

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

   if -7.0000000000000001e97 < l < -4.999999999999985e-310

   1. Initial program 64.9%

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

   3. Simplified 63.8%

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

      1. associate-*r/ 67.5%

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

      2. clear-num 67.5%

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

      3. *-commutative 67.5%

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

      4. *-commutative 67.5%

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

      5. clear-num 67.5%

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

      6. frac-times 68.6%

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

      7. *-commutative 68.6%

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

      8. *-un-lft-identity 68.6%

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

   6. Applied egg-rr 68.6%

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

      1. associate-/r/ 68.6%

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

      2. associate-/l/ 68.6%

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

      3. remove-double-neg 68.6%

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

      4. distribute-frac-neg 68.6%

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

      5. *-rgt-identity 68.6%

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

      6. associate-/l* 68.6%

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

      7. associate-/r/ 68.6%

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

      8. metadata-eval 68.6%

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

      9. distribute-neg-frac 68.6%

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

      10. associate-/l* 67.4%

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

      11. *-commutative 67.4%

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

      12. associate-*r/ 67.4%

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

      13. distribute-lft-neg-in 67.4%

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

      14. remove-double-neg 67.4%

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

   8. Simplified 67.4%

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

      1. frac-2neg 67.4%

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

      2. sqrt-div 81.9%

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

   10. Applied egg-rr 81.9%

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

   if -4.999999999999985e-310 < l < 1.49999999999999996e-21

   1. Initial program 69.6%

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

   3. Simplified 71.4%

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

      1. associate-*r/ 78.8%

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

      2. clear-num 78.8%

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

      3. *-commutative 78.8%

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

      4. *-commutative 78.8%

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

      5. clear-num 78.8%

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

      6. frac-times 77.0%

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

      7. *-commutative 77.0%

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

      8. *-un-lft-identity 77.0%

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

   6. Applied egg-rr 77.0%

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

      1. associate-/r/ 77.0%

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

      2. associate-/l/ 77.0%

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

      3. remove-double-neg 77.0%

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

      4. distribute-frac-neg 77.0%

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

      5. *-rgt-identity 77.0%

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

      6. associate-/l* 77.0%

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

      7. associate-/r/ 77.0%

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

      8. metadata-eval 77.0%

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

      9. distribute-neg-frac 77.0%

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

      10. associate-/l* 78.9%

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

      11. *-commutative 78.9%

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

      12. associate-*r/ 78.8%

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

      13. distribute-lft-neg-in 78.8%

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

      14. remove-double-neg 78.8%

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

   8. Simplified 78.8%

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

      1. sqrt-div 92.7%

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

      2. div-inv 92.7%

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

   10. Applied egg-rr 92.7%

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

       1. associate-*r/ 92.7%

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

       2. *-rgt-identity 92.7%

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

   12. Simplified 92.7%

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

   if 1.49999999999999996e-21 < l

   1. Initial program 63.0%

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

   3. Simplified 63.0%

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

      1. sqrt-div 68.8%

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

      2. div-inv 68.7%

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

   6. Applied egg-rr 68.7%

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

      1. associate-*r/ 68.8%

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

      2. *-rgt-identity 68.8%

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

   8. Simplified 68.8%

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

      1. sqrt-div 66.8%

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

      2. div-inv 66.7%

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

   10. Applied egg-rr 80.2%

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

       1. associate-*r/ 66.8%

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

       2. *-rgt-identity 66.8%

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

   12. Simplified 80.2%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7 \cdot 10^{+97}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(\left(h \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right) \cdot \frac{-1}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-21}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(\left(h \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right) \cdot \frac{-1}{\ell}\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M\_m}{d \cdot 2}\right)}^{2}\right)\right)\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-220}:\\ \;\;\;\;\left(t\_0 \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \cdot t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-163} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+208}\right):\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot t\_2\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* D M_m) (* d 2.0)) 2.0))))))
        (t_2 (sqrt (/ d l))))
   (if (<= t_1 -2e-220)
     (*
      (* t_0 (+ 1.0 (* (/ h l) (* (pow (* D (/ (/ M_m 2.0) d)) 2.0) -0.5))))
      t_2)
     (if (or (<= t_1 1e-163) (not (<= t_1 2e+208)))
       (fabs (/ d (sqrt (* l h))))
       (* t_0 t_2)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((D * M_m) / (d * 2.0)), 2.0))));
	double t_2 = sqrt((d / l));
	double tmp;
	if (t_1 <= -2e-220) {
		tmp = (t_0 * (1.0 + ((h / l) * (pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5)))) * t_2;
	} else if ((t_1 <= 1e-163) || !(t_1 <= 2e+208)) {
		tmp = fabs((d / sqrt((l * h))));
	} else {
		tmp = t_0 * t_2;
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt((d / h))
    t_1 = (((d / h) ** 0.5d0) * ((d / l) ** 0.5d0)) * (1.0d0 - ((h / l) * (0.5d0 * (((d_1 * m_m) / (d * 2.0d0)) ** 2.0d0))))
    t_2 = sqrt((d / l))
    if (t_1 <= (-2d-220)) then
        tmp = (t_0 * (1.0d0 + ((h / l) * (((d_1 * ((m_m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0))))) * t_2
    else if ((t_1 <= 1d-163) .or. (.not. (t_1 <= 2d+208))) then
        tmp = abs((d / sqrt((l * h))))
    else
        tmp = t_0 * t_2
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = Math.sqrt((d / h));
	double t_1 = (Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * Math.pow(((D * M_m) / (d * 2.0)), 2.0))));
	double t_2 = Math.sqrt((d / l));
	double tmp;
	if (t_1 <= -2e-220) {
		tmp = (t_0 * (1.0 + ((h / l) * (Math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5)))) * t_2;
	} else if ((t_1 <= 1e-163) || !(t_1 <= 2e+208)) {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	} else {
		tmp = t_0 * t_2;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = math.sqrt((d / h))
	t_1 = (math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * math.pow(((D * M_m) / (d * 2.0)), 2.0))))
	t_2 = math.sqrt((d / l))
	tmp = 0
	if t_1 <= -2e-220:
		tmp = (t_0 * (1.0 + ((h / l) * (math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5)))) * t_2
	elif (t_1 <= 1e-163) or not (t_1 <= 2e+208):
		tmp = math.fabs((d / math.sqrt((l * h))))
	else:
		tmp = t_0 * t_2
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(D * M_m) / Float64(d * 2.0)) ^ 2.0)))))
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (t_1 <= -2e-220)
		tmp = Float64(Float64(t_0 * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5)))) * t_2);
	elseif ((t_1 <= 1e-163) || !(t_1 <= 2e+208))
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(t_0 * t_2);
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = sqrt((d / h));
	t_1 = (((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((h / l) * (0.5 * (((D * M_m) / (d * 2.0)) ^ 2.0))));
	t_2 = sqrt((d / l));
	tmp = 0.0;
	if (t_1 <= -2e-220)
		tmp = (t_0 * (1.0 + ((h / l) * (((D * ((M_m / 2.0) / d)) ^ 2.0) * -0.5)))) * t_2;
	elseif ((t_1 <= 1e-163) || ~((t_1 <= 2e+208)))
		tmp = abs((d / sqrt((l * h))));
	else
		tmp = t_0 * t_2;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(D * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -2e-220], N[(N[(t$95$0 * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[Or[LessEqual[t$95$1, 1e-163], N[Not[LessEqual[t$95$1, 2e+208]], $MachinePrecision]], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$0 * t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.99999999999999998e-220

   1. Initial program 84.6%

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

   3. Simplified 85.6%

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

   if -1.99999999999999998e-220 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.99999999999999923e-164 or 2e208 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 21.8%

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

   3. Simplified 22.7%

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

   5. Taylor expanded in M around 0 30.4%

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

      1. pow1 30.4%

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

      2. pow1/2 30.4%

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

      3. *-rgt-identity 30.4%

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

      4. pow1/2 30.4%

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

      5. pow-prod-down 24.3%

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

   7. Applied egg-rr 24.3%

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

      1. unpow1 24.3%

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

      2. unpow1/2 24.3%

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

   9. Simplified 24.3%

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

       1. sqrt-prod 30.4%

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

       2. sqrt-undiv 25.7%

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

       3. *-rgt-identity 25.7%

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

       4. add-sqr-sqrt 25.5%

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

       5. sqrt-prod 13.4%

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

       6. rem-sqrt-square 25.7%

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

       7. sqrt-undiv 30.4%

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

       8. *-rgt-identity 30.4%

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

       9. sqrt-prod 24.3%

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

       10. frac-times 24.3%

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

       11. sqrt-div 29.6%

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

       12. sqrt-unprod 28.6%

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

       13. add-sqr-sqrt 52.8%

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

   11. Applied egg-rr 52.8%

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

   if 9.99999999999999923e-164 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2e208

