Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.8% → 81.3%

Time: 23.1s

Alternatives: 20

Speedup: 3.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 81.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{0.5 \cdot \left(M\_m \cdot D\_m\right)}{d}\\ t_1 := 0.5 \cdot t\_0\\ t_2 := \frac{t\_0}{\ell}\\ \mathbf{if}\;d \leq -3.1 \cdot 10^{-140}:\\ \;\;\;\;\left(\frac{\sqrt{0 - d}}{\sqrt{0 - h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{h \cdot t\_0}{\ell} \cdot t\_1\right)\\ \mathbf{elif}\;d \leq -1.5 \cdot 10^{-287}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(t\_2 \cdot \frac{t\_1}{\frac{1}{h}} + -1\right)\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{-230}:\\ \;\;\;\;\mathsf{fma}\left(D\_m \cdot D\_m, \frac{\sqrt{\frac{h}{\ell}}}{\ell} \cdot \frac{\left(M\_m \cdot M\_m\right) \cdot -0.125}{d}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t\_2 \cdot \frac{t\_1}{\frac{-1}{h}}\right) \cdot \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ (* 0.5 (* M_m D_m)) d)) (t_1 (* 0.5 t_0)) (t_2 (/ t_0 l)))
   (if (<= d -3.1e-140)
     (*
      (* (/ (sqrt (- 0.0 d)) (sqrt (- 0.0 h))) (sqrt (/ d l)))
      (- 1.0 (* (/ (* h t_0) l) t_1)))
     (if (<= d -1.5e-287)
       (* (* d (sqrt (/ 1.0 (* h l)))) (+ (* t_2 (/ t_1 (/ 1.0 h))) -1.0))
       (if (<= d 3.6e-230)
         (fma
          (* D_m D_m)
          (* (/ (sqrt (/ h l)) l) (/ (* (* M_m M_m) -0.125) d))
          0.0)
         (*
          (+ 1.0 (* t_2 (/ t_1 (/ -1.0 h))))
          (* (/ 1.0 (sqrt (/ h d))) (/ (sqrt d) (sqrt l)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (0.5 * (M_m * D_m)) / d;
	double t_1 = 0.5 * t_0;
	double t_2 = t_0 / l;
	double tmp;
	if (d <= -3.1e-140) {
		tmp = ((sqrt((0.0 - d)) / sqrt((0.0 - h))) * sqrt((d / l))) * (1.0 - (((h * t_0) / l) * t_1));
	} else if (d <= -1.5e-287) {
		tmp = (d * sqrt((1.0 / (h * l)))) * ((t_2 * (t_1 / (1.0 / h))) + -1.0);
	} else if (d <= 3.6e-230) {
		tmp = fma((D_m * D_m), ((sqrt((h / l)) / l) * (((M_m * M_m) * -0.125) / d)), 0.0);
	} else {
		tmp = (1.0 + (t_2 * (t_1 / (-1.0 / h)))) * ((1.0 / sqrt((h / d))) * (sqrt(d) / sqrt(l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(0.5 * Float64(M_m * D_m)) / d)
	t_1 = Float64(0.5 * t_0)
	t_2 = Float64(t_0 / l)
	tmp = 0.0
	if (d <= -3.1e-140)
		tmp = Float64(Float64(Float64(sqrt(Float64(0.0 - d)) / sqrt(Float64(0.0 - h))) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(Float64(Float64(h * t_0) / l) * t_1)));
	elseif (d <= -1.5e-287)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(Float64(t_2 * Float64(t_1 / Float64(1.0 / h))) + -1.0));
	elseif (d <= 3.6e-230)
		tmp = fma(Float64(D_m * D_m), Float64(Float64(sqrt(Float64(h / l)) / l) * Float64(Float64(Float64(M_m * M_m) * -0.125) / d)), 0.0);
	else
		tmp = Float64(Float64(1.0 + Float64(t_2 * Float64(t_1 / Float64(-1.0 / h)))) * Float64(Float64(1.0 / sqrt(Float64(h / d))) * Float64(sqrt(d) / sqrt(l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(0.5 * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / l), $MachinePrecision]}, If[LessEqual[d, -3.1e-140], N[(N[(N[(N[Sqrt[N[(0.0 - d), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(0.0 - h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(h * t$95$0), $MachinePrecision] / l), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.5e-287], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$2 * N[(t$95$1 / N[(1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.6e-230], N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * -0.125), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision], N[(N[(1.0 + N[(t$95$2 * N[(t$95$1 / N[(-1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.0999999999999999e-140

   1. Initial program 82.9%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 88.3%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. /-lowering-/.f64 88.3

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

   6. Applied egg-rr 88.3%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. neg-sub0 N/A

         \[\leadsto \left(\frac{\sqrt{\color{blue}{0 - d}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \left(\frac{\sqrt{\color{blue}{0 - d}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{\sqrt{0 - d}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \left(\frac{\sqrt{0 - d}}{\sqrt{\color{blue}{0 - h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}\right)\right) \]

      11. --lowering--.f64 96.1

          \[\leadsto \left(\frac{\sqrt{0 - d}}{\sqrt{\color{blue}{0 - h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)\right) \]

   8. Applied egg-rr 96.1%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{0 - d}}{\sqrt{0 - h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)\right) \]

   if -3.0999999999999999e-140 < d < -1.49999999999999996e-287

   1. Initial program 55.5%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. /-lowering-/.f64 N/A

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

      12. clear-num N/A

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

      13. associate-/r/ N/A

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 58.5%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. /-lowering-/.f64 58.5

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

   6. Applied egg-rr 58.5%

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

      1. metadata-eval N/A

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

      2. pow1/2 N/A

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

      3. frac-2neg N/A

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

      4. div-inv N/A

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

      5. sqrt-prod N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{\color{blue}{0 - d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{\color{blue}{0 - d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      12. neg-sub0 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{\color{blue}{0 - \ell}}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      13. --lowering--.f64 77.4

          \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{\color{blue}{0 - \ell}}}\right)\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   8. Applied egg-rr 77.4%

      \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   9. Taylor expanded in h around 0

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. rem-square-sqrt N/A

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. sqrt-lowering-sqrt.f64 N/A

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

       7. /-lowering-/.f64 N/A

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

       8. *-lowering-*.f64 N/A

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

       9. mul-1-neg N/A

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

       10. neg-sub0 N/A

           \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(0 - d\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

       11. --lowering--.f64 91.7

           \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(0 - d\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   11. Simplified 91.7%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(0 - d\right)\right)} \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   if -1.49999999999999996e-287 < d < 3.5999999999999998e-230

   1. Initial program 38.2%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. /-lowering-/.f64 N/A

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

      12. clear-num N/A

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

      13. associate-/r/ N/A

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 42.1%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. /-lowering-/.f64 42.1

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

   6. Applied egg-rr 42.1%

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

   7. Taylor expanded in h around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8}} \]

      2. associate-/l* N/A

         \[\leadsto \left(\color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left({D}^{2} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \cdot \frac{-1}{8} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{{D}^{2} \cdot \left(\left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8}\right)} \]

      5. *-commutative N/A

         \[\leadsto {D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]

      6. +-rgt-identity N/A

         \[\leadsto \color{blue}{{D}^{2} \cdot \left(\frac{-1}{8} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) + 0} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({D}^{2}, \frac{-1}{8} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right), 0\right)} \]

   9. Simplified 30.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(D \cdot D, \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}, 0\right)} \]
   10. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \sqrt{\color{blue}{\frac{\frac{h}{\ell}}{\ell \cdot \ell}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       2. sqrt-div N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \color{blue}{\frac{\sqrt{\frac{h}{\ell}}}{\sqrt{\ell \cdot \ell}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       3. pow2 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\sqrt{\frac{h}{\ell}}}{\sqrt{\color{blue}{{\ell}^{2}}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       4. sqrt-pow1 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\sqrt{\frac{h}{\ell}}}{\color{blue}{{\ell}^{\left(\frac{2}{2}\right)}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\sqrt{\frac{h}{\ell}}}{{\ell}^{\color{blue}{1}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       6. unpow1 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\sqrt{\frac{h}{\ell}}}{\color{blue}{\ell}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \color{blue}{\frac{\sqrt{\frac{h}{\ell}}}{\ell}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\color{blue}{\sqrt{\frac{h}{\ell}}}}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       9. /-lowering-/.f64 60.2

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\sqrt{\color{blue}{\frac{h}{\ell}}}}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}, 0\right) \]

   11. Applied egg-rr 60.2%

       \[\leadsto \mathsf{fma}\left(D \cdot D, \color{blue}{\frac{\sqrt{\frac{h}{\ell}}}{\ell}} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}, 0\right) \]

   if 3.5999999999999998e-230 < d

   1. Initial program 71.3%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. /-lowering-/.f64 N/A

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

      12. clear-num N/A

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

      13. associate-/r/ N/A

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 79.2%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. /-lowering-/.f64 79.6

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

   6. Applied egg-rr 79.6%

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

      1. metadata-eval N/A

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

      2. pow1/2 N/A

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

      3. sqrt-div N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 87.7

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

   8. Applied egg-rr 87.7%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.1 \cdot 10^{-140}:\\ \;\;\;\;\left(\frac{\sqrt{0 - d}}{\sqrt{0 - h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{h \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \left(0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)\right)\\ \mathbf{elif}\;d \leq -1.5 \cdot 10^{-287}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(\frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}} + -1\right)\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{-230}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot D, \frac{\sqrt{\frac{h}{\ell}}}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{-1}{h}}\right) \cdot \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 66.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\left(M\_m \cdot M\_m\right) \cdot -0.125}{d}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(D\_m \cdot D\_m, \frac{\sqrt{\frac{h}{\ell}}}{\ell} \cdot t\_0, 0\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(D\_m \cdot D\_m, t\_0 \cdot \frac{1}{\ell \cdot \sqrt{\frac{\ell}{h}}}, 0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ (* (* M_m M_m) -0.125) d))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (+
           1.0
           (* (/ h l) (* (pow (/ (* M_m D_m) (* d 2.0)) 2.0) (/ -1.0 2.0)))))))
   (if (<= t_1 -1e-40)
     (fma (* D_m D_m) (* (/ (sqrt (/ h l)) l) t_0) 0.0)
     (if (<= t_1 INFINITY)
       (/ (sqrt (/ d l)) (sqrt (/ h d)))
       (fma (* D_m D_m) (* t_0 (/ 1.0 (* l (sqrt (/ l h))))) 0.0)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = ((M_m * M_m) * -0.125) / d;
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double tmp;
	if (t_1 <= -1e-40) {
		tmp = fma((D_m * D_m), ((sqrt((h / l)) / l) * t_0), 0.0);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt((d / l)) / sqrt((h / d));
	} else {
		tmp = fma((D_m * D_m), (t_0 * (1.0 / (l * sqrt((l / h))))), 0.0);
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(Float64(M_m * M_m) * -0.125) / d)
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M_m * D_m) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))))
	tmp = 0.0
	if (t_1 <= -1e-40)
		tmp = fma(Float64(D_m * D_m), Float64(Float64(sqrt(Float64(h / l)) / l) * t_0), 0.0);
	elseif (t_1 <= Inf)
		tmp = Float64(sqrt(Float64(d / l)) / sqrt(Float64(h / d)));
	else
		tmp = fma(Float64(D_m * D_m), Float64(t_0 * Float64(1.0 / Float64(l * sqrt(Float64(l / h))))), 0.0);
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * -0.125), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = 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[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-40], N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] / l), $MachinePrecision] * t$95$0), $MachinePrecision] + 0.0), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(t$95$0 * N[(1.0 / N[(l * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -9.9999999999999993e-41

   1. Initial program 89.0%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. /-lowering-/.f64 N/A

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

      12. clear-num N/A

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

      13. associate-/r/ N/A

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 92.8%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. /-lowering-/.f64 91.8

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

   6. Applied egg-rr 91.8%

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

   7. Taylor expanded in h around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8}} \]

      2. associate-/l* N/A

         \[\leadsto \left(\color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left({D}^{2} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \cdot \frac{-1}{8} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{{D}^{2} \cdot \left(\left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8}\right)} \]

      5. *-commutative N/A

         \[\leadsto {D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]

      6. +-rgt-identity N/A

         \[\leadsto \color{blue}{{D}^{2} \cdot \left(\frac{-1}{8} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) + 0} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({D}^{2}, \frac{-1}{8} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right), 0\right)} \]

   9. Simplified 30.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(D \cdot D, \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}, 0\right)} \]
   10. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \sqrt{\color{blue}{\frac{\frac{h}{\ell}}{\ell \cdot \ell}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       2. sqrt-div N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \color{blue}{\frac{\sqrt{\frac{h}{\ell}}}{\sqrt{\ell \cdot \ell}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       3. pow2 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\sqrt{\frac{h}{\ell}}}{\sqrt{\color{blue}{{\ell}^{2}}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       4. sqrt-pow1 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\sqrt{\frac{h}{\ell}}}{\color{blue}{{\ell}^{\left(\frac{2}{2}\right)}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\sqrt{\frac{h}{\ell}}}{{\ell}^{\color{blue}{1}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       6. unpow1 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\sqrt{\frac{h}{\ell}}}{\color{blue}{\ell}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \color{blue}{\frac{\sqrt{\frac{h}{\ell}}}{\ell}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\color{blue}{\sqrt{\frac{h}{\ell}}}}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       9. /-lowering-/.f64 73.4

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\sqrt{\color{blue}{\frac{h}{\ell}}}}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}, 0\right) \]

   11. Applied egg-rr 73.4%

       \[\leadsto \mathsf{fma}\left(D \cdot D, \color{blue}{\frac{\sqrt{\frac{h}{\ell}}}{\ell}} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}, 0\right) \]

   if -9.9999999999999993e-41 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 82.7%

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

   3. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. *-lowering-*.f64 56.9

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

   5. Simplified 56.9%

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

   6. Taylor expanded in d around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}} + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}, 0\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}, 0\right) \]

      5. *-lowering-*.f64 43.4

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}, 0\right) \]

   8. Simplified 43.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]
   9. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. *-lowering-*.f64 43.3

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

   10. Applied egg-rr 43.3%

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

       1. rem-square-sqrt N/A

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

       2. sqrt-prod N/A

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

       3. frac-times N/A

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

       4. sqrt-div N/A

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

       5. sqrt-div N/A

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

       6. frac-2neg N/A

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

       7. sub0-neg N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{\mathsf{neg}\left(d\right)}{\color{blue}{0 - \ell}}} \]

       8. div-inv N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{0 - \ell}}} \]

       9. sqrt-prod N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{0 - \ell}}\right)} \]

       10. sub0-neg N/A

           \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\color{blue}{0 - d}} \cdot \sqrt{\frac{1}{0 - \ell}}\right) \]

       11. *-commutative N/A

           \[\leadsto \color{blue}{\left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

       12. clear-num N/A

           \[\leadsto \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \]

       13. sqrt-div N/A

           \[\leadsto \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \]

       14. metadata-eval N/A

           \[\leadsto \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right) \cdot \frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \]

       15. un-div-inv N/A

           \[\leadsto \color{blue}{\frac{\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}}{\sqrt{\frac{h}{d}}}} \]

       16. /-lowering-/.f64 N/A

           \[\leadsto \color{blue}{\frac{\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}}{\sqrt{\frac{h}{d}}}} \]

   12. Applied egg-rr 81.4%

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

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

   1. Initial program 0.0%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. /-lowering-/.f64 N/A

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

      12. clear-num N/A

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

      13. associate-/r/ N/A

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 22.9%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. /-lowering-/.f64 22.9

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

   6. Applied egg-rr 22.9%

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

   7. Taylor expanded in h around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8}} \]

      2. associate-/l* N/A

         \[\leadsto \left(\color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left({D}^{2} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \cdot \frac{-1}{8} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{{D}^{2} \cdot \left(\left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8}\right)} \]

      5. *-commutative N/A

         \[\leadsto {D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]

