Henrywood and Agarwal, Equation (12)

Percentage Accurate: 67.5% → 84.9%

Time: 28.0s

Alternatives: 16

Speedup: 1.0×

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: 84.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := M_m \cdot \left(\frac{0.5}{d} \cdot D\right)\\ t_1 := \frac{t_0}{\ell}\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{t_2}{\sqrt{-h}} \cdot \left(\frac{t_2}{\sqrt{-\ell}} \cdot \left(1 + h \cdot \left(t_1 \cdot \frac{t_0}{-2}\right)\right)\right)\\ \mathbf{elif}\;h \leq 2.5 \cdot 10^{+195}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 + h \cdot \left(t_1 \cdot \frac{\frac{0.5}{\frac{d}{M_m \cdot D}}}{-2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M_m}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \end{array} \]

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 (* M_m (* (/ 0.5 d) D))) (t_1 (/ t_0 l)) (t_2 (sqrt (- d))))
   (if (<= h -5e-310)
     (*
      (/ t_2 (sqrt (- h)))
      (* (/ t_2 (sqrt (- l))) (+ 1.0 (* h (* t_1 (/ t_0 -2.0))))))
     (if (<= h 2.5e+195)
       (*
        (sqrt (/ d h))
        (*
         (/ (sqrt d) (sqrt l))
         (+ 1.0 (* h (* t_1 (/ (/ 0.5 (/ d (* M_m D))) -2.0))))))
       (*
        (/ (sqrt d) (sqrt h))
        (*
         (+ 1.0 (* (/ h l) (* (pow (* (/ M_m 2.0) (/ D d)) 2.0) -0.5)))
         (sqrt (/ d 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 = M_m * ((0.5 / d) * D);
	double t_1 = t_0 / l;
	double t_2 = sqrt(-d);
	double tmp;
	if (h <= -5e-310) {
		tmp = (t_2 / sqrt(-h)) * ((t_2 / sqrt(-l)) * (1.0 + (h * (t_1 * (t_0 / -2.0)))));
	} else if (h <= 2.5e+195) {
		tmp = sqrt((d / h)) * ((sqrt(d) / sqrt(l)) * (1.0 + (h * (t_1 * ((0.5 / (d / (M_m * D))) / -2.0)))));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * ((1.0 + ((h / l) * (pow(((M_m / 2.0) * (D / d)), 2.0) * -0.5))) * sqrt((d / 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) :: tmp
    t_0 = m_m * ((0.5d0 / d) * d_1)
    t_1 = t_0 / l
    t_2 = sqrt(-d)
    if (h <= (-5d-310)) then
        tmp = (t_2 / sqrt(-h)) * ((t_2 / sqrt(-l)) * (1.0d0 + (h * (t_1 * (t_0 / (-2.0d0))))))
    else if (h <= 2.5d+195) then
        tmp = sqrt((d / h)) * ((sqrt(d) / sqrt(l)) * (1.0d0 + (h * (t_1 * ((0.5d0 / (d / (m_m * d_1))) / (-2.0d0))))))
    else
        tmp = (sqrt(d) / sqrt(h)) * ((1.0d0 + ((h / l) * ((((m_m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (-0.5d0)))) * sqrt((d / 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 = M_m * ((0.5 / d) * D);
	double t_1 = t_0 / l;
	double t_2 = Math.sqrt(-d);
	double tmp;
	if (h <= -5e-310) {
		tmp = (t_2 / Math.sqrt(-h)) * ((t_2 / Math.sqrt(-l)) * (1.0 + (h * (t_1 * (t_0 / -2.0)))));
	} else if (h <= 2.5e+195) {
		tmp = Math.sqrt((d / h)) * ((Math.sqrt(d) / Math.sqrt(l)) * (1.0 + (h * (t_1 * ((0.5 / (d / (M_m * D))) / -2.0)))));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * ((1.0 + ((h / l) * (Math.pow(((M_m / 2.0) * (D / d)), 2.0) * -0.5))) * Math.sqrt((d / 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 = M_m * ((0.5 / d) * D)
	t_1 = t_0 / l
	t_2 = math.sqrt(-d)
	tmp = 0
	if h <= -5e-310:
		tmp = (t_2 / math.sqrt(-h)) * ((t_2 / math.sqrt(-l)) * (1.0 + (h * (t_1 * (t_0 / -2.0)))))
	elif h <= 2.5e+195:
		tmp = math.sqrt((d / h)) * ((math.sqrt(d) / math.sqrt(l)) * (1.0 + (h * (t_1 * ((0.5 / (d / (M_m * D))) / -2.0)))))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * ((1.0 + ((h / l) * (math.pow(((M_m / 2.0) * (D / d)), 2.0) * -0.5))) * math.sqrt((d / 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(M_m * Float64(Float64(0.5 / d) * D))
	t_1 = Float64(t_0 / l)
	t_2 = sqrt(Float64(-d))
	tmp = 0.0
	if (h <= -5e-310)
		tmp = Float64(Float64(t_2 / sqrt(Float64(-h))) * Float64(Float64(t_2 / sqrt(Float64(-l))) * Float64(1.0 + Float64(h * Float64(t_1 * Float64(t_0 / -2.0))))));
	elseif (h <= 2.5e+195)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(sqrt(d) / sqrt(l)) * Float64(1.0 + Float64(h * Float64(t_1 * Float64(Float64(0.5 / Float64(d / Float64(M_m * D))) / -2.0))))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0) * -0.5))) * sqrt(Float64(d / 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 = M_m * ((0.5 / d) * D);
	t_1 = t_0 / l;
	t_2 = sqrt(-d);
	tmp = 0.0;
	if (h <= -5e-310)
		tmp = (t_2 / sqrt(-h)) * ((t_2 / sqrt(-l)) * (1.0 + (h * (t_1 * (t_0 / -2.0)))));
	elseif (h <= 2.5e+195)
		tmp = sqrt((d / h)) * ((sqrt(d) / sqrt(l)) * (1.0 + (h * (t_1 * ((0.5 / (d / (M_m * D))) / -2.0)))));
	else
		tmp = (sqrt(d) / sqrt(h)) * ((1.0 + ((h / l) * ((((M_m / 2.0) * (D / d)) ^ 2.0) * -0.5))) * sqrt((d / 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[(M$95$m * N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / l), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[h, -5e-310], N[(N[(t$95$2 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$2 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(h * N[(t$95$1 * N[(t$95$0 / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 2.5e+195], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(h * N[(t$95$1 * N[(N[(0.5 / N[(d / N[(M$95$m * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 66.4%

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

   3. Simplified 64.3%

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

   6. Applied egg-rr 30.6%

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

      1. expm1-def 30.6%

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

      2. expm1-log1p 64.3%

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

      3. associate-*l/ 65.9%

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

      4. *-commutative 65.9%

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

      5. associate-*l/ 66.6%

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

      6. *-commutative 66.6%

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

      7. associate-/l* 66.6%

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

      8. metadata-eval 66.6%

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

      9. times-frac 66.6%

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

      10. *-commutative 66.6%

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

      11. associate-/l* 66.6%

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

      12. associate-*l/ 66.6%

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

      13. *-commutative 66.6%

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

      14. associate-/r* 66.6%

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

      15. metadata-eval 66.6%

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

   8. Simplified 66.6%

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

      1. unpow2 66.6%

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

      2. div-inv 66.6%

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

      3. metadata-eval 66.6%

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

      4. metadata-eval 66.6%

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

      5. times-frac 68.0%

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

      6. associate-/r/ 68.0%

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

      7. associate-/r/ 68.0%

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

      8. metadata-eval 68.0%

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

   10. Applied egg-rr 68.0%

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

       1. frac-2neg 68.0%

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

       2. sqrt-div 75.2%

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

   12. Applied egg-rr 75.2%

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

       1. frac-2neg 75.2%

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

       2. sqrt-div 89.9%

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

   14. Applied egg-rr 89.9%

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

   if -4.999999999999985e-310 < h < 2.4999999999999999e195

   1. Initial program 75.3%

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

   3. Simplified 75.3%

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

   6. Applied egg-rr 44.4%

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

      1. expm1-def 44.4%

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

      2. expm1-log1p 75.3%

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

      3. associate-*l/ 75.6%

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

      4. *-commutative 75.6%

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

      5. associate-*l/ 74.6%

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

      6. *-commutative 74.6%

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

      7. associate-/l* 74.6%

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

      8. metadata-eval 74.6%

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

      9. times-frac 74.6%

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

      10. *-commutative 74.6%

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

      11. associate-/l* 74.5%

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

      12. associate-*l/ 74.5%

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

      13. *-commutative 74.5%

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

      14. associate-/r* 74.5%

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

      15. metadata-eval 74.5%

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

   8. Simplified 74.5%

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

      1. unpow2 74.5%

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

      2. div-inv 74.5%

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

      3. metadata-eval 74.5%

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

      4. metadata-eval 74.5%

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

      5. times-frac 75.5%

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

      6. associate-/r/ 75.5%

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

      7. associate-/r/ 75.6%

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

      8. metadata-eval 75.6%

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

   10. Applied egg-rr 75.6%

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

       1. *-commutative 75.6%

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

       2. associate-*l/ 75.6%

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

       3. associate-/r/ 75.6%

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

       4. associate-/l* 75.6%

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

       5. associate-/l/ 75.6%

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

   12. Applied egg-rr 75.6%

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

       1. sqrt-div 88.0%

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

       2. div-inv 88.0%

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

   14. Applied egg-rr 88.0%

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

       1. associate-*r/ 88.0%

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

       2. *-rgt-identity 88.0%

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

   16. Simplified 88.0%

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

   if 2.4999999999999999e195 < h

   1. Initial program 46.1%

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

   3. Simplified 46.0%

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

      1. sqrt-div 83.4%

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

      2. div-inv 83.3%

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

   6. Applied egg-rr 83.3%

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

      1. associate-*r/ 83.4%

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

      2. *-rgt-identity 83.4%

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

   8. Simplified 83.4%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 + h \cdot \left(\frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{\ell} \cdot \frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{-2}\right)\right)\right)\\ \mathbf{elif}\;h \leq 2.5 \cdot 10^{+195}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 + h \cdot \left(\frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{\ell} \cdot \frac{\frac{0.5}{\frac{d}{M \cdot D}}}{-2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M_m \cdot D}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{-120} \lor \neg \left(t_0 \leq 0\right) \land t_0 \leq 5 \cdot 10^{+263}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(\frac{M_m \cdot \left(\frac{0.5}{d} \cdot D\right)}{\ell} \cdot \frac{0.5 \cdot \left(M_m \cdot \frac{D}{d}\right)}{-2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

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
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* M_m D) (* d 2.0)) 2.0)))))))
   (if (or (<= t_0 -4e-120) (and (not (<= t_0 0.0)) (<= t_0 5e+263)))
     (*
      (sqrt (/ d h))
      (*
       (sqrt (/ d l))
       (+
        1.0
        (*
         h
         (* (/ (* M_m (* (/ 0.5 d) D)) l) (/ (* 0.5 (* M_m (/ D d))) -2.0))))))
     (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) {
	double t_0 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((M_m * D) / (d * 2.0)), 2.0))));
	double tmp;
	if ((t_0 <= -4e-120) || (!(t_0 <= 0.0) && (t_0 <= 5e+263))) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + (h * (((M_m * ((0.5 / d) * D)) / l) * ((0.5 * (M_m * (D / d))) / -2.0)))));
	} else {
		tmp = fabs((d / sqrt((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) :: t_0
    real(8) :: tmp
    t_0 = (((d / h) ** 0.5d0) * ((d / l) ** 0.5d0)) * (1.0d0 - ((h / l) * (0.5d0 * (((m_m * d_1) / (d * 2.0d0)) ** 2.0d0))))
    if ((t_0 <= (-4d-120)) .or. (.not. (t_0 <= 0.0d0)) .and. (t_0 <= 5d+263)) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 + (h * (((m_m * ((0.5d0 / d) * d_1)) / l) * ((0.5d0 * (m_m * (d_1 / d))) / (-2.0d0))))))
    else
        tmp = abs((d / sqrt((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 t_0 = (Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * Math.pow(((M_m * D) / (d * 2.0)), 2.0))));
	double tmp;
	if ((t_0 <= -4e-120) || (!(t_0 <= 0.0) && (t_0 <= 5e+263))) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 + (h * (((M_m * ((0.5 / d) * D)) / l) * ((0.5 * (M_m * (D / d))) / -2.0)))));
	} else {
		tmp = Math.abs((d / Math.sqrt((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):
	t_0 = (math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * math.pow(((M_m * D) / (d * 2.0)), 2.0))))
	tmp = 0
	if (t_0 <= -4e-120) or (not (t_0 <= 0.0) and (t_0 <= 5e+263)):
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 + (h * (((M_m * ((0.5 / d) * D)) / l) * ((0.5 * (M_m * (D / d))) / -2.0)))))
	else:
		tmp = math.fabs((d / math.sqrt((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)
	t_0 = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(M_m * D) / Float64(d * 2.0)) ^ 2.0)))))
	tmp = 0.0
	if ((t_0 <= -4e-120) || (!(t_0 <= 0.0) && (t_0 <= 5e+263)))
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(h * Float64(Float64(Float64(M_m * Float64(Float64(0.5 / d) * D)) / l) * Float64(Float64(0.5 * Float64(M_m * Float64(D / d))) / -2.0))))));
	else
		tmp = abs(Float64(d / sqrt(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)
	t_0 = (((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((h / l) * (0.5 * (((M_m * D) / (d * 2.0)) ^ 2.0))));
	tmp = 0.0;
	if ((t_0 <= -4e-120) || (~((t_0 <= 0.0)) && (t_0 <= 5e+263)))
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + (h * (((M_m * ((0.5 / d) * D)) / l) * ((0.5 * (M_m * (D / d))) / -2.0)))));
	else
		tmp = abs((d / sqrt((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_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -4e-120], And[N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision], LessEqual[t$95$0, 5e+263]]], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h * N[(N[(N[(M$95$m * N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(0.5 * N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < -3.99999999999999991e-120 or 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 5.00000000000000022e263