   1. Initial program 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in M around 0 99.0%

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

      1. pow1 99.0%

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

      2. pow1/2 99.0%

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

      3. *-rgt-identity 99.0%

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

      4. pow1/2 99.0%

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

      5. pow-prod-down 89.8%

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

   7. Applied egg-rr 89.8%

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

      1. unpow1 89.8%

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

      2. unpow1/2 89.8%

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

   9. Simplified 89.8%

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

       1. *-commutative 89.8%

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

       2. sqrt-prod 99.0%

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

   11. Applied egg-rr 99.0%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq -2 \cdot 10^{-220}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq 10^{-163} \lor \neg \left(\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq 2 \cdot 10^{+208}\right):\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M\_m}{d \cdot 2}\right)}^{2}\right)\right)\\ t_1 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-220}:\\ \;\;\;\;t\_1 \cdot \left(1 - 0.5 \cdot {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\frac{D}{d} \cdot \left(M\_m \cdot 0.5\right)\right)\right)}^{2}\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-163} \lor \neg \left(t\_0 \leq 2 \cdot 10^{+208}\right):\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* D M_m) (* d 2.0)) 2.0))))))
        (t_1 (* (sqrt (/ d h)) (sqrt (/ d l)))))
   (if (<= t_0 -2e-220)
     (*
      t_1
      (- 1.0 (* 0.5 (pow (* (sqrt (/ h l)) (* (/ D d) (* M_m 0.5))) 2.0))))
     (if (or (<= t_0 1e-163) (not (<= t_0 2e+208)))
       (fabs (/ d (sqrt (* l h))))
       t_1))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((D * M_m) / (d * 2.0)), 2.0))));
	double t_1 = sqrt((d / h)) * sqrt((d / l));
	double tmp;
	if (t_0 <= -2e-220) {
		tmp = t_1 * (1.0 - (0.5 * pow((sqrt((h / l)) * ((D / d) * (M_m * 0.5))), 2.0)));
	} else if ((t_0 <= 1e-163) || !(t_0 <= 2e+208)) {
		tmp = fabs((d / sqrt((l * h))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((d / h) ** 0.5d0) * ((d / l) ** 0.5d0)) * (1.0d0 - ((h / l) * (0.5d0 * (((d_1 * m_m) / (d * 2.0d0)) ** 2.0d0))))
    t_1 = sqrt((d / h)) * sqrt((d / l))
    if (t_0 <= (-2d-220)) then
        tmp = t_1 * (1.0d0 - (0.5d0 * ((sqrt((h / l)) * ((d_1 / d) * (m_m * 0.5d0))) ** 2.0d0)))
    else if ((t_0 <= 1d-163) .or. (.not. (t_0 <= 2d+208))) then
        tmp = abs((d / sqrt((l * h))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * Math.pow(((D * M_m) / (d * 2.0)), 2.0))));
	double t_1 = Math.sqrt((d / h)) * Math.sqrt((d / l));
	double tmp;
	if (t_0 <= -2e-220) {
		tmp = t_1 * (1.0 - (0.5 * Math.pow((Math.sqrt((h / l)) * ((D / d) * (M_m * 0.5))), 2.0)));
	} else if ((t_0 <= 1e-163) || !(t_0 <= 2e+208)) {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = (math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * math.pow(((D * M_m) / (d * 2.0)), 2.0))))
	t_1 = math.sqrt((d / h)) * math.sqrt((d / l))
	tmp = 0
	if t_0 <= -2e-220:
		tmp = t_1 * (1.0 - (0.5 * math.pow((math.sqrt((h / l)) * ((D / d) * (M_m * 0.5))), 2.0)))
	elif (t_0 <= 1e-163) or not (t_0 <= 2e+208):
		tmp = math.fabs((d / math.sqrt((l * h))))
	else:
		tmp = t_1
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(D * M_m) / Float64(d * 2.0)) ^ 2.0)))))
	t_1 = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)))
	tmp = 0.0
	if (t_0 <= -2e-220)
		tmp = Float64(t_1 * Float64(1.0 - Float64(0.5 * (Float64(sqrt(Float64(h / l)) * Float64(Float64(D / d) * Float64(M_m * 0.5))) ^ 2.0))));
	elseif ((t_0 <= 1e-163) || !(t_0 <= 2e+208))
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = (((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((h / l) * (0.5 * (((D * M_m) / (d * 2.0)) ^ 2.0))));
	t_1 = sqrt((d / h)) * sqrt((d / l));
	tmp = 0.0;
	if (t_0 <= -2e-220)
		tmp = t_1 * (1.0 - (0.5 * ((sqrt((h / l)) * ((D / d) * (M_m * 0.5))) ^ 2.0)));
	elseif ((t_0 <= 1e-163) || ~((t_0 <= 2e+208)))
		tmp = abs((d / sqrt((l * h))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(D * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-220], N[(t$95$1 * N[(1.0 - N[(0.5 * N[Power[N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, 1e-163], N[Not[LessEqual[t$95$0, 2e+208]], $MachinePrecision]], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.99999999999999998e-220

   1. Initial program 84.6%

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

   3. Simplified 85.8%

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

      1. add-sqr-sqrt 85.7%

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

      2. pow2 85.7%

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

      3. sqrt-prod 85.7%

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

      4. sqrt-pow1 88.0%

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

      5. metadata-eval 88.0%

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

      6. pow1 88.0%

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

      7. *-commutative 88.0%

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

      8. clear-num 87.9%

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

      9. frac-times 90.1%

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

      10. *-commutative 90.1%

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

      11. *-un-lft-identity 90.1%

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

   6. Applied egg-rr 90.1%

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

   7. Taylor expanded in D around 0 89.2%

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

      1. *-commutative 89.2%

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

      2. associate-*l/ 89.2%

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

      3. associate-*r* 89.2%

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

      4. associate-*l/ 88.0%

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

      5. *-commutative 88.0%

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

   9. Simplified 88.0%

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

   if -1.99999999999999998e-220 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.99999999999999923e-164 or 2e208 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 21.8%

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

   3. Simplified 22.7%

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

   5. Taylor expanded in M around 0 30.4%

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

      1. pow1 30.4%

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

      2. pow1/2 30.4%

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

      3. *-rgt-identity 30.4%

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

      4. pow1/2 30.4%

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

      5. pow-prod-down 24.3%

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

   7. Applied egg-rr 24.3%

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

      1. unpow1 24.3%

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

      2. unpow1/2 24.3%

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

   9. Simplified 24.3%

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

       1. sqrt-prod 30.4%

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

       2. sqrt-undiv 25.7%

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

       3. *-rgt-identity 25.7%

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

       4. add-sqr-sqrt 25.5%

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

       5. sqrt-prod 13.4%

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

       6. rem-sqrt-square 25.7%

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

       7. sqrt-undiv 30.4%

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

       8. *-rgt-identity 30.4%

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

       9. sqrt-prod 24.3%

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

       10. frac-times 24.3%

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

       11. sqrt-div 29.6%

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

       12. sqrt-unprod 28.6%

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

       13. add-sqr-sqrt 52.8%

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

   11. Applied egg-rr 52.8%

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

   if 9.99999999999999923e-164 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2e208

   1. Initial program 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in M around 0 99.0%

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

      1. pow1 99.0%

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

      2. pow1/2 99.0%

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

      3. *-rgt-identity 99.0%

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

      4. pow1/2 99.0%

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

      5. pow-prod-down 89.8%

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

   7. Applied egg-rr 89.8%

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

      1. unpow1 89.8%

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

      2. unpow1/2 89.8%

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

   9. Simplified 89.8%

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

       1. *-commutative 89.8%

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

       2. sqrt-prod 99.0%

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

   11. Applied egg-rr 99.0%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq -2 \cdot 10^{-220}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)\right)}^{2}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq 10^{-163} \lor \neg \left(\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq 2 \cdot 10^{+208}\right):\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M\_m}{d \cdot 2}\right)}^{2}\right)\right)\\ t_1 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-220}:\\ \;\;\;\;t\_1 \cdot \left(1 - 0.5 \cdot {\left(\frac{D}{d \cdot \frac{2}{M\_m}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-163} \lor \neg \left(t\_0 \leq 2 \cdot 10^{+208}\right):\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* D M_m) (* d 2.0)) 2.0))))))
        (t_1 (* (sqrt (/ d h)) (sqrt (/ d l)))))
   (if (<= t_0 -2e-220)
     (*
      t_1
      (- 1.0 (* 0.5 (pow (* (/ D (* d (/ 2.0 M_m))) (sqrt (/ h l))) 2.0))))
     (if (or (<= t_0 1e-163) (not (<= t_0 2e+208)))
       (fabs (/ d (sqrt (* l h))))
       t_1))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((D * M_m) / (d * 2.0)), 2.0))));
	double t_1 = sqrt((d / h)) * sqrt((d / l));
	double tmp;
	if (t_0 <= -2e-220) {
		tmp = t_1 * (1.0 - (0.5 * pow(((D / (d * (2.0 / M_m))) * sqrt((h / l))), 2.0)));
	} else if ((t_0 <= 1e-163) || !(t_0 <= 2e+208)) {
		tmp = fabs((d / sqrt((l * h))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((d / h) ** 0.5d0) * ((d / l) ** 0.5d0)) * (1.0d0 - ((h / l) * (0.5d0 * (((d_1 * m_m) / (d * 2.0d0)) ** 2.0d0))))
    t_1 = sqrt((d / h)) * sqrt((d / l))
    if (t_0 <= (-2d-220)) then
        tmp = t_1 * (1.0d0 - (0.5d0 * (((d_1 / (d * (2.0d0 / m_m))) * sqrt((h / l))) ** 2.0d0)))
    else if ((t_0 <= 1d-163) .or. (.not. (t_0 <= 2d+208))) then
        tmp = abs((d / sqrt((l * h))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * Math.pow(((D * M_m) / (d * 2.0)), 2.0))));
	double t_1 = Math.sqrt((d / h)) * Math.sqrt((d / l));
	double tmp;
	if (t_0 <= -2e-220) {
		tmp = t_1 * (1.0 - (0.5 * Math.pow(((D / (d * (2.0 / M_m))) * Math.sqrt((h / l))), 2.0)));
	} else if ((t_0 <= 1e-163) || !(t_0 <= 2e+208)) {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = (math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * math.pow(((D * M_m) / (d * 2.0)), 2.0))))
	t_1 = math.sqrt((d / h)) * math.sqrt((d / l))
	tmp = 0
	if t_0 <= -2e-220:
		tmp = t_1 * (1.0 - (0.5 * math.pow(((D / (d * (2.0 / M_m))) * math.sqrt((h / l))), 2.0)))
	elif (t_0 <= 1e-163) or not (t_0 <= 2e+208):
		tmp = math.fabs((d / math.sqrt((l * h))))
	else:
		tmp = t_1
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(D * M_m) / Float64(d * 2.0)) ^ 2.0)))))
	t_1 = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)))
	tmp = 0.0
	if (t_0 <= -2e-220)
		tmp = Float64(t_1 * Float64(1.0 - Float64(0.5 * (Float64(Float64(D / Float64(d * Float64(2.0 / M_m))) * sqrt(Float64(h / l))) ^ 2.0))));
	elseif ((t_0 <= 1e-163) || !(t_0 <= 2e+208))
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = (((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((h / l) * (0.5 * (((D * M_m) / (d * 2.0)) ^ 2.0))));
	t_1 = sqrt((d / h)) * sqrt((d / l));
	tmp = 0.0;
	if (t_0 <= -2e-220)
		tmp = t_1 * (1.0 - (0.5 * (((D / (d * (2.0 / M_m))) * sqrt((h / l))) ^ 2.0)));
	elseif ((t_0 <= 1e-163) || ~((t_0 <= 2e+208)))
		tmp = abs((d / sqrt((l * h))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(D * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-220], N[(t$95$1 * N[(1.0 - N[(0.5 * N[Power[N[(N[(D / N[(d * N[(2.0 / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, 1e-163], N[Not[LessEqual[t$95$0, 2e+208]], $MachinePrecision]], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.99999999999999998e-220

   1. Initial program 84.6%

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

   3. Simplified 85.8%

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

      1. add-sqr-sqrt 85.7%

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

      2. pow2 85.7%

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

      3. sqrt-prod 85.7%

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

      4. sqrt-pow1 88.0%

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

      5. metadata-eval 88.0%

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

      6. pow1 88.0%

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

      7. *-commutative 88.0%

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

      8. clear-num 87.9%

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

      9. frac-times 90.1%

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

      10. *-commutative 90.1%

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

      11. *-un-lft-identity 90.1%

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

   6. Applied egg-rr 90.1%

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

   if -1.99999999999999998e-220 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.99999999999999923e-164 or 2e208 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 21.8%

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

   3. Simplified 22.7%

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

   5. Taylor expanded in M around 0 30.4%

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

      1. pow1 30.4%

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

      2. pow1/2 30.4%

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

      3. *-rgt-identity 30.4%

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

      4. pow1/2 30.4%

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

      5. pow-prod-down 24.3%

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

   7. Applied egg-rr 24.3%

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

      1. unpow1 24.3%

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

      2. unpow1/2 24.3%

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

   9. Simplified 24.3%

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

       1. sqrt-prod 30.4%

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

       2. sqrt-undiv 25.7%

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

       3. *-rgt-identity 25.7%

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

       4. add-sqr-sqrt 25.5%

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

       5. sqrt-prod 13.4%

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

       6. rem-sqrt-square 25.7%

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

       7. sqrt-undiv 30.4%

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

       8. *-rgt-identity 30.4%

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

       9. sqrt-prod 24.3%

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

       10. frac-times 24.3%

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

       11. sqrt-div 29.6%

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

       12. sqrt-unprod 28.6%

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

       13. add-sqr-sqrt 52.8%

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

   11. Applied egg-rr 52.8%

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

   if 9.99999999999999923e-164 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2e208