      6. +-rgt-identity N/A

         \[\leadsto \color{blue}{{D}^{2} \cdot \left(\frac{-1}{8} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) + 0} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({D}^{2}, \frac{-1}{8} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right), 0\right)} \]

   9. Simplified 7.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(D \cdot D, \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}, 0\right)} \]
   10. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \sqrt{\color{blue}{\frac{1}{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       2. sqrt-div N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\color{blue}{1}}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \color{blue}{\frac{1}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{1}{\sqrt{\frac{\color{blue}{\left(\ell \cdot \ell\right) \cdot \ell}}{h}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       6. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{1}{\sqrt{\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\ell}{h}}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       7. un-div-inv N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{1}{\sqrt{\left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot \frac{1}{h}\right)}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       8. sqrt-prod N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{1}{\color{blue}{\sqrt{\ell \cdot \ell} \cdot \sqrt{\ell \cdot \frac{1}{h}}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       9. pow2 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{1}{\sqrt{\color{blue}{{\ell}^{2}}} \cdot \sqrt{\ell \cdot \frac{1}{h}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       10. sqrt-pow1 N/A

           \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{1}{\color{blue}{{\ell}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\ell \cdot \frac{1}{h}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{1}{{\ell}^{\color{blue}{1}} \cdot \sqrt{\ell \cdot \frac{1}{h}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       12. unpow1 N/A

           \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{1}{\color{blue}{\ell} \cdot \sqrt{\ell \cdot \frac{1}{h}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{1}{\color{blue}{\ell \cdot \sqrt{\ell \cdot \frac{1}{h}}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       14. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{1}{\ell \cdot \color{blue}{\sqrt{\ell \cdot \frac{1}{h}}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       15. un-div-inv N/A

           \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{1}{\ell \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       16. /-lowering-/.f64 27.1

           \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{1}{\ell \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}, 0\right) \]

   11. Applied egg-rr 27.1%

       \[\leadsto \mathsf{fma}\left(D \cdot D, \color{blue}{\frac{1}{\ell \cdot \sqrt{\frac{\ell}{h}}}} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}, 0\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq -1 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot D, \frac{\sqrt{\frac{h}{\ell}}}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}, 0\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot D, \frac{\left(M \cdot M\right) \cdot -0.125}{d} \cdot \frac{1}{\ell \cdot \sqrt{\frac{\ell}{h}}}, 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(D\_m \cdot D\_m, \frac{\sqrt{\frac{h}{\ell}}}{\ell} \cdot \frac{\left(M\_m \cdot M\_m\right) \cdot -0.125}{d}, 0\right)\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (fma
          (* D_m D_m)
          (* (/ (sqrt (/ h l)) l) (/ (* (* M_m M_m) -0.125) d))
          0.0))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (+
           1.0
           (* (/ h l) (* (pow (/ (* M_m D_m) (* d 2.0)) 2.0) (/ -1.0 2.0)))))))
   (if (<= t_1 -1e-40)
     t_0
     (if (<= t_1 INFINITY) (/ (sqrt (/ d l)) (sqrt (/ h d))) t_0))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = fma((D_m * D_m), ((sqrt((h / l)) / l) * (((M_m * M_m) * -0.125) / d)), 0.0);
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double tmp;
	if (t_1 <= -1e-40) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt((d / l)) / sqrt((h / d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = fma(Float64(D_m * D_m), Float64(Float64(sqrt(Float64(h / l)) / l) * Float64(Float64(Float64(M_m * M_m) * -0.125) / d)), 0.0)
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M_m * D_m) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))))
	tmp = 0.0
	if (t_1 <= -1e-40)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(sqrt(Float64(d / l)) / sqrt(Float64(h / d)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * -0.125), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]}, Block[{t$95$1 = 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[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-40], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -9.9999999999999993e-41 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 60.4%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. /-lowering-/.f64 N/A

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

      12. clear-num N/A

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

      13. associate-/r/ N/A

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 70.3%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. /-lowering-/.f64 69.6

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

   6. Applied egg-rr 69.6%

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

   7. Taylor expanded in h around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8}} \]

      2. associate-/l* N/A

         \[\leadsto \left(\color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left({D}^{2} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \cdot \frac{-1}{8} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{{D}^{2} \cdot \left(\left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8}\right)} \]

      5. *-commutative N/A

         \[\leadsto {D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]

      6. +-rgt-identity N/A

         \[\leadsto \color{blue}{{D}^{2} \cdot \left(\frac{-1}{8} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) + 0} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({D}^{2}, \frac{-1}{8} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right), 0\right)} \]

   9. Simplified 22.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(D \cdot D, \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}, 0\right)} \]
   10. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \sqrt{\color{blue}{\frac{\frac{h}{\ell}}{\ell \cdot \ell}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       2. sqrt-div N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \color{blue}{\frac{\sqrt{\frac{h}{\ell}}}{\sqrt{\ell \cdot \ell}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       3. pow2 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\sqrt{\frac{h}{\ell}}}{\sqrt{\color{blue}{{\ell}^{2}}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       4. sqrt-pow1 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\sqrt{\frac{h}{\ell}}}{\color{blue}{{\ell}^{\left(\frac{2}{2}\right)}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\sqrt{\frac{h}{\ell}}}{{\ell}^{\color{blue}{1}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       6. unpow1 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\sqrt{\frac{h}{\ell}}}{\color{blue}{\ell}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \color{blue}{\frac{\sqrt{\frac{h}{\ell}}}{\ell}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\color{blue}{\sqrt{\frac{h}{\ell}}}}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       9. /-lowering-/.f64 58.5

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\sqrt{\color{blue}{\frac{h}{\ell}}}}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}, 0\right) \]

   11. Applied egg-rr 58.5%

       \[\leadsto \mathsf{fma}\left(D \cdot D, \color{blue}{\frac{\sqrt{\frac{h}{\ell}}}{\ell}} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}, 0\right) \]

   if -9.9999999999999993e-41 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 82.7%

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

   3. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. *-lowering-*.f64 56.9

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

   5. Simplified 56.9%

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

   6. Taylor expanded in d around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}} + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}, 0\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}, 0\right) \]

      5. *-lowering-*.f64 43.4

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}, 0\right) \]

   8. Simplified 43.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]
   9. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. *-lowering-*.f64 43.3

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

   10. Applied egg-rr 43.3%

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

       1. rem-square-sqrt N/A

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

       2. sqrt-prod N/A

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

       3. frac-times N/A

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

       4. sqrt-div N/A

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

       5. sqrt-div N/A

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

       6. frac-2neg N/A

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

       7. sub0-neg N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{\mathsf{neg}\left(d\right)}{\color{blue}{0 - \ell}}} \]

       8. div-inv N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{0 - \ell}}} \]

       9. sqrt-prod N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{0 - \ell}}\right)} \]

       10. sub0-neg N/A

           \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\color{blue}{0 - d}} \cdot \sqrt{\frac{1}{0 - \ell}}\right) \]

       11. *-commutative N/A

           \[\leadsto \color{blue}{\left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

       12. clear-num N/A

           \[\leadsto \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \]

       13. sqrt-div N/A

           \[\leadsto \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \]

       14. metadata-eval N/A

           \[\leadsto \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right) \cdot \frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \]

       15. un-div-inv N/A

           \[\leadsto \color{blue}{\frac{\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}}{\sqrt{\frac{h}{d}}}} \]

       16. /-lowering-/.f64 N/A

           \[\leadsto \color{blue}{\frac{\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}}{\sqrt{\frac{h}{d}}}} \]

   12. Applied egg-rr 81.4%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq -1 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot D, \frac{\sqrt{\frac{h}{\ell}}}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}, 0\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot D, \frac{\sqrt{\frac{h}{\ell}}}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}, 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m \cdot D\_m}{d \cdot 2}\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({t\_0}^{2} \cdot \frac{-1}{2}\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - t\_0 \cdot \left(\frac{h}{\ell} \cdot \frac{D\_m \cdot \left(M\_m \cdot 0.25\right)}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(D\_m \cdot D\_m, \frac{\left(M\_m \cdot M\_m\right) \cdot -0.125}{d} \cdot \frac{1}{\ell \cdot \sqrt{\frac{\ell}{h}}}, 0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ (* M_m D_m) (* d 2.0))))
   (if (<=
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (+ 1.0 (* (/ h l) (* (pow t_0 2.0) (/ -1.0 2.0)))))
        INFINITY)
     (*
      (sqrt (/ d l))
      (*
       (sqrt (/ d h))
       (- 1.0 (* t_0 (* (/ h l) (/ (* D_m (* M_m 0.25)) d))))))
     (fma
      (* D_m D_m)
      (* (/ (* (* M_m M_m) -0.125) d) (/ 1.0 (* l (sqrt (/ l h)))))
      0.0))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (M_m * D_m) / (d * 2.0);
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (pow(t_0, 2.0) * (-1.0 / 2.0))))) <= ((double) INFINITY)) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 - (t_0 * ((h / l) * ((D_m * (M_m * 0.25)) / d)))));
	} else {
		tmp = fma((D_m * D_m), ((((M_m * M_m) * -0.125) / d) * (1.0 / (l * sqrt((l / h))))), 0.0);
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(M_m * D_m) / Float64(d * 2.0))
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 + Float64(Float64(h / l) * Float64((t_0 ^ 2.0) * Float64(-1.0 / 2.0))))) <= Inf)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 - Float64(t_0 * Float64(Float64(h / l) * Float64(Float64(D_m * Float64(M_m * 0.25)) / d))))));
	else
		tmp = fma(Float64(D_m * D_m), Float64(Float64(Float64(Float64(M_m * M_m) * -0.125) / d) * Float64(1.0 / Float64(l * sqrt(Float64(l / h))))), 0.0);
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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[(h / l), $MachinePrecision] * N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * N[(N[(D$95$m * N[(M$95$m * 0.25), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * -0.125), $MachinePrecision] / d), $MachinePrecision] * N[(1.0 / N[(l * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 85.6%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. /-lowering-/.f64 N/A

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

      12. clear-num N/A

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

      13. associate-/r/ N/A

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 87.3%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. /-lowering-/.f64 87.1

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

   6. Applied egg-rr 87.1%

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

   8. Applied egg-rr 70.8%

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

      1. associate-*l/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. frac-times N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

   10. Applied egg-rr 87.4%

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

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

   1. Initial program 0.0%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. /-lowering-/.f64 N/A

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

      12. clear-num N/A

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

      13. associate-/r/ N/A

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 22.9%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. /-lowering-/.f64 22.9

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

   6. Applied egg-rr 22.9%

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

   7. Taylor expanded in h around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8}} \]

      2. associate-/l* N/A

         \[\leadsto \left(\color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left({D}^{2} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \cdot \frac{-1}{8} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{{D}^{2} \cdot \left(\left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8}\right)} \]

      5. *-commutative N/A

         \[\leadsto {D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]

      6. +-rgt-identity N/A

         \[\leadsto \color{blue}{{D}^{2} \cdot \left(\frac{-1}{8} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) + 0} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({D}^{2}, \frac{-1}{8} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right), 0\right)} \]

   9. Simplified 7.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(D \cdot D, \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}, 0\right)} \]
   10. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \sqrt{\color{blue}{\frac{1}{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       2. sqrt-div N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\color{blue}{1}}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \color{blue}{\frac{1}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{1}{\sqrt{\frac{\color{blue}{\left(\ell \cdot \ell\right) \cdot \ell}}{h}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       6. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{1}{\sqrt{\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\ell}{h}}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       7. un-div-inv N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{1}{\sqrt{\left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot \frac{1}{h}\right)}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       8. sqrt-prod N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{1}{\color{blue}{\sqrt{\ell \cdot \ell} \cdot \sqrt{\ell \cdot \frac{1}{h}}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       9. pow2 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{1}{\sqrt{\color{blue}{{\ell}^{2}}} \cdot \sqrt{\ell \cdot \frac{1}{h}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       10. sqrt-pow1 N/A

           \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{1}{\color{blue}{{\ell}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\ell \cdot \frac{1}{h}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{1}{{\ell}^{\color{blue}{1}} \cdot \sqrt{\ell \cdot \frac{1}{h}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       12. unpow1 N/A

           \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{1}{\color{blue}{\ell} \cdot \sqrt{\ell \cdot \frac{1}{h}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{1}{\color{blue}{\ell \cdot \sqrt{\ell \cdot \frac{1}{h}}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       14. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{1}{\ell \cdot \color{blue}{\sqrt{\ell \cdot \frac{1}{h}}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       15. un-div-inv N/A

           \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{1}{\ell \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       16. /-lowering-/.f64 27.1

           \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{1}{\ell \cdot \sqrt{\color{blue}{\frac{\ell}{h}}}} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}, 0\right) \]

   11. Applied egg-rr 27.1%

       \[\leadsto \mathsf{fma}\left(D \cdot D, \color{blue}{\frac{1}{\ell \cdot \sqrt{\frac{\ell}{h}}}} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}, 0\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - \frac{M \cdot D}{d \cdot 2} \cdot \left(\frac{h}{\ell} \cdot \frac{D \cdot \left(M \cdot 0.25\right)}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot D, \frac{\left(M \cdot M\right) \cdot -0.125}{d} \cdot \frac{1}{\ell \cdot \sqrt{\frac{\ell}{h}}}, 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq -4 \cdot 10^{+47}:\\ \;\;\;\;D\_m \cdot \left(D\_m \cdot \frac{\sqrt{h} \cdot \left(M\_m \cdot \left(M\_m \cdot -0.125\right)\right)}{d \cdot \left(\ell \cdot \sqrt{\ell}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
       (+
        1.0
        (* (/ h l) (* (pow (/ (* M_m D_m) (* d 2.0)) 2.0) (/ -1.0 2.0)))))
      -4e+47)
   (* D_m (* D_m (/ (* (sqrt h) (* M_m (* M_m -0.125))) (* d (* l (sqrt l))))))
   (/ (sqrt (/ d l)) (sqrt (/ h d)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))))) <= -4e+47) {
		tmp = D_m * (D_m * ((sqrt(h) * (M_m * (M_m * -0.125))) / (d * (l * sqrt(l)))));
	} else {
		tmp = sqrt((d / l)) / sqrt((h / d));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (((((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 + ((h / l) * ((((m_m * d_m) / (d * 2.0d0)) ** 2.0d0) * ((-1.0d0) / 2.0d0))))) <= (-4d+47)) then
        tmp = d_m * (d_m * ((sqrt(h) * (m_m * (m_m * (-0.125d0)))) / (d * (l * sqrt(l)))))
    else
        tmp = sqrt((d / l)) / sqrt((h / d))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (((Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (Math.pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))))) <= -4e+47) {
		tmp = D_m * (D_m * ((Math.sqrt(h) * (M_m * (M_m * -0.125))) / (d * (l * Math.sqrt(l)))));
	} else {
		tmp = Math.sqrt((d / l)) / Math.sqrt((h / d));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if ((math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (math.pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))))) <= -4e+47:
		tmp = D_m * (D_m * ((math.sqrt(h) * (M_m * (M_m * -0.125))) / (d * (l * math.sqrt(l)))))
	else:
		tmp = math.sqrt((d / l)) / math.sqrt((h / d))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M_m * D_m) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0))))) <= -4e+47)
		tmp = Float64(D_m * Float64(D_m * Float64(Float64(sqrt(h) * Float64(M_m * Float64(M_m * -0.125))) / Float64(d * Float64(l * sqrt(l))))));
	else
		tmp = Float64(sqrt(Float64(d / l)) / sqrt(Float64(h / d)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (((((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 + ((h / l) * ((((M_m * D_m) / (d * 2.0)) ^ 2.0) * (-1.0 / 2.0))))) <= -4e+47)
		tmp = D_m * (D_m * ((sqrt(h) * (M_m * (M_m * -0.125))) / (d * (l * sqrt(l)))));
	else
		tmp = sqrt((d / l)) / sqrt((h / d));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[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[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e+47], N[(D$95$m * N[(D$95$m * N[(N[(N[Sqrt[h], $MachinePrecision] * N[(M$95$m * N[(M$95$m * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(l * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.0000000000000002e47