   1. Initial program 93.8%

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

   3. Simplified 91.9%

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

   6. Applied egg-rr 41.4%

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

      1. expm1-def 41.4%

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

      2. expm1-log1p 91.9%

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

      3. associate-*l/ 92.0%

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

      4. *-commutative 92.0%

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

      5. associate-*l/ 92.0%

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

      6. *-commutative 92.0%

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

      7. associate-/l* 92.0%

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

      8. metadata-eval 92.0%

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

      9. times-frac 92.0%

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

      10. *-commutative 92.0%

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

      11. associate-/l* 91.9%

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

      12. associate-*l/ 91.9%

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

      13. *-commutative 91.9%

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

      14. associate-/r* 91.9%

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

      15. metadata-eval 91.9%

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

   8. Simplified 91.9%

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

      1. unpow2 91.9%

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

      2. div-inv 91.9%

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

      3. metadata-eval 91.9%

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

      4. metadata-eval 91.9%

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

      5. times-frac 93.1%

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

      6. associate-/r/ 93.1%

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

      7. associate-/r/ 93.2%

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

      8. metadata-eval 93.2%

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

   10. Applied egg-rr 93.2%

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

   11. Taylor expanded in M around 0 92.6%

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

       1. *-commutative 92.6%

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

       2. associate-*r/ 93.2%

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

   13. Simplified 93.2%

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

   if -3.99999999999999991e-120 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 0.0 or 5.00000000000000022e263 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))

   1. Initial program 23.4%

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

   3. Simplified 23.4%

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

   5. Taylor expanded in h around 0 26.5%

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

      1. *-rgt-identity 26.5%

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

      2. sqrt-unprod 25.6%

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

   7. Applied egg-rr 25.6%

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

      1. frac-times 30.7%

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

      2. unpow2 30.7%

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

      3. associate-/l/ 27.6%

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

      4. add-sqr-sqrt 27.6%

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

      5. rem-sqrt-square 27.6%

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

      6. associate-/l/ 30.7%

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

      7. sqrt-div 36.5%

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

      8. unpow2 36.5%

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

      9. sqrt-prod 24.5%

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

      10. add-sqr-sqrt 56.7%

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

      11. *-commutative 56.7%

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

   9. Applied egg-rr 56.7%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq -4 \cdot 10^{-120} \lor \neg \left(\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq 0\right) \land \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq 5 \cdot 10^{+263}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(\frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{\ell} \cdot \frac{0.5 \cdot \left(M \cdot \frac{D}{d}\right)}{-2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.9% accurate, 0.6× speedup

Math

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

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 (* M_m (* (/ 0.5 d) D))) (t_1 (sqrt (- d))))
   (if (<= l -1e-310)
     (*
      (/ t_1 (sqrt (- h)))
      (* (/ t_1 (sqrt (- l))) (+ 1.0 (* h (* (/ t_0 l) (/ t_0 -2.0))))))
     (*
      (/ 1.0 (/ (sqrt h) (sqrt d)))
      (*
       (/ (sqrt d) (sqrt l))
       (+ 1.0 (* (/ h l) (* (pow (* (/ M_m 2.0) (/ D d)) 2.0) -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 t_0 = M_m * ((0.5 / d) * D);
	double t_1 = sqrt(-d);
	double tmp;
	if (l <= -1e-310) {
		tmp = (t_1 / sqrt(-h)) * ((t_1 / sqrt(-l)) * (1.0 + (h * ((t_0 / l) * (t_0 / -2.0)))));
	} else {
		tmp = (1.0 / (sqrt(h) / sqrt(d))) * ((sqrt(d) / sqrt(l)) * (1.0 + ((h / l) * (pow(((M_m / 2.0) * (D / d)), 2.0) * -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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = m_m * ((0.5d0 / d) * d_1)
    t_1 = sqrt(-d)
    if (l <= (-1d-310)) then
        tmp = (t_1 / sqrt(-h)) * ((t_1 / sqrt(-l)) * (1.0d0 + (h * ((t_0 / l) * (t_0 / (-2.0d0))))))
    else
        tmp = (1.0d0 / (sqrt(h) / sqrt(d))) * ((sqrt(d) / sqrt(l)) * (1.0d0 + ((h / l) * ((((m_m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (-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 t_0 = M_m * ((0.5 / d) * D);
	double t_1 = Math.sqrt(-d);
	double tmp;
	if (l <= -1e-310) {
		tmp = (t_1 / Math.sqrt(-h)) * ((t_1 / Math.sqrt(-l)) * (1.0 + (h * ((t_0 / l) * (t_0 / -2.0)))));
	} else {
		tmp = (1.0 / (Math.sqrt(h) / Math.sqrt(d))) * ((Math.sqrt(d) / Math.sqrt(l)) * (1.0 + ((h / l) * (Math.pow(((M_m / 2.0) * (D / d)), 2.0) * -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):
	t_0 = M_m * ((0.5 / d) * D)
	t_1 = math.sqrt(-d)
	tmp = 0
	if l <= -1e-310:
		tmp = (t_1 / math.sqrt(-h)) * ((t_1 / math.sqrt(-l)) * (1.0 + (h * ((t_0 / l) * (t_0 / -2.0)))))
	else:
		tmp = (1.0 / (math.sqrt(h) / math.sqrt(d))) * ((math.sqrt(d) / math.sqrt(l)) * (1.0 + ((h / l) * (math.pow(((M_m / 2.0) * (D / d)), 2.0) * -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)
	t_0 = Float64(M_m * Float64(Float64(0.5 / d) * D))
	t_1 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -1e-310)
		tmp = Float64(Float64(t_1 / sqrt(Float64(-h))) * Float64(Float64(t_1 / sqrt(Float64(-l))) * Float64(1.0 + Float64(h * Float64(Float64(t_0 / l) * Float64(t_0 / -2.0))))));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(h) / sqrt(d))) * Float64(Float64(sqrt(d) / sqrt(l)) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0) * -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)
	t_0 = M_m * ((0.5 / d) * D);
	t_1 = sqrt(-d);
	tmp = 0.0;
	if (l <= -1e-310)
		tmp = (t_1 / sqrt(-h)) * ((t_1 / sqrt(-l)) * (1.0 + (h * ((t_0 / l) * (t_0 / -2.0)))));
	else
		tmp = (1.0 / (sqrt(h) / sqrt(d))) * ((sqrt(d) / sqrt(l)) * (1.0 + ((h / l) * ((((M_m / 2.0) * (D / d)) ^ 2.0) * -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_] := Block[{t$95$0 = N[(M$95$m * N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -1e-310], N[(N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(h * N[(N[(t$95$0 / l), $MachinePrecision] * N[(t$95$0 / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[h], $MachinePrecision] / N[Sqrt[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if l < -9.999999999999969e-311

   1. Initial program 66.4%

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

   3. Simplified 64.3%

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

   6. Applied egg-rr 30.6%

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

      1. expm1-def 30.6%

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

      2. expm1-log1p 64.3%

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

      3. associate-*l/ 65.9%

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

      4. *-commutative 65.9%

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

      5. associate-*l/ 66.6%

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

      6. *-commutative 66.6%

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

      7. associate-/l* 66.6%

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

      8. metadata-eval 66.6%

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

      9. times-frac 66.6%

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

      10. *-commutative 66.6%

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

      11. associate-/l* 66.6%

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

      12. associate-*l/ 66.6%

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

      13. *-commutative 66.6%

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

      14. associate-/r* 66.6%

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

      15. metadata-eval 66.6%

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

   8. Simplified 66.6%

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

      1. unpow2 66.6%

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

      2. div-inv 66.6%

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

      3. metadata-eval 66.6%

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

      4. metadata-eval 66.6%

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

      5. times-frac 68.0%

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

      6. associate-/r/ 68.0%

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

      7. associate-/r/ 68.0%

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

      8. metadata-eval 68.0%

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

   10. Applied egg-rr 68.0%

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

       1. frac-2neg 68.0%

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

       2. sqrt-div 75.2%

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

   12. Applied egg-rr 75.2%

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

       1. frac-2neg 75.2%

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

       2. sqrt-div 89.9%

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

   14. Applied egg-rr 89.9%

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

   if -9.999999999999969e-311 < l

   1. Initial program 69.2%

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

   3. Simplified 69.2%

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

      1. sqrt-div 80.2%

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

      2. clear-num 80.3%

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

   6. Applied egg-rr 80.3%

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

      1. sqrt-div 80.1%

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

      2. div-inv 80.1%

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

   8. Applied egg-rr 87.0%

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

      1. associate-*r/ 80.1%

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

      2. *-rgt-identity 80.1%

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

   10. Simplified 87.0%

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

4. Final simplification 88.6%

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

Alternative 4: 80.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := M_m \cdot \left(\frac{0.5}{d} \cdot D\right)\\ t_1 := \frac{t_0}{\ell}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\left(1 + h \cdot \left(t_1 \cdot \frac{t_0}{-2}\right)\right) \cdot t_2\right)\\ \mathbf{elif}\;h \leq 6.5 \cdot 10^{+193}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 + h \cdot \left(t_1 \cdot \frac{\frac{0.5}{\frac{d}{M_m \cdot D}}}{-2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M_m}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot t_2\right)\\ \end{array} \end{array} \]

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 (* M_m (* (/ 0.5 d) D))) (t_1 (/ t_0 l)) (t_2 (sqrt (/ d l))))
   (if (<= h -5e-310)
     (*
      (/ (sqrt (- d)) (sqrt (- h)))
      (* (+ 1.0 (* h (* t_1 (/ t_0 -2.0)))) t_2))
     (if (<= h 6.5e+193)
       (*
        (sqrt (/ d h))
        (*
         (/ (sqrt d) (sqrt l))
         (+ 1.0 (* h (* t_1 (/ (/ 0.5 (/ d (* M_m D))) -2.0))))))
       (*
        (/ (sqrt d) (sqrt h))
        (*
         (+ 1.0 (* (/ h l) (* (pow (* (/ M_m 2.0) (/ D d)) 2.0) -0.5)))
         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 = M_m * ((0.5 / d) * D);
	double t_1 = t_0 / l;
	double t_2 = sqrt((d / l));
	double tmp;
	if (h <= -5e-310) {
		tmp = (sqrt(-d) / sqrt(-h)) * ((1.0 + (h * (t_1 * (t_0 / -2.0)))) * t_2);
	} else if (h <= 6.5e+193) {
		tmp = sqrt((d / h)) * ((sqrt(d) / sqrt(l)) * (1.0 + (h * (t_1 * ((0.5 / (d / (M_m * D))) / -2.0)))));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * ((1.0 + ((h / l) * (pow(((M_m / 2.0) * (D / d)), 2.0) * -0.5))) * 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 = m_m * ((0.5d0 / d) * d_1)
    t_1 = t_0 / l
    t_2 = sqrt((d / l))
    if (h <= (-5d-310)) then
        tmp = (sqrt(-d) / sqrt(-h)) * ((1.0d0 + (h * (t_1 * (t_0 / (-2.0d0))))) * t_2)
    else if (h <= 6.5d+193) then
        tmp = sqrt((d / h)) * ((sqrt(d) / sqrt(l)) * (1.0d0 + (h * (t_1 * ((0.5d0 / (d / (m_m * d_1))) / (-2.0d0))))))
    else
        tmp = (sqrt(d) / sqrt(h)) * ((1.0d0 + ((h / l) * ((((m_m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (-0.5d0)))) * 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 = M_m * ((0.5 / d) * D);
	double t_1 = t_0 / l;
	double t_2 = Math.sqrt((d / l));
	double tmp;
	if (h <= -5e-310) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * ((1.0 + (h * (t_1 * (t_0 / -2.0)))) * t_2);
	} else if (h <= 6.5e+193) {
		tmp = Math.sqrt((d / h)) * ((Math.sqrt(d) / Math.sqrt(l)) * (1.0 + (h * (t_1 * ((0.5 / (d / (M_m * D))) / -2.0)))));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * ((1.0 + ((h / l) * (Math.pow(((M_m / 2.0) * (D / d)), 2.0) * -0.5))) * 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 = M_m * ((0.5 / d) * D)
	t_1 = t_0 / l
	t_2 = math.sqrt((d / l))
	tmp = 0
	if h <= -5e-310:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * ((1.0 + (h * (t_1 * (t_0 / -2.0)))) * t_2)
	elif h <= 6.5e+193:
		tmp = math.sqrt((d / h)) * ((math.sqrt(d) / math.sqrt(l)) * (1.0 + (h * (t_1 * ((0.5 / (d / (M_m * D))) / -2.0)))))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * ((1.0 + ((h / l) * (math.pow(((M_m / 2.0) * (D / d)), 2.0) * -0.5))) * 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 = Float64(M_m * Float64(Float64(0.5 / d) * D))
	t_1 = Float64(t_0 / l)
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (h <= -5e-310)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(Float64(1.0 + Float64(h * Float64(t_1 * Float64(t_0 / -2.0)))) * t_2));
	elseif (h <= 6.5e+193)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(sqrt(d) / sqrt(l)) * Float64(1.0 + Float64(h * Float64(t_1 * Float64(Float64(0.5 / Float64(d / Float64(M_m * D))) / -2.0))))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0) * -0.5))) * 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 = M_m * ((0.5 / d) * D);
	t_1 = t_0 / l;
	t_2 = sqrt((d / l));
	tmp = 0.0;
	if (h <= -5e-310)
		tmp = (sqrt(-d) / sqrt(-h)) * ((1.0 + (h * (t_1 * (t_0 / -2.0)))) * t_2);
	elseif (h <= 6.5e+193)
		tmp = sqrt((d / h)) * ((sqrt(d) / sqrt(l)) * (1.0 + (h * (t_1 * ((0.5 / (d / (M_m * D))) / -2.0)))));
	else
		tmp = (sqrt(d) / sqrt(h)) * ((1.0 + ((h / l) * ((((M_m / 2.0) * (D / d)) ^ 2.0) * -0.5))) * 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[(M$95$m * N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / l), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -5e-310], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(h * N[(t$95$1 * N[(t$95$0 / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 6.5e+193], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(h * N[(t$95$1 * N[(N[(0.5 / N[(d / N[(M$95$m * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 66.4%