   1. Initial program 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in M around 0 99.0%

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

      1. pow1 99.0%

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

      2. pow1/2 99.0%

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

      3. *-rgt-identity 99.0%

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

      4. pow1/2 99.0%

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

      5. pow-prod-down 89.8%

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

   7. Applied egg-rr 89.8%

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

      1. unpow1 89.8%

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

      2. unpow1/2 89.8%

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

   9. Simplified 89.8%

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

       1. *-commutative 89.8%

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

       2. sqrt-prod 99.0%

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

   11. Applied egg-rr 99.0%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq -2 \cdot 10^{-220}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D}{d \cdot \frac{2}{M}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq 10^{-163} \lor \neg \left(\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq 2 \cdot 10^{+208}\right):\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := 1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\\ t_3 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -5.1 \cdot 10^{+194}:\\ \;\;\;\;\left(\frac{t\_3}{\sqrt{-\ell}} \cdot t\_0\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(0.5 \cdot \frac{D \cdot M\_m}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_1 \cdot \left(t\_2 \cdot \frac{t\_3}{\sqrt{-h}}\right)\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+227}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(\left(h \cdot {\left(D \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2}\right) \cdot \frac{-1}{\ell}\right)\right) \cdot \left(t\_1 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot t\_2\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (sqrt (/ d l)))
        (t_2 (+ 1.0 (* (/ h l) (* (pow (* D (/ (/ M_m 2.0) d)) 2.0) -0.5))))
        (t_3 (sqrt (- d))))
   (if (<= l -5.1e+194)
     (*
      (* (/ t_3 (sqrt (- l))) t_0)
      (- 1.0 (* 0.5 (* h (/ (pow (* 0.5 (/ (* D M_m) d)) 2.0) l)))))
     (if (<= l -5e-310)
       (* t_1 (* t_2 (/ t_3 (sqrt (- h)))))
       (if (<= l 8e+227)
         (*
          (+
           1.0
           (* 0.5 (* (* h (pow (* D (* M_m (/ 0.5 d))) 2.0)) (/ -1.0 l))))
          (* t_1 (/ (sqrt d) (sqrt h))))
         (* (* t_0 t_2) (/ (sqrt d) (sqrt l))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = sqrt((d / l));
	double t_2 = 1.0 + ((h / l) * (pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5));
	double t_3 = sqrt(-d);
	double tmp;
	if (l <= -5.1e+194) {
		tmp = ((t_3 / sqrt(-l)) * t_0) * (1.0 - (0.5 * (h * (pow((0.5 * ((D * M_m) / d)), 2.0) / l))));
	} else if (l <= -5e-310) {
		tmp = t_1 * (t_2 * (t_3 / sqrt(-h)));
	} else if (l <= 8e+227) {
		tmp = (1.0 + (0.5 * ((h * pow((D * (M_m * (0.5 / d))), 2.0)) * (-1.0 / l)))) * (t_1 * (sqrt(d) / sqrt(h)));
	} else {
		tmp = (t_0 * t_2) * (sqrt(d) / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sqrt((d / h))
    t_1 = sqrt((d / l))
    t_2 = 1.0d0 + ((h / l) * (((d_1 * ((m_m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))
    t_3 = sqrt(-d)
    if (l <= (-5.1d+194)) then
        tmp = ((t_3 / sqrt(-l)) * t_0) * (1.0d0 - (0.5d0 * (h * (((0.5d0 * ((d_1 * m_m) / d)) ** 2.0d0) / l))))
    else if (l <= (-5d-310)) then
        tmp = t_1 * (t_2 * (t_3 / sqrt(-h)))
    else if (l <= 8d+227) then
        tmp = (1.0d0 + (0.5d0 * ((h * ((d_1 * (m_m * (0.5d0 / d))) ** 2.0d0)) * ((-1.0d0) / l)))) * (t_1 * (sqrt(d) / sqrt(h)))
    else
        tmp = (t_0 * t_2) * (sqrt(d) / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = Math.sqrt((d / h));
	double t_1 = Math.sqrt((d / l));
	double t_2 = 1.0 + ((h / l) * (Math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5));
	double t_3 = Math.sqrt(-d);
	double tmp;
	if (l <= -5.1e+194) {
		tmp = ((t_3 / Math.sqrt(-l)) * t_0) * (1.0 - (0.5 * (h * (Math.pow((0.5 * ((D * M_m) / d)), 2.0) / l))));
	} else if (l <= -5e-310) {
		tmp = t_1 * (t_2 * (t_3 / Math.sqrt(-h)));
	} else if (l <= 8e+227) {
		tmp = (1.0 + (0.5 * ((h * Math.pow((D * (M_m * (0.5 / d))), 2.0)) * (-1.0 / l)))) * (t_1 * (Math.sqrt(d) / Math.sqrt(h)));
	} else {
		tmp = (t_0 * t_2) * (Math.sqrt(d) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = math.sqrt((d / h))
	t_1 = math.sqrt((d / l))
	t_2 = 1.0 + ((h / l) * (math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))
	t_3 = math.sqrt(-d)
	tmp = 0
	if l <= -5.1e+194:
		tmp = ((t_3 / math.sqrt(-l)) * t_0) * (1.0 - (0.5 * (h * (math.pow((0.5 * ((D * M_m) / d)), 2.0) / l))))
	elif l <= -5e-310:
		tmp = t_1 * (t_2 * (t_3 / math.sqrt(-h)))
	elif l <= 8e+227:
		tmp = (1.0 + (0.5 * ((h * math.pow((D * (M_m * (0.5 / d))), 2.0)) * (-1.0 / l)))) * (t_1 * (math.sqrt(d) / math.sqrt(h)))
	else:
		tmp = (t_0 * t_2) * (math.sqrt(d) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(d / l))
	t_2 = Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5)))
	t_3 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -5.1e+194)
		tmp = Float64(Float64(Float64(t_3 / sqrt(Float64(-l))) * t_0) * Float64(1.0 - Float64(0.5 * Float64(h * Float64((Float64(0.5 * Float64(Float64(D * M_m) / d)) ^ 2.0) / l)))));
	elseif (l <= -5e-310)
		tmp = Float64(t_1 * Float64(t_2 * Float64(t_3 / sqrt(Float64(-h)))));
	elseif (l <= 8e+227)
		tmp = Float64(Float64(1.0 + Float64(0.5 * Float64(Float64(h * (Float64(D * Float64(M_m * Float64(0.5 / d))) ^ 2.0)) * Float64(-1.0 / l)))) * Float64(t_1 * Float64(sqrt(d) / sqrt(h))));
	else
		tmp = Float64(Float64(t_0 * t_2) * Float64(sqrt(d) / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = sqrt((d / h));
	t_1 = sqrt((d / l));
	t_2 = 1.0 + ((h / l) * (((D * ((M_m / 2.0) / d)) ^ 2.0) * -0.5));
	t_3 = sqrt(-d);
	tmp = 0.0;
	if (l <= -5.1e+194)
		tmp = ((t_3 / sqrt(-l)) * t_0) * (1.0 - (0.5 * (h * (((0.5 * ((D * M_m) / d)) ^ 2.0) / l))));
	elseif (l <= -5e-310)
		tmp = t_1 * (t_2 * (t_3 / sqrt(-h)));
	elseif (l <= 8e+227)
		tmp = (1.0 + (0.5 * ((h * ((D * (M_m * (0.5 / d))) ^ 2.0)) * (-1.0 / l)))) * (t_1 * (sqrt(d) / sqrt(h)));
	else
		tmp = (t_0 * t_2) * (sqrt(d) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -5.1e+194], N[(N[(N[(t$95$3 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(N[Power[N[(0.5 * N[(N[(D * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(t$95$1 * N[(t$95$2 * N[(t$95$3 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8e+227], N[(N[(1.0 + N[(0.5 * N[(N[(h * N[Power[N[(D * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * t$95$2), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -5.1000000000000002e194