   1. Initial program 89.0%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. /-lowering-/.f64 N/A

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

      12. clear-num N/A

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

      13. associate-/r/ N/A

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 92.8%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. /-lowering-/.f64 91.7

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

   6. Applied egg-rr 91.7%

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

   7. Taylor expanded in h around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8}} \]

      2. associate-/l* N/A

         \[\leadsto \left(\color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left({D}^{2} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \cdot \frac{-1}{8} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{{D}^{2} \cdot \left(\left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8}\right)} \]

      5. *-commutative N/A

         \[\leadsto {D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]

      6. +-rgt-identity N/A

         \[\leadsto \color{blue}{{D}^{2} \cdot \left(\frac{-1}{8} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) + 0} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({D}^{2}, \frac{-1}{8} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right), 0\right)} \]

   9. Simplified 30.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(D \cdot D, \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}, 0\right)} \]
   10. Step-by-step derivation

       1. +-rgt-identity N/A

          \[\leadsto \color{blue}{\left(D \cdot D\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}\right)} \]

       2. associate-*l* N/A

          \[\leadsto \color{blue}{D \cdot \left(D \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}\right)\right)} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{\left(D \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}\right)\right) \cdot D} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(D \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}\right)\right) \cdot D} \]

   11. Applied egg-rr 35.7%

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

   if -4.0000000000000002e47 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 59.0%

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

   3. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. *-lowering-*.f64 44.8

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

   5. Simplified 44.8%

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

   6. Taylor expanded in d around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}} + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}, 0\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}, 0\right) \]

      5. *-lowering-*.f64 33.8

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}, 0\right) \]

   8. Simplified 33.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]
   9. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. *-lowering-*.f64 33.7

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

   10. Applied egg-rr 33.7%

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

       1. rem-square-sqrt N/A

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

       2. sqrt-prod N/A

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

       3. frac-times N/A

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

       4. sqrt-div N/A

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

       5. sqrt-div N/A

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

       6. frac-2neg N/A

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

       7. sub0-neg N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{\mathsf{neg}\left(d\right)}{\color{blue}{0 - \ell}}} \]

       8. div-inv N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{0 - \ell}}} \]

       9. sqrt-prod N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{0 - \ell}}\right)} \]

       10. sub0-neg N/A

           \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\color{blue}{0 - d}} \cdot \sqrt{\frac{1}{0 - \ell}}\right) \]

       11. *-commutative N/A

           \[\leadsto \color{blue}{\left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

       12. clear-num N/A

           \[\leadsto \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \]

       13. sqrt-div N/A

           \[\leadsto \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \]

       14. metadata-eval N/A

           \[\leadsto \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right) \cdot \frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \]

       15. un-div-inv N/A

           \[\leadsto \color{blue}{\frac{\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}}{\sqrt{\frac{h}{d}}}} \]

       16. /-lowering-/.f64 N/A

           \[\leadsto \color{blue}{\frac{\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}}{\sqrt{\frac{h}{d}}}} \]

   12. Applied egg-rr 61.9%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq -4 \cdot 10^{+47}:\\ \;\;\;\;D \cdot \left(D \cdot \frac{\sqrt{h} \cdot \left(M \cdot \left(M \cdot -0.125\right)\right)}{d \cdot \left(\ell \cdot \sqrt{\ell}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 0:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
       (+
        1.0
        (* (/ h l) (* (pow (/ (* M_m D_m) (* d 2.0)) 2.0) (/ -1.0 2.0)))))
      0.0)
   (/ d (sqrt (* h l)))
   (/ (sqrt (/ d l)) (sqrt (/ h d)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))))) <= 0.0) {
		tmp = d / sqrt((h * l));
	} else {
		tmp = sqrt((d / l)) / sqrt((h / d));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (((((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 + ((h / l) * ((((m_m * d_m) / (d * 2.0d0)) ** 2.0d0) * ((-1.0d0) / 2.0d0))))) <= 0.0d0) then
        tmp = d / sqrt((h * l))
    else
        tmp = sqrt((d / l)) / sqrt((h / d))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (((Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (Math.pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))))) <= 0.0) {
		tmp = d / Math.sqrt((h * l));
	} else {
		tmp = Math.sqrt((d / l)) / Math.sqrt((h / d));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if ((math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (math.pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))))) <= 0.0:
		tmp = d / math.sqrt((h * l))
	else:
		tmp = math.sqrt((d / l)) / math.sqrt((h / d))
	return tmp

Julia

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

MATLAB

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[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[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 84.2%

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

   3. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. *-lowering-*.f64 58.3

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

   5. Simplified 58.3%

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

   6. Taylor expanded in d around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}} + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}, 0\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}, 0\right) \]

      5. *-lowering-*.f64 19.1

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}, 0\right) \]

   8. Simplified 19.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]
   9. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. *-lowering-*.f64 18.3

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

   10. Applied egg-rr 18.3%

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

       1. frac-2neg N/A

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

       2. distribute-frac-neg N/A

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

       3. neg-lowering-neg.f64 N/A

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

       4. /-lowering-/.f64 N/A

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

       5. neg-sub0 N/A

          \[\leadsto \mathsf{neg}\left(\frac{d}{\color{blue}{0 - \sqrt{h \cdot \ell}}}\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{d}{\color{blue}{0 - \sqrt{h \cdot \ell}}}\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{d}{0 - \color{blue}{\sqrt{h \cdot \ell}}}\right) \]

       8. *-lowering-*.f64 20.7

          \[\leadsto -\frac{d}{0 - \sqrt{\color{blue}{h \cdot \ell}}} \]

   12. Applied egg-rr 20.7%

       \[\leadsto \color{blue}{-\frac{d}{0 - \sqrt{h \cdot \ell}}} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 58.6%

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

   3. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. *-lowering-*.f64 46.7

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

   5. Simplified 46.7%

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

   6. Taylor expanded in d around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}} + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}, 0\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}, 0\right) \]

      5. *-lowering-*.f64 29.9

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}, 0\right) \]

   8. Simplified 29.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]
   9. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. *-lowering-*.f64 29.8

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

   10. Applied egg-rr 29.8%

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

       1. rem-square-sqrt N/A

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

       2. sqrt-prod N/A

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

       3. frac-times N/A

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

       4. sqrt-div N/A

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

       5. sqrt-div N/A

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

       6. frac-2neg N/A

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

       7. sub0-neg N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{\mathsf{neg}\left(d\right)}{\color{blue}{0 - \ell}}} \]

       8. div-inv N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{0 - \ell}}} \]

       9. sqrt-prod N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{0 - \ell}}\right)} \]

       10. sub0-neg N/A

           \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\color{blue}{0 - d}} \cdot \sqrt{\frac{1}{0 - \ell}}\right) \]

       11. *-commutative N/A

           \[\leadsto \color{blue}{\left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

       12. clear-num N/A

           \[\leadsto \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \]

       13. sqrt-div N/A

           \[\leadsto \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \]

       14. metadata-eval N/A

           \[\leadsto \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right) \cdot \frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \]

       15. un-div-inv N/A

           \[\leadsto \color{blue}{\frac{\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}}{\sqrt{\frac{h}{d}}}} \]

       16. /-lowering-/.f64 N/A

           \[\leadsto \color{blue}{\frac{\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}}{\sqrt{\frac{h}{d}}}} \]

   12. Applied egg-rr 64.0%

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

4. Final simplification 43.3%

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

Alternative 7: 43.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 0:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
       (+
        1.0
        (* (/ h l) (* (pow (/ (* M_m D_m) (* d 2.0)) 2.0) (/ -1.0 2.0)))))
      0.0)
   (/ d (sqrt (* h l)))
   (* (sqrt (/ d l)) (sqrt (/ d h)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))))) <= 0.0) {
		tmp = d / sqrt((h * l));
	} else {
		tmp = sqrt((d / l)) * sqrt((d / h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (((((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 + ((h / l) * ((((m_m * d_m) / (d * 2.0d0)) ** 2.0d0) * ((-1.0d0) / 2.0d0))))) <= 0.0d0) then
        tmp = d / sqrt((h * l))
    else
        tmp = sqrt((d / l)) * sqrt((d / h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (((Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (Math.pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))))) <= 0.0) {
		tmp = d / Math.sqrt((h * l));
	} else {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if ((math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (math.pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))))) <= 0.0:
		tmp = d / math.sqrt((h * l))
	else:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	return tmp

Julia

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

MATLAB

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[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[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 84.2%

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

   3. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. *-lowering-*.f64 58.3

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

   5. Simplified 58.3%

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

   6. Taylor expanded in d around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}} + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}, 0\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}, 0\right) \]

      5. *-lowering-*.f64 19.1

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}, 0\right) \]

   8. Simplified 19.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]
   9. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. *-lowering-*.f64 18.3

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

   10. Applied egg-rr 18.3%

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

       1. frac-2neg N/A

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

       2. distribute-frac-neg N/A

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

       3. neg-lowering-neg.f64 N/A

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

       4. /-lowering-/.f64 N/A

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

       5. neg-sub0 N/A

          \[\leadsto \mathsf{neg}\left(\frac{d}{\color{blue}{0 - \sqrt{h \cdot \ell}}}\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{d}{\color{blue}{0 - \sqrt{h \cdot \ell}}}\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{d}{0 - \color{blue}{\sqrt{h \cdot \ell}}}\right) \]

       8. *-lowering-*.f64 20.7

          \[\leadsto -\frac{d}{0 - \sqrt{\color{blue}{h \cdot \ell}}} \]

   12. Applied egg-rr 20.7%

       \[\leadsto \color{blue}{-\frac{d}{0 - \sqrt{h \cdot \ell}}} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 58.6%

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

   3. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. *-lowering-*.f64 46.7

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

   5. Simplified 46.7%

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

   6. Taylor expanded in d around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}} + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}, 0\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}, 0\right) \]

      5. *-lowering-*.f64 29.9

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}, 0\right) \]

   8. Simplified 29.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]
   9. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. *-lowering-*.f64 29.8

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

   10. Applied egg-rr 29.8%

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

       1. rem-square-sqrt N/A

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

       2. sqrt-prod N/A

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

       3. frac-times N/A

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

       4. sqrt-div N/A

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

       5. sqrt-div N/A

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

       6. frac-2neg N/A

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

       7. sub0-neg N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{\mathsf{neg}\left(d\right)}{\color{blue}{0 - \ell}}} \]

       8. div-inv N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{0 - \ell}}} \]

       9. sqrt-prod N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{0 - \ell}}\right)} \]

       10. sub0-neg N/A

           \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\color{blue}{0 - d}} \cdot \sqrt{\frac{1}{0 - \ell}}\right) \]

       11. *-commutative N/A

           \[\leadsto \color{blue}{\left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \color{blue}{\left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

   12. Applied egg-rr 63.5%

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

4. Final simplification 43.0%

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

Alternative 8: 78.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{0.5 \cdot \left(M\_m \cdot D\_m\right)}{d}\\ t_1 := \frac{h \cdot t\_0}{\ell}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := 0.5 \cdot t\_0\\ \mathbf{if}\;d \leq -9.4 \cdot 10^{-108}:\\ \;\;\;\;\left(1 - t\_1 \cdot \left(0.5 \cdot \left(\left(M\_m \cdot D\_m\right) \cdot \frac{0.5}{d}\right)\right)\right) \cdot \left(t\_2 \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;d \leq -3 \cdot 10^{-287}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(\frac{t\_0}{\ell} \cdot \frac{t\_3}{\frac{1}{h}} + -1\right)\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{-220}:\\ \;\;\;\;\mathsf{fma}\left(D\_m \cdot D\_m, \frac{\sqrt{\frac{h}{\ell}}}{\ell} \cdot \frac{\left(M\_m \cdot M\_m\right) \cdot -0.125}{d}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - t\_1 \cdot t\_3\right) \cdot \left(t\_2 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ (* 0.5 (* M_m D_m)) d))
        (t_1 (/ (* h t_0) l))
        (t_2 (sqrt (/ d l)))
        (t_3 (* 0.5 t_0)))
   (if (<= d -9.4e-108)
     (*
      (- 1.0 (* t_1 (* 0.5 (* (* M_m D_m) (/ 0.5 d)))))
      (* t_2 (sqrt (/ d h))))
     (if (<= d -3e-287)
       (*
        (* d (sqrt (/ 1.0 (* h l))))
        (+ (* (/ t_0 l) (/ t_3 (/ 1.0 h))) -1.0))
       (if (<= d 4.6e-220)
         (fma
          (* D_m D_m)
          (* (/ (sqrt (/ h l)) l) (/ (* (* M_m M_m) -0.125) d))
          0.0)
         (* (- 1.0 (* t_1 t_3)) (* t_2 (/ (sqrt d) (sqrt h)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (0.5 * (M_m * D_m)) / d;
	double t_1 = (h * t_0) / l;
	double t_2 = sqrt((d / l));
	double t_3 = 0.5 * t_0;
	double tmp;
	if (d <= -9.4e-108) {
		tmp = (1.0 - (t_1 * (0.5 * ((M_m * D_m) * (0.5 / d))))) * (t_2 * sqrt((d / h)));
	} else if (d <= -3e-287) {
		tmp = (d * sqrt((1.0 / (h * l)))) * (((t_0 / l) * (t_3 / (1.0 / h))) + -1.0);
	} else if (d <= 4.6e-220) {
		tmp = fma((D_m * D_m), ((sqrt((h / l)) / l) * (((M_m * M_m) * -0.125) / d)), 0.0);
	} else {
		tmp = (1.0 - (t_1 * t_3)) * (t_2 * (sqrt(d) / sqrt(h)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(0.5 * Float64(M_m * D_m)) / d)
	t_1 = Float64(Float64(h * t_0) / l)
	t_2 = sqrt(Float64(d / l))
	t_3 = Float64(0.5 * t_0)
	tmp = 0.0
	if (d <= -9.4e-108)
		tmp = Float64(Float64(1.0 - Float64(t_1 * Float64(0.5 * Float64(Float64(M_m * D_m) * Float64(0.5 / d))))) * Float64(t_2 * sqrt(Float64(d / h))));
	elseif (d <= -3e-287)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(Float64(Float64(t_0 / l) * Float64(t_3 / Float64(1.0 / h))) + -1.0));
	elseif (d <= 4.6e-220)
		tmp = fma(Float64(D_m * D_m), Float64(Float64(sqrt(Float64(h / l)) / l) * Float64(Float64(Float64(M_m * M_m) * -0.125) / d)), 0.0);
	else
		tmp = Float64(Float64(1.0 - Float64(t_1 * t_3)) * Float64(t_2 * Float64(sqrt(d) / sqrt(h))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(0.5 * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[(h * t$95$0), $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(0.5 * t$95$0), $MachinePrecision]}, If[LessEqual[d, -9.4e-108], N[(N[(1.0 - N[(t$95$1 * N[(0.5 * N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -3e-287], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 / l), $MachinePrecision] * N[(t$95$3 / N[(1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.6e-220], N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * -0.125), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision], N[(N[(1.0 - N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -9.40000000000000026e-108

   1. Initial program 83.5%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 89.2%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. /-lowering-/.f64 89.2

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

   6. Applied egg-rr 89.2%

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. metadata-eval 89.2

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

   8. Applied egg-rr 89.2%

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

      1. metadata-eval N/A

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

      2. pow1/2 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. /-lowering-/.f64 89.2

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

   10. Applied egg-rr 89.2%

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

   if -9.40000000000000026e-108 < d < -2.99999999999999992e-287

   1. Initial program 57.4%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. /-lowering-/.f64 N/A

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

      12. clear-num N/A

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

      13. associate-/r/ N/A

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 57.9%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. /-lowering-/.f64 57.8