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

   3. Simplified 64.3%

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

   6. Applied egg-rr 30.6%

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

      1. expm1-def 30.6%

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

      2. expm1-log1p 64.3%

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

      3. associate-*l/ 65.9%

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

      4. *-commutative 65.9%

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

      5. associate-*l/ 66.6%

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

      6. *-commutative 66.6%

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

      7. associate-/l* 66.6%

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

      8. metadata-eval 66.6%

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

      9. times-frac 66.6%

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

      10. *-commutative 66.6%

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

      11. associate-/l* 66.6%

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

      12. associate-*l/ 66.6%

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

      13. *-commutative 66.6%

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

      14. associate-/r* 66.6%

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

      15. metadata-eval 66.6%

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

   8. Simplified 66.6%

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

      1. unpow2 66.6%

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

      2. div-inv 66.6%

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

      3. metadata-eval 66.6%

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

      4. metadata-eval 66.6%

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

      5. times-frac 68.0%

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

      6. associate-/r/ 68.0%

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

      7. associate-/r/ 68.0%

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

      8. metadata-eval 68.0%

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

   10. Applied egg-rr 68.0%

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

       1. frac-2neg 75.2%

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

       2. sqrt-div 89.9%

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

   12. Applied egg-rr 80.5%

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

   if -4.999999999999985e-310 < h < 6.4999999999999997e193

   1. Initial program 75.3%

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

   3. Simplified 75.3%

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

   6. Applied egg-rr 44.4%

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

      1. expm1-def 44.4%

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

      2. expm1-log1p 75.3%

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

      3. associate-*l/ 75.6%

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

      4. *-commutative 75.6%

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

      5. associate-*l/ 74.6%

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

      6. *-commutative 74.6%

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

      7. associate-/l* 74.6%

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

      8. metadata-eval 74.6%

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

      9. times-frac 74.6%

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

      10. *-commutative 74.6%

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

      11. associate-/l* 74.5%

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

      12. associate-*l/ 74.5%

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

      13. *-commutative 74.5%

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

      14. associate-/r* 74.5%

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

      15. metadata-eval 74.5%

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

   8. Simplified 74.5%

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

      1. unpow2 74.5%

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

      2. div-inv 74.5%

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

      3. metadata-eval 74.5%

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

      4. metadata-eval 74.5%

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

      5. times-frac 75.5%

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

      6. associate-/r/ 75.5%

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

      7. associate-/r/ 75.6%

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

      8. metadata-eval 75.6%

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

   10. Applied egg-rr 75.6%

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

       1. *-commutative 75.6%

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

       2. associate-*l/ 75.6%

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

       3. associate-/r/ 75.6%

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

       4. associate-/l* 75.6%

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

       5. associate-/l/ 75.6%

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

   12. Applied egg-rr 75.6%

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

       1. sqrt-div 88.0%

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

       2. div-inv 88.0%

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

   14. Applied egg-rr 88.0%

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

       1. associate-*r/ 88.0%

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

       2. *-rgt-identity 88.0%

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

   16. Simplified 88.0%

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

   if 6.4999999999999997e193 < h

   1. Initial program 46.1%

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

   3. Simplified 46.0%

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

      1. sqrt-div 83.4%

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

      2. div-inv 83.3%

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

   6. Applied egg-rr 83.3%

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

      1. associate-*r/ 83.4%

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

      2. *-rgt-identity 83.4%

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

   8. Simplified 83.4%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\left(1 + h \cdot \left(\frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{\ell} \cdot \frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{-2}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;h \leq 6.5 \cdot 10^{+193}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 + h \cdot \left(\frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{\ell} \cdot \frac{\frac{0.5}{\frac{d}{M \cdot D}}}{-2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := M_m \cdot \left(\frac{0.5}{d} \cdot D\right)\\ t_1 := \frac{t_0}{\ell}\\ t_2 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 + h \cdot \left(t_1 \cdot \frac{t_0}{-2}\right)\right)\right) \cdot t_2\\ \mathbf{elif}\;h \leq 1.7 \cdot 10^{+188}:\\ \;\;\;\;t_2 \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 + h \cdot \left(t_1 \cdot \frac{\frac{0.5}{\frac{d}{M_m \cdot D}}}{-2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(t_1 \cdot \frac{0.5 \cdot \left(M_m \cdot \frac{D}{d}\right)}{-2}\right)\right)\right)\\ \end{array} \end{array} \]

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 (* M_m (* (/ 0.5 d) D))) (t_1 (/ t_0 l)) (t_2 (sqrt (/ d h))))
   (if (<= h -5e-310)
     (*
      (* (/ (sqrt (- d)) (sqrt (- l))) (+ 1.0 (* h (* t_1 (/ t_0 -2.0)))))
      t_2)
     (if (<= h 1.7e+188)
       (*
        t_2
        (*
         (/ (sqrt d) (sqrt l))
         (+ 1.0 (* h (* t_1 (/ (/ 0.5 (/ d (* M_m D))) -2.0))))))
       (*
        (/ (sqrt d) (sqrt h))
        (*
         (sqrt (/ d l))
         (+ 1.0 (* h (* t_1 (/ (* 0.5 (* M_m (/ D d))) -2.0))))))))))

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 = M_m * ((0.5 / d) * D);
	double t_1 = t_0 / l;
	double t_2 = sqrt((d / h));
	double tmp;
	if (h <= -5e-310) {
		tmp = ((sqrt(-d) / sqrt(-l)) * (1.0 + (h * (t_1 * (t_0 / -2.0))))) * t_2;
	} else if (h <= 1.7e+188) {
		tmp = t_2 * ((sqrt(d) / sqrt(l)) * (1.0 + (h * (t_1 * ((0.5 / (d / (M_m * D))) / -2.0)))));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * (1.0 + (h * (t_1 * ((0.5 * (M_m * (D / d))) / -2.0)))));
	}
	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 = m_m * ((0.5d0 / d) * d_1)
    t_1 = t_0 / l
    t_2 = sqrt((d / h))
    if (h <= (-5d-310)) then
        tmp = ((sqrt(-d) / sqrt(-l)) * (1.0d0 + (h * (t_1 * (t_0 / (-2.0d0)))))) * t_2
    else if (h <= 1.7d+188) then
        tmp = t_2 * ((sqrt(d) / sqrt(l)) * (1.0d0 + (h * (t_1 * ((0.5d0 / (d / (m_m * d_1))) / (-2.0d0))))))
    else
        tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * (1.0d0 + (h * (t_1 * ((0.5d0 * (m_m * (d_1 / d))) / (-2.0d0))))))
    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 = M_m * ((0.5 / d) * D);
	double t_1 = t_0 / l;
	double t_2 = Math.sqrt((d / h));
	double tmp;
	if (h <= -5e-310) {
		tmp = ((Math.sqrt(-d) / Math.sqrt(-l)) * (1.0 + (h * (t_1 * (t_0 / -2.0))))) * t_2;
	} else if (h <= 1.7e+188) {
		tmp = t_2 * ((Math.sqrt(d) / Math.sqrt(l)) * (1.0 + (h * (t_1 * ((0.5 / (d / (M_m * D))) / -2.0)))));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * (Math.sqrt((d / l)) * (1.0 + (h * (t_1 * ((0.5 * (M_m * (D / d))) / -2.0)))));
	}
	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 = M_m * ((0.5 / d) * D)
	t_1 = t_0 / l
	t_2 = math.sqrt((d / h))
	tmp = 0
	if h <= -5e-310:
		tmp = ((math.sqrt(-d) / math.sqrt(-l)) * (1.0 + (h * (t_1 * (t_0 / -2.0))))) * t_2
	elif h <= 1.7e+188:
		tmp = t_2 * ((math.sqrt(d) / math.sqrt(l)) * (1.0 + (h * (t_1 * ((0.5 / (d / (M_m * D))) / -2.0)))))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * (math.sqrt((d / l)) * (1.0 + (h * (t_1 * ((0.5 * (M_m * (D / d))) / -2.0)))))
	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(M_m * Float64(Float64(0.5 / d) * D))
	t_1 = Float64(t_0 / l)
	t_2 = sqrt(Float64(d / h))
	tmp = 0.0
	if (h <= -5e-310)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * Float64(1.0 + Float64(h * Float64(t_1 * Float64(t_0 / -2.0))))) * t_2);
	elseif (h <= 1.7e+188)
		tmp = Float64(t_2 * Float64(Float64(sqrt(d) / sqrt(l)) * Float64(1.0 + Float64(h * Float64(t_1 * Float64(Float64(0.5 / Float64(d / Float64(M_m * D))) / -2.0))))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(h * Float64(t_1 * Float64(Float64(0.5 * Float64(M_m * Float64(D / d))) / -2.0))))));
	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 = M_m * ((0.5 / d) * D);
	t_1 = t_0 / l;
	t_2 = sqrt((d / h));
	tmp = 0.0;
	if (h <= -5e-310)
		tmp = ((sqrt(-d) / sqrt(-l)) * (1.0 + (h * (t_1 * (t_0 / -2.0))))) * t_2;
	elseif (h <= 1.7e+188)
		tmp = t_2 * ((sqrt(d) / sqrt(l)) * (1.0 + (h * (t_1 * ((0.5 / (d / (M_m * D))) / -2.0)))));
	else
		tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * (1.0 + (h * (t_1 * ((0.5 * (M_m * (D / d))) / -2.0)))));
	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[(M$95$m * N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / l), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -5e-310], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(h * N[(t$95$1 * N[(t$95$0 / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[h, 1.7e+188], N[(t$95$2 * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(h * N[(t$95$1 * N[(N[(0.5 / N[(d / N[(M$95$m * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h * N[(t$95$1 * N[(N[(0.5 * N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 66.4%