   1. Initial program 45.5%

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

   3. Simplified 45.2%

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

      1. associate-*r/ 44.8%

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

      2. clear-num 44.8%

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

      3. *-commutative 44.8%

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

      4. *-commutative 44.8%

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

      5. clear-num 44.8%

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

      6. frac-times 45.1%

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

      7. *-commutative 45.1%

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

      8. *-un-lft-identity 45.1%

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

   6. Applied egg-rr 45.1%

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

      1. associate-/r/ 45.1%

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

      2. associate-/l/ 45.1%

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

      3. remove-double-neg 45.1%

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

      4. distribute-frac-neg 45.1%

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

      5. *-rgt-identity 45.1%

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

      6. associate-/l* 45.1%

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

      7. associate-/r/ 45.1%

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

      8. metadata-eval 45.1%

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

      9. distribute-neg-frac 45.1%

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

      10. associate-/l* 45.1%

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

      11. *-commutative 45.1%

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

      12. associate-*r/ 45.1%

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

      13. distribute-lft-neg-in 45.1%

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

      14. remove-double-neg 45.1%

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

   8. Simplified 45.1%

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

      1. frac-2neg 45.5%

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

      2. sqrt-div 66.1%

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

   10. Applied egg-rr 62.2%

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

   11. Taylor expanded in l around 0 40.7%

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

   13. Simplified 66.6%

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

   if -5.1000000000000002e194 < l < -4.999999999999985e-310

   1. Initial program 62.2%

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

   3. Simplified 62.0%

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

      1. frac-2neg 64.3%

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

      2. sqrt-div 78.2%

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

   6. Applied egg-rr 75.7%

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

   if -4.999999999999985e-310 < l < 8.0000000000000007e227

   1. Initial program 68.2%

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

   3. Simplified 69.1%

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

      1. associate-*r/ 74.4%

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

      2. clear-num 74.4%

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

      3. *-commutative 74.4%

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

      4. *-commutative 74.4%

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

      5. clear-num 74.4%

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

      6. frac-times 73.5%

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

      7. *-commutative 73.5%

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

      8. *-un-lft-identity 73.5%

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

   6. Applied egg-rr 73.5%

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

      1. associate-/r/ 73.5%

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

      2. associate-/l/ 73.6%

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

      3. remove-double-neg 73.6%

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

      4. distribute-frac-neg 73.6%

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

      5. *-rgt-identity 73.6%

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

      6. associate-/l* 73.5%

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

      7. associate-/r/ 73.6%

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

      8. metadata-eval 73.6%

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

      9. distribute-neg-frac 73.6%

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

      10. associate-/l* 74.4%

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

      11. *-commutative 74.4%

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

      12. associate-*r/ 74.4%

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

      13. distribute-lft-neg-in 74.4%

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

      14. remove-double-neg 74.4%

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

   8. Simplified 74.4%

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

      1. sqrt-div 83.4%

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

      2. div-inv 83.3%

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

   10. Applied egg-rr 83.3%

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

       1. associate-*r/ 83.4%

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

       2. *-rgt-identity 83.4%

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

   12. Simplified 83.4%

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

   if 8.0000000000000007e227 < l

   1. Initial program 47.8%

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

   3. Simplified 47.5%

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

      1. sqrt-div 68.4%

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

      2. div-inv 68.3%

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

   6. Applied egg-rr 68.3%

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

      1. associate-*r/ 68.4%

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

      2. *-rgt-identity 68.4%

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

   8. Simplified 68.4%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.1 \cdot 10^{+194}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(0.5 \cdot \frac{D \cdot M}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+227}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(\left(h \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right) \cdot \frac{-1}{\ell}\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := 1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\\ t_1 := \sqrt{\frac{d}{h}} \cdot t\_0\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -1.65 \cdot 10^{+98}:\\ \;\;\;\;\frac{t\_3}{\sqrt{-\ell}} \cdot t\_1\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_2 \cdot \left(t\_0 \cdot \frac{t\_3}{\sqrt{-h}}\right)\\ \mathbf{elif}\;\ell \leq 10^{+227}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(\left(h \cdot {\left(D \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2}\right) \cdot \frac{-1}{\ell}\right)\right) \cdot \left(t\_2 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (/ h l) (* (pow (* D (/ (/ M_m 2.0) d)) 2.0) -0.5))))
        (t_1 (* (sqrt (/ d h)) t_0))
        (t_2 (sqrt (/ d l)))
        (t_3 (sqrt (- d))))
   (if (<= l -1.65e+98)
     (* (/ t_3 (sqrt (- l))) t_1)
     (if (<= l -5e-310)
       (* t_2 (* t_0 (/ t_3 (sqrt (- h)))))
       (if (<= l 1e+227)
         (*
          (+
           1.0
           (* 0.5 (* (* h (pow (* D (* M_m (/ 0.5 d))) 2.0)) (/ -1.0 l))))
          (* t_2 (/ (sqrt d) (sqrt h))))
         (* t_1 (/ (sqrt d) (sqrt l))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = 1.0 + ((h / l) * (pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5));
	double t_1 = sqrt((d / h)) * t_0;
	double t_2 = sqrt((d / l));
	double t_3 = sqrt(-d);
	double tmp;
	if (l <= -1.65e+98) {
		tmp = (t_3 / sqrt(-l)) * t_1;
	} else if (l <= -5e-310) {
		tmp = t_2 * (t_0 * (t_3 / sqrt(-h)));
	} else if (l <= 1e+227) {
		tmp = (1.0 + (0.5 * ((h * pow((D * (M_m * (0.5 / d))), 2.0)) * (-1.0 / l)))) * (t_2 * (sqrt(d) / sqrt(h)));
	} else {
		tmp = t_1 * (sqrt(d) / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 1.0d0 + ((h / l) * (((d_1 * ((m_m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))
    t_1 = sqrt((d / h)) * t_0
    t_2 = sqrt((d / l))
    t_3 = sqrt(-d)
    if (l <= (-1.65d+98)) then
        tmp = (t_3 / sqrt(-l)) * t_1
    else if (l <= (-5d-310)) then
        tmp = t_2 * (t_0 * (t_3 / sqrt(-h)))
    else if (l <= 1d+227) then
        tmp = (1.0d0 + (0.5d0 * ((h * ((d_1 * (m_m * (0.5d0 / d))) ** 2.0d0)) * ((-1.0d0) / l)))) * (t_2 * (sqrt(d) / sqrt(h)))
    else
        tmp = t_1 * (sqrt(d) / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = 1.0 + ((h / l) * (Math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5));
	double t_1 = Math.sqrt((d / h)) * t_0;
	double t_2 = Math.sqrt((d / l));
	double t_3 = Math.sqrt(-d);
	double tmp;
	if (l <= -1.65e+98) {
		tmp = (t_3 / Math.sqrt(-l)) * t_1;
	} else if (l <= -5e-310) {
		tmp = t_2 * (t_0 * (t_3 / Math.sqrt(-h)));
	} else if (l <= 1e+227) {
		tmp = (1.0 + (0.5 * ((h * Math.pow((D * (M_m * (0.5 / d))), 2.0)) * (-1.0 / l)))) * (t_2 * (Math.sqrt(d) / Math.sqrt(h)));
	} else {
		tmp = t_1 * (Math.sqrt(d) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = 1.0 + ((h / l) * (math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))
	t_1 = math.sqrt((d / h)) * t_0
	t_2 = math.sqrt((d / l))
	t_3 = math.sqrt(-d)
	tmp = 0
	if l <= -1.65e+98:
		tmp = (t_3 / math.sqrt(-l)) * t_1
	elif l <= -5e-310:
		tmp = t_2 * (t_0 * (t_3 / math.sqrt(-h)))
	elif l <= 1e+227:
		tmp = (1.0 + (0.5 * ((h * math.pow((D * (M_m * (0.5 / d))), 2.0)) * (-1.0 / l)))) * (t_2 * (math.sqrt(d) / math.sqrt(h)))
	else:
		tmp = t_1 * (math.sqrt(d) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5)))
	t_1 = Float64(sqrt(Float64(d / h)) * t_0)
	t_2 = sqrt(Float64(d / l))
	t_3 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -1.65e+98)
		tmp = Float64(Float64(t_3 / sqrt(Float64(-l))) * t_1);
	elseif (l <= -5e-310)
		tmp = Float64(t_2 * Float64(t_0 * Float64(t_3 / sqrt(Float64(-h)))));
	elseif (l <= 1e+227)
		tmp = Float64(Float64(1.0 + Float64(0.5 * Float64(Float64(h * (Float64(D * Float64(M_m * Float64(0.5 / d))) ^ 2.0)) * Float64(-1.0 / l)))) * Float64(t_2 * Float64(sqrt(d) / sqrt(h))));
	else
		tmp = Float64(t_1 * Float64(sqrt(d) / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = 1.0 + ((h / l) * (((D * ((M_m / 2.0) / d)) ^ 2.0) * -0.5));
	t_1 = sqrt((d / h)) * t_0;
	t_2 = sqrt((d / l));
	t_3 = sqrt(-d);
	tmp = 0.0;
	if (l <= -1.65e+98)
		tmp = (t_3 / sqrt(-l)) * t_1;
	elseif (l <= -5e-310)
		tmp = t_2 * (t_0 * (t_3 / sqrt(-h)));
	elseif (l <= 1e+227)
		tmp = (1.0 + (0.5 * ((h * ((D * (M_m * (0.5 / d))) ^ 2.0)) * (-1.0 / l)))) * (t_2 * (sqrt(d) / sqrt(h)));
	else
		tmp = t_1 * (sqrt(d) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -1.65e+98], N[(N[(t$95$3 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[l, -5e-310], N[(t$95$2 * N[(t$95$0 * N[(t$95$3 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1e+227], N[(N[(1.0 + N[(0.5 * N[(N[(h * N[Power[N[(D * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.65000000000000014e98