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

   6. Applied egg-rr 57.8%

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

      1. metadata-eval N/A

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

      2. pow1/2 N/A

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

      3. frac-2neg N/A

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

      4. div-inv N/A

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

      5. sqrt-prod N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{\color{blue}{0 - d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{\color{blue}{0 - d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      12. neg-sub0 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{\color{blue}{0 - \ell}}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      13. --lowering--.f64 75.5

          \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{\color{blue}{0 - \ell}}}\right)\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   8. Applied egg-rr 75.5%

      \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   9. Taylor expanded in h around 0

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. rem-square-sqrt N/A

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. sqrt-lowering-sqrt.f64 N/A

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

       7. /-lowering-/.f64 N/A

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

       8. *-lowering-*.f64 N/A

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

       9. mul-1-neg N/A

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

       10. neg-sub0 N/A

           \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(0 - d\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

       11. --lowering--.f64 90.4

           \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(0 - d\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   11. Simplified 90.4%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(0 - d\right)\right)} \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   if -2.99999999999999992e-287 < d < 4.59999999999999961e-220

   1. Initial program 37.6%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. /-lowering-/.f64 N/A

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

      12. clear-num N/A

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

      13. associate-/r/ N/A

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 41.2%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. /-lowering-/.f64 41.1

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

   6. Applied egg-rr 41.1%

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

   7. Taylor expanded in h around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8}} \]

      2. associate-/l* N/A

         \[\leadsto \left(\color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left({D}^{2} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \cdot \frac{-1}{8} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{{D}^{2} \cdot \left(\left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8}\right)} \]

      5. *-commutative N/A

         \[\leadsto {D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]

      6. +-rgt-identity N/A

         \[\leadsto \color{blue}{{D}^{2} \cdot \left(\frac{-1}{8} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) + 0} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({D}^{2}, \frac{-1}{8} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right), 0\right)} \]

   9. Simplified 30.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(D \cdot D, \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}, 0\right)} \]
   10. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \sqrt{\color{blue}{\frac{\frac{h}{\ell}}{\ell \cdot \ell}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       2. sqrt-div N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \color{blue}{\frac{\sqrt{\frac{h}{\ell}}}{\sqrt{\ell \cdot \ell}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       3. pow2 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\sqrt{\frac{h}{\ell}}}{\sqrt{\color{blue}{{\ell}^{2}}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       4. sqrt-pow1 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\sqrt{\frac{h}{\ell}}}{\color{blue}{{\ell}^{\left(\frac{2}{2}\right)}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\sqrt{\frac{h}{\ell}}}{{\ell}^{\color{blue}{1}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       6. unpow1 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\sqrt{\frac{h}{\ell}}}{\color{blue}{\ell}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \color{blue}{\frac{\sqrt{\frac{h}{\ell}}}{\ell}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\color{blue}{\sqrt{\frac{h}{\ell}}}}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       9. /-lowering-/.f64 60.9

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\sqrt{\color{blue}{\frac{h}{\ell}}}}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}, 0\right) \]

   11. Applied egg-rr 60.9%

       \[\leadsto \mathsf{fma}\left(D \cdot D, \color{blue}{\frac{\sqrt{\frac{h}{\ell}}}{\ell}} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}, 0\right) \]

   if 4.59999999999999961e-220 < d

   1. Initial program 72.5%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 80.5%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. /-lowering-/.f64 80.5

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

   6. Applied egg-rr 80.5%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-div N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 86.2

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

   8. Applied egg-rr 86.2%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.4 \cdot 10^{-108}:\\ \;\;\;\;\left(1 - \frac{h \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \left(0.5 \cdot \left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;d \leq -3 \cdot 10^{-287}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(\frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}} + -1\right)\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{-220}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot D, \frac{\sqrt{\frac{h}{\ell}}}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{h \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \left(0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \frac{0.5 \cdot \left(M\_m \cdot D\_m\right)}{d}\\ t_2 := 0.5 \cdot t\_1\\ t_3 := 1 - \frac{h \cdot t\_1}{\ell} \cdot t\_2\\ \mathbf{if}\;h \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\left(\frac{\sqrt{0 - d}}{\sqrt{0 - h}} \cdot t\_0\right) \cdot t\_3\\ \mathbf{elif}\;h \leq 1.35 \cdot 10^{+114}:\\ \;\;\;\;\left(1 + \frac{t\_1}{\ell} \cdot \frac{t\_2}{\frac{-1}{h}}\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot \left(t\_0 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1 (/ (* 0.5 (* M_m D_m)) d))
        (t_2 (* 0.5 t_1))
        (t_3 (- 1.0 (* (/ (* h t_1) l) t_2))))
   (if (<= h -2e-311)
     (* (* (/ (sqrt (- 0.0 d)) (sqrt (- 0.0 h))) t_0) t_3)
     (if (<= h 1.35e+114)
       (* (+ 1.0 (* (/ t_1 l) (/ t_2 (/ -1.0 h)))) (/ d (sqrt (* h l))))
       (* t_3 (* t_0 (/ (sqrt d) (sqrt h))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / l));
	double t_1 = (0.5 * (M_m * D_m)) / d;
	double t_2 = 0.5 * t_1;
	double t_3 = 1.0 - (((h * t_1) / l) * t_2);
	double tmp;
	if (h <= -2e-311) {
		tmp = ((sqrt((0.0 - d)) / sqrt((0.0 - h))) * t_0) * t_3;
	} else if (h <= 1.35e+114) {
		tmp = (1.0 + ((t_1 / l) * (t_2 / (-1.0 / h)))) * (d / sqrt((h * l)));
	} else {
		tmp = t_3 * (t_0 * (sqrt(d) / sqrt(h)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = (0.5d0 * (m_m * d_m)) / d
    t_2 = 0.5d0 * t_1
    t_3 = 1.0d0 - (((h * t_1) / l) * t_2)
    if (h <= (-2d-311)) then
        tmp = ((sqrt((0.0d0 - d)) / sqrt((0.0d0 - h))) * t_0) * t_3
    else if (h <= 1.35d+114) then
        tmp = (1.0d0 + ((t_1 / l) * (t_2 / ((-1.0d0) / h)))) * (d / sqrt((h * l)))
    else
        tmp = t_3 * (t_0 * (sqrt(d) / sqrt(h)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = (0.5 * (M_m * D_m)) / d;
	double t_2 = 0.5 * t_1;
	double t_3 = 1.0 - (((h * t_1) / l) * t_2);
	double tmp;
	if (h <= -2e-311) {
		tmp = ((Math.sqrt((0.0 - d)) / Math.sqrt((0.0 - h))) * t_0) * t_3;
	} else if (h <= 1.35e+114) {
		tmp = (1.0 + ((t_1 / l) * (t_2 / (-1.0 / h)))) * (d / Math.sqrt((h * l)));
	} else {
		tmp = t_3 * (t_0 * (Math.sqrt(d) / Math.sqrt(h)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((d / l))
	t_1 = (0.5 * (M_m * D_m)) / d
	t_2 = 0.5 * t_1
	t_3 = 1.0 - (((h * t_1) / l) * t_2)
	tmp = 0
	if h <= -2e-311:
		tmp = ((math.sqrt((0.0 - d)) / math.sqrt((0.0 - h))) * t_0) * t_3
	elif h <= 1.35e+114:
		tmp = (1.0 + ((t_1 / l) * (t_2 / (-1.0 / h)))) * (d / math.sqrt((h * l)))
	else:
		tmp = t_3 * (t_0 * (math.sqrt(d) / math.sqrt(h)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(Float64(0.5 * Float64(M_m * D_m)) / d)
	t_2 = Float64(0.5 * t_1)
	t_3 = Float64(1.0 - Float64(Float64(Float64(h * t_1) / l) * t_2))
	tmp = 0.0
	if (h <= -2e-311)
		tmp = Float64(Float64(Float64(sqrt(Float64(0.0 - d)) / sqrt(Float64(0.0 - h))) * t_0) * t_3);
	elseif (h <= 1.35e+114)
		tmp = Float64(Float64(1.0 + Float64(Float64(t_1 / l) * Float64(t_2 / Float64(-1.0 / h)))) * Float64(d / sqrt(Float64(h * l))));
	else
		tmp = Float64(t_3 * Float64(t_0 * Float64(sqrt(d) / sqrt(h))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((d / l));
	t_1 = (0.5 * (M_m * D_m)) / d;
	t_2 = 0.5 * t_1;
	t_3 = 1.0 - (((h * t_1) / l) * t_2);
	tmp = 0.0;
	if (h <= -2e-311)
		tmp = ((sqrt((0.0 - d)) / sqrt((0.0 - h))) * t_0) * t_3;
	elseif (h <= 1.35e+114)
		tmp = (1.0 + ((t_1 / l) * (t_2 / (-1.0 / h)))) * (d / sqrt((h * l)));
	else
		tmp = t_3 * (t_0 * (sqrt(d) / sqrt(h)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - N[(N[(N[(h * t$95$1), $MachinePrecision] / l), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -2e-311], N[(N[(N[(N[Sqrt[N[(0.0 - d), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(0.0 - h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[h, 1.35e+114], N[(N[(1.0 + N[(N[(t$95$1 / l), $MachinePrecision] * N[(t$95$2 / N[(-1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 * N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.9999999999999e-311

   1. Initial program 73.6%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 78.2%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. /-lowering-/.f64 78.2

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

   6. Applied egg-rr 78.2%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. neg-sub0 N/A

         \[\leadsto \left(\frac{\sqrt{\color{blue}{0 - d}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \left(\frac{\sqrt{\color{blue}{0 - d}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{\sqrt{0 - d}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \left(\frac{\sqrt{0 - d}}{\sqrt{\color{blue}{0 - h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}\right)\right) \]

      11. --lowering--.f64 87.9

          \[\leadsto \left(\frac{\sqrt{0 - d}}{\sqrt{\color{blue}{0 - h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)\right) \]

   8. Applied egg-rr 87.9%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{0 - d}}{\sqrt{0 - h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)\right) \]

   if -1.9999999999999e-311 < h < 1.35e114

   1. Initial program 73.1%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. /-lowering-/.f64 N/A

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

      12. clear-num N/A

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

      13. associate-/r/ N/A

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 81.2%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. /-lowering-/.f64 81.7

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

   6. Applied egg-rr 81.7%

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

      1. metadata-eval N/A

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

      2. pow1/2 N/A

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

      3. frac-2neg N/A

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

      4. div-inv N/A

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

      5. sqrt-prod N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{\color{blue}{0 - d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{\color{blue}{0 - d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      12. neg-sub0 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{\color{blue}{0 - \ell}}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      13. --lowering--.f64 0.0

          \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{\color{blue}{0 - \ell}}}\right)\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   8. Applied egg-rr 0.0%

      \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]
   9. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{1}}}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      2. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{\frac{h}{d}}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      4. sqrt-unprod N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\left(0 - d\right) \cdot \frac{1}{0 - \ell}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      5. sub0-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \frac{1}{0 - \ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      6. div-inv N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{0 - \ell}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      7. sub0-neg N/A

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

      8. frac-2neg N/A

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

      9. sqrt-div N/A

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

      10. sqrt-div N/A

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

      11. frac-times N/A

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

      12. rem-square-sqrt N/A

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

      13. sqrt-prod N/A

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

      14. /-lowering-/.f64 N/A

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

      15. sqrt-lowering-sqrt.f64 N/A

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

      16. *-lowering-*.f64 93.3

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

   10. Applied egg-rr 93.3%

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

   if 1.35e114 < h

   1. Initial program 47.7%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 50.6%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. /-lowering-/.f64 50.6

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

   6. Applied egg-rr 50.6%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-div N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 69.3

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

   8. Applied egg-rr 69.3%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\left(\frac{\sqrt{0 - d}}{\sqrt{0 - h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{h \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \left(0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)\right)\\ \mathbf{elif}\;h \leq 1.35 \cdot 10^{+114}:\\ \;\;\;\;\left(1 + \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{-1}{h}}\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{h \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \left(0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := 0.5 \cdot \left(M\_m \cdot D\_m\right)\\ t_2 := \frac{t\_1}{d}\\ \mathbf{if}\;\ell \leq -2.25 \cdot 10^{+263}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq -5.6 \cdot 10^{-129}:\\ \;\;\;\;t\_0 \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - \frac{M\_m \cdot D\_m}{d \cdot 2} \cdot \left(\frac{h}{\ell} \cdot \frac{D\_m \cdot \left(M\_m \cdot 0.25\right)}{d}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t\_0 \cdot \left(\frac{\sqrt{0 - d}}{\sqrt{0 - h}} \cdot \left(1 - h \cdot \frac{t\_1 \cdot \left(\left(M\_m \cdot D\_m\right) \cdot 0.25\right)}{d \cdot \left(d \cdot \ell\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{t\_2}{\ell} \cdot \frac{0.5 \cdot t\_2}{\frac{-1}{h}}\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))) (t_1 (* 0.5 (* M_m D_m))) (t_2 (/ t_1 d)))
   (if (<= l -2.25e+263)
     (* (- 0.0 d) (sqrt (/ 1.0 (* h l))))
     (if (<= l -5.6e-129)
       (*
        t_0
        (*
         (sqrt (/ d h))
         (-
          1.0
          (*
           (/ (* M_m D_m) (* d 2.0))
           (* (/ h l) (/ (* D_m (* M_m 0.25)) d))))))
       (if (<= l -2e-310)
         (*
          t_0
          (*
           (/ (sqrt (- 0.0 d)) (sqrt (- 0.0 h)))
           (- 1.0 (* h (/ (* t_1 (* (* M_m D_m) 0.25)) (* d (* d l)))))))
         (*
          (+ 1.0 (* (/ t_2 l) (/ (* 0.5 t_2) (/ -1.0 h))))
          (/ d (sqrt (* h l)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / l));
	double t_1 = 0.5 * (M_m * D_m);
	double t_2 = t_1 / d;
	double tmp;
	if (l <= -2.25e+263) {
		tmp = (0.0 - d) * sqrt((1.0 / (h * l)));
	} else if (l <= -5.6e-129) {
		tmp = t_0 * (sqrt((d / h)) * (1.0 - (((M_m * D_m) / (d * 2.0)) * ((h / l) * ((D_m * (M_m * 0.25)) / d)))));
	} else if (l <= -2e-310) {
		tmp = t_0 * ((sqrt((0.0 - d)) / sqrt((0.0 - h))) * (1.0 - (h * ((t_1 * ((M_m * D_m) * 0.25)) / (d * (d * l))))));
	} else {
		tmp = (1.0 + ((t_2 / l) * ((0.5 * t_2) / (-1.0 / h)))) * (d / sqrt((h * l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = 0.5d0 * (m_m * d_m)
    t_2 = t_1 / d
    if (l <= (-2.25d+263)) then
        tmp = (0.0d0 - d) * sqrt((1.0d0 / (h * l)))
    else if (l <= (-5.6d-129)) then
        tmp = t_0 * (sqrt((d / h)) * (1.0d0 - (((m_m * d_m) / (d * 2.0d0)) * ((h / l) * ((d_m * (m_m * 0.25d0)) / d)))))
    else if (l <= (-2d-310)) then
        tmp = t_0 * ((sqrt((0.0d0 - d)) / sqrt((0.0d0 - h))) * (1.0d0 - (h * ((t_1 * ((m_m * d_m) * 0.25d0)) / (d * (d * l))))))
    else
        tmp = (1.0d0 + ((t_2 / l) * ((0.5d0 * t_2) / ((-1.0d0) / h)))) * (d / sqrt((h * l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = 0.5 * (M_m * D_m);
	double t_2 = t_1 / d;
	double tmp;
	if (l <= -2.25e+263) {
		tmp = (0.0 - d) * Math.sqrt((1.0 / (h * l)));
	} else if (l <= -5.6e-129) {
		tmp = t_0 * (Math.sqrt((d / h)) * (1.0 - (((M_m * D_m) / (d * 2.0)) * ((h / l) * ((D_m * (M_m * 0.25)) / d)))));
	} else if (l <= -2e-310) {
		tmp = t_0 * ((Math.sqrt((0.0 - d)) / Math.sqrt((0.0 - h))) * (1.0 - (h * ((t_1 * ((M_m * D_m) * 0.25)) / (d * (d * l))))));
	} else {
		tmp = (1.0 + ((t_2 / l) * ((0.5 * t_2) / (-1.0 / h)))) * (d / Math.sqrt((h * l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((d / l))
	t_1 = 0.5 * (M_m * D_m)
	t_2 = t_1 / d
	tmp = 0
	if l <= -2.25e+263:
		tmp = (0.0 - d) * math.sqrt((1.0 / (h * l)))
	elif l <= -5.6e-129:
		tmp = t_0 * (math.sqrt((d / h)) * (1.0 - (((M_m * D_m) / (d * 2.0)) * ((h / l) * ((D_m * (M_m * 0.25)) / d)))))
	elif l <= -2e-310:
		tmp = t_0 * ((math.sqrt((0.0 - d)) / math.sqrt((0.0 - h))) * (1.0 - (h * ((t_1 * ((M_m * D_m) * 0.25)) / (d * (d * l))))))
	else:
		tmp = (1.0 + ((t_2 / l) * ((0.5 * t_2) / (-1.0 / h)))) * (d / math.sqrt((h * l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(0.5 * Float64(M_m * D_m))
	t_2 = Float64(t_1 / d)
	tmp = 0.0
	if (l <= -2.25e+263)
		tmp = Float64(Float64(0.0 - d) * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (l <= -5.6e-129)
		tmp = Float64(t_0 * Float64(sqrt(Float64(d / h)) * Float64(1.0 - Float64(Float64(Float64(M_m * D_m) / Float64(d * 2.0)) * Float64(Float64(h / l) * Float64(Float64(D_m * Float64(M_m * 0.25)) / d))))));
	elseif (l <= -2e-310)
		tmp = Float64(t_0 * Float64(Float64(sqrt(Float64(0.0 - d)) / sqrt(Float64(0.0 - h))) * Float64(1.0 - Float64(h * Float64(Float64(t_1 * Float64(Float64(M_m * D_m) * 0.25)) / Float64(d * Float64(d * l)))))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(t_2 / l) * Float64(Float64(0.5 * t_2) / Float64(-1.0 / h)))) * Float64(d / sqrt(Float64(h * l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((d / l));
	t_1 = 0.5 * (M_m * D_m);
	t_2 = t_1 / d;
	tmp = 0.0;
	if (l <= -2.25e+263)
		tmp = (0.0 - d) * sqrt((1.0 / (h * l)));
	elseif (l <= -5.6e-129)
		tmp = t_0 * (sqrt((d / h)) * (1.0 - (((M_m * D_m) / (d * 2.0)) * ((h / l) * ((D_m * (M_m * 0.25)) / d)))));
	elseif (l <= -2e-310)
		tmp = t_0 * ((sqrt((0.0 - d)) / sqrt((0.0 - h))) * (1.0 - (h * ((t_1 * ((M_m * D_m) * 0.25)) / (d * (d * l))))));
	else
		tmp = (1.0 + ((t_2 / l) * ((0.5 * t_2) / (-1.0 / h)))) * (d / sqrt((h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / d), $MachinePrecision]}, If[LessEqual[l, -2.25e+263], N[(N[(0.0 - d), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5.6e-129], N[(t$95$0 * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(N[(D$95$m * N[(M$95$m * 0.25), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(t$95$0 * N[(N[(N[Sqrt[N[(0.0 - d), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(0.0 - h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(h * N[(N[(t$95$1 * N[(N[(M$95$m * D$95$m), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(t$95$2 / l), $MachinePrecision] * N[(N[(0.5 * t$95$2), $MachinePrecision] / N[(-1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.25000000000000007e263