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

   3. Simplified 64.3%

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

   6. Applied egg-rr 30.6%

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

      1. expm1-def 30.6%

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

      2. expm1-log1p 64.3%

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

      3. associate-*l/ 65.9%

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

      4. *-commutative 65.9%

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

      5. associate-*l/ 66.6%

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

      6. *-commutative 66.6%

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

      7. associate-/l* 66.6%

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

      8. metadata-eval 66.6%

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

      9. times-frac 66.6%

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

      10. *-commutative 66.6%

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

      11. associate-/l* 66.6%

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

      12. associate-*l/ 66.6%

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

      13. *-commutative 66.6%

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

      14. associate-/r* 66.6%

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

      15. metadata-eval 66.6%

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

   8. Simplified 66.6%

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

      1. unpow2 66.6%

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

      2. div-inv 66.6%

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

      3. metadata-eval 66.6%

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

      4. metadata-eval 66.6%

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

      5. times-frac 68.0%

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

      6. associate-/r/ 68.0%

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

      7. associate-/r/ 68.0%

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

      8. metadata-eval 68.0%

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

   10. Applied egg-rr 68.0%

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

       1. frac-2neg 68.0%

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

       2. sqrt-div 75.2%

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

   12. Applied egg-rr 75.2%

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

   if -4.999999999999985e-310 < h < 1.69999999999999998e188

   1. Initial program 75.3%

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

   3. Simplified 75.3%

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

   6. Applied egg-rr 44.4%

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

      1. expm1-def 44.4%

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

      2. expm1-log1p 75.3%

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

      3. associate-*l/ 75.6%

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

      4. *-commutative 75.6%

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

      5. associate-*l/ 74.6%

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

      6. *-commutative 74.6%

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

      7. associate-/l* 74.6%

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

      8. metadata-eval 74.6%

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

      9. times-frac 74.6%

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

      10. *-commutative 74.6%

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

      11. associate-/l* 74.5%

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

      12. associate-*l/ 74.5%

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

      13. *-commutative 74.5%

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

      14. associate-/r* 74.5%

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

      15. metadata-eval 74.5%

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

   8. Simplified 74.5%

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

      1. unpow2 74.5%

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

      2. div-inv 74.5%

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

      3. metadata-eval 74.5%

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

      4. metadata-eval 74.5%

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

      5. times-frac 75.5%

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

      6. associate-/r/ 75.5%

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

      7. associate-/r/ 75.6%

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

      8. metadata-eval 75.6%

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

   10. Applied egg-rr 75.6%

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

       1. *-commutative 75.6%

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

       2. associate-*l/ 75.6%

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

       3. associate-/r/ 75.6%

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

       4. associate-/l* 75.6%

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

       5. associate-/l/ 75.6%

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

   12. Applied egg-rr 75.6%

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

       1. sqrt-div 88.0%

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

       2. div-inv 88.0%

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

   14. Applied egg-rr 88.0%

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

       1. associate-*r/ 88.0%

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

       2. *-rgt-identity 88.0%

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

   16. Simplified 88.0%

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

   if 1.69999999999999998e188 < h

   1. Initial program 46.1%

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

   3. Simplified 46.0%

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

   6. Applied egg-rr 20.9%

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

      1. expm1-def 20.9%

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

      2. expm1-log1p 46.0%

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

      3. associate-*l/ 46.0%

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

      4. *-commutative 46.0%

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

      5. associate-*l/ 46.0%

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

      6. *-commutative 46.0%

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

      7. associate-/l* 46.0%

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

      8. metadata-eval 46.0%

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

      9. times-frac 46.0%

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

      10. *-commutative 46.0%

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

      11. associate-/l* 46.0%

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

      12. associate-*l/ 46.0%

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

      13. *-commutative 46.0%

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

      14. associate-/r* 46.0%

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

      15. metadata-eval 46.0%

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

   8. Simplified 46.0%

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

      1. unpow2 46.0%

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

      2. div-inv 46.0%

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

      3. metadata-eval 46.0%

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

      4. metadata-eval 46.0%

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

      5. times-frac 46.0%

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

      6. associate-/r/ 46.0%

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

      7. associate-/r/ 46.0%

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

      8. metadata-eval 46.0%

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

   10. Applied egg-rr 46.0%

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

   11. Taylor expanded in M around 0 46.0%

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

       1. *-commutative 46.0%

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

       2. associate-*r/ 46.0%

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

   13. Simplified 46.0%

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

       1. sqrt-div 83.4%

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

       2. div-inv 83.3%

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

   15. Applied egg-rr 83.2%

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

       1. associate-*r/ 83.4%

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

       2. *-rgt-identity 83.4%

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

   17. Simplified 83.2%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 + h \cdot \left(\frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{\ell} \cdot \frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{-2}\right)\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;h \leq 1.7 \cdot 10^{+188}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 + h \cdot \left(\frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{\ell} \cdot \frac{\frac{0.5}{\frac{d}{M \cdot D}}}{-2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(\frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{\ell} \cdot \frac{0.5 \cdot \left(M \cdot \frac{D}{d}\right)}{-2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \frac{M_m \cdot \left(\frac{0.5}{d} \cdot D\right)}{\ell}\\ t_1 := \sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(t_0 \cdot \frac{0.5 \cdot \left(M_m \cdot \frac{D}{d}\right)}{-2}\right)\right)\\ \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t_1\\ \mathbf{elif}\;h \leq 1.35 \cdot 10^{+188}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 + h \cdot \left(t_0 \cdot \frac{\frac{0.5}{\frac{d}{M_m \cdot D}}}{-2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot t_1\\ \end{array} \end{array} \]

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 (/ (* M_m (* (/ 0.5 d) D)) l))
        (t_1
         (*
          (sqrt (/ d l))
          (+ 1.0 (* h (* t_0 (/ (* 0.5 (* M_m (/ D d))) -2.0)))))))
   (if (<= h -5e-310)
     (* (/ (sqrt (- d)) (sqrt (- h))) t_1)
     (if (<= h 1.35e+188)
       (*
        (sqrt (/ d h))
        (*
         (/ (sqrt d) (sqrt l))
         (+ 1.0 (* h (* t_0 (/ (/ 0.5 (/ d (* M_m D))) -2.0))))))
       (* (/ (sqrt d) (sqrt 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 = (M_m * ((0.5 / d) * D)) / l;
	double t_1 = sqrt((d / l)) * (1.0 + (h * (t_0 * ((0.5 * (M_m * (D / d))) / -2.0))));
	double tmp;
	if (h <= -5e-310) {
		tmp = (sqrt(-d) / sqrt(-h)) * t_1;
	} else if (h <= 1.35e+188) {
		tmp = sqrt((d / h)) * ((sqrt(d) / sqrt(l)) * (1.0 + (h * (t_0 * ((0.5 / (d / (M_m * D))) / -2.0)))));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * 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 = (m_m * ((0.5d0 / d) * d_1)) / l
    t_1 = sqrt((d / l)) * (1.0d0 + (h * (t_0 * ((0.5d0 * (m_m * (d_1 / d))) / (-2.0d0)))))
    if (h <= (-5d-310)) then
        tmp = (sqrt(-d) / sqrt(-h)) * t_1
    else if (h <= 1.35d+188) then
        tmp = sqrt((d / h)) * ((sqrt(d) / sqrt(l)) * (1.0d0 + (h * (t_0 * ((0.5d0 / (d / (m_m * d_1))) / (-2.0d0))))))
    else
        tmp = (sqrt(d) / sqrt(h)) * 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 = (M_m * ((0.5 / d) * D)) / l;
	double t_1 = Math.sqrt((d / l)) * (1.0 + (h * (t_0 * ((0.5 * (M_m * (D / d))) / -2.0))));
	double tmp;
	if (h <= -5e-310) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * t_1;
	} else if (h <= 1.35e+188) {
		tmp = Math.sqrt((d / h)) * ((Math.sqrt(d) / Math.sqrt(l)) * (1.0 + (h * (t_0 * ((0.5 / (d / (M_m * D))) / -2.0)))));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * 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 = (M_m * ((0.5 / d) * D)) / l
	t_1 = math.sqrt((d / l)) * (1.0 + (h * (t_0 * ((0.5 * (M_m * (D / d))) / -2.0))))
	tmp = 0
	if h <= -5e-310:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * t_1
	elif h <= 1.35e+188:
		tmp = math.sqrt((d / h)) * ((math.sqrt(d) / math.sqrt(l)) * (1.0 + (h * (t_0 * ((0.5 / (d / (M_m * D))) / -2.0)))))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * 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(M_m * Float64(Float64(0.5 / d) * D)) / l)
	t_1 = Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(h * Float64(t_0 * Float64(Float64(0.5 * Float64(M_m * Float64(D / d))) / -2.0)))))
	tmp = 0.0
	if (h <= -5e-310)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_1);
	elseif (h <= 1.35e+188)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(sqrt(d) / sqrt(l)) * Float64(1.0 + Float64(h * Float64(t_0 * Float64(Float64(0.5 / Float64(d / Float64(M_m * D))) / -2.0))))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * 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 = (M_m * ((0.5 / d) * D)) / l;
	t_1 = sqrt((d / l)) * (1.0 + (h * (t_0 * ((0.5 * (M_m * (D / d))) / -2.0))));
	tmp = 0.0;
	if (h <= -5e-310)
		tmp = (sqrt(-d) / sqrt(-h)) * t_1;
	elseif (h <= 1.35e+188)
		tmp = sqrt((d / h)) * ((sqrt(d) / sqrt(l)) * (1.0 + (h * (t_0 * ((0.5 / (d / (M_m * D))) / -2.0)))));
	else
		tmp = (sqrt(d) / sqrt(h)) * 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[(M$95$m * N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h * N[(t$95$0 * N[(N[(0.5 * N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -5e-310], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[h, 1.35e+188], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(h * N[(t$95$0 * N[(N[(0.5 / N[(d / N[(M$95$m * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 66.4%

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

   3. Simplified 64.3%

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

   6. Applied egg-rr 30.6%

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

      1. expm1-def 30.6%

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

      2. expm1-log1p 64.3%

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

      3. associate-*l/ 65.9%

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

      4. *-commutative 65.9%

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

      5. associate-*l/ 66.6%

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

      6. *-commutative 66.6%

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

      7. associate-/l* 66.6%

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

      8. metadata-eval 66.6%

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

      9. times-frac 66.6%

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

      10. *-commutative 66.6%

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

      11. associate-/l* 66.6%

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

      12. associate-*l/ 66.6%

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

      13. *-commutative 66.6%

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

      14. associate-/r* 66.6%

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

      15. metadata-eval 66.6%

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

   8. Simplified 66.6%

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

      1. unpow2 66.6%

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

      2. div-inv 66.6%

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

      3. metadata-eval 66.6%

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

      4. metadata-eval 66.6%

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

      5. times-frac 68.0%

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

      6. associate-/r/ 68.0%

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

      7. associate-/r/ 68.0%

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

      8. metadata-eval 68.0%

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

   10. Applied egg-rr 68.0%

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

   11. Taylor expanded in M around 0 67.3%

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

       1. *-commutative 67.3%

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

       2. associate-*r/ 68.0%

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

   13. Simplified 68.0%

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

       1. frac-2neg 75.2%

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

       2. sqrt-div 89.9%

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

   15. Applied egg-rr 80.5%

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

   if -4.999999999999985e-310 < h < 1.35e188

   1. Initial program 75.3%

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

   3. Simplified 75.3%

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

   6. Applied egg-rr 44.4%

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

      1. expm1-def 44.4%

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

      2. expm1-log1p 75.3%

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

      3. associate-*l/ 75.6%

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

      4. *-commutative 75.6%

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

      5. associate-*l/ 74.6%

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

      6. *-commutative 74.6%

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

      7. associate-/l* 74.6%

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

      8. metadata-eval 74.6%

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

      9. times-frac 74.6%

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

      10. *-commutative 74.6%

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

      11. associate-/l* 74.5%

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

      12. associate-*l/ 74.5%

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

      13. *-commutative 74.5%

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

      14. associate-/r* 74.5%

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

      15. metadata-eval 74.5%

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

   8. Simplified 74.5%

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

      1. unpow2 74.5%

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

      2. div-inv 74.5%

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

      3. metadata-eval 74.5%

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

      4. metadata-eval 74.5%

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

      5. times-frac 75.5%

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

      6. associate-/r/ 75.5%

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

      7. associate-/r/ 75.6%

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

      8. metadata-eval 75.6%

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

   10. Applied egg-rr 75.6%

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

       1. *-commutative 75.6%

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

       2. associate-*l/ 75.6%

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

       3. associate-/r/ 75.6%

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

       4. associate-/l* 75.6%

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

       5. associate-/l/ 75.6%

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

   12. Applied egg-rr 75.6%

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

       1. sqrt-div 88.0%

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

       2. div-inv 88.0%

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

   14. Applied egg-rr 88.0%

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

       1. associate-*r/ 88.0%

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

       2. *-rgt-identity 88.0%

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

   16. Simplified 88.0%

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

   if 1.35e188 < h

   1. Initial program 46.1%

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

   3. Simplified 46.0%

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

   6. Applied egg-rr 20.9%

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

      1. expm1-def 20.9%

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

      2. expm1-log1p 46.0%

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

      3. associate-*l/ 46.0%

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

      4. *-commutative 46.0%

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

      5. associate-*l/ 46.0%

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

      6. *-commutative 46.0%

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

      7. associate-/l* 46.0%

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

      8. metadata-eval 46.0%

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

      9. times-frac 46.0%

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

      10. *-commutative 46.0%

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

      11. associate-/l* 46.0%

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

      12. associate-*l/ 46.0%

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

      13. *-commutative 46.0%

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

      14. associate-/r* 46.0%

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

      15. metadata-eval 46.0%

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

   8. Simplified 46.0%

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

      1. unpow2 46.0%

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

      2. div-inv 46.0%

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

      3. metadata-eval 46.0%

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

      4. metadata-eval 46.0%

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

      5. times-frac 46.0%

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

      6. associate-/r/ 46.0%

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

      7. associate-/r/ 46.0%

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

      8. metadata-eval 46.0%

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

   10. Applied egg-rr 46.0%

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

   11. Taylor expanded in M around 0 46.0%

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

       1. *-commutative 46.0%

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

       2. associate-*r/ 46.0%

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

   13. Simplified 46.0%

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

       1. sqrt-div 83.4%

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

       2. div-inv 83.3%

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

   15. Applied egg-rr 83.2%

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

       1. associate-*r/ 83.4%

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

       2. *-rgt-identity 83.4%

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

   17. Simplified 83.2%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(\frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{\ell} \cdot \frac{0.5 \cdot \left(M \cdot \frac{D}{d}\right)}{-2}\right)\right)\right)\\ \mathbf{elif}\;h \leq 1.35 \cdot 10^{+188}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 + h \cdot \left(\frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{\ell} \cdot \frac{\frac{0.5}{\frac{d}{M \cdot D}}}{-2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(\frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{\ell} \cdot \frac{0.5 \cdot \left(M \cdot \frac{D}{d}\right)}{-2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := M_m \cdot \left(\frac{0.5}{d} \cdot D\right)\\ t_1 := \frac{t_0}{\ell}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\left(1 + h \cdot \left(t_1 \cdot \frac{t_0}{-2}\right)\right) \cdot t_2\right)\\ \mathbf{elif}\;h \leq 1.3 \cdot 10^{+188}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 + h \cdot \left(t_1 \cdot \frac{\frac{0.5}{\frac{d}{M_m \cdot D}}}{-2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t_2 \cdot \left(1 + h \cdot \left(t_1 \cdot \frac{0.5 \cdot \left(M_m \cdot \frac{D}{d}\right)}{-2}\right)\right)\right)\\ \end{array} \end{array} \]