   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 48.9%

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

      1. frac-2neg 48.9%

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

      2. sqrt-div 63.5%

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

   6. Applied egg-rr 63.5%

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

   if -1.65000000000000014e98 < l < -4.999999999999985e-310

   1. Initial program 64.9%

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

   3. Simplified 63.7%

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

      1. frac-2neg 67.4%

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

      2. sqrt-div 81.9%

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

   6. Applied egg-rr 78.0%

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

   if -4.999999999999985e-310 < l < 1.0000000000000001e227

   1. Initial program 68.2%

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

   3. Simplified 69.1%

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

      1. associate-*r/ 74.4%

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

      2. clear-num 74.4%

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

      3. *-commutative 74.4%

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

      4. *-commutative 74.4%

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

      5. clear-num 74.4%

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

      6. frac-times 73.5%

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

      7. *-commutative 73.5%

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

      8. *-un-lft-identity 73.5%

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

   6. Applied egg-rr 73.5%

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

      1. associate-/r/ 73.5%

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

      2. associate-/l/ 73.6%

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

      3. remove-double-neg 73.6%

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

      4. distribute-frac-neg 73.6%

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

      5. *-rgt-identity 73.6%

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

      6. associate-/l* 73.5%

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

      7. associate-/r/ 73.6%

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

      8. metadata-eval 73.6%

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

      9. distribute-neg-frac 73.6%

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

      10. associate-/l* 74.4%

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

      11. *-commutative 74.4%

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

      12. associate-*r/ 74.4%

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

      13. distribute-lft-neg-in 74.4%

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

      14. remove-double-neg 74.4%

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

   8. Simplified 74.4%

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

      1. sqrt-div 83.4%

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

      2. div-inv 83.3%

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

   10. Applied egg-rr 83.3%

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

       1. associate-*r/ 83.4%

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

       2. *-rgt-identity 83.4%

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

   12. Simplified 83.4%

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

   if 1.0000000000000001e227 < l

   1. Initial program 47.8%

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

   3. Simplified 47.5%

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

      1. sqrt-div 68.4%

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

      2. div-inv 68.3%

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

   6. Applied egg-rr 68.3%

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

      1. associate-*r/ 68.4%

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

      2. *-rgt-identity 68.4%

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

   8. Simplified 68.4%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.65 \cdot 10^{+98}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;\ell \leq 10^{+227}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(\left(h \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right) \cdot \frac{-1}{\ell}\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{-d}\\ t_3 := 1 + 0.5 \cdot \left(\left(h \cdot {\left(D \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2}\right) \cdot \frac{-1}{\ell}\right)\\ \mathbf{if}\;\ell \leq -4.1 \cdot 10^{+93}:\\ \;\;\;\;\frac{t\_2}{\sqrt{-\ell}} \cdot t\_0\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{t\_2}{\sqrt{-h}} \cdot t\_1\right) \cdot t\_3\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+226}:\\ \;\;\;\;t\_3 \cdot \left(t\_1 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0
         (*
          (sqrt (/ d h))
          (+ 1.0 (* (/ h l) (* (pow (* D (/ (/ M_m 2.0) d)) 2.0) -0.5)))))
        (t_1 (sqrt (/ d l)))
        (t_2 (sqrt (- d)))
        (t_3
         (+
          1.0
          (* 0.5 (* (* h (pow (* D (* M_m (/ 0.5 d))) 2.0)) (/ -1.0 l))))))
   (if (<= l -4.1e+93)
     (* (/ t_2 (sqrt (- l))) t_0)
     (if (<= l -5e-310)
       (* (* (/ t_2 (sqrt (- h))) t_1) t_3)
       (if (<= l 1.8e+226)
         (* t_3 (* t_1 (/ (sqrt d) (sqrt h))))
         (* t_0 (/ (sqrt d) (sqrt l))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt((d / h)) * (1.0 + ((h / l) * (pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5)));
	double t_1 = sqrt((d / l));
	double t_2 = sqrt(-d);
	double t_3 = 1.0 + (0.5 * ((h * pow((D * (M_m * (0.5 / d))), 2.0)) * (-1.0 / l)));
	double tmp;
	if (l <= -4.1e+93) {
		tmp = (t_2 / sqrt(-l)) * t_0;
	} else if (l <= -5e-310) {
		tmp = ((t_2 / sqrt(-h)) * t_1) * t_3;
	} else if (l <= 1.8e+226) {
		tmp = t_3 * (t_1 * (sqrt(d) / sqrt(h)));
	} else {
		tmp = t_0 * (sqrt(d) / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sqrt((d / h)) * (1.0d0 + ((h / l) * (((d_1 * ((m_m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0))))
    t_1 = sqrt((d / l))
    t_2 = sqrt(-d)
    t_3 = 1.0d0 + (0.5d0 * ((h * ((d_1 * (m_m * (0.5d0 / d))) ** 2.0d0)) * ((-1.0d0) / l)))
    if (l <= (-4.1d+93)) then
        tmp = (t_2 / sqrt(-l)) * t_0
    else if (l <= (-5d-310)) then
        tmp = ((t_2 / sqrt(-h)) * t_1) * t_3
    else if (l <= 1.8d+226) then
        tmp = t_3 * (t_1 * (sqrt(d) / sqrt(h)))
    else
        tmp = t_0 * (sqrt(d) / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = Math.sqrt((d / h)) * (1.0 + ((h / l) * (Math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5)));
	double t_1 = Math.sqrt((d / l));
	double t_2 = Math.sqrt(-d);
	double t_3 = 1.0 + (0.5 * ((h * Math.pow((D * (M_m * (0.5 / d))), 2.0)) * (-1.0 / l)));
	double tmp;
	if (l <= -4.1e+93) {
		tmp = (t_2 / Math.sqrt(-l)) * t_0;
	} else if (l <= -5e-310) {
		tmp = ((t_2 / Math.sqrt(-h)) * t_1) * t_3;
	} else if (l <= 1.8e+226) {
		tmp = t_3 * (t_1 * (Math.sqrt(d) / Math.sqrt(h)));
	} else {
		tmp = t_0 * (Math.sqrt(d) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = math.sqrt((d / h)) * (1.0 + ((h / l) * (math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5)))
	t_1 = math.sqrt((d / l))
	t_2 = math.sqrt(-d)
	t_3 = 1.0 + (0.5 * ((h * math.pow((D * (M_m * (0.5 / d))), 2.0)) * (-1.0 / l)))
	tmp = 0
	if l <= -4.1e+93:
		tmp = (t_2 / math.sqrt(-l)) * t_0
	elif l <= -5e-310:
		tmp = ((t_2 / math.sqrt(-h)) * t_1) * t_3
	elif l <= 1.8e+226:
		tmp = t_3 * (t_1 * (math.sqrt(d) / math.sqrt(h)))
	else:
		tmp = t_0 * (math.sqrt(d) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5))))
	t_1 = sqrt(Float64(d / l))
	t_2 = sqrt(Float64(-d))
	t_3 = Float64(1.0 + Float64(0.5 * Float64(Float64(h * (Float64(D * Float64(M_m * Float64(0.5 / d))) ^ 2.0)) * Float64(-1.0 / l))))
	tmp = 0.0
	if (l <= -4.1e+93)
		tmp = Float64(Float64(t_2 / sqrt(Float64(-l))) * t_0);
	elseif (l <= -5e-310)
		tmp = Float64(Float64(Float64(t_2 / sqrt(Float64(-h))) * t_1) * t_3);
	elseif (l <= 1.8e+226)
		tmp = Float64(t_3 * Float64(t_1 * Float64(sqrt(d) / sqrt(h))));
	else
		tmp = Float64(t_0 * Float64(sqrt(d) / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = sqrt((d / h)) * (1.0 + ((h / l) * (((D * ((M_m / 2.0) / d)) ^ 2.0) * -0.5)));
	t_1 = sqrt((d / l));
	t_2 = sqrt(-d);
	t_3 = 1.0 + (0.5 * ((h * ((D * (M_m * (0.5 / d))) ^ 2.0)) * (-1.0 / l)));
	tmp = 0.0;
	if (l <= -4.1e+93)
		tmp = (t_2 / sqrt(-l)) * t_0;
	elseif (l <= -5e-310)
		tmp = ((t_2 / sqrt(-h)) * t_1) * t_3;
	elseif (l <= 1.8e+226)
		tmp = t_3 * (t_1 * (sqrt(d) / sqrt(h)));
	else
		tmp = t_0 * (sqrt(d) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + N[(0.5 * N[(N[(h * N[Power[N[(D * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4.1e+93], N[(N[(t$95$2 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[l, -5e-310], N[(N[(N[(t$95$2 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[l, 1.8e+226], N[(t$95$3 * N[(t$95$1 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.1000000000000001e93

   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 48.9%

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

      1. frac-2neg 48.9%

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

      2. sqrt-div 63.5%

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

   6. Applied egg-rr 63.5%

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

   if -4.1000000000000001e93 < l < -4.999999999999985e-310

   1. Initial program 64.9%

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

   3. Simplified 63.8%

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

      1. associate-*r/ 67.5%

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

      2. clear-num 67.5%

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

      3. *-commutative 67.5%

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

      4. *-commutative 67.5%

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

      5. clear-num 67.5%

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

      6. frac-times 68.6%

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

      7. *-commutative 68.6%

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

      8. *-un-lft-identity 68.6%

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

   6. Applied egg-rr 68.6%

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

      1. associate-/r/ 68.6%

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

      2. associate-/l/ 68.6%

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

      3. remove-double-neg 68.6%

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

      4. distribute-frac-neg 68.6%

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

      5. *-rgt-identity 68.6%

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

      6. associate-/l* 68.6%

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

      7. associate-/r/ 68.6%

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

      8. metadata-eval 68.6%

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

      9. distribute-neg-frac 68.6%

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

      10. associate-/l* 67.4%

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

      11. *-commutative 67.4%

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

      12. associate-*r/ 67.4%

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

      13. distribute-lft-neg-in 67.4%

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

      14. remove-double-neg 67.4%

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

   8. Simplified 67.4%

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

      1. frac-2neg 67.4%

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

      2. sqrt-div 81.9%

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

   10. Applied egg-rr 81.9%

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

   if -4.999999999999985e-310 < l < 1.7999999999999999e226

   1. Initial program 68.2%

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

   3. Simplified 69.1%

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

      1. associate-*r/ 74.4%

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

      2. clear-num 74.4%

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

      3. *-commutative 74.4%

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

      4. *-commutative 74.4%

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

      5. clear-num 74.4%

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

      6. frac-times 73.5%

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

      7. *-commutative 73.5%

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

      8. *-un-lft-identity 73.5%

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

   6. Applied egg-rr 73.5%

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

      1. associate-/r/ 73.5%

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

      2. associate-/l/ 73.6%

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

      3. remove-double-neg 73.6%

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

      4. distribute-frac-neg 73.6%

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

      5. *-rgt-identity 73.6%

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

      6. associate-/l* 73.5%

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

      7. associate-/r/ 73.6%

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

      8. metadata-eval 73.6%

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

      9. distribute-neg-frac 73.6%

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

      10. associate-/l* 74.4%

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

      11. *-commutative 74.4%

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

      12. associate-*r/ 74.4%

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

      13. distribute-lft-neg-in 74.4%

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

      14. remove-double-neg 74.4%

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

   8. Simplified 74.4%

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

      1. sqrt-div 83.4%

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

      2. div-inv 83.3%

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

   10. Applied egg-rr 83.3%

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

       1. associate-*r/ 83.4%

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

       2. *-rgt-identity 83.4%

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

   12. Simplified 83.4%

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

   if 1.7999999999999999e226 < l

   1. Initial program 47.8%

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

   3. Simplified 47.5%

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

      1. sqrt-div 68.4%

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

      2. div-inv 68.3%

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

   6. Applied egg-rr 68.3%

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

      1. associate-*r/ 68.4%

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

      2. *-rgt-identity 68.4%

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

   8. Simplified 68.4%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.1 \cdot 10^{+93}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(\left(h \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right) \cdot \frac{-1}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+226}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(\left(h \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right) \cdot \frac{-1}{\ell}\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := 1 - 0.5 \cdot \left(h \cdot \frac{{\left(0.5 \cdot \frac{D \cdot M\_m}{d}\right)}^{2}}{\ell}\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;d \leq -6 \cdot 10^{-8}:\\ \;\;\;\;\left(\frac{t\_2}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot t\_0\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;t\_1 \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \frac{t\_2}{\sqrt{-h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (- 1.0 (* 0.5 (* h (/ (pow (* 0.5 (/ (* D M_m) d)) 2.0) l)))))
        (t_1 (sqrt (/ d l)))
        (t_2 (sqrt (- d))))
   (if (<= d -6e-8)
     (* (* (/ t_2 (sqrt (- l))) (sqrt (/ d h))) t_0)
     (if (<= d -1e-309)
       (*
        t_1
        (*
         (+ 1.0 (* (/ h l) (* (pow (* D (/ (/ M_m 2.0) d)) 2.0) -0.5)))
         (/ t_2 (sqrt (- h)))))
       (* (* t_1 (/ (sqrt d) (sqrt h))) t_0)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = 1.0 - (0.5 * (h * (pow((0.5 * ((D * M_m) / d)), 2.0) / l)));
	double t_1 = sqrt((d / l));
	double t_2 = sqrt(-d);
	double tmp;
	if (d <= -6e-8) {
		tmp = ((t_2 / sqrt(-l)) * sqrt((d / h))) * t_0;
	} else if (d <= -1e-309) {
		tmp = t_1 * ((1.0 + ((h / l) * (pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))) * (t_2 / sqrt(-h)));
	} else {
		tmp = (t_1 * (sqrt(d) / sqrt(h))) * t_0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 - (0.5d0 * (h * (((0.5d0 * ((d_1 * m_m) / d)) ** 2.0d0) / l)))
    t_1 = sqrt((d / l))
    t_2 = sqrt(-d)
    if (d <= (-6d-8)) then
        tmp = ((t_2 / sqrt(-l)) * sqrt((d / h))) * t_0
    else if (d <= (-1d-309)) then
        tmp = t_1 * ((1.0d0 + ((h / l) * (((d_1 * ((m_m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))) * (t_2 / sqrt(-h)))
    else
        tmp = (t_1 * (sqrt(d) / sqrt(h))) * t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = 1.0 - (0.5 * (h * (Math.pow((0.5 * ((D * M_m) / d)), 2.0) / l)));
	double t_1 = Math.sqrt((d / l));
	double t_2 = Math.sqrt(-d);
	double tmp;
	if (d <= -6e-8) {
		tmp = ((t_2 / Math.sqrt(-l)) * Math.sqrt((d / h))) * t_0;
	} else if (d <= -1e-309) {
		tmp = t_1 * ((1.0 + ((h / l) * (Math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))) * (t_2 / Math.sqrt(-h)));
	} else {
		tmp = (t_1 * (Math.sqrt(d) / Math.sqrt(h))) * t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = 1.0 - (0.5 * (h * (math.pow((0.5 * ((D * M_m) / d)), 2.0) / l)))
	t_1 = math.sqrt((d / l))
	t_2 = math.sqrt(-d)
	tmp = 0
	if d <= -6e-8:
		tmp = ((t_2 / math.sqrt(-l)) * math.sqrt((d / h))) * t_0
	elif d <= -1e-309:
		tmp = t_1 * ((1.0 + ((h / l) * (math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))) * (t_2 / math.sqrt(-h)))
	else:
		tmp = (t_1 * (math.sqrt(d) / math.sqrt(h))) * t_0
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(1.0 - Float64(0.5 * Float64(h * Float64((Float64(0.5 * Float64(Float64(D * M_m) / d)) ^ 2.0) / l))))
	t_1 = sqrt(Float64(d / l))
	t_2 = sqrt(Float64(-d))
	tmp = 0.0
	if (d <= -6e-8)
		tmp = Float64(Float64(Float64(t_2 / sqrt(Float64(-l))) * sqrt(Float64(d / h))) * t_0);
	elseif (d <= -1e-309)
		tmp = Float64(t_1 * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5))) * Float64(t_2 / sqrt(Float64(-h)))));
	else
		tmp = Float64(Float64(t_1 * Float64(sqrt(d) / sqrt(h))) * t_0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = 1.0 - (0.5 * (h * (((0.5 * ((D * M_m) / d)) ^ 2.0) / l)));
	t_1 = sqrt((d / l));
	t_2 = sqrt(-d);
	tmp = 0.0;
	if (d <= -6e-8)
		tmp = ((t_2 / sqrt(-l)) * sqrt((d / h))) * t_0;
	elseif (d <= -1e-309)
		tmp = t_1 * ((1.0 + ((h / l) * (((D * ((M_m / 2.0) / d)) ^ 2.0) * -0.5))) * (t_2 / sqrt(-h)));
	else
		tmp = (t_1 * (sqrt(d) / sqrt(h))) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(1.0 - N[(0.5 * N[(h * N[(N[Power[N[(0.5 * N[(N[(D * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[d, -6e-8], N[(N[(N[(t$95$2 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[d, -1e-309], N[(t$95$1 * N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.99999999999999946e-8