   1. Initial program 33.8%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. /-lowering-/.f64 N/A

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

      12. clear-num N/A

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

      13. associate-/r/ N/A

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 34.9%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. /-lowering-/.f64 34.9

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

   6. Applied egg-rr 34.9%

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

   8. Applied egg-rr 34.9%

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

   9. Taylor expanded in l around -inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. rem-square-sqrt N/A

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. sqrt-lowering-sqrt.f64 N/A

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

       7. /-lowering-/.f64 N/A

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

       8. *-lowering-*.f64 N/A

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

       9. mul-1-neg N/A

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

       10. neg-sub0 N/A

           \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(0 - d\right)} \]

       11. --lowering--.f64 83.3

           \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(0 - d\right)} \]

   11. Simplified 83.3%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(0 - d\right)} \]

   if -2.25000000000000007e263 < l < -5.5999999999999998e-129

   1. Initial program 79.3%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. /-lowering-/.f64 N/A

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

      12. clear-num N/A

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

      13. associate-/r/ N/A

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 83.3%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. /-lowering-/.f64 81.8

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

   6. Applied egg-rr 81.8%

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

   8. Applied egg-rr 63.1%

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

      1. associate-*l/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. frac-times N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

   10. Applied egg-rr 83.3%

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

   if -5.5999999999999998e-129 < l < -1.999999999999994e-310

   1. Initial program 74.8%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. /-lowering-/.f64 N/A

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

      12. clear-num N/A

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

      13. associate-/r/ N/A

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 80.8%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. /-lowering-/.f64 81.4

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

   6. Applied egg-rr 81.4%

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

   8. Applied egg-rr 75.0%

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

      1. frac-2neg N/A

         \[\leadsto \left(\left(1 - \frac{\left(\frac{1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(\frac{1}{4} \cdot \left(M \cdot D\right)\right)}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(h\right)}}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. sqrt-div N/A

         \[\leadsto \left(\left(1 - \frac{\left(\frac{1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(\frac{1}{4} \cdot \left(M \cdot D\right)\right)}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. sub0-neg N/A

         \[\leadsto \left(\left(1 - \frac{\left(\frac{1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(\frac{1}{4} \cdot \left(M \cdot D\right)\right)}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \frac{\sqrt{\color{blue}{0 - d}}}{\sqrt{\mathsf{neg}\left(h\right)}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(1 - \frac{\left(\frac{1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(\frac{1}{4} \cdot \left(M \cdot D\right)\right)}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \color{blue}{\frac{\sqrt{0 - d}}{\sqrt{\mathsf{neg}\left(h\right)}}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. sub0-neg N/A

         \[\leadsto \left(\left(1 - \frac{\left(\frac{1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(\frac{1}{4} \cdot \left(M \cdot D\right)\right)}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \frac{\sqrt{\color{blue}{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(h\right)}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 - \frac{\left(\frac{1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(\frac{1}{4} \cdot \left(M \cdot D\right)\right)}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \frac{\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(h\right)}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. sub0-neg N/A

         \[\leadsto \left(\left(1 - \frac{\left(\frac{1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(\frac{1}{4} \cdot \left(M \cdot D\right)\right)}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \frac{\sqrt{\color{blue}{0 - d}}}{\sqrt{\mathsf{neg}\left(h\right)}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. --lowering--.f64 N/A

         \[\leadsto \left(\left(1 - \frac{\left(\frac{1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(\frac{1}{4} \cdot \left(M \cdot D\right)\right)}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \frac{\sqrt{\color{blue}{0 - d}}}{\sqrt{\mathsf{neg}\left(h\right)}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\left(1 - \frac{\left(\frac{1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(\frac{1}{4} \cdot \left(M \cdot D\right)\right)}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \frac{\sqrt{0 - d}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. neg-sub0 N/A

          \[\leadsto \left(\left(1 - \frac{\left(\frac{1}{2} \cdot \left(M \cdot D\right)\right) \cdot \left(\frac{1}{4} \cdot \left(M \cdot D\right)\right)}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \frac{\sqrt{0 - d}}{\sqrt{\color{blue}{0 - h}}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. --lowering--.f64 88.5

          \[\leadsto \left(\left(1 - \frac{\left(0.5 \cdot \left(M \cdot D\right)\right) \cdot \left(0.25 \cdot \left(M \cdot D\right)\right)}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \frac{\sqrt{0 - d}}{\sqrt{\color{blue}{0 - h}}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   10. Applied egg-rr 88.5%

       \[\leadsto \left(\left(1 - \frac{\left(0.5 \cdot \left(M \cdot D\right)\right) \cdot \left(0.25 \cdot \left(M \cdot D\right)\right)}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \color{blue}{\frac{\sqrt{0 - d}}{\sqrt{0 - h}}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   if -1.999999999999994e-310 < l

   1. Initial program 66.3%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. /-lowering-/.f64 N/A

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

      12. clear-num N/A

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

      13. associate-/r/ N/A

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 73.0%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      4. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      8. /-lowering-/.f64 73.3

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   6. Applied egg-rr 73.3%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      2. pow1/2 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      3. frac-2neg N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      4. div-inv N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      5. sqrt-prod N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      8. neg-sub0 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{\color{blue}{0 - d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{\color{blue}{0 - d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      12. neg-sub0 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{\color{blue}{0 - \ell}}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      13. --lowering--.f64 0.0

          \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{\color{blue}{0 - \ell}}}\right)\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   8. Applied egg-rr 0.0%

      \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]
   9. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{1}}}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      2. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{\frac{h}{d}}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      4. sqrt-unprod N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\left(0 - d\right) \cdot \frac{1}{0 - \ell}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      5. sub0-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \frac{1}{0 - \ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      6. div-inv N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{0 - \ell}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      7. sub0-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{\mathsf{neg}\left(d\right)}{\color{blue}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      8. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      9. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      10. sqrt-div N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      11. frac-times N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}} \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      12. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      13. sqrt-prod N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      16. *-lowering-*.f64 81.1

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   10. Applied egg-rr 81.1%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.25 \cdot 10^{+263}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq -5.6 \cdot 10^{-129}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - \frac{M \cdot D}{d \cdot 2} \cdot \left(\frac{h}{\ell} \cdot \frac{D \cdot \left(M \cdot 0.25\right)}{d}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{0 - d}}{\sqrt{0 - h}} \cdot \left(1 - h \cdot \frac{\left(0.5 \cdot \left(M \cdot D\right)\right) \cdot \left(\left(M \cdot D\right) \cdot 0.25\right)}{d \cdot \left(d \cdot \ell\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{-1}{h}}\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{0.5 \cdot \left(M\_m \cdot D\_m\right)}{d}\\ t_1 := \frac{t\_0}{\ell}\\ t_2 := 0.5 \cdot t\_0\\ \mathbf{if}\;d \leq -5.9 \cdot 10^{-102}:\\ \;\;\;\;\left(1 - \frac{h \cdot t\_0}{\ell} \cdot \left(0.5 \cdot \left(\left(M\_m \cdot D\_m\right) \cdot \frac{0.5}{d}\right)\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-299}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(t\_1 \cdot \frac{t\_2}{\frac{1}{h}} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t\_1 \cdot \frac{t\_2}{\frac{-1}{h}}\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ (* 0.5 (* M_m D_m)) d)) (t_1 (/ t_0 l)) (t_2 (* 0.5 t_0)))
   (if (<= d -5.9e-102)
     (*
      (- 1.0 (* (/ (* h t_0) l) (* 0.5 (* (* M_m D_m) (/ 0.5 d)))))
      (* (sqrt (/ d l)) (sqrt (/ d h))))
     (if (<= d -5e-299)
       (* (* d (sqrt (/ 1.0 (* h l)))) (+ (* t_1 (/ t_2 (/ 1.0 h))) -1.0))
       (* (+ 1.0 (* t_1 (/ t_2 (/ -1.0 h)))) (/ d (sqrt (* h l))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (0.5 * (M_m * D_m)) / d;
	double t_1 = t_0 / l;
	double t_2 = 0.5 * t_0;
	double tmp;
	if (d <= -5.9e-102) {
		tmp = (1.0 - (((h * t_0) / l) * (0.5 * ((M_m * D_m) * (0.5 / d))))) * (sqrt((d / l)) * sqrt((d / h)));
	} else if (d <= -5e-299) {
		tmp = (d * sqrt((1.0 / (h * l)))) * ((t_1 * (t_2 / (1.0 / h))) + -1.0);
	} else {
		tmp = (1.0 + (t_1 * (t_2 / (-1.0 / h)))) * (d / sqrt((h * l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (0.5d0 * (m_m * d_m)) / d
    t_1 = t_0 / l
    t_2 = 0.5d0 * t_0
    if (d <= (-5.9d-102)) then
        tmp = (1.0d0 - (((h * t_0) / l) * (0.5d0 * ((m_m * d_m) * (0.5d0 / d))))) * (sqrt((d / l)) * sqrt((d / h)))
    else if (d <= (-5d-299)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((t_1 * (t_2 / (1.0d0 / h))) + (-1.0d0))
    else
        tmp = (1.0d0 + (t_1 * (t_2 / ((-1.0d0) / h)))) * (d / sqrt((h * l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (0.5 * (M_m * D_m)) / d;
	double t_1 = t_0 / l;
	double t_2 = 0.5 * t_0;
	double tmp;
	if (d <= -5.9e-102) {
		tmp = (1.0 - (((h * t_0) / l) * (0.5 * ((M_m * D_m) * (0.5 / d))))) * (Math.sqrt((d / l)) * Math.sqrt((d / h)));
	} else if (d <= -5e-299) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * ((t_1 * (t_2 / (1.0 / h))) + -1.0);
	} else {
		tmp = (1.0 + (t_1 * (t_2 / (-1.0 / h)))) * (d / Math.sqrt((h * l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (0.5 * (M_m * D_m)) / d
	t_1 = t_0 / l
	t_2 = 0.5 * t_0
	tmp = 0
	if d <= -5.9e-102:
		tmp = (1.0 - (((h * t_0) / l) * (0.5 * ((M_m * D_m) * (0.5 / d))))) * (math.sqrt((d / l)) * math.sqrt((d / h)))
	elif d <= -5e-299:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * ((t_1 * (t_2 / (1.0 / h))) + -1.0)
	else:
		tmp = (1.0 + (t_1 * (t_2 / (-1.0 / h)))) * (d / math.sqrt((h * l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(0.5 * Float64(M_m * D_m)) / d)
	t_1 = Float64(t_0 / l)
	t_2 = Float64(0.5 * t_0)
	tmp = 0.0
	if (d <= -5.9e-102)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(h * t_0) / l) * Float64(0.5 * Float64(Float64(M_m * D_m) * Float64(0.5 / d))))) * Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))));
	elseif (d <= -5e-299)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(Float64(t_1 * Float64(t_2 / Float64(1.0 / h))) + -1.0));
	else
		tmp = Float64(Float64(1.0 + Float64(t_1 * Float64(t_2 / Float64(-1.0 / h)))) * Float64(d / sqrt(Float64(h * l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (0.5 * (M_m * D_m)) / d;
	t_1 = t_0 / l;
	t_2 = 0.5 * t_0;
	tmp = 0.0;
	if (d <= -5.9e-102)
		tmp = (1.0 - (((h * t_0) / l) * (0.5 * ((M_m * D_m) * (0.5 / d))))) * (sqrt((d / l)) * sqrt((d / h)));
	elseif (d <= -5e-299)
		tmp = (d * sqrt((1.0 / (h * l)))) * ((t_1 * (t_2 / (1.0 / h))) + -1.0);
	else
		tmp = (1.0 + (t_1 * (t_2 / (-1.0 / h)))) * (d / sqrt((h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(0.5 * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / l), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * t$95$0), $MachinePrecision]}, If[LessEqual[d, -5.9e-102], N[(N[(1.0 - N[(N[(N[(h * t$95$0), $MachinePrecision] / l), $MachinePrecision] * N[(0.5 * N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-299], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * N[(t$95$2 / N[(1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(t$95$1 * N[(t$95$2 / N[(-1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.9000000000000003e-102

   1. Initial program 83.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

   4. Applied egg-rr 89.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)}\right) \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}\right)\right) \]