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 (* M_m (* (/ 0.5 d) D))) (t_1 (/ t_0 l)) (t_2 (sqrt (/ d l))))
   (if (<= h -5e-310)
     (*
      (/ (sqrt (- d)) (sqrt (- h)))
      (* (+ 1.0 (* h (* t_1 (/ t_0 -2.0)))) t_2))
     (if (<= h 1.3e+188)
       (*
        (sqrt (/ d h))
        (*
         (/ (sqrt d) (sqrt l))
         (+ 1.0 (* h (* t_1 (/ (/ 0.5 (/ d (* M_m D))) -2.0))))))
       (*
        (/ (sqrt d) (sqrt h))
        (* t_2 (+ 1.0 (* h (* t_1 (/ (* 0.5 (* M_m (/ D d))) -2.0))))))))))

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 = M_m * ((0.5 / d) * D);
	double t_1 = t_0 / l;
	double t_2 = sqrt((d / l));
	double tmp;
	if (h <= -5e-310) {
		tmp = (sqrt(-d) / sqrt(-h)) * ((1.0 + (h * (t_1 * (t_0 / -2.0)))) * t_2);
	} else if (h <= 1.3e+188) {
		tmp = sqrt((d / h)) * ((sqrt(d) / sqrt(l)) * (1.0 + (h * (t_1 * ((0.5 / (d / (M_m * D))) / -2.0)))));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (t_2 * (1.0 + (h * (t_1 * ((0.5 * (M_m * (D / d))) / -2.0)))));
	}
	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 = m_m * ((0.5d0 / d) * d_1)
    t_1 = t_0 / l
    t_2 = sqrt((d / l))
    if (h <= (-5d-310)) then
        tmp = (sqrt(-d) / sqrt(-h)) * ((1.0d0 + (h * (t_1 * (t_0 / (-2.0d0))))) * t_2)
    else if (h <= 1.3d+188) then
        tmp = sqrt((d / h)) * ((sqrt(d) / sqrt(l)) * (1.0d0 + (h * (t_1 * ((0.5d0 / (d / (m_m * d_1))) / (-2.0d0))))))
    else
        tmp = (sqrt(d) / sqrt(h)) * (t_2 * (1.0d0 + (h * (t_1 * ((0.5d0 * (m_m * (d_1 / d))) / (-2.0d0))))))
    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 = M_m * ((0.5 / d) * D);
	double t_1 = t_0 / l;
	double t_2 = Math.sqrt((d / l));
	double tmp;
	if (h <= -5e-310) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * ((1.0 + (h * (t_1 * (t_0 / -2.0)))) * t_2);
	} else if (h <= 1.3e+188) {
		tmp = Math.sqrt((d / h)) * ((Math.sqrt(d) / Math.sqrt(l)) * (1.0 + (h * (t_1 * ((0.5 / (d / (M_m * D))) / -2.0)))));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * (t_2 * (1.0 + (h * (t_1 * ((0.5 * (M_m * (D / d))) / -2.0)))));
	}
	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 = M_m * ((0.5 / d) * D)
	t_1 = t_0 / l
	t_2 = math.sqrt((d / l))
	tmp = 0
	if h <= -5e-310:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * ((1.0 + (h * (t_1 * (t_0 / -2.0)))) * t_2)
	elif h <= 1.3e+188:
		tmp = math.sqrt((d / h)) * ((math.sqrt(d) / math.sqrt(l)) * (1.0 + (h * (t_1 * ((0.5 / (d / (M_m * D))) / -2.0)))))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * (t_2 * (1.0 + (h * (t_1 * ((0.5 * (M_m * (D / d))) / -2.0)))))
	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(M_m * Float64(Float64(0.5 / d) * D))
	t_1 = Float64(t_0 / l)
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (h <= -5e-310)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(Float64(1.0 + Float64(h * Float64(t_1 * Float64(t_0 / -2.0)))) * t_2));
	elseif (h <= 1.3e+188)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(sqrt(d) / sqrt(l)) * Float64(1.0 + Float64(h * Float64(t_1 * Float64(Float64(0.5 / Float64(d / Float64(M_m * D))) / -2.0))))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(t_2 * Float64(1.0 + Float64(h * Float64(t_1 * Float64(Float64(0.5 * Float64(M_m * Float64(D / d))) / -2.0))))));
	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 = M_m * ((0.5 / d) * D);
	t_1 = t_0 / l;
	t_2 = sqrt((d / l));
	tmp = 0.0;
	if (h <= -5e-310)
		tmp = (sqrt(-d) / sqrt(-h)) * ((1.0 + (h * (t_1 * (t_0 / -2.0)))) * t_2);
	elseif (h <= 1.3e+188)
		tmp = sqrt((d / h)) * ((sqrt(d) / sqrt(l)) * (1.0 + (h * (t_1 * ((0.5 / (d / (M_m * D))) / -2.0)))));
	else
		tmp = (sqrt(d) / sqrt(h)) * (t_2 * (1.0 + (h * (t_1 * ((0.5 * (M_m * (D / d))) / -2.0)))));
	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[(M$95$m * N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / l), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -5e-310], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(h * N[(t$95$1 * N[(t$95$0 / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 1.3e+188], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(h * N[(t$95$1 * N[(N[(0.5 / N[(d / N[(M$95$m * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(1.0 + N[(h * N[(t$95$1 * N[(N[(0.5 * N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 66.4%

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

   3. Simplified 64.3%

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

   6. Applied egg-rr 30.6%

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

      1. expm1-def 30.6%

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

      2. expm1-log1p 64.3%

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

      3. associate-*l/ 65.9%

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

      4. *-commutative 65.9%

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

      5. associate-*l/ 66.6%

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

      6. *-commutative 66.6%

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

      7. associate-/l* 66.6%

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

      8. metadata-eval 66.6%

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

      9. times-frac 66.6%

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

      10. *-commutative 66.6%

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

      11. associate-/l* 66.6%

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

      12. associate-*l/ 66.6%

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

      13. *-commutative 66.6%

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

      14. associate-/r* 66.6%

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

      15. metadata-eval 66.6%

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

   8. Simplified 66.6%

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

      1. unpow2 66.6%

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

      2. div-inv 66.6%

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

      3. metadata-eval 66.6%

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

      4. metadata-eval 66.6%

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

      5. times-frac 68.0%

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

      6. associate-/r/ 68.0%

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

      7. associate-/r/ 68.0%

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

      8. metadata-eval 68.0%

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

   10. Applied egg-rr 68.0%

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

       1. frac-2neg 75.2%

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

       2. sqrt-div 89.9%

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

   12. Applied egg-rr 80.5%

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

   if -4.999999999999985e-310 < h < 1.29999999999999994e188

   1. Initial program 75.3%

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

   3. Simplified 75.3%

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

   6. Applied egg-rr 44.4%

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

      1. expm1-def 44.4%

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

      2. expm1-log1p 75.3%

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

      3. associate-*l/ 75.6%

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

      4. *-commutative 75.6%

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

      5. associate-*l/ 74.6%

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

      6. *-commutative 74.6%

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

      7. associate-/l* 74.6%

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

      8. metadata-eval 74.6%

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

      9. times-frac 74.6%

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

      10. *-commutative 74.6%

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

      11. associate-/l* 74.5%

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

      12. associate-*l/ 74.5%

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

      13. *-commutative 74.5%

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

      14. associate-/r* 74.5%

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

      15. metadata-eval 74.5%

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

   8. Simplified 74.5%

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

      1. unpow2 74.5%

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

      2. div-inv 74.5%

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

      3. metadata-eval 74.5%

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

      4. metadata-eval 74.5%

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

      5. times-frac 75.5%

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

      6. associate-/r/ 75.5%

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

      7. associate-/r/ 75.6%

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

      8. metadata-eval 75.6%

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

   10. Applied egg-rr 75.6%

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

       1. *-commutative 75.6%

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

       2. associate-*l/ 75.6%

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

       3. associate-/r/ 75.6%

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

       4. associate-/l* 75.6%

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

       5. associate-/l/ 75.6%

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

   12. Applied egg-rr 75.6%

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

       1. sqrt-div 88.0%

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

       2. div-inv 88.0%

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

   14. Applied egg-rr 88.0%

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

       1. associate-*r/ 88.0%

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

       2. *-rgt-identity 88.0%

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

   16. Simplified 88.0%

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

   if 1.29999999999999994e188 < h

   1. Initial program 46.1%

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

   3. Simplified 46.0%

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

   6. Applied egg-rr 20.9%

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

      1. expm1-def 20.9%

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

      2. expm1-log1p 46.0%

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

      3. associate-*l/ 46.0%

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

      4. *-commutative 46.0%

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

      5. associate-*l/ 46.0%

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

      6. *-commutative 46.0%

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

      7. associate-/l* 46.0%

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

      8. metadata-eval 46.0%

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

      9. times-frac 46.0%

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

      10. *-commutative 46.0%

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

      11. associate-/l* 46.0%

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

      12. associate-*l/ 46.0%

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

      13. *-commutative 46.0%

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

      14. associate-/r* 46.0%

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

      15. metadata-eval 46.0%

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

   8. Simplified 46.0%

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

      1. unpow2 46.0%

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

      2. div-inv 46.0%

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

      3. metadata-eval 46.0%

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

      4. metadata-eval 46.0%

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

      5. times-frac 46.0%

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

      6. associate-/r/ 46.0%

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

      7. associate-/r/ 46.0%

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

      8. metadata-eval 46.0%

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

   10. Applied egg-rr 46.0%

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

   11. Taylor expanded in M around 0 46.0%

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

       1. *-commutative 46.0%

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

       2. associate-*r/ 46.0%

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

   13. Simplified 46.0%

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

       1. sqrt-div 83.4%

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

       2. div-inv 83.3%

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

   15. Applied egg-rr 83.2%

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

       1. associate-*r/ 83.4%

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

       2. *-rgt-identity 83.4%

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

   17. Simplified 83.2%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\left(1 + h \cdot \left(\frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{\ell} \cdot \frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{-2}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;h \leq 1.3 \cdot 10^{+188}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 + h \cdot \left(\frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{\ell} \cdot \frac{\frac{0.5}{\frac{d}{M \cdot D}}}{-2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(\frac{M \cdot \left(\frac{0.5}{d} \cdot D\right)}{\ell} \cdot \frac{0.5 \cdot \left(M \cdot \frac{D}{d}\right)}{-2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;M_m \leq 1.25 \cdot 10^{-226} \lor \neg \left(M_m \leq 2.7 \cdot 10^{-169}\right) \land M_m \leq 4.4 \cdot 10^{-124}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot M_m}{\frac{\ell \cdot d}{0.5 \cdot D}} \cdot \left(-0.25 \cdot \frac{D}{\frac{d}{M_m}}\right)\right)\right)\\ \end{array} \end{array} \]

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 (or (<= M_m 1.25e-226) (and (not (<= M_m 2.7e-169)) (<= M_m 4.4e-124)))
   (fabs (/ d (sqrt (* l h))))
   (*
    (sqrt (/ d h))
    (*
     (sqrt (/ d l))
     (+
      1.0
      (* (/ (* h M_m) (/ (* l d) (* 0.5 D))) (* -0.25 (/ D (/ d M_m)))))))))