   1. Initial program 71.1%

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

   3. Simplified 74.5%

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

      1. associate-*r/ 78.8%

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

      2. clear-num 78.8%

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

      3. *-commutative 78.8%

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

      4. *-commutative 78.8%

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

      5. clear-num 78.8%

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

      6. frac-times 78.8%

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

      7. *-commutative 78.8%

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

      8. *-un-lft-identity 78.8%

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

   6. Applied egg-rr 78.8%

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

      1. associate-/r/ 78.8%

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

      2. associate-/l/ 76.9%

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

      3. remove-double-neg 76.9%

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

      4. distribute-frac-neg 76.9%

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

      5. *-rgt-identity 76.9%

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

      6. associate-/l* 76.9%

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

      7. associate-/r/ 76.9%

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

      8. metadata-eval 76.9%

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

      9. distribute-neg-frac 76.9%

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

      10. associate-/l* 78.8%

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

      11. *-commutative 78.8%

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

      12. associate-*r/ 78.8%

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

      13. distribute-lft-neg-in 78.8%

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

      14. remove-double-neg 78.8%

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

   8. Simplified 78.8%

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

      1. frac-2neg 74.5%

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

      2. sqrt-div 79.7%

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

   10. Applied egg-rr 89.3%

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

   11. Taylor expanded in l around 0 47.3%

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

   13. Simplified 87.4%

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

   if -5.99999999999999946e-8 < d < -1.000000000000002e-309

   1. Initial program 50.1%

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

   3. Simplified 47.3%

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

      1. frac-2neg 47.2%

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

      2. sqrt-div 63.1%

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

   6. Applied egg-rr 63.2%

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

   if -1.000000000000002e-309 < d

   1. Initial program 65.8%

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

   3. Simplified 66.5%

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

      1. associate-*r/ 69.1%

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

      2. clear-num 69.1%

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

      3. *-commutative 69.1%

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

      4. *-commutative 69.1%

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

      5. clear-num 69.1%

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

      6. frac-times 68.3%

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

      7. *-commutative 68.3%

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

      8. *-un-lft-identity 68.3%

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

   6. Applied egg-rr 68.3%

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

      1. associate-/r/ 68.3%

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

      2. associate-/l/ 68.3%

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

      3. remove-double-neg 68.3%

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

      4. distribute-frac-neg 68.3%

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

      5. *-rgt-identity 68.3%

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

      6. associate-/l* 68.3%

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

      7. associate-/r/ 68.3%

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

      8. metadata-eval 68.3%

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

      9. distribute-neg-frac 68.3%

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

      10. associate-/l* 69.0%

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

      11. *-commutative 69.0%

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

      12. associate-*r/ 69.0%

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

      13. distribute-lft-neg-in 69.0%

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

      14. remove-double-neg 69.0%

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

   8. Simplified 69.0%

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

      1. sqrt-div 77.6%

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

      2. div-inv 77.6%

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

   10. Applied egg-rr 77.6%

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

       1. associate-*r/ 77.6%

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

       2. *-rgt-identity 77.6%

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

   12. Simplified 77.6%

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

   13. Taylor expanded in l around 0 49.2%

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

   15. Simplified 78.7%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6 \cdot 10^{-8}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(0.5 \cdot \frac{D \cdot M}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(0.5 \cdot \frac{D \cdot M}{d}\right)}^{2}}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((d / l))
    if (d <= (-1d-309)) then
        tmp = t_0 * ((1.0d0 + ((h / l) * (((d_1 * ((m_m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))) * (sqrt(-d) / sqrt(-h)))
    else
        tmp = (t_0 * (sqrt(d) / sqrt(h))) * (1.0d0 - (0.5d0 * (h * (((0.5d0 * ((d_1 * m_m) / d)) ** 2.0d0) / l))))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if d < -1.000000000000002e-309

   1. Initial program 59.1%

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

   3. Simplified 58.9%

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

      1. frac-2neg 60.7%

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

      2. sqrt-div 72.0%

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

   6. Applied egg-rr 70.3%

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

   if -1.000000000000002e-309 < d

   1. Initial program 65.8%

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

   3. Simplified 66.5%

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

      1. associate-*r/ 69.1%

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

      2. clear-num 69.1%

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

      3. *-commutative 69.1%

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

      4. *-commutative 69.1%

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

      5. clear-num 69.1%

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

      6. frac-times 68.3%

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

      7. *-commutative 68.3%

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

      8. *-un-lft-identity 68.3%

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

   6. Applied egg-rr 68.3%

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

      1. associate-/r/ 68.3%

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

      2. associate-/l/ 68.3%

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

      3. remove-double-neg 68.3%

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

      4. distribute-frac-neg 68.3%

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

      5. *-rgt-identity 68.3%

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

      6. associate-/l* 68.3%

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

      7. associate-/r/ 68.3%

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

      8. metadata-eval 68.3%

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

      9. distribute-neg-frac 68.3%

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

      10. associate-/l* 69.0%

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

      11. *-commutative 69.0%

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

      12. associate-*r/ 69.0%

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

      13. distribute-lft-neg-in 69.0%

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

      14. remove-double-neg 69.0%

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

   8. Simplified 69.0%

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

      1. sqrt-div 77.6%

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

      2. div-inv 77.6%

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

   10. Applied egg-rr 77.6%

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

       1. associate-*r/ 77.6%

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

       2. *-rgt-identity 77.6%

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

   12. Simplified 77.6%

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

   13. Taylor expanded in l around 0 49.2%

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

   15. Simplified 78.7%

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

4. Final simplification 74.7%

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

Alternative 10: 60.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(\frac{0.5}{d} \cdot \left(D \cdot M\_m\right)\right)}^{2}}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= D 6.2e-6)
   (fabs (/ d (sqrt (* l h))))
   (*
    (sqrt (* (/ d h) (/ d l)))
    (+ 1.0 (* -0.5 (* h (/ (pow (* (/ 0.5 d) (* D M_m)) 2.0) l)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (D <= 6.2e-6) {
		tmp = fabs((d / sqrt((l * h))));
	} else {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * (h * (pow(((0.5 / d) * (D * M_m)), 2.0) / l))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: tmp
    if (d_1 <= 6.2d-6) then
        tmp = abs((d / sqrt((l * h))))
    else
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 + ((-0.5d0) * (h * ((((0.5d0 / d) * (d_1 * m_m)) ** 2.0d0) / l))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (D <= 6.2e-6) {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	} else {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * (h * (Math.pow(((0.5 / d) * (D * M_m)), 2.0) / l))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if D <= 6.2e-6:
		tmp = math.fabs((d / math.sqrt((l * h))))
	else:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * (h * (math.pow(((0.5 / d) * (D * M_m)), 2.0) / l))))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (D <= 6.2e-6)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(-0.5 * Float64(h * Float64((Float64(Float64(0.5 / d) * Float64(D * M_m)) ^ 2.0) / l)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (D <= 6.2e-6)
		tmp = abs((d / sqrt((l * h))));
	else
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * (h * ((((0.5 / d) * (D * M_m)) ^ 2.0) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[D, 6.2e-6], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(h * N[(N[Power[N[(N[(0.5 / d), $MachinePrecision] * N[(D * M$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 6.1999999999999999e-6

   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 63.5%

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

   5. Taylor expanded in M around 0 45.0%

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

      1. pow1 45.0%

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

      2. pow1/2 45.0%

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

      3. *-rgt-identity 45.0%

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

      4. pow1/2 45.0%

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

      5. pow-prod-down 38.5%

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

   7. Applied egg-rr 38.5%

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

      1. unpow1 38.5%

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

      2. unpow1/2 38.5%

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

   9. Simplified 38.5%

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

       1. sqrt-prod 45.0%

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

       2. sqrt-undiv 26.6%

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

       3. *-rgt-identity 26.6%

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

       4. add-sqr-sqrt 26.5%

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

       5. sqrt-prod 19.1%

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

       6. rem-sqrt-square 26.6%

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

       7. sqrt-undiv 45.0%

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

       8. *-rgt-identity 45.0%

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

       9. sqrt-prod 38.5%

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

       10. frac-times 29.2%

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

       11. sqrt-div 32.9%

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

       12. sqrt-unprod 25.6%

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

       13. add-sqr-sqrt 48.2%

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

   11. Applied egg-rr 48.2%

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

   if 6.1999999999999999e-6 < D

   1. Initial program 59.0%

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

   3. Simplified 60.8%

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

      1. associate-*r/ 62.6%

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

      2. clear-num 62.6%

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

      3. *-commutative 62.6%

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

      4. *-commutative 62.6%

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

      5. clear-num 62.6%

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

      6. frac-times 64.2%

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

      7. *-commutative 64.2%

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

      8. *-un-lft-identity 64.2%

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

   6. Applied egg-rr 64.2%

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

      1. associate-/r/ 64.2%

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

      2. associate-/l/ 60.8%

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

      3. remove-double-neg 60.8%

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

      4. distribute-frac-neg 60.8%

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

      5. *-rgt-identity 60.8%

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

      6. associate-/l* 60.8%

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

      7. associate-/r/ 60.8%

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

      8. metadata-eval 60.8%

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

      9. distribute-neg-frac 60.8%

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

      10. associate-/l* 64.3%

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

      11. *-commutative 64.3%

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

      12. associate-*r/ 64.3%

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

      13. distribute-lft-neg-in 64.3%

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

      14. remove-double-neg 64.3%

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

   8. Simplified 64.3%

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

      1. frac-2neg 62.5%

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

      2. sqrt-div 35.5%

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

   10. Applied egg-rr 35.6%

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

       1. pow1 35.6%

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

       2. sqrt-undiv 64.3%

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

       3. frac-2neg 64.3%

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

       4. sqrt-unprod 55.7%

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

       5. cancel-sign-sub-inv 55.7%

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

       6. metadata-eval 55.7%

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

       7. associate-*l/ 55.7%

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

       8. *-un-lft-identity 55.7%

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

   12. Applied egg-rr 55.7%

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

       1. unpow1 55.7%

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

       2. associate-/l* 55.7%

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

       3. associate-*r* 52.1%

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

   14. Simplified 52.1%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(\frac{0.5}{d} \cdot \left(D \cdot M\right)\right)}^{2}}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 45.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.9 \cdot 10^{-241}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -4.9e-241)
   (fabs (/ d (sqrt (* l h))))
   (if (<= l -5e-310)
     (/ d (cbrt (pow (* l h) 1.5)))
     (/ d (* (sqrt h) (sqrt l))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -4.9e-241) {
		tmp = fabs((d / sqrt((l * h))));
	} else if (l <= -5e-310) {
		tmp = d / cbrt(pow((l * h), 1.5));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -4.9e-241) {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	} else if (l <= -5e-310) {
		tmp = d / Math.cbrt(Math.pow((l * h), 1.5));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -4.9e-241)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	elseif (l <= -5e-310)
		tmp = Float64(d / cbrt((Float64(l * h) ^ 1.5)));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -4.9e-241], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, -5e-310], N[(d / N[Power[N[Power[N[(l * h), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.8999999999999998e-241