      2. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}\right)\right) \]

      4. /-lowering-/.f64 89.2

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)\right) \]

   6. Applied egg-rr 89.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(\frac{1}{2} \cdot \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{d}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(\frac{1}{2} \cdot \frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{\left(M \cdot D\right)} \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(\frac{1}{2} \cdot \left(\left(M \cdot D\right) \cdot \color{blue}{\frac{\frac{1}{2}}{d}}\right)\right)\right) \]

      7. metadata-eval 89.2

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(0.5 \cdot \left(\left(M \cdot D\right) \cdot \frac{\color{blue}{0.5}}{d}\right)\right)\right) \]

   8. Applied egg-rr 89.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(0.5 \cdot \color{blue}{\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(\frac{1}{2} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \]

      2. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(\frac{1}{2} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(\frac{1}{2} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \]

      4. /-lowering-/.f64 89.2

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(0.5 \cdot \left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)\right)\right) \]

   10. Applied egg-rr 89.2%

       \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(0.5 \cdot \left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)\right)\right) \]

   if -5.9000000000000003e-102 < d < -4.99999999999999956e-299

   1. Initial program 58.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      2. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\frac{\ell}{h}}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\frac{\ell}{h}}\right) \]

      6. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      7. times-frac N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\frac{M \cdot D}{2}}{d}}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\frac{M \cdot D}{2}}{d}}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      12. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{\frac{2}{M \cdot D}}}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      13. associate-/r/ N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

   4. Applied egg-rr 60.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      2. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      4. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      8. /-lowering-/.f64 60.6

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   6. Applied egg-rr 60.6%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      2. pow1/2 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      3. frac-2neg N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      4. div-inv N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      5. sqrt-prod N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      8. neg-sub0 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{\color{blue}{0 - d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{\color{blue}{0 - d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      12. neg-sub0 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{\color{blue}{0 - \ell}}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      13. --lowering--.f64 77.1

          \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{\color{blue}{0 - \ell}}}\right)\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   8. Applied egg-rr 77.1%

      \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   9. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

       2. unpow2 N/A

          \[\leadsto \left(\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

       3. rem-square-sqrt N/A

          \[\leadsto \left(\left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)\right)} \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)\right)} \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

       9. mul-1-neg N/A

          \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

       10. neg-sub0 N/A

           \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(0 - d\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

       11. --lowering--.f64 88.8

           \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(0 - d\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   11. Simplified 88.8%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(0 - d\right)\right)} \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   if -4.99999999999999956e-299 < d

   1. Initial program 64.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      2. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\frac{\ell}{h}}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\frac{\ell}{h}}\right) \]

      6. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      7. times-frac N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\frac{M \cdot D}{2}}{d}}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\frac{M \cdot D}{2}}{d}}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      12. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{\frac{2}{M \cdot D}}}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      13. associate-/r/ N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

   4. Applied egg-rr 71.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      2. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      4. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      8. /-lowering-/.f64 71.8

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   6. Applied egg-rr 71.8%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      2. pow1/2 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      3. frac-2neg N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      4. div-inv N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      5. sqrt-prod N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      8. neg-sub0 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{\color{blue}{0 - d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{\color{blue}{0 - d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      12. neg-sub0 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{\color{blue}{0 - \ell}}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      13. --lowering--.f64 0.8

          \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{\color{blue}{0 - \ell}}}\right)\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   8. Applied egg-rr 0.8%

      \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]
   9. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{1}}}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      2. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{\frac{h}{d}}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      4. sqrt-unprod N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\left(0 - d\right) \cdot \frac{1}{0 - \ell}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      5. sub0-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \frac{1}{0 - \ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      6. div-inv N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{0 - \ell}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      7. sub0-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{\mathsf{neg}\left(d\right)}{\color{blue}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      8. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      9. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      10. sqrt-div N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      11. frac-times N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}} \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      12. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      13. sqrt-prod N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      16. *-lowering-*.f64 79.2

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   10. Applied egg-rr 79.2%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.9 \cdot 10^{-102}:\\ \;\;\;\;\left(1 - \frac{h \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \left(0.5 \cdot \left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-299}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(\frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{-1}{h}}\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{0.5 \cdot \left(M\_m \cdot D\_m\right)}{d}\\ \mathbf{if}\;\ell \leq -4.3 \cdot 10^{+262}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{-259}:\\ \;\;\;\;\left(1 - \frac{h \cdot t\_0}{\ell} \cdot \left(0.5 \cdot \left(\left(M\_m \cdot D\_m\right) \cdot \frac{0.5}{d}\right)\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{t\_0}{\ell} \cdot \frac{0.5 \cdot t\_0}{\frac{-1}{h}}\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ (* 0.5 (* M_m D_m)) d)))
   (if (<= l -4.3e+262)
     (* (- 0.0 d) (sqrt (/ 1.0 (* h l))))
     (if (<= l 8e-259)
       (*
        (- 1.0 (* (/ (* h t_0) l) (* 0.5 (* (* M_m D_m) (/ 0.5 d)))))
        (* (sqrt (/ d l)) (sqrt (/ d h))))
       (*
        (+ 1.0 (* (/ t_0 l) (/ (* 0.5 t_0) (/ -1.0 h))))
        (/ d (sqrt (* h l))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (0.5 * (M_m * D_m)) / d;
	double tmp;
	if (l <= -4.3e+262) {
		tmp = (0.0 - d) * sqrt((1.0 / (h * l)));
	} else if (l <= 8e-259) {
		tmp = (1.0 - (((h * t_0) / l) * (0.5 * ((M_m * D_m) * (0.5 / d))))) * (sqrt((d / l)) * sqrt((d / h)));
	} else {
		tmp = (1.0 + ((t_0 / l) * ((0.5 * t_0) / (-1.0 / h)))) * (d / sqrt((h * l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 * (m_m * d_m)) / d
    if (l <= (-4.3d+262)) then
        tmp = (0.0d0 - d) * sqrt((1.0d0 / (h * l)))
    else if (l <= 8d-259) then
        tmp = (1.0d0 - (((h * t_0) / l) * (0.5d0 * ((m_m * d_m) * (0.5d0 / d))))) * (sqrt((d / l)) * sqrt((d / h)))
    else
        tmp = (1.0d0 + ((t_0 / l) * ((0.5d0 * t_0) / ((-1.0d0) / h)))) * (d / sqrt((h * l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (0.5 * (M_m * D_m)) / d;
	double tmp;
	if (l <= -4.3e+262) {
		tmp = (0.0 - d) * Math.sqrt((1.0 / (h * l)));
	} else if (l <= 8e-259) {
		tmp = (1.0 - (((h * t_0) / l) * (0.5 * ((M_m * D_m) * (0.5 / d))))) * (Math.sqrt((d / l)) * Math.sqrt((d / h)));
	} else {
		tmp = (1.0 + ((t_0 / l) * ((0.5 * t_0) / (-1.0 / h)))) * (d / Math.sqrt((h * l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (0.5 * (M_m * D_m)) / d
	tmp = 0
	if l <= -4.3e+262:
		tmp = (0.0 - d) * math.sqrt((1.0 / (h * l)))
	elif l <= 8e-259:
		tmp = (1.0 - (((h * t_0) / l) * (0.5 * ((M_m * D_m) * (0.5 / d))))) * (math.sqrt((d / l)) * math.sqrt((d / h)))
	else:
		tmp = (1.0 + ((t_0 / l) * ((0.5 * t_0) / (-1.0 / h)))) * (d / math.sqrt((h * l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(0.5 * Float64(M_m * D_m)) / d)
	tmp = 0.0
	if (l <= -4.3e+262)
		tmp = Float64(Float64(0.0 - d) * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (l <= 8e-259)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(h * t_0) / l) * Float64(0.5 * Float64(Float64(M_m * D_m) * Float64(0.5 / d))))) * Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(t_0 / l) * Float64(Float64(0.5 * t_0) / Float64(-1.0 / h)))) * Float64(d / sqrt(Float64(h * l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (0.5 * (M_m * D_m)) / d;
	tmp = 0.0;
	if (l <= -4.3e+262)
		tmp = (0.0 - d) * sqrt((1.0 / (h * l)));
	elseif (l <= 8e-259)
		tmp = (1.0 - (((h * t_0) / l) * (0.5 * ((M_m * D_m) * (0.5 / d))))) * (sqrt((d / l)) * sqrt((d / h)));
	else
		tmp = (1.0 + ((t_0 / l) * ((0.5 * t_0) / (-1.0 / h)))) * (d / sqrt((h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(0.5 * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[l, -4.3e+262], N[(N[(0.0 - d), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8e-259], N[(N[(1.0 - N[(N[(N[(h * t$95$0), $MachinePrecision] / l), $MachinePrecision] * N[(0.5 * N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(t$95$0 / l), $MachinePrecision] * N[(N[(0.5 * t$95$0), $MachinePrecision] / N[(-1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.3e262

   1. Initial program 33.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      2. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\frac{\ell}{h}}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\frac{\ell}{h}}\right) \]

      6. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      7. times-frac N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\frac{M \cdot D}{2}}{d}}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\frac{M \cdot D}{2}}{d}}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      12. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{\frac{2}{M \cdot D}}}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      13. associate-/r/ N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

   4. Applied egg-rr 34.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      2. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      4. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      8. /-lowering-/.f64 34.9

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   6. Applied egg-rr 34.9%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   8. Applied egg-rr 34.9%

      \[\leadsto \color{blue}{\left(\left(1 - \frac{\left(0.5 \cdot \left(M \cdot D\right)\right) \cdot \left(0.25 \cdot \left(M \cdot D\right)\right)}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]

   9. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       2. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       3. rem-square-sqrt N/A

          \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

       9. mul-1-neg N/A

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

       10. neg-sub0 N/A

           \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(0 - d\right)} \]

       11. --lowering--.f64 83.3

           \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(0 - d\right)} \]

   11. Simplified 83.3%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(0 - d\right)} \]

   if -4.3e262 < l < 8.0000000000000006e-259

   1. Initial program 77.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

   4. Applied egg-rr 83.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)}\right) \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}\right)\right) \]

      2. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}\right)\right) \]

      4. /-lowering-/.f64 83.8

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)\right) \]

   6. Applied egg-rr 83.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(\frac{1}{2} \cdot \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{d}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(\frac{1}{2} \cdot \frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{\left(M \cdot D\right)} \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(\frac{1}{2} \cdot \left(\left(M \cdot D\right) \cdot \color{blue}{\frac{\frac{1}{2}}{d}}\right)\right)\right) \]

      7. metadata-eval 83.8

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(0.5 \cdot \left(\left(M \cdot D\right) \cdot \frac{\color{blue}{0.5}}{d}\right)\right)\right) \]

   8. Applied egg-rr 83.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(0.5 \cdot \color{blue}{\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(\frac{1}{2} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \]

      2. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(\frac{1}{2} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(\frac{1}{2} \cdot \left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)\right)\right) \]

      4. /-lowering-/.f64 83.8

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(0.5 \cdot \left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)\right)\right) \]

   10. Applied egg-rr 83.8%

       \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell} \cdot \left(0.5 \cdot \left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)\right)\right) \]

   if 8.0000000000000006e-259 < l

   1. Initial program 64.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      2. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\frac{\ell}{h}}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\frac{\ell}{h}}\right) \]

      6. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      7. times-frac N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\frac{M \cdot D}{2}}{d}}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\frac{M \cdot D}{2}}{d}}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      12. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{\frac{2}{M \cdot D}}}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      13. associate-/r/ N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

   4. Applied egg-rr 70.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      2. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      4. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      8. /-lowering-/.f64 70.6

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   6. Applied egg-rr 70.6%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      2. pow1/2 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      3. frac-2neg N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      4. div-inv N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      5. sqrt-prod N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      8. neg-sub0 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{\color{blue}{0 - d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{\color{blue}{0 - d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      12. neg-sub0 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{\color{blue}{0 - \ell}}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      13. --lowering--.f64 0.0

          \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{\color{blue}{0 - \ell}}}\right)\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   8. Applied egg-rr 0.0%

      \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]
   9. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{1}}}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      2. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{\frac{h}{d}}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      4. sqrt-unprod N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\left(0 - d\right) \cdot \frac{1}{0 - \ell}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      5. sub0-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \frac{1}{0 - \ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      6. div-inv N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{0 - \ell}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      7. sub0-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{\mathsf{neg}\left(d\right)}{\color{blue}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      8. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      9. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      10. sqrt-div N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      11. frac-times N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}} \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      12. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      13. sqrt-prod N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      16. *-lowering-*.f64 80.0

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   10. Applied egg-rr 80.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.3 \cdot 10^{+262}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{-259}:\\ \;\;\;\;\left(1 - \frac{h \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \left(0.5 \cdot \left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{-1}{h}}\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{0.5 \cdot \left(M\_m \cdot D\_m\right)}{d}\\ \mathbf{if}\;\ell \leq -6.2 \cdot 10^{+260}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{-261}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - \frac{\frac{h \cdot \left(D\_m \cdot \left(M\_m \cdot 0.25\right)\right)}{d}}{\ell \cdot \frac{d \cdot 2}{M\_m \cdot D\_m}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{t\_0}{\ell} \cdot \frac{0.5 \cdot t\_0}{\frac{-1}{h}}\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ (* 0.5 (* M_m D_m)) d)))
   (if (<= l -6.2e+260)
     (* (- 0.0 d) (sqrt (/ 1.0 (* h l))))
     (if (<= l 1.25e-261)
       (*
        (sqrt (/ d l))
        (*
         (sqrt (/ d h))
         (-
          1.0
          (/
           (/ (* h (* D_m (* M_m 0.25))) d)
           (* l (/ (* d 2.0) (* M_m D_m)))))))
       (*
        (+ 1.0 (* (/ t_0 l) (/ (* 0.5 t_0) (/ -1.0 h))))
        (/ d (sqrt (* h l))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (0.5 * (M_m * D_m)) / d;
	double tmp;
	if (l <= -6.2e+260) {
		tmp = (0.0 - d) * sqrt((1.0 / (h * l)));
	} else if (l <= 1.25e-261) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 - (((h * (D_m * (M_m * 0.25))) / d) / (l * ((d * 2.0) / (M_m * D_m))))));
	} else {
		tmp = (1.0 + ((t_0 / l) * ((0.5 * t_0) / (-1.0 / h)))) * (d / sqrt((h * l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 * (m_m * d_m)) / d
    if (l <= (-6.2d+260)) then
        tmp = (0.0d0 - d) * sqrt((1.0d0 / (h * l)))
    else if (l <= 1.25d-261) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 - (((h * (d_m * (m_m * 0.25d0))) / d) / (l * ((d * 2.0d0) / (m_m * d_m))))))
    else
        tmp = (1.0d0 + ((t_0 / l) * ((0.5d0 * t_0) / ((-1.0d0) / h)))) * (d / sqrt((h * l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (0.5 * (M_m * D_m)) / d;
	double tmp;
	if (l <= -6.2e+260) {
		tmp = (0.0 - d) * Math.sqrt((1.0 / (h * l)));
	} else if (l <= 1.25e-261) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 - (((h * (D_m * (M_m * 0.25))) / d) / (l * ((d * 2.0) / (M_m * D_m))))));
	} else {
		tmp = (1.0 + ((t_0 / l) * ((0.5 * t_0) / (-1.0 / h)))) * (d / Math.sqrt((h * l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (0.5 * (M_m * D_m)) / d
	tmp = 0
	if l <= -6.2e+260:
		tmp = (0.0 - d) * math.sqrt((1.0 / (h * l)))
	elif l <= 1.25e-261:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 - (((h * (D_m * (M_m * 0.25))) / d) / (l * ((d * 2.0) / (M_m * D_m))))))
	else:
		tmp = (1.0 + ((t_0 / l) * ((0.5 * t_0) / (-1.0 / h)))) * (d / math.sqrt((h * l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(0.5 * Float64(M_m * D_m)) / d)
	tmp = 0.0
	if (l <= -6.2e+260)
		tmp = Float64(Float64(0.0 - d) * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (l <= 1.25e-261)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 - Float64(Float64(Float64(h * Float64(D_m * Float64(M_m * 0.25))) / d) / Float64(l * Float64(Float64(d * 2.0) / Float64(M_m * D_m)))))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(t_0 / l) * Float64(Float64(0.5 * t_0) / Float64(-1.0 / h)))) * Float64(d / sqrt(Float64(h * l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (0.5 * (M_m * D_m)) / d;
	tmp = 0.0;
	if (l <= -6.2e+260)
		tmp = (0.0 - d) * sqrt((1.0 / (h * l)));
	elseif (l <= 1.25e-261)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 - (((h * (D_m * (M_m * 0.25))) / d) / (l * ((d * 2.0) / (M_m * D_m))))));
	else
		tmp = (1.0 + ((t_0 / l) * ((0.5 * t_0) / (-1.0 / h)))) * (d / sqrt((h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(0.5 * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[l, -6.2e+260], N[(N[(0.0 - d), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.25e-261], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(N[(h * N[(D$95$m * N[(M$95$m * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / N[(l * N[(N[(d * 2.0), $MachinePrecision] / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(t$95$0 / l), $MachinePrecision] * N[(N[(0.5 * t$95$0), $MachinePrecision] / N[(-1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.1999999999999996e260