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 ((M_m <= 1.25e-226) || (!(M_m <= 2.7e-169) && (M_m <= 4.4e-124))) {
		tmp = fabs((d / sqrt((l * h))));
	} else {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + (((h * M_m) / ((l * d) / (0.5 * D))) * (-0.25 * (D / (d / M_m))))));
	}
	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 ((m_m <= 1.25d-226) .or. (.not. (m_m <= 2.7d-169)) .and. (m_m <= 4.4d-124)) then
        tmp = abs((d / sqrt((l * h))))
    else
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 + (((h * m_m) / ((l * d) / (0.5d0 * d_1))) * ((-0.25d0) * (d_1 / (d / m_m))))))
    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 ((M_m <= 1.25e-226) || (!(M_m <= 2.7e-169) && (M_m <= 4.4e-124))) {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	} else {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 + (((h * M_m) / ((l * d) / (0.5 * D))) * (-0.25 * (D / (d / M_m))))));
	}
	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 (M_m <= 1.25e-226) or (not (M_m <= 2.7e-169) and (M_m <= 4.4e-124)):
		tmp = math.fabs((d / math.sqrt((l * h))))
	else:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 + (((h * M_m) / ((l * d) / (0.5 * D))) * (-0.25 * (D / (d / M_m))))))
	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 ((M_m <= 1.25e-226) || (!(M_m <= 2.7e-169) && (M_m <= 4.4e-124)))
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64(Float64(h * M_m) / Float64(Float64(l * d) / Float64(0.5 * D))) * Float64(-0.25 * Float64(D / Float64(d / M_m)))))));
	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 ((M_m <= 1.25e-226) || (~((M_m <= 2.7e-169)) && (M_m <= 4.4e-124)))
		tmp = abs((d / sqrt((l * h))));
	else
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + (((h * M_m) / ((l * d) / (0.5 * D))) * (-0.25 * (D / (d / M_m))))));
	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[Or[LessEqual[M$95$m, 1.25e-226], And[N[Not[LessEqual[M$95$m, 2.7e-169]], $MachinePrecision], LessEqual[M$95$m, 4.4e-124]]], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(N[(h * M$95$m), $MachinePrecision] / N[(N[(l * d), $MachinePrecision] / N[(0.5 * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * N[(D / N[(d / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.2499999999999999e-226 or 2.7000000000000002e-169 < M < 4.3999999999999998e-124

   1. Initial program 69.4%

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

   3. Simplified 69.7%

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

   5. Taylor expanded in h around 0 37.6%

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

      1. *-rgt-identity 37.6%

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

      2. sqrt-unprod 26.8%

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

   7. Applied egg-rr 26.8%

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

      1. frac-times 19.2%

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

      2. unpow2 19.2%

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

      3. associate-/l/ 20.0%

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

      4. add-sqr-sqrt 20.0%

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

      5. rem-sqrt-square 20.0%

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

      6. associate-/l/ 19.2%

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

      7. sqrt-div 24.0%

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

      8. unpow2 24.0%

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

      9. sqrt-prod 20.3%

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

      10. add-sqr-sqrt 40.4%

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

      11. *-commutative 40.4%

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

   9. Applied egg-rr 40.4%

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

   if 1.2499999999999999e-226 < M < 2.7000000000000002e-169 or 4.3999999999999998e-124 < M

   1. Initial program 65.0%

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

   3. Simplified 64.0%

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

   6. Applied egg-rr 32.2%

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

      1. expm1-def 32.2%

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

      2. expm1-log1p 64.0%

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

      3. associate-*l/ 64.4%

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

      4. *-commutative 64.4%

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

      5. associate-*l/ 64.4%

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

      6. *-commutative 64.4%

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

      7. associate-/l* 64.4%

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

      8. metadata-eval 64.4%

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

      9. times-frac 64.4%

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

      10. *-commutative 64.4%

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

      11. associate-/l* 64.3%

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

      12. associate-*l/ 64.3%

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

      13. *-commutative 64.3%

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

      14. associate-/r* 64.3%

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

      15. metadata-eval 64.3%

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

   8. Simplified 64.3%

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

      1. unpow2 64.3%

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

      2. div-inv 64.3%

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

      3. metadata-eval 64.3%

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

      4. metadata-eval 64.3%

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

      5. times-frac 66.2%

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

      6. associate-/r/ 66.2%

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

      7. associate-/r/ 66.3%

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

      8. metadata-eval 66.3%

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

   10. Applied egg-rr 66.3%

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

   11. Taylor expanded in M around 0 66.3%

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

       1. *-commutative 66.3%

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

       2. associate-*r/ 66.3%

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

   13. Simplified 66.3%

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

       1. expm1-log1p-u 33.3%

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

       2. expm1-udef 33.3%

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

       3. associate-*r* 33.3%

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

       4. associate-/l* 33.3%

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

       5. associate-*l/ 33.3%

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

       6. div-inv 33.3%

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

       7. associate-*r* 33.3%

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

       8. metadata-eval 33.3%

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

   15. Applied egg-rr 33.3%

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

       1. expm1-def 33.3%

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

       2. expm1-log1p 64.9%

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

       3. *-commutative 64.9%

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

       4. associate-*l/ 60.9%

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

       5. associate-/r/ 59.9%

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

       6. associate-*l/ 60.8%

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

       7. *-commutative 60.8%

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

       8. *-commutative 60.8%

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

       9. associate-*l* 60.8%

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

       10. associate-*r* 60.8%

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

       11. metadata-eval 60.8%

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

       12. associate-*r/ 61.8%

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

       13. *-commutative 61.8%

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

       14. associate-/l* 61.3%

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

   17. Simplified 61.3%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.25 \cdot 10^{-226} \lor \neg \left(M \leq 2.7 \cdot 10^{-169}\right) \land M \leq 4.4 \cdot 10^{-124}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot M}{\frac{\ell \cdot d}{0.5 \cdot D}} \cdot \left(-0.25 \cdot \frac{D}{\frac{d}{M}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{+126}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;d \leq -1.25 \cdot 10^{-87}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{-244}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

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 (<= d -7.2e+126)
   (fabs (/ d (sqrt (* l h))))
   (if (<= d -1.25e-87)
     (* (sqrt (/ d h)) (sqrt (/ d l)))
     (if (<= d 1.6e-244)
       (* d (- (sqrt (/ 1.0 (* l h)))))
       (/ (/ d (sqrt l)) (sqrt 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 (d <= -7.2e+126) {
		tmp = fabs((d / sqrt((l * h))));
	} else if (d <= -1.25e-87) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (d <= 1.6e-244) {
		tmp = d * -sqrt((1.0 / (l * h)));
	} else {
		tmp = (d / sqrt(l)) / sqrt(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 (d <= (-7.2d+126)) then
        tmp = abs((d / sqrt((l * h))))
    else if (d <= (-1.25d-87)) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else if (d <= 1.6d-244) then
        tmp = d * -sqrt((1.0d0 / (l * h)))
    else
        tmp = (d / sqrt(l)) / sqrt(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 (d <= -7.2e+126) {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	} else if (d <= -1.25e-87) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (d <= 1.6e-244) {
		tmp = d * -Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = (d / Math.sqrt(l)) / Math.sqrt(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 d <= -7.2e+126:
		tmp = math.fabs((d / math.sqrt((l * h))))
	elif d <= -1.25e-87:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif d <= 1.6e-244:
		tmp = d * -math.sqrt((1.0 / (l * h)))
	else:
		tmp = (d / math.sqrt(l)) / math.sqrt(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 (d <= -7.2e+126)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	elseif (d <= -1.25e-87)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (d <= 1.6e-244)
		tmp = Float64(d * Float64(-sqrt(Float64(1.0 / Float64(l * h)))));
	else
		tmp = Float64(Float64(d / sqrt(l)) / sqrt(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 (d <= -7.2e+126)
		tmp = abs((d / sqrt((l * h))));
	elseif (d <= -1.25e-87)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (d <= 1.6e-244)
		tmp = d * -sqrt((1.0 / (l * h)));
	else
		tmp = (d / sqrt(l)) / sqrt(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[d, -7.2e+126], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[d, -1.25e-87], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.6e-244], N[(d * (-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -7.2000000000000001e126

   1. Initial program 61.0%

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

   3. Simplified 63.3%

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

   5. Taylor expanded in h around 0 36.9%

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

      1. *-rgt-identity 36.9%

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

      2. sqrt-unprod 27.8%

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

   7. Applied egg-rr 27.8%

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

      1. frac-times 25.3%

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

      2. unpow2 25.3%

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

      3. associate-/l/ 25.5%

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

      4. add-sqr-sqrt 25.5%

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

      5. rem-sqrt-square 25.5%

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

      6. associate-/l/ 25.3%

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

      7. sqrt-div 25.3%

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

      8. unpow2 25.3%

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

      9. sqrt-prod 0.0%

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

      10. add-sqr-sqrt 68.5%

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

      11. *-commutative 68.5%

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

   9. Applied egg-rr 68.5%

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

   if -7.2000000000000001e126 < d < -1.25000000000000011e-87

   1. Initial program 83.6%

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

   3. Simplified 83.6%

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

   5. Taylor expanded in h around 0 50.1%

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

      1. *-rgt-identity 50.1%

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

      2. sqrt-unprod 36.6%

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

   7. Applied egg-rr 36.6%

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

      1. *-commutative 36.6%

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

      2. sqrt-prod 50.1%

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

   9. Applied egg-rr 50.1%

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

   if -1.25000000000000011e-87 < d < 1.5999999999999999e-244

   1. Initial program 53.0%

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

   3. Simplified 52.3%

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

   5. Taylor expanded in h around 0 12.6%

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

      1. *-rgt-identity 12.6%

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

      2. sqrt-unprod 9.8%

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

   7. Applied egg-rr 9.8%

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

   8. Taylor expanded in d around -inf 21.5%

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

   if 1.5999999999999999e-244 < d

   1. Initial program 72.2%

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

   3. Simplified 72.2%

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

   5. Taylor expanded in h around 0 44.2%

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

      1. expm1-log1p-u 42.4%

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

      2. expm1-udef 34.1%

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

      3. sqrt-div 37.3%

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

      4. *-rgt-identity 37.3%

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

      5. sqrt-div 42.3%

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

      6. frac-times 42.3%

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

      7. add-sqr-sqrt 42.3%

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

   7. Applied egg-rr 42.3%

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

      1. expm1-def 52.3%

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

      2. expm1-log1p 55.0%

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

      3. associate-/l/ 53.3%

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

   9. Simplified 53.3%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{+126}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;d \leq -1.25 \cdot 10^{-87}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{-244}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 46.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2.2 \cdot 10^{+116}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;d \leq -5.8 \cdot 10^{-87}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq 7.1 \cdot 10^{-245}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

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 (<= d -2.2e+116)
   (fabs (/ d (sqrt (* l h))))
   (if (<= d -5.8e-87)
     (* (sqrt (/ d l)) (/ 1.0 (sqrt (/ h d))))
     (if (<= d 7.1e-245)
       (* d (- (sqrt (/ 1.0 (* l h)))))
       (/ (/ d (sqrt l)) (sqrt 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 (d <= -2.2e+116) {
		tmp = fabs((d / sqrt((l * h))));
	} else if (d <= -5.8e-87) {
		tmp = sqrt((d / l)) * (1.0 / sqrt((h / d)));
	} else if (d <= 7.1e-245) {
		tmp = d * -sqrt((1.0 / (l * h)));
	} else {
		tmp = (d / sqrt(l)) / sqrt(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 (d <= (-2.2d+116)) then
        tmp = abs((d / sqrt((l * h))))
    else if (d <= (-5.8d-87)) then
        tmp = sqrt((d / l)) * (1.0d0 / sqrt((h / d)))
    else if (d <= 7.1d-245) then
        tmp = d * -sqrt((1.0d0 / (l * h)))
    else
        tmp = (d / sqrt(l)) / sqrt(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 (d <= -2.2e+116) {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	} else if (d <= -5.8e-87) {
		tmp = Math.sqrt((d / l)) * (1.0 / Math.sqrt((h / d)));
	} else if (d <= 7.1e-245) {
		tmp = d * -Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = (d / Math.sqrt(l)) / Math.sqrt(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 d <= -2.2e+116:
		tmp = math.fabs((d / math.sqrt((l * h))))
	elif d <= -5.8e-87:
		tmp = math.sqrt((d / l)) * (1.0 / math.sqrt((h / d)))
	elif d <= 7.1e-245:
		tmp = d * -math.sqrt((1.0 / (l * h)))
	else:
		tmp = (d / math.sqrt(l)) / math.sqrt(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 (d <= -2.2e+116)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	elseif (d <= -5.8e-87)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(1.0 / sqrt(Float64(h / d))));
	elseif (d <= 7.1e-245)
		tmp = Float64(d * Float64(-sqrt(Float64(1.0 / Float64(l * h)))));
	else
		tmp = Float64(Float64(d / sqrt(l)) / sqrt(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 (d <= -2.2e+116)
		tmp = abs((d / sqrt((l * h))));
	elseif (d <= -5.8e-87)
		tmp = sqrt((d / l)) * (1.0 / sqrt((h / d)));
	elseif (d <= 7.1e-245)
		tmp = d * -sqrt((1.0 / (l * h)));
	else
		tmp = (d / sqrt(l)) / sqrt(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[d, -2.2e+116], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[d, -5.8e-87], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.1e-245], N[(d * (-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.2e116