   1. Initial program 57.6%

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

   3. Simplified 58.4%

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

   5. Taylor expanded in M around 0 41.9%

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

      1. pow1 41.9%

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

      2. pow1/2 41.9%

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

      3. *-rgt-identity 41.9%

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

      4. pow1/2 41.9%

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

      5. pow-prod-down 38.6%

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

   7. Applied egg-rr 38.6%

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

      1. unpow1 38.6%

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

      2. unpow1/2 38.6%

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

   9. Simplified 38.6%

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

       1. sqrt-prod 41.9%

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

       2. sqrt-undiv 0.0%

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

       3. *-rgt-identity 0.0%

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

       4. add-sqr-sqrt 0.0%

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

       5. sqrt-prod 0.0%

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

       6. rem-sqrt-square 0.0%

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

       7. sqrt-undiv 41.9%

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

       8. *-rgt-identity 41.9%

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

       9. sqrt-prod 38.6%

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

       10. frac-times 28.8%

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

       11. sqrt-div 31.4%

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

       12. sqrt-unprod 0.0%

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

       13. add-sqr-sqrt 47.0%

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

   11. Applied egg-rr 47.0%

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

   if -4.8999999999999998e-241 < l < -4.999999999999985e-310

   1. Initial program 69.1%

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

   3. Simplified 69.1%

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

   5. Taylor expanded in M around 0 14.1%

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

      1. pow1 14.1%

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

      2. pow1/2 14.1%

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

      3. *-rgt-identity 14.1%

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

      4. pow1/2 14.1%

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

      5. pow-prod-down 14.0%

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

   7. Applied egg-rr 14.0%

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

      1. unpow1 14.0%

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

      2. unpow1/2 14.0%

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

   9. Simplified 14.0%

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

       1. *-un-lft-identity 14.0%

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

       2. frac-times 2.3%

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

       3. sqrt-div 14.0%

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

       4. sqrt-unprod 0.0%

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

       5. add-sqr-sqrt 26.8%

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

   11. Applied egg-rr 26.8%

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

       1. *-lft-identity 26.8%

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

   13. Simplified 26.8%

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

       1. add-cbrt-cube 51.1%

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

       2. pow1/3 51.1%

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

       3. add-sqr-sqrt 51.1%

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

       4. pow1 51.1%

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

       5. pow1/2 51.1%

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

       6. pow-prod-up 51.1%

          \[\leadsto \frac{d}{{\color{blue}{\left({\left(h \cdot \ell\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}} \]

       7. *-commutative 51.1%

          \[\leadsto \frac{d}{{\left({\color{blue}{\left(\ell \cdot h\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333}} \]

       8. metadata-eval 51.1%

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

   15. Applied egg-rr 51.1%

       \[\leadsto \frac{d}{\color{blue}{{\left({\left(\ell \cdot h\right)}^{1.5}\right)}^{0.3333333333333333}}} \]
   16. Step-by-step derivation

       1. unpow1/3 51.1%

          \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}} \]

   17. Simplified 51.1%

       \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}} \]

   if -4.999999999999985e-310 < l

   1. Initial program 65.8%

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

   3. Simplified 65.8%

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

   5. Taylor expanded in M around 0 42.2%

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

      1. pow1 42.2%

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

      2. pow1/2 42.2%

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

      3. *-rgt-identity 42.2%

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

      4. pow1/2 42.2%

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

      5. pow-prod-down 35.2%

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

   7. Applied egg-rr 35.2%

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

      1. unpow1 35.2%

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

      2. unpow1/2 35.2%

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

   9. Simplified 35.2%

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

       1. sqrt-prod 42.2%

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

       2. sqrt-undiv 47.8%

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

       3. *-commutative 47.8%

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

       4. sqrt-div 51.8%

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

       5. frac-times 51.7%

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

       6. add-sqr-sqrt 51.8%

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

   11. Applied egg-rr 51.8%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.9 \cdot 10^{-241}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 45.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.4 \cdot 10^{-292}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l 5.4e-292)
   (* d (- (sqrt (/ 1.0 (* l h)))))
   (/ d (* (sqrt h) (sqrt l)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= 5.4e-292) {
		tmp = d * -sqrt((1.0 / (l * h)));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: tmp
    if (l <= 5.4d-292) then
        tmp = d * -sqrt((1.0d0 / (l * h)))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= 5.4e-292) {
		tmp = d * -Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= 5.4e-292:
		tmp = d * -math.sqrt((1.0 / (l * h)))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= 5.4e-292)
		tmp = Float64(d * Float64(-sqrt(Float64(1.0 / Float64(l * h)))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= 5.4e-292)
		tmp = d * -sqrt((1.0 / (l * h)));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, 5.4e-292], N[(d * (-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.3999999999999998e-292

   1. Initial program 58.8%

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

   3. Simplified 59.5%

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

   5. Taylor expanded in M around 0 37.2%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 41.5%

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

      4. neg-mul-1 41.5%

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

   8. Simplified 41.5%

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

   if 5.3999999999999998e-292 < l

   1. Initial program 66.3%

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

   3. Simplified 66.3%

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

   5. Taylor expanded in M around 0 43.5%

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

      1. pow1 43.5%

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

      2. pow1/2 43.5%

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

      3. *-rgt-identity 43.5%

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

      4. pow1/2 43.5%

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

      5. pow-prod-down 36.3%

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

   7. Applied egg-rr 36.3%

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

      1. unpow1 36.3%

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

      2. unpow1/2 36.3%

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

   9. Simplified 36.3%

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

       1. sqrt-prod 43.5%

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

       2. sqrt-undiv 49.3%

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

       3. *-commutative 49.3%

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

       4. sqrt-div 53.4%

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

       5. frac-times 53.3%

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

       6. add-sqr-sqrt 53.4%

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

   11. Applied egg-rr 53.4%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.4 \cdot 10^{-292}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 42.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D) :precision binary64 (fabs (/ d (sqrt (* l h)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	return fabs((d / sqrt((l * h))));
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    code = abs((d / sqrt((l * h))))
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	return Math.abs((d / Math.sqrt((l * h))));
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	return math.fabs((d / math.sqrt((l * h))))

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	return abs(Float64(d / sqrt(Float64(l * h))))
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp = code(d, h, l, M_m, D)
	tmp = abs((d / sqrt((l * h))));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 62.5%

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

3. Simplified 62.9%

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

5. Taylor expanded in M around 0 40.3%

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

   1. pow1 40.3%

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

   2. pow1/2 40.3%

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

   3. *-rgt-identity 40.3%

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

   4. pow1/2 40.3%

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

   5. pow-prod-down 35.3%

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

7. Applied egg-rr 35.3%

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

   1. unpow1 35.3%

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

   2. unpow1/2 35.3%

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

9. Simplified 35.3%

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

    1. sqrt-prod 40.3%

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

    2. sqrt-undiv 24.7%

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

    3. *-rgt-identity 24.7%

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

    4. add-sqr-sqrt 24.6%

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

    5. sqrt-prod 18.1%

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

    6. rem-sqrt-square 24.7%

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

    7. sqrt-undiv 40.3%

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

    8. *-rgt-identity 40.3%

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

    9. sqrt-prod 35.3%

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

    10. frac-times 26.9%

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

    11. sqrt-div 30.2%

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

    12. sqrt-unprod 23.1%

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

    13. add-sqr-sqrt 43.9%

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

11. Applied egg-rr 43.9%

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

12. Final simplification 43.9%

    \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \]
13. Add Preprocessing

Alternative 14: 42.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \cdot 10^{-241}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -2.5e-241)
   (* d (- (sqrt (/ 1.0 (* l h)))))
   (* d (sqrt (/ (/ 1.0 h) l)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -2.5e-241) {
		tmp = d * -sqrt((1.0 / (l * h)));
	} else {
		tmp = d * sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: tmp
    if (l <= (-2.5d-241)) then
        tmp = d * -sqrt((1.0d0 / (l * h)))
    else
        tmp = d * sqrt(((1.0d0 / h) / l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -2.5e-241) {
		tmp = d * -Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= -2.5e-241:
		tmp = d * -math.sqrt((1.0 / (l * h)))
	else:
		tmp = d * math.sqrt(((1.0 / h) / l))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -2.5e-241)
		tmp = Float64(d * Float64(-sqrt(Float64(1.0 / Float64(l * h)))));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= -2.5e-241)
		tmp = d * -sqrt((1.0 / (l * h)));
	else
		tmp = d * sqrt(((1.0 / h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -2.5e-241], N[(d * (-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.4999999999999999e-241

   1. Initial program 57.6%

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

   3. Simplified 58.4%

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

   5. Taylor expanded in M around 0 41.9%

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

   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 47.0%

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

      4. neg-mul-1 47.0%

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

   8. Simplified 47.0%

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

   if -2.4999999999999999e-241 < l

   1. Initial program 66.1%

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

   3. Simplified 66.2%

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

   5. Taylor expanded in M around 0 39.2%

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

   6. Taylor expanded in d around 0 43.5%

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

      1. associate-/r* 43.7%

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

   8. Simplified 43.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \cdot 10^{-241}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 37.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{-210}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -4e-210) (sqrt (* (/ d h) (/ d l))) (* d (pow (* l h) -0.5))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -4e-210) {
		tmp = sqrt(((d / h) * (d / l)));
	} else {
		tmp = d * pow((l * h), -0.5);
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: tmp
    if (l <= (-4d-210)) then
        tmp = sqrt(((d / h) * (d / l)))
    else
        tmp = d * ((l * h) ** (-0.5d0))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -4e-210) {
		tmp = Math.sqrt(((d / h) * (d / l)));
	} else {
		tmp = d * Math.pow((l * h), -0.5);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= -4e-210:
		tmp = math.sqrt(((d / h) * (d / l)))
	else:
		tmp = d * math.pow((l * h), -0.5)
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -4e-210)
		tmp = sqrt(Float64(Float64(d / h) * Float64(d / l)));
	else
		tmp = Float64(d * (Float64(l * h) ^ -0.5));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= -4e-210)
		tmp = sqrt(((d / h) * (d / l)));
	else
		tmp = d * ((l * h) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -4e-210], N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $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 < -4.0000000000000002e-210