   1. Initial program 33.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      2. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\frac{\ell}{h}}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\frac{\ell}{h}}\right) \]

      6. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      7. times-frac N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\frac{M \cdot D}{2}}{d}}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\frac{M \cdot D}{2}}{d}}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      12. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{\frac{2}{M \cdot D}}}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      13. associate-/r/ N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

   4. Applied egg-rr 34.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      2. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      4. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      8. /-lowering-/.f64 34.9

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   6. Applied egg-rr 34.9%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   8. Applied egg-rr 34.9%

      \[\leadsto \color{blue}{\left(\left(1 - \frac{\left(0.5 \cdot \left(M \cdot D\right)\right) \cdot \left(0.25 \cdot \left(M \cdot D\right)\right)}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]

   9. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       2. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       3. rem-square-sqrt N/A

          \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

       9. mul-1-neg N/A

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

       10. neg-sub0 N/A

           \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(0 - d\right)} \]

       11. --lowering--.f64 83.3

           \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(0 - d\right)} \]

   11. Simplified 83.3%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(0 - d\right)} \]

   if -6.1999999999999996e260 < l < 1.24999999999999995e-261

   1. Initial program 77.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      2. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\frac{\ell}{h}}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\frac{\ell}{h}}\right) \]

      6. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      7. times-frac N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\frac{M \cdot D}{2}}{d}}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\frac{M \cdot D}{2}}{d}}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      12. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{\frac{2}{M \cdot D}}}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      13. associate-/r/ N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

   4. Applied egg-rr 83.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      2. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      4. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      8. /-lowering-/.f64 83.1

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   6. Applied egg-rr 83.1%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   8. Applied egg-rr 70.6%

      \[\leadsto \color{blue}{\left(\left(1 - \frac{\left(0.5 \cdot \left(M \cdot D\right)\right) \cdot \left(0.25 \cdot \left(M \cdot D\right)\right)}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]

   10. Applied egg-rr 80.9%

       \[\leadsto \left(\left(1 - \color{blue}{\frac{\frac{\left(D \cdot \left(M \cdot 0.25\right)\right) \cdot h}{d}}{\ell \cdot \frac{d \cdot 2}{M \cdot D}}}\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   if 1.24999999999999995e-261 < l

   1. Initial program 64.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      2. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\frac{\ell}{h}}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\frac{\ell}{h}}\right) \]

      6. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      7. times-frac N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\frac{M \cdot D}{2}}{d}}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\frac{M \cdot D}{2}}{d}}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      12. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{\frac{2}{M \cdot D}}}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      13. associate-/r/ N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

   4. Applied egg-rr 70.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      2. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      4. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      8. /-lowering-/.f64 70.6

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   6. Applied egg-rr 70.6%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      2. pow1/2 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      3. frac-2neg N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      4. div-inv N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      5. sqrt-prod N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      8. neg-sub0 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{\color{blue}{0 - d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{\color{blue}{0 - d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      12. neg-sub0 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{\color{blue}{0 - \ell}}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      13. --lowering--.f64 0.0

          \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{\color{blue}{0 - \ell}}}\right)\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   8. Applied egg-rr 0.0%

      \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]
   9. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{1}}}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      2. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{\frac{h}{d}}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      4. sqrt-unprod N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\left(0 - d\right) \cdot \frac{1}{0 - \ell}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      5. sub0-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \frac{1}{0 - \ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      6. div-inv N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{0 - \ell}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      7. sub0-neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{\mathsf{neg}\left(d\right)}{\color{blue}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      8. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      9. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      10. sqrt-div N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      11. frac-times N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}} \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      12. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      13. sqrt-prod N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      16. *-lowering-*.f64 80.0

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   10. Applied egg-rr 80.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.2 \cdot 10^{+260}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{-261}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - \frac{\frac{h \cdot \left(D \cdot \left(M \cdot 0.25\right)\right)}{d}}{\ell \cdot \frac{d \cdot 2}{M \cdot D}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{-1}{h}}\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 66.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;M\_m \cdot D\_m \leq 10^{-216}:\\ \;\;\;\;\frac{t\_0}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;M\_m \cdot D\_m \leq 5 \cdot 10^{+129}:\\ \;\;\;\;t\_0 \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - h \cdot \frac{\left(M\_m \cdot D\_m\right) \cdot \left(\left(M\_m \cdot D\_m\right) \cdot 0.125\right)}{d \cdot \left(d \cdot \ell\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(D\_m \cdot D\_m, \frac{\sqrt{\frac{h}{\ell}}}{\ell} \cdot \frac{\left(M\_m \cdot M\_m\right) \cdot -0.125}{d}, 0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))))
   (if (<= (* M_m D_m) 1e-216)
     (/ t_0 (sqrt (/ h d)))
     (if (<= (* M_m D_m) 5e+129)
       (*
        t_0
        (*
         (sqrt (/ d h))
         (-
          1.0
          (* h (/ (* (* M_m D_m) (* (* M_m D_m) 0.125)) (* d (* d l)))))))
       (fma
        (* D_m D_m)
        (* (/ (sqrt (/ h l)) l) (/ (* (* M_m M_m) -0.125) d))
        0.0)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / l));
	double tmp;
	if ((M_m * D_m) <= 1e-216) {
		tmp = t_0 / sqrt((h / d));
	} else if ((M_m * D_m) <= 5e+129) {
		tmp = t_0 * (sqrt((d / h)) * (1.0 - (h * (((M_m * D_m) * ((M_m * D_m) * 0.125)) / (d * (d * l))))));
	} else {
		tmp = fma((D_m * D_m), ((sqrt((h / l)) / l) * (((M_m * M_m) * -0.125) / d)), 0.0);
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / l))
	tmp = 0.0
	if (Float64(M_m * D_m) <= 1e-216)
		tmp = Float64(t_0 / sqrt(Float64(h / d)));
	elseif (Float64(M_m * D_m) <= 5e+129)
		tmp = Float64(t_0 * Float64(sqrt(Float64(d / h)) * Float64(1.0 - Float64(h * Float64(Float64(Float64(M_m * D_m) * Float64(Float64(M_m * D_m) * 0.125)) / Float64(d * Float64(d * l)))))));
	else
		tmp = fma(Float64(D_m * D_m), Float64(Float64(sqrt(Float64(h / l)) / l) * Float64(Float64(Float64(M_m * M_m) * -0.125) / d)), 0.0);
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 1e-216], N[(t$95$0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 5e+129], N[(t$95$0 * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(h * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(N[(M$95$m * D$95$m), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * -0.125), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 1e-216

   1. Initial program 70.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. *-lowering-*.f64 53.4

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 53.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}} + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}, 0\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}, 0\right) \]

      5. *-lowering-*.f64 31.4

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}, 0\right) \]

   8. Simplified 31.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]
   9. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      3. metadata-eval N/A

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      7. *-lowering-*.f64 30.7

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   10. Applied egg-rr 30.7%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       3. frac-times N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}} \]

       4. sqrt-div N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       5. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       6. frac-2neg N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}} \]

       7. sub0-neg N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{\mathsf{neg}\left(d\right)}{\color{blue}{0 - \ell}}} \]

       8. div-inv N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{0 - \ell}}} \]

       9. sqrt-prod N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{0 - \ell}}\right)} \]

       10. sub0-neg N/A

           \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\color{blue}{0 - d}} \cdot \sqrt{\frac{1}{0 - \ell}}\right) \]

       11. *-commutative N/A

           \[\leadsto \color{blue}{\left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

       12. clear-num N/A

           \[\leadsto \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \]

       13. sqrt-div N/A

           \[\leadsto \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \]

       14. metadata-eval N/A

           \[\leadsto \left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right) \cdot \frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \]

       15. un-div-inv N/A

           \[\leadsto \color{blue}{\frac{\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}}{\sqrt{\frac{h}{d}}}} \]

       16. /-lowering-/.f64 N/A

           \[\leadsto \color{blue}{\frac{\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}}{\sqrt{\frac{h}{d}}}} \]

   12. Applied egg-rr 46.4%

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}} \]

   if 1e-216 < (*.f64 M D) < 5.0000000000000003e129

   1. Initial program 67.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      2. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\frac{\ell}{h}}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\frac{\ell}{h}}\right) \]

      6. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      7. times-frac N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\frac{M \cdot D}{2}}{d}}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\frac{M \cdot D}{2}}{d}}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      12. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{\frac{2}{M \cdot D}}}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      13. associate-/r/ N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

   4. Applied egg-rr 67.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      2. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      4. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      8. /-lowering-/.f64 68.3

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   6. Applied egg-rr 68.3%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   8. Applied egg-rr 69.4%

      \[\leadsto \color{blue}{\left(\left(1 - \frac{\left(0.5 \cdot \left(M \cdot D\right)\right) \cdot \left(0.25 \cdot \left(M \cdot D\right)\right)}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\left(1 - \frac{\color{blue}{\left(\left(\frac{1}{2} \cdot \left(M \cdot D\right)\right) \cdot \frac{1}{4}\right) \cdot \left(M \cdot D\right)}}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(1 - \frac{\color{blue}{\left(\left(\frac{1}{2} \cdot \left(M \cdot D\right)\right) \cdot \frac{1}{4}\right) \cdot \left(M \cdot D\right)}}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(1 - \frac{\left(\left(\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)\right) \cdot \frac{1}{4}\right) \cdot \left(M \cdot D\right)}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. *-commutative N/A

         \[\leadsto \left(\left(1 - \frac{\left(\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{1}{2}\right)} \cdot \frac{1}{4}\right) \cdot \left(M \cdot D\right)}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. associate-*l* N/A

         \[\leadsto \left(\left(1 - \frac{\color{blue}{\left(\left(M \cdot D\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{4}\right)\right)} \cdot \left(M \cdot D\right)}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(1 - \frac{\left(\left(M \cdot D\right) \cdot \left(\color{blue}{\frac{1}{2}} \cdot \frac{1}{4}\right)\right) \cdot \left(M \cdot D\right)}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. metadata-eval N/A

         \[\leadsto \left(\left(1 - \frac{\left(\left(M \cdot D\right) \cdot \color{blue}{\frac{1}{8}}\right) \cdot \left(M \cdot D\right)}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. metadata-eval N/A

         \[\leadsto \left(\left(1 - \frac{\left(\left(M \cdot D\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{8}\right)\right)}\right) \cdot \left(M \cdot D\right)}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(1 - \frac{\color{blue}{\left(\left(M \cdot D\right) \cdot \left(\mathsf{neg}\left(\frac{-1}{8}\right)\right)\right)} \cdot \left(M \cdot D\right)}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(\left(1 - \frac{\left(\color{blue}{\left(M \cdot D\right)} \cdot \left(\mathsf{neg}\left(\frac{-1}{8}\right)\right)\right) \cdot \left(M \cdot D\right)}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(1 - \frac{\left(\left(M \cdot D\right) \cdot \color{blue}{\frac{1}{8}}\right) \cdot \left(M \cdot D\right)}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. *-lowering-*.f64 69.4

          \[\leadsto \left(\left(1 - \frac{\left(\left(M \cdot D\right) \cdot 0.125\right) \cdot \color{blue}{\left(M \cdot D\right)}}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   10. Applied egg-rr 69.4%

       \[\leadsto \left(\left(1 - \frac{\color{blue}{\left(\left(M \cdot D\right) \cdot 0.125\right) \cdot \left(M \cdot D\right)}}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   if 5.0000000000000003e129 < (*.f64 M D)

   1. Initial program 71.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      2. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\frac{\ell}{h}}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\frac{\ell}{h}}\right) \]

      6. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      7. times-frac N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\frac{M \cdot D}{2}}{d}}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\frac{M \cdot D}{2}}{d}}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      12. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{\frac{2}{M \cdot D}}}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      13. associate-/r/ N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

   4. Applied egg-rr 79.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      2. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      4. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      8. /-lowering-/.f64 77.3

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   6. Applied egg-rr 77.3%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   7. Taylor expanded in h around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8}} \]

      2. associate-/l* N/A

         \[\leadsto \left(\color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left({D}^{2} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \cdot \frac{-1}{8} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{{D}^{2} \cdot \left(\left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8}\right)} \]

      5. *-commutative N/A

         \[\leadsto {D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]

      6. +-rgt-identity N/A

         \[\leadsto \color{blue}{{D}^{2} \cdot \left(\frac{-1}{8} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) + 0} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({D}^{2}, \frac{-1}{8} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right), 0\right)} \]

   9. Simplified 34.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(D \cdot D, \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}, 0\right)} \]
   10. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \sqrt{\color{blue}{\frac{\frac{h}{\ell}}{\ell \cdot \ell}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       2. sqrt-div N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \color{blue}{\frac{\sqrt{\frac{h}{\ell}}}{\sqrt{\ell \cdot \ell}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       3. pow2 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\sqrt{\frac{h}{\ell}}}{\sqrt{\color{blue}{{\ell}^{2}}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       4. sqrt-pow1 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\sqrt{\frac{h}{\ell}}}{\color{blue}{{\ell}^{\left(\frac{2}{2}\right)}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\sqrt{\frac{h}{\ell}}}{{\ell}^{\color{blue}{1}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       6. unpow1 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\sqrt{\frac{h}{\ell}}}{\color{blue}{\ell}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \color{blue}{\frac{\sqrt{\frac{h}{\ell}}}{\ell}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\color{blue}{\sqrt{\frac{h}{\ell}}}}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       9. /-lowering-/.f64 73.8

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\sqrt{\color{blue}{\frac{h}{\ell}}}}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}, 0\right) \]

   11. Applied egg-rr 73.8%

       \[\leadsto \mathsf{fma}\left(D \cdot D, \color{blue}{\frac{\sqrt{\frac{h}{\ell}}}{\ell}} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}, 0\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{-216}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;M \cdot D \leq 5 \cdot 10^{+129}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - h \cdot \frac{\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot 0.125\right)}{d \cdot \left(d \cdot \ell\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot D, \frac{\sqrt{\frac{h}{\ell}}}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}, 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 69.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - h \cdot \left(\frac{\left(M\_m \cdot D\_m\right) \cdot 0.125}{d \cdot \ell} \cdot \frac{M\_m \cdot D\_m}{d}\right)\right)\right)\\ \mathbf{if}\;d \leq -2.6 \cdot 10^{-240}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{-170}:\\ \;\;\;\;\mathsf{fma}\left(D\_m \cdot D\_m, \frac{\sqrt{\frac{h}{\ell}}}{\ell} \cdot \frac{\left(M\_m \cdot M\_m\right) \cdot -0.125}{d}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (sqrt (/ d l))
          (*
           (sqrt (/ d h))
           (-
            1.0
            (* h (* (/ (* (* M_m D_m) 0.125) (* d l)) (/ (* M_m D_m) d))))))))
   (if (<= d -2.6e-240)
     t_0
     (if (<= d 6.5e-170)
       (fma
        (* D_m D_m)
        (* (/ (sqrt (/ h l)) l) (/ (* (* M_m M_m) -0.125) d))
        0.0)
       t_0))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / l)) * (sqrt((d / h)) * (1.0 - (h * ((((M_m * D_m) * 0.125) / (d * l)) * ((M_m * D_m) / d)))));
	double tmp;
	if (d <= -2.6e-240) {
		tmp = t_0;
	} else if (d <= 6.5e-170) {
		tmp = fma((D_m * D_m), ((sqrt((h / l)) / l) * (((M_m * M_m) * -0.125) / d)), 0.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 - Float64(h * Float64(Float64(Float64(Float64(M_m * D_m) * 0.125) / Float64(d * l)) * Float64(Float64(M_m * D_m) / d))))))
	tmp = 0.0
	if (d <= -2.6e-240)
		tmp = t_0;
	elseif (d <= 6.5e-170)
		tmp = fma(Float64(D_m * D_m), Float64(Float64(sqrt(Float64(h / l)) / l) * Float64(Float64(Float64(M_m * M_m) * -0.125) / d)), 0.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(h * N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * 0.125), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.6e-240], t$95$0, If[LessEqual[d, 6.5e-170], N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * -0.125), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -2.59999999999999992e-240 or 6.50000000000000035e-170 < d

   1. Initial program 75.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      2. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\frac{\ell}{h}}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\frac{\ell}{h}}\right) \]

      6. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      7. times-frac N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\frac{M \cdot D}{2}}{d}}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\frac{M \cdot D}{2}}{d}}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      12. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{\frac{2}{M \cdot D}}}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      13. associate-/r/ N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

   4. Applied egg-rr 81.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      2. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      4. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      8. /-lowering-/.f64 80.9

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   6. Applied egg-rr 80.9%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   8. Applied egg-rr 66.6%

      \[\leadsto \color{blue}{\left(\left(1 - \frac{\left(0.5 \cdot \left(M \cdot D\right)\right) \cdot \left(0.25 \cdot \left(M \cdot D\right)\right)}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\left(1 - \frac{\color{blue}{\left(\left(\frac{1}{2} \cdot \left(M \cdot D\right)\right) \cdot \frac{1}{4}\right) \cdot \left(M \cdot D\right)}}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(1 - \frac{\left(\left(\frac{1}{2} \cdot \left(M \cdot D\right)\right) \cdot \frac{1}{4}\right) \cdot \left(M \cdot D\right)}{\color{blue}{\left(d \cdot \ell\right) \cdot d}} \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. times-frac N/A

         \[\leadsto \left(\left(1 - \color{blue}{\left(\frac{\left(\frac{1}{2} \cdot \left(M \cdot D\right)\right) \cdot \frac{1}{4}}{d \cdot \ell} \cdot \frac{M \cdot D}{d}\right)} \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(1 - \color{blue}{\left(\frac{\left(\frac{1}{2} \cdot \left(M \cdot D\right)\right) \cdot \frac{1}{4}}{d \cdot \ell} \cdot \frac{M \cdot D}{d}\right)} \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(1 - \left(\color{blue}{\frac{\left(\frac{1}{2} \cdot \left(M \cdot D\right)\right) \cdot \frac{1}{4}}{d \cdot \ell}} \cdot \frac{M \cdot D}{d}\right) \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(1 - \left(\frac{\left(\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)\right) \cdot \frac{1}{4}}{d \cdot \ell} \cdot \frac{M \cdot D}{d}\right) \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. *-commutative N/A

         \[\leadsto \left(\left(1 - \left(\frac{\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{1}{2}\right)} \cdot \frac{1}{4}}{d \cdot \ell} \cdot \frac{M \cdot D}{d}\right) \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. associate-*l* N/A

         \[\leadsto \left(\left(1 - \left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{4}\right)}}{d \cdot \ell} \cdot \frac{M \cdot D}{d}\right) \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(1 - \left(\frac{\left(M \cdot D\right) \cdot \left(\color{blue}{\frac{1}{2}} \cdot \frac{1}{4}\right)}{d \cdot \ell} \cdot \frac{M \cdot D}{d}\right) \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. metadata-eval N/A

          \[\leadsto \left(\left(1 - \left(\frac{\left(M \cdot D\right) \cdot \color{blue}{\frac{1}{8}}}{d \cdot \ell} \cdot \frac{M \cdot D}{d}\right) \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(1 - \left(\frac{\left(M \cdot D\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{8}\right)\right)}}{d \cdot \ell} \cdot \frac{M \cdot D}{d}\right) \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \left(\left(1 - \left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \left(\mathsf{neg}\left(\frac{-1}{8}\right)\right)}}{d \cdot \ell} \cdot \frac{M \cdot D}{d}\right) \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \left(\left(1 - \left(\frac{\color{blue}{\left(M \cdot D\right)} \cdot \left(\mathsf{neg}\left(\frac{-1}{8}\right)\right)}{d \cdot \ell} \cdot \frac{M \cdot D}{d}\right) \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      14. metadata-eval N/A

          \[\leadsto \left(\left(1 - \left(\frac{\left(M \cdot D\right) \cdot \color{blue}{\frac{1}{8}}}{d \cdot \ell} \cdot \frac{M \cdot D}{d}\right) \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \left(\left(1 - \left(\frac{\left(M \cdot D\right) \cdot \frac{1}{8}}{\color{blue}{d \cdot \ell}} \cdot \frac{M \cdot D}{d}\right) \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \left(\left(1 - \left(\frac{\left(M \cdot D\right) \cdot \frac{1}{8}}{d \cdot \ell} \cdot \color{blue}{\frac{M \cdot D}{d}}\right) \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      17. *-lowering-*.f64 77.3

          \[\leadsto \left(\left(1 - \left(\frac{\left(M \cdot D\right) \cdot 0.125}{d \cdot \ell} \cdot \frac{\color{blue}{M \cdot D}}{d}\right) \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   10. Applied egg-rr 77.3%

       \[\leadsto \left(\left(1 - \color{blue}{\left(\frac{\left(M \cdot D\right) \cdot 0.125}{d \cdot \ell} \cdot \frac{M \cdot D}{d}\right)} \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   if -2.59999999999999992e-240 < d < 6.50000000000000035e-170

   1. Initial program 49.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      2. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\frac{\ell}{h}}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\frac{\ell}{h}}\right) \]

      6. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      7. times-frac N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\frac{M \cdot D}{2}}{d}}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\frac{M \cdot D}{2}}{d}}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      12. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{\frac{2}{M \cdot D}}}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      13. associate-/r/ N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

   4. Applied egg-rr 53.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      2. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      4. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      8. /-lowering-/.f64 53.3

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   6. Applied egg-rr 53.3%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   7. Taylor expanded in h around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8}} \]

      2. associate-/l* N/A

         \[\leadsto \left(\color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left({D}^{2} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \cdot \frac{-1}{8} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{{D}^{2} \cdot \left(\left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8}\right)} \]

      5. *-commutative N/A

         \[\leadsto {D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]

      6. +-rgt-identity N/A

         \[\leadsto \color{blue}{{D}^{2} \cdot \left(\frac{-1}{8} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) + 0} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({D}^{2}, \frac{-1}{8} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right), 0\right)} \]

   9. Simplified 35.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(D \cdot D, \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}, 0\right)} \]
   10. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \sqrt{\color{blue}{\frac{\frac{h}{\ell}}{\ell \cdot \ell}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       2. sqrt-div N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \color{blue}{\frac{\sqrt{\frac{h}{\ell}}}{\sqrt{\ell \cdot \ell}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       3. pow2 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\sqrt{\frac{h}{\ell}}}{\sqrt{\color{blue}{{\ell}^{2}}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       4. sqrt-pow1 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\sqrt{\frac{h}{\ell}}}{\color{blue}{{\ell}^{\left(\frac{2}{2}\right)}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\sqrt{\frac{h}{\ell}}}{{\ell}^{\color{blue}{1}}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       6. unpow1 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\sqrt{\frac{h}{\ell}}}{\color{blue}{\ell}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \color{blue}{\frac{\sqrt{\frac{h}{\ell}}}{\ell}} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\color{blue}{\sqrt{\frac{h}{\ell}}}}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot \frac{-1}{8}}{d}, 0\right) \]

       9. /-lowering-/.f64 60.2

          \[\leadsto \mathsf{fma}\left(D \cdot D, \frac{\sqrt{\color{blue}{\frac{h}{\ell}}}}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}, 0\right) \]

   11. Applied egg-rr 60.2%

       \[\leadsto \mathsf{fma}\left(D \cdot D, \color{blue}{\frac{\sqrt{\frac{h}{\ell}}}{\ell}} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}, 0\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.6 \cdot 10^{-240}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - h \cdot \left(\frac{\left(M \cdot D\right) \cdot 0.125}{d \cdot \ell} \cdot \frac{M \cdot D}{d}\right)\right)\right)\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{-170}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot D, \frac{\sqrt{\frac{h}{\ell}}}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - h \cdot \left(\frac{\left(M \cdot D\right) \cdot 0.125}{d \cdot \ell} \cdot \frac{M \cdot D}{d}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 43.8% accurate, 6.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.45 \cdot 10^{+264}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{-302}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 3.9 \cdot 10^{-149}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(d, \frac{\frac{1}{\sqrt{\ell}}}{\sqrt{h}}, 0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -1.45e+264)
   (* (- 0.0 d) (sqrt (/ 1.0 (* h l))))
   (if (<= l 3.6e-302)
     (* (sqrt (/ d l)) (sqrt (/ d h)))
     (if (<= l 3.9e-149)
       (/ d (sqrt (* h l)))
       (fma d (/ (/ 1.0 (sqrt l)) (sqrt h)) 0.0)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1.45e+264) {
		tmp = (0.0 - d) * sqrt((1.0 / (h * l)));
	} else if (l <= 3.6e-302) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (l <= 3.9e-149) {
		tmp = d / sqrt((h * l));
	} else {
		tmp = fma(d, ((1.0 / sqrt(l)) / sqrt(h)), 0.0);
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -1.45e+264)
		tmp = Float64(Float64(0.0 - d) * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (l <= 3.6e-302)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (l <= 3.9e-149)
		tmp = Float64(d / sqrt(Float64(h * l)));
	else
		tmp = fma(d, Float64(Float64(1.0 / sqrt(l)) / sqrt(h)), 0.0);
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -1.45e+264], N[(N[(0.0 - d), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.6e-302], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.9e-149], N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.4499999999999999e264

   1. Initial program 33.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      2. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\frac{\ell}{h}}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\frac{\ell}{h}}\right) \]

      6. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      7. times-frac N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\frac{M \cdot D}{2}}{d}}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\frac{M \cdot D}{2}}{d}}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      12. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{\frac{2}{M \cdot D}}}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      13. associate-/r/ N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

   4. Applied egg-rr 34.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      2. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      4. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      8. /-lowering-/.f64 34.9

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   6. Applied egg-rr 34.9%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   8. Applied egg-rr 34.9%

      \[\leadsto \color{blue}{\left(\left(1 - \frac{\left(0.5 \cdot \left(M \cdot D\right)\right) \cdot \left(0.25 \cdot \left(M \cdot D\right)\right)}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]

   9. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       2. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       3. rem-square-sqrt N/A

          \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

       9. mul-1-neg N/A

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

       10. neg-sub0 N/A

           \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(0 - d\right)} \]

       11. --lowering--.f64 83.3

           \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(0 - d\right)} \]

   11. Simplified 83.3%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(0 - d\right)} \]

   if -1.4499999999999999e264 < l < 3.6000000000000001e-302

   1. Initial program 77.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. *-lowering-*.f64 56.6

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 56.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}} + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}, 0\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}, 0\right) \]

      5. *-lowering-*.f64 13.9

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}, 0\right) \]

   8. Simplified 13.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]
   9. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      3. metadata-eval N/A

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      7. *-lowering-*.f64 13.1

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   10. Applied egg-rr 13.1%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       3. frac-times N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}} \]

       4. sqrt-div N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       5. sqrt-div N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       6. frac-2neg N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}} \]

       7. sub0-neg N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{\mathsf{neg}\left(d\right)}{\color{blue}{0 - \ell}}} \]

       8. div-inv N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{0 - \ell}}} \]

       9. sqrt-prod N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{0 - \ell}}\right)} \]

       10. sub0-neg N/A

           \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\color{blue}{0 - d}} \cdot \sqrt{\frac{1}{0 - \ell}}\right) \]

       11. *-commutative N/A

           \[\leadsto \color{blue}{\left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \color{blue}{\left(\sqrt{0 - d} \cdot \sqrt{\frac{1}{0 - \ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

   12. Applied egg-rr 43.7%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}} \]

   if 3.6000000000000001e-302 < l < 3.9000000000000002e-149

   1. Initial program 76.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. *-lowering-*.f64 62.6

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 62.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}} + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}, 0\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}, 0\right) \]

      5. *-lowering-*.f64 24.2

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}, 0\right) \]

   8. Simplified 24.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]
   9. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      3. metadata-eval N/A

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      7. *-lowering-*.f64 24.1

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   10. Applied egg-rr 24.1%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. frac-2neg N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)}} \]

       2. distribute-frac-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{d}{\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)}\right)} \]

       3. neg-lowering-neg.f64 N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{d}{\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)}\right)} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{d}{\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)}}\right) \]

       5. neg-sub0 N/A

          \[\leadsto \mathsf{neg}\left(\frac{d}{\color{blue}{0 - \sqrt{h \cdot \ell}}}\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{d}{\color{blue}{0 - \sqrt{h \cdot \ell}}}\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{d}{0 - \color{blue}{\sqrt{h \cdot \ell}}}\right) \]

       8. *-lowering-*.f64 47.3

          \[\leadsto -\frac{d}{0 - \sqrt{\color{blue}{h \cdot \ell}}} \]

   12. Applied egg-rr 47.3%

       \[\leadsto \color{blue}{-\frac{d}{0 - \sqrt{h \cdot \ell}}} \]

   if 3.9000000000000002e-149 < l

   1. Initial program 61.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. *-lowering-*.f64 44.0

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 44.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}} + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}, 0\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}, 0\right) \]

      5. *-lowering-*.f64 41.7

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}, 0\right) \]

   8. Simplified 41.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}, 0\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}, 0\right) \]

      3. sqrt-div N/A

         \[\leadsto \mathsf{fma}\left(d, \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}}, 0\right) \]

      4. pow1/2 N/A

         \[\leadsto \mathsf{fma}\left(d, \frac{\color{blue}{{\left(\frac{1}{\ell}\right)}^{\frac{1}{2}}}}{\sqrt{h}}, 0\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(d, \frac{{\left(\frac{1}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{h}}, 0\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \color{blue}{\frac{{\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{h}}}, 0\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(d, \frac{{\left(\frac{1}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}}{\sqrt{h}}, 0\right) \]

      8. pow1/2 N/A

         \[\leadsto \mathsf{fma}\left(d, \frac{\color{blue}{\sqrt{\frac{1}{\ell}}}}{\sqrt{h}}, 0\right) \]

      9. sqrt-div N/A

         \[\leadsto \mathsf{fma}\left(d, \frac{\color{blue}{\frac{\sqrt{1}}{\sqrt{\ell}}}}{\sqrt{h}}, 0\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(d, \frac{\frac{\color{blue}{1}}{\sqrt{\ell}}}{\sqrt{h}}, 0\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(d, \frac{\color{blue}{\frac{1}{\sqrt{\ell}}}}{\sqrt{h}}, 0\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(d, \frac{\frac{1}{\color{blue}{\sqrt{\ell}