   1. Initial program 64.5%

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

   3. Simplified 66.5%

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

   5. Taylor expanded in h 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. *-rgt-identity 40.3%

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

      2. sqrt-unprod 32.0%

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

   7. Applied egg-rr 32.0%

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

      1. frac-times 29.8%

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

      2. unpow2 29.8%

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

      3. associate-/l/ 29.9%

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

      4. add-sqr-sqrt 29.9%

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

      5. rem-sqrt-square 29.9%

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

      6. associate-/l/ 29.8%

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

      7. sqrt-div 29.8%

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

      8. unpow2 29.8%

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

      9. sqrt-prod 0.0%

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

      10. add-sqr-sqrt 69.1%

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

      11. *-commutative 69.1%

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

   9. Applied egg-rr 69.1%

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

   if -2.2e116 < d < -5.7999999999999998e-87

   1. Initial program 82.1%

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

   3. Simplified 82.1%

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

      1. clear-num 82.2%

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

      2. sqrt-div 83.4%

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

      3. metadata-eval 83.4%

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

   6. Applied egg-rr 83.4%

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

   7. Taylor expanded in h around 0 49.2%

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

   if -5.7999999999999998e-87 < d < 7.10000000000000016e-245

   1. Initial program 53.0%

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

   3. Simplified 52.3%

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

   5. Taylor expanded in h around 0 12.6%

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

      1. *-rgt-identity 12.6%

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

      2. sqrt-unprod 9.8%

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

   7. Applied egg-rr 9.8%

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

   8. Taylor expanded in d around -inf 21.5%

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

   if 7.10000000000000016e-245 < d

   1. Initial program 72.2%

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

   3. Simplified 72.2%

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

   5. Taylor expanded in h around 0 44.2%

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

      1. expm1-log1p-u 42.4%

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

      2. expm1-udef 34.1%

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

      3. sqrt-div 37.3%

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

      4. *-rgt-identity 37.3%

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

      5. sqrt-div 42.3%

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

      6. frac-times 42.3%

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

      7. add-sqr-sqrt 42.3%

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

   7. Applied egg-rr 42.3%

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

      1. expm1-def 52.3%

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

      2. expm1-log1p 55.0%

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

      3. associate-/l/ 53.3%

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

   9. Simplified 53.3%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.2 \cdot 10^{+116}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;d \leq -5.8 \cdot 10^{-87}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq 7.1 \cdot 10^{-245}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{-239}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{-307}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

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 -4.5e-239)
   (fabs (/ d (sqrt (* l h))))
   (if (<= l 4.2e-307)
     (/ d (cbrt (pow (* l h) 1.5)))
     (/ (/ d (sqrt l)) (sqrt 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 <= -4.5e-239) {
		tmp = fabs((d / sqrt((l * h))));
	} else if (l <= 4.2e-307) {
		tmp = d / cbrt(pow((l * h), 1.5));
	} else {
		tmp = (d / sqrt(l)) / sqrt(h);
	}
	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 <= -4.5e-239) {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	} else if (l <= 4.2e-307) {
		tmp = d / Math.cbrt(Math.pow((l * h), 1.5));
	} else {
		tmp = (d / Math.sqrt(l)) / Math.sqrt(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 <= -4.5e-239)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	elseif (l <= 4.2e-307)
		tmp = Float64(d / cbrt((Float64(l * h) ^ 1.5)));
	else
		tmp = Float64(Float64(d / sqrt(l)) / sqrt(h));
	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, -4.5e-239], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 4.2e-307], N[(d / N[Power[N[Power[N[(l * h), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.50000000000000013e-239

   1. Initial program 63.7%

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

   3. Simplified 64.1%

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

   5. Taylor expanded in h around 0 33.7%

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

      1. *-rgt-identity 33.7%

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

      2. sqrt-unprod 25.8%

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

   7. Applied egg-rr 25.8%

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

      1. frac-times 22.7%

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

      2. unpow2 22.7%

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

      3. associate-/l/ 22.9%

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

      4. add-sqr-sqrt 22.9%

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

      5. rem-sqrt-square 22.9%

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

      6. associate-/l/ 22.7%

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

      7. sqrt-div 25.6%

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

      8. unpow2 25.6%

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

      9. sqrt-prod 0.0%

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

      10. add-sqr-sqrt 42.3%

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

      11. *-commutative 42.3%

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

   9. Applied egg-rr 42.3%

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

   if -4.50000000000000013e-239 < l < 4.2000000000000002e-307

   1. Initial program 86.0%

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

   3. Simplified 86.1%

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

   5. Taylor expanded in h around 0 21.8%

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

      1. *-rgt-identity 21.8%

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

      2. sqrt-unprod 8.2%

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

   7. Applied egg-rr 8.2%

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

      1. frac-times 8.2%

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

      2. unpow2 8.2%

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

      3. associate-/l/ 8.3%

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

      4. expm1-log1p-u 8.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\frac{{d}^{2}}{\ell}}{h}}\right)\right)} \]

      5. expm1-udef 8.4%

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

      6. associate-/l/ 8.2%

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

      7. unpow2 8.2%

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

      8. frac-times 8.3%

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

      9. frac-times 8.2%

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

      10. unpow2 8.2%

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

      11. sqrt-div 8.2%

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

      12. unpow2 8.2%

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

      13. sqrt-prod 0.0%

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

      14. add-sqr-sqrt 0.3%

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

      15. *-commutative 0.3%

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

   9. Applied egg-rr 0.3%

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

       1. expm1-def 0.4%

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

       2. expm1-log1p 37.1%

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

   11. Simplified 37.1%

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

       1. add-cbrt-cube 44.0%

          \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{\left(\sqrt{\ell \cdot h} \cdot \sqrt{\ell \cdot h}\right) \cdot \sqrt{\ell \cdot h}}}} \]

       2. pow1/3 44.0%

          \[\leadsto \frac{d}{\color{blue}{{\left(\left(\sqrt{\ell \cdot h} \cdot \sqrt{\ell \cdot h}\right) \cdot \sqrt{\ell \cdot h}\right)}^{0.3333333333333333}}} \]

       3. add-sqr-sqrt 44.0%

          \[\leadsto \frac{d}{{\left(\color{blue}{\left(\ell \cdot h\right)} \cdot \sqrt{\ell \cdot h}\right)}^{0.3333333333333333}} \]

       4. pow1 44.0%

          \[\leadsto \frac{d}{{\left(\color{blue}{{\left(\ell \cdot h\right)}^{1}} \cdot \sqrt{\ell \cdot h}\right)}^{0.3333333333333333}} \]

       5. pow1/2 44.0%

          \[\leadsto \frac{d}{{\left({\left(\ell \cdot h\right)}^{1} \cdot \color{blue}{{\left(\ell \cdot h\right)}^{0.5}}\right)}^{0.3333333333333333}} \]

       6. pow-prod-up 44.0%

          \[\leadsto \frac{d}{{\color{blue}{\left({\left(\ell \cdot h\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}} \]

       7. *-commutative 44.0%

          \[\leadsto \frac{d}{{\left({\color{blue}{\left(h \cdot \ell\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333}} \]

       8. metadata-eval 44.0%

          \[\leadsto \frac{d}{{\left({\left(h \cdot \ell\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \]

   13. Applied egg-rr 44.0%

       \[\leadsto \frac{d}{\color{blue}{{\left({\left(h \cdot \ell\right)}^{1.5}\right)}^{0.3333333333333333}}} \]
   14. Step-by-step derivation

       1. unpow1/3 44.0%

          \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}} \]

   15. Simplified 44.0%

       \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}} \]

   if 4.2000000000000002e-307 < l

   1. Initial program 69.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 69.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{D}{\frac{2}{M} \cdot d}\right)}^{2} \cdot -0.5, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0 40.6%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
   6. Step-by-step derivation

      1. expm1-log1p-u 39.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right)\right)\right)} \]

      2. expm1-udef 30.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right)\right)} - 1} \]

      3. sqrt-div 33.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right)\right)} - 1 \]

      4. *-rgt-identity 33.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right)} - 1 \]

      5. sqrt-div 38.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right)} - 1 \]

      6. frac-times 38.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}}\right)} - 1 \]

      7. add-sqr-sqrt 38.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}}\right)} - 1 \]

   7. Applied egg-rr 38.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 47.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\right)\right)} \]

      2. expm1-log1p 50.2%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      3. associate-/l/ 48.7%

         \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]

   9. Simplified 48.7%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{-239}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{-307}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;\ell \leq -1.05 \cdot 10^{-243}:\\ \;\;\;\;\left|\frac{d}{t_0}\right|\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{reciprocal}\left(\left(\frac{t_0}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (sqrt (* l h))))
   (if (<= l -1.05e-243)
     (fabs (/ d t_0))
     (if (<= l -1e-310) (reciprocal (/ t_0 d)) (/ (/ d (sqrt l)) (sqrt h))))))

Derivation

1. Split input into 3 regimes

2. if l < -1.05e-243

   1. Initial program 63.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 64.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{D}{\frac{2}{M} \cdot d}\right)}^{2} \cdot -0.5, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0 33.7%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
   6. Step-by-step derivation

      1. *-rgt-identity 33.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-unprod 25.8%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   7. Applied egg-rr 25.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]
   8. Step-by-step derivation

      1. frac-times 22.7%

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \]

      2. unpow2 22.7%

         \[\leadsto \sqrt{\frac{\color{blue}{{d}^{2}}}{h \cdot \ell}} \]

      3. associate-/l/ 22.9%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{{d}^{2}}{\ell}}{h}}} \]

      4. add-sqr-sqrt 22.9%

         \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{\frac{{d}^{2}}{\ell}}{h}} \cdot \sqrt{\frac{\frac{{d}^{2}}{\ell}}{h}}}} \]

      5. rem-sqrt-square 22.9%

         \[\leadsto \color{blue}{\left|\sqrt{\frac{\frac{{d}^{2}}{\ell}}{h}}\right|} \]

      6. associate-/l/ 22.7%

         \[\leadsto \left|\sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}}\right| \]

      7. sqrt-div 25.6%

         \[\leadsto \left|\color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}}}\right| \]

      8. unpow2 25.6%

         \[\leadsto \left|\frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}}\right| \]

      9. sqrt-prod 0.0%

         \[\leadsto \left|\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}}\right| \]

      10. add-sqr-sqrt 42.3%

          \[\leadsto \left|\frac{\color{blue}{d}}{\sqrt{h \cdot \ell}}\right| \]

      11. *-commutative 42.3%

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \]

   9. Applied egg-rr 42.3%

      \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \]

   if -1.05e-243 < l < -9.999999999999969e-311

   1. Initial program 92.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 92.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{D}{\frac{2}{M} \cdot d}\right)}^{2} \cdot -0.5, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0 23.4%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
   6. Step-by-step derivation

      1. *-rgt-identity 23.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-unprod 8.9%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   7. Applied egg-rr 8.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]
   8. Step-by-step derivation

      1. frac-times 8.8%

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \]

      2. unpow2 8.8%

         \[\leadsto \sqrt{\frac{\color{blue}{{d}^{2}}}{h \cdot \ell}} \]

      3. associate-/l/ 8.9%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{{d}^{2}}{\ell}}{h}}} \]

      4. expm1-log1p-u 8.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\frac{{d}^{2}}{\ell}}{h}}\right)\right)} \]

      5. expm1-udef 9.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{\frac{{d}^{2}}{\ell}}{h}}\right)} - 1} \]

      6. associate-/l/ 8.8%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}}\right)} - 1 \]

      7. unpow2 8.8%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}}\right)} - 1 \]

      8. frac-times 8.9%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}}\right)} - 1 \]

      9. frac-times 8.8%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right)} - 1 \]

      10. unpow2 8.8%

          \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\frac{\color{blue}{{d}^{2}}}{h \cdot \ell}}\right)} - 1 \]

      11. sqrt-div 8.8%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      12. unpow2 8.8%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

      13. sqrt-prod 0.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

      14. add-sqr-sqrt 0.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{d}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

      15. *-commutative 0.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right)} - 1 \]

   9. Applied egg-rr 0.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 0.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right)} \]

       2. expm1-log1p 40.0%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

   11. Simplified 40.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   12. Step-by-step derivation

       1. clear-num 40.0%

          \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell \cdot h}}{d}}} \]

       2. reciprocal-define 40.0%

          \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(\frac{\sqrt{\ell \cdot h}}{d}\right)\right)} \]

       3. *-commutative 40.0%

          \[\leadsto \mathsf{reciprocal}\left(\left(\frac{\sqrt{\color{blue}{h \cdot \ell}}}{d}\right)\right) \]

   13. Applied egg-rr 40.0%

       \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(\frac{\sqrt{h \cdot \ell}}{d}\right)\right)} \]

   if -9.999999999999969e-311 < l

   1. Initial program 69.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 69.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{D}{\frac{2}{M} \cdot d}\right)}^{2} \cdot -0.5, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0 40.2%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
   6. Step-by-step derivation

      1. expm1-log1p-u 38.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right)\right)\right)} \]

      2. expm1-udef 30.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right)\right)} - 1} \]