   1. Initial program 57.8%

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

   3. Simplified 59.6%

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

   5. Taylor expanded in M around 0 44.4%

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

      1. pow1 44.4%

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

      2. pow1/2 44.4%

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

      3. *-rgt-identity 44.4%

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

      4. pow1/2 44.4%

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

      5. pow-prod-down 40.7%

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

   7. Applied egg-rr 40.7%

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

      1. unpow1 40.7%

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

      2. unpow1/2 40.7%

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

   9. Simplified 40.7%

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

   if -4.0000000000000002e-210 < l

   1. Initial program 65.4%

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

   3. Simplified 64.9%

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

   5. Taylor expanded in M around 0 37.9%

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

      1. pow1 37.9%

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

      2. pow1/2 37.9%

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

      3. *-rgt-identity 37.9%

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

      4. pow1/2 37.9%

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

      5. pow-prod-down 32.0%

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

   7. Applied egg-rr 32.0%

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

      1. unpow1 32.0%

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

      2. unpow1/2 32.0%

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

   9. Simplified 32.0%

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

       1. *-un-lft-identity 32.0%

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

       2. frac-times 25.2%

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

       3. sqrt-div 28.7%

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

       4. sqrt-unprod 37.3%

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

       5. add-sqr-sqrt 41.5%

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

   11. Applied egg-rr 41.5%

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

       1. *-lft-identity 41.5%

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

   13. Simplified 41.5%

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

       1. div-inv 41.5%

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

       2. pow1/2 41.5%

          \[\leadsto d \cdot \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \]

       3. pow-flip 41.6%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-0.5\right)}} \]

       4. *-commutative 41.6%

          \[\leadsto d \cdot {\color{blue}{\left(\ell \cdot h\right)}}^{\left(-0.5\right)} \]

       5. metadata-eval 41.6%

          \[\leadsto d \cdot {\left(\ell \cdot h\right)}^{\color{blue}{-0.5}} \]

   15. Applied egg-rr 41.6%

       \[\leadsto \color{blue}{d \cdot {\left(\ell \cdot h\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{-210}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 37.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.5 \cdot 10^{-210}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -5.5e-210) (sqrt (* (/ d h) (/ d l))) (* d (sqrt (/ 1.0 (* l h))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -5.5e-210) {
		tmp = sqrt(((d / h) * (d / l)));
	} else {
		tmp = d * sqrt((1.0 / (l * h)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: tmp
    if (l <= (-5.5d-210)) then
        tmp = sqrt(((d / h) * (d / l)))
    else
        tmp = d * sqrt((1.0d0 / (l * h)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -5.5e-210) {
		tmp = Math.sqrt(((d / h) * (d / l)));
	} else {
		tmp = d * Math.sqrt((1.0 / (l * h)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= -5.5e-210:
		tmp = math.sqrt(((d / h) * (d / l)))
	else:
		tmp = d * math.sqrt((1.0 / (l * h)))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -5.5e-210)
		tmp = sqrt(Float64(Float64(d / h) * Float64(d / l)));
	else
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(l * h))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= -5.5e-210)
		tmp = sqrt(((d / h) * (d / l)));
	else
		tmp = d * sqrt((1.0 / (l * h)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -5.5e-210], N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -5.50000000000000024e-210

   1. Initial program 57.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 59.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in M around 0 44.4%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
   6. Step-by-step derivation

      1. pow1 44.4%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right)\right)}^{1}} \]

      2. pow1/2 44.4%

         \[\leadsto {\left(\color{blue}{{\left(\frac{d}{h}\right)}^{0.5}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right)\right)}^{1} \]

      3. *-rgt-identity 44.4%

         \[\leadsto {\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right)}^{1} \]

      4. pow1/2 44.4%

         \[\leadsto {\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}}\right)}^{1} \]

      5. pow-prod-down 40.7%

         \[\leadsto {\color{blue}{\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}\right)}}^{1} \]

   7. Applied egg-rr 40.7%

      \[\leadsto \color{blue}{{\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 40.7%

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}} \]

      2. unpow1/2 40.7%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   9. Simplified 40.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   if -5.50000000000000024e-210 < l

   1. Initial program 65.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 64.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in M around 0 37.9%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]

   6. Taylor expanded in d around 0 41.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.5 \cdot 10^{-210}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 37.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -8 \cdot 10^{-211}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -8e-211) (sqrt (* (/ d h) (/ d l))) (* d (sqrt (/ (/ 1.0 h) l)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -8e-211) {
		tmp = sqrt(((d / h) * (d / l)));
	} else {
		tmp = d * sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    real(8) :: tmp
    if (l <= (-8d-211)) then
        tmp = sqrt(((d / h) * (d / l)))
    else
        tmp = d * sqrt(((1.0d0 / h) / l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -8e-211) {
		tmp = Math.sqrt(((d / h) * (d / l)));
	} else {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= -8e-211:
		tmp = math.sqrt(((d / h) * (d / l)))
	else:
		tmp = d * math.sqrt(((1.0 / h) / l))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -8e-211)
		tmp = sqrt(Float64(Float64(d / h) * Float64(d / l)));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= -8e-211)
		tmp = sqrt(((d / h) * (d / l)));
	else
		tmp = d * sqrt(((1.0 / h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -8e-211], N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -8.00000000000000069e-211

   1. Initial program 57.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 59.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in M around 0 44.4%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
   6. Step-by-step derivation

      1. pow1 44.4%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right)\right)}^{1}} \]

      2. pow1/2 44.4%

         \[\leadsto {\left(\color{blue}{{\left(\frac{d}{h}\right)}^{0.5}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right)\right)}^{1} \]

      3. *-rgt-identity 44.4%

         \[\leadsto {\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right)}^{1} \]

      4. pow1/2 44.4%

         \[\leadsto {\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}}\right)}^{1} \]

      5. pow-prod-down 40.7%

         \[\leadsto {\color{blue}{\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}\right)}}^{1} \]

   7. Applied egg-rr 40.7%

      \[\leadsto \color{blue}{{\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 40.7%

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}} \]

      2. unpow1/2 40.7%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   9. Simplified 40.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   if -8.00000000000000069e-211 < l

   1. Initial program 65.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 64.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in M around 0 37.9%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]

   6. Taylor expanded in d around 0 41.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. associate-/r* 42.1%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   8. Simplified 42.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8 \cdot 10^{-211}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 25.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ d \cdot {\left(\ell \cdot h\right)}^{-0.5} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D) :precision binary64 (* d (pow (* l h) -0.5)))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	return d * pow((l * h), -0.5);
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    code = d * ((l * h) ** (-0.5d0))
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	return d * Math.pow((l * h), -0.5);
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	return d * math.pow((l * h), -0.5)

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	return Float64(d * (Float64(l * h) ^ -0.5))
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp = code(d, h, l, M_m, D)
	tmp = d * ((l * h) ^ -0.5);
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.5%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 62.9%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in M around 0 40.3%

   \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
6. Step-by-step derivation

   1. pow1 40.3%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right)\right)}^{1}} \]

   2. pow1/2 40.3%

      \[\leadsto {\left(\color{blue}{{\left(\frac{d}{h}\right)}^{0.5}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right)\right)}^{1} \]

   3. *-rgt-identity 40.3%

      \[\leadsto {\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right)}^{1} \]

   4. pow1/2 40.3%

      \[\leadsto {\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}}\right)}^{1} \]

   5. pow-prod-down 35.3%

      \[\leadsto {\color{blue}{\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}\right)}}^{1} \]

7. Applied egg-rr 35.3%

   \[\leadsto \color{blue}{{\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}\right)}^{1}} \]
8. Step-by-step derivation

   1. unpow1 35.3%

      \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}} \]

   2. unpow1/2 35.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

9. Simplified 35.3%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]
10. Step-by-step derivation

    1. *-un-lft-identity 35.3%

       \[\leadsto \color{blue}{1 \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

    2. frac-times 26.9%

       \[\leadsto 1 \cdot \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \]

    3. sqrt-div 30.2%

       \[\leadsto 1 \cdot \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \]

    4. sqrt-unprod 23.1%

       \[\leadsto 1 \cdot \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

    5. add-sqr-sqrt 27.0%

       \[\leadsto 1 \cdot \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

11. Applied egg-rr 27.0%

    \[\leadsto \color{blue}{1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]
12. Step-by-step derivation

    1. *-lft-identity 27.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

13. Simplified 27.0%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
14. Step-by-step derivation

    1. div-inv 27.0%

       \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

    2. pow1/2 27.0%

       \[\leadsto d \cdot \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \]

    3. pow-flip 27.1%

       \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-0.5\right)}} \]

    4. *-commutative 27.1%

       \[\leadsto d \cdot {\color{blue}{\left(\ell \cdot h\right)}}^{\left(-0.5\right)} \]

    5. metadata-eval 27.1%

       \[\leadsto d \cdot {\left(\ell \cdot h\right)}^{\color{blue}{-0.5}} \]

15. Applied egg-rr 27.1%

    \[\leadsto \color{blue}{d \cdot {\left(\ell \cdot h\right)}^{-0.5}} \]

16. Final simplification 27.1%

    \[\leadsto d \cdot {\left(\ell \cdot h\right)}^{-0.5} \]
17. Add Preprocessing

Alternative 19: 25.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D) :precision binary64 (/ d (sqrt (* l h))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	return d / sqrt((l * h));
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    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_1
    code = d / sqrt((l * h))
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	return d / Math.sqrt((l * h));
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	return d / math.sqrt((l * h))

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp = code(d, h, l, M_m, D)
	tmp = d / sqrt((l * h));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.5%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 62.9%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in M around 0 40.3%

   \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
6. Step-by-step derivation

   1. pow1 40.3%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right)\right)}^{1}} \]

   2. pow1/2 40.3%

      \[\leadsto {\left(\color{blue}{{\left(\frac{d}{h}\right)}^{0.5}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right)\right)}^{1} \]

   3. *-rgt-identity 40.3%

      \[\leadsto {\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right)}^{1} \]

   4. pow1/2 40.3%

      \[\leadsto {\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}}\right)}^{1} \]

   5. pow-prod-down 35.3%

      \[\leadsto {\color{blue}{\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}\right)}}^{1} \]

7. Applied egg-rr 35.3%

   \[\leadsto \color{blue}{{\left({\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}\right)}^{1}} \]
8. Step-by-step derivation

   1. unpow1 35.3%

      \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}} \]

   2. unpow1/2 35.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

9. Simplified 35.3%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]
10. Step-by-step derivation

    1. *-un-lft-identity 35.3%

       \[\leadsto \color{blue}{1 \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

    2. frac-times 26.9%

       \[\leadsto 1 \cdot \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \]

    3. sqrt-div 30.2%

       \[\leadsto 1 \cdot \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \]

    4. sqrt-unprod 23.1%

       \[\leadsto 1 \cdot \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

    5. add-sqr-sqrt 27.0%

       \[\leadsto 1 \cdot \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

11. Applied egg-rr 27.0%

    \[\leadsto \color{blue}{1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]
12. Step-by-step derivation

    1. *-lft-identity 27.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

13. Simplified 27.0%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

14. Final simplification 27.0%

    \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024080 
(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)))))