      3. sqrt-div 33.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right)\right)} - 1 \]

      4. *-rgt-identity 33.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right)} - 1 \]

      5. sqrt-div 37.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right)} - 1 \]

      6. frac-times 37.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}}\right)} - 1 \]

      7. add-sqr-sqrt 37.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}}\right)} - 1 \]

   7. Applied egg-rr 37.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 47.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\right)\right)} \]

      2. expm1-log1p 49.8%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      3. associate-/l/ 48.3%

         \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]

   9. Simplified 48.3%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.05 \cdot 10^{-243}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{reciprocal}\left(\left(\frac{\sqrt{\ell \cdot h}}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 43.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;M_m \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\left|t_0\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

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 (/ d (sqrt (* l h))))) (if (<= M_m 2e-7) (fabs t_0) 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 = d / sqrt((l * h));
	double tmp;
	if (M_m <= 2e-7) {
		tmp = fabs(t_0);
	} else {
		tmp = 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) :: tmp
    t_0 = d / sqrt((l * h))
    if (m_m <= 2d-7) then
        tmp = abs(t_0)
    else
        tmp = 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 = d / Math.sqrt((l * h));
	double tmp;
	if (M_m <= 2e-7) {
		tmp = Math.abs(t_0);
	} else {
		tmp = 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 = d / math.sqrt((l * h))
	tmp = 0
	if M_m <= 2e-7:
		tmp = math.fabs(t_0)
	else:
		tmp = 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(d / sqrt(Float64(l * h)))
	tmp = 0.0
	if (M_m <= 2e-7)
		tmp = abs(t_0);
	else
		tmp = 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 = d / sqrt((l * h));
	tmp = 0.0;
	if (M_m <= 2e-7)
		tmp = abs(t_0);
	else
		tmp = 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[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M$95$m, 2e-7], N[Abs[t$95$0], $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if M < 1.9999999999999999e-7

   1. Initial program 70.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 70.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{D}{\frac{2}{M} \cdot d}\right)}^{2} \cdot -0.5, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0 40.6%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
   6. Step-by-step derivation

      1. *-rgt-identity 40.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-unprod 29.4%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   7. Applied egg-rr 29.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]
   8. Step-by-step derivation

      1. frac-times 22.2%

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \]

      2. unpow2 22.2%

         \[\leadsto \sqrt{\frac{\color{blue}{{d}^{2}}}{h \cdot \ell}} \]

      3. associate-/l/ 22.4%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{{d}^{2}}{\ell}}{h}}} \]

      4. add-sqr-sqrt 22.4%

         \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{\frac{{d}^{2}}{\ell}}{h}} \cdot \sqrt{\frac{\frac{{d}^{2}}{\ell}}{h}}}} \]

      5. rem-sqrt-square 22.4%

         \[\leadsto \color{blue}{\left|\sqrt{\frac{\frac{{d}^{2}}{\ell}}{h}}\right|} \]

      6. associate-/l/ 22.2%

         \[\leadsto \left|\sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}}\right| \]

      7. sqrt-div 27.0%

         \[\leadsto \left|\color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}}}\right| \]

      8. unpow2 27.0%

         \[\leadsto \left|\frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}}\right| \]

      9. sqrt-prod 21.2%

         \[\leadsto \left|\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}}\right| \]

      10. add-sqr-sqrt 44.2%

          \[\leadsto \left|\frac{\color{blue}{d}}{\sqrt{h \cdot \ell}}\right| \]

      11. *-commutative 44.2%

          \[\leadsto \left|\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right| \]

   9. Applied egg-rr 44.2%

      \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \]

   if 1.9999999999999999e-7 < M

   1. Initial program 59.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 59.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{D}{\frac{2}{M} \cdot d}\right)}^{2} \cdot -0.5, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0 23.1%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
   6. Step-by-step derivation

      1. *-rgt-identity 23.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-unprod 23.2%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   7. Applied egg-rr 23.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]
   8. Step-by-step derivation

      1. frac-times 21.7%

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \]

      2. unpow2 21.7%

         \[\leadsto \sqrt{\frac{\color{blue}{{d}^{2}}}{h \cdot \ell}} \]

      3. associate-/l/ 23.2%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{{d}^{2}}{\ell}}{h}}} \]

      4. expm1-log1p-u 22.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\frac{{d}^{2}}{\ell}}{h}}\right)\right)} \]

      5. expm1-udef 18.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{\frac{{d}^{2}}{\ell}}{h}}\right)} - 1} \]

      6. associate-/l/ 18.2%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}}\right)} - 1 \]

      7. unpow2 18.2%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}}\right)} - 1 \]

      8. frac-times 18.2%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}}\right)} - 1 \]

      9. frac-times 18.2%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right)} - 1 \]

      10. unpow2 18.2%

          \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\frac{\color{blue}{{d}^{2}}}{h \cdot \ell}}\right)} - 1 \]

      11. sqrt-div 18.2%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      12. unpow2 18.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

      13. sqrt-prod 10.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

      14. add-sqr-sqrt 13.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{d}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

      15. *-commutative 13.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right)} - 1 \]

   9. Applied egg-rr 13.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 13.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right)} \]

       2. expm1-log1p 18.3%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

   11. Simplified 18.3%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 42.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.02 \cdot 10^{-243}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

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 -1.02e-243) (* d (- (sqrt (/ 1.0 (* l h))))) (/ 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) {
	double tmp;
	if (l <= -1.02e-243) {
		tmp = d * -sqrt((1.0 / (l * h)));
	} else {
		tmp = d / sqrt((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 <= (-1.02d-243)) then
        tmp = d * -sqrt((1.0d0 / (l * h)))
    else
        tmp = d / sqrt((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 <= -1.02e-243) {
		tmp = d * -Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = d / Math.sqrt((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 <= -1.02e-243:
		tmp = d * -math.sqrt((1.0 / (l * h)))
	else:
		tmp = d / math.sqrt((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 <= -1.02e-243)
		tmp = Float64(d * Float64(-sqrt(Float64(1.0 / Float64(l * h)))));
	else
		tmp = Float64(d / sqrt(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 <= -1.02e-243)
		tmp = d * -sqrt((1.0 / (l * h)));
	else
		tmp = d / sqrt((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, -1.02e-243], N[(d * (-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.01999999999999996e-243

   1. Initial program 63.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 64.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{D}{\frac{2}{M} \cdot d}\right)}^{2} \cdot -0.5, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0 33.7%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
   6. Step-by-step derivation

      1. *-rgt-identity 33.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-unprod 25.8%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   7. Applied egg-rr 25.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   8. Taylor expanded in d around -inf 41.6%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   if -1.01999999999999996e-243 < l

   1. Initial program 71.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 71.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{D}{\frac{2}{M} \cdot d}\right)}^{2} \cdot -0.5, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0 38.5%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
   6. Step-by-step derivation

      1. *-rgt-identity 38.5%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-unprod 29.8%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   7. Applied egg-rr 29.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]
   8. Step-by-step derivation

      1. frac-times 21.4%

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \]

      2. unpow2 21.4%

         \[\leadsto \sqrt{\frac{\color{blue}{{d}^{2}}}{h \cdot \ell}} \]

      3. associate-/l/ 22.4%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{{d}^{2}}{\ell}}{h}}} \]

      4. expm1-log1p-u 21.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\frac{{d}^{2}}{\ell}}{h}}\right)\right)} \]

      5. expm1-udef 19.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{\frac{{d}^{2}}{\ell}}{h}}\right)} - 1} \]

      6. associate-/l/ 18.1%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}}\right)} - 1 \]

      7. unpow2 18.1%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}}\right)} - 1 \]

      8. frac-times 23.0%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}}\right)} - 1 \]

      9. frac-times 18.1%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right)} - 1 \]

      10. unpow2 18.1%

          \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\frac{\color{blue}{{d}^{2}}}{h \cdot \ell}}\right)} - 1 \]

      11. sqrt-div 20.7%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      12. unpow2 20.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

      13. sqrt-prod 29.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

      14. add-sqr-sqrt 29.8%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{d}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

      15. *-commutative 29.8%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right)} - 1 \]

   9. Applied egg-rr 29.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 35.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right)} \]

       2. expm1-log1p 41.4%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

   11. Simplified 41.4%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.02 \cdot 10^{-243}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 36.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.7 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

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 (<= d -1.7e-68) (sqrt (* (/ d h) (/ d l))) (/ 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) {
	double tmp;
	if (d <= -1.7e-68) {
		tmp = sqrt(((d / h) * (d / l)));
	} else {
		tmp = d / sqrt((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 (d <= (-1.7d-68)) then
        tmp = sqrt(((d / h) * (d / l)))
    else
        tmp = d / sqrt((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 (d <= -1.7e-68) {
		tmp = Math.sqrt(((d / h) * (d / l)));
	} else {
		tmp = d / Math.sqrt((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 d <= -1.7e-68:
		tmp = math.sqrt(((d / h) * (d / l)))
	else:
		tmp = d / math.sqrt((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 (d <= -1.7e-68)
		tmp = sqrt(Float64(Float64(d / h) * Float64(d / l)));
	else
		tmp = Float64(d / sqrt(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 (d <= -1.7e-68)
		tmp = sqrt(((d / h) * (d / l)));
	else
		tmp = d / sqrt((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[d, -1.7e-68], N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.70000000000000009e-68

   1. Initial program 71.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 72.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{D}{\frac{2}{M} \cdot d}\right)}^{2} \cdot -0.5, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0 44.3%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
   6. Step-by-step derivation

      1. *-rgt-identity 44.3%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-unprod 34.4%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   7. Applied egg-rr 34.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   if -1.70000000000000009e-68 < d

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 65.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{D}{\frac{2}{M} \cdot d}\right)}^{2} \cdot -0.5, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0 32.2%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
   6. Step-by-step derivation

      1. *-rgt-identity 32.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-unprod 24.6%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   7. Applied egg-rr 24.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]
   8. Step-by-step derivation

      1. frac-times 17.8%

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \]

      2. unpow2 17.8%

         \[\leadsto \sqrt{\frac{\color{blue}{{d}^{2}}}{h \cdot \ell}} \]

      3. associate-/l/ 18.0%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{{d}^{2}}{\ell}}{h}}} \]

      4. expm1-log1p-u 17.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\frac{{d}^{2}}{\ell}}{h}}\right)\right)} \]

      5. expm1-udef 15.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{\frac{{d}^{2}}{\ell}}{h}}\right)} - 1} \]

      6. associate-/l/ 14.8%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}}\right)} - 1 \]

      7. unpow2 14.8%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}}\right)} - 1 \]

      8. frac-times 18.5%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}}\right)} - 1 \]

      9. frac-times 14.8%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right)} - 1 \]

      10. unpow2 14.8%

          \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\frac{\color{blue}{{d}^{2}}}{h \cdot \ell}}\right)} - 1 \]

      11. sqrt-div 16.8%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      12. unpow2 16.8%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

      13. sqrt-prod 22.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

      14. add-sqr-sqrt 24.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{d}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

      15. *-commutative 24.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right)} - 1 \]

   9. Applied egg-rr 24.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 28.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right)} \]

       2. expm1-log1p 34.5%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

   11. Simplified 34.5%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.7 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 26.2% 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 67.6%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 67.9%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{D}{\frac{2}{M} \cdot d}\right)}^{2} \cdot -0.5, 1\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in h around 0 36.1%

   \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
6. Step-by-step derivation

   1. *-rgt-identity 36.1%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

   2. sqrt-unprod 27.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

7. Applied egg-rr 27.8%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]
8. Step-by-step derivation

   1. frac-times 22.0%

      \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \]

   2. unpow2 22.0%

      \[\leadsto \sqrt{\frac{\color{blue}{{d}^{2}}}{h \cdot \ell}} \]

   3. associate-/l/ 22.6%

      \[\leadsto \sqrt{\color{blue}{\frac{\frac{{d}^{2}}{\ell}}{h}}} \]

   4. expm1-log1p-u 21.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\frac{{d}^{2}}{\ell}}{h}}\right)\right)} \]

   5. expm1-udef 17.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{\frac{{d}^{2}}{\ell}}{h}}\right)} - 1} \]

   6. associate-/l/ 17.4%

      \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{{d}^{2}}{h \cdot \ell}}}\right)} - 1 \]

   7. unpow2 17.4%

      \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}}\right)} - 1 \]

   8. frac-times 20.6%

      \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}}\right)} - 1 \]

   9. frac-times 17.4%

      \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right)} - 1 \]

   10. unpow2 17.4%

       \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\frac{\color{blue}{{d}^{2}}}{h \cdot \ell}}\right)} - 1 \]

   11. sqrt-div 18.7%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

   12. unpow2 18.7%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   13. sqrt-prod 14.9%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   14. add-sqr-sqrt 16.5%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{d}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   15. *-commutative 16.5%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\color{blue}{\ell \cdot h}}}\right)} - 1 \]

9. Applied egg-rr 16.5%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-def 19.3%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right)} \]

    2. expm1-log1p 25.4%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

11. Simplified 25.4%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

12. Final simplification 25.4%

    \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(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)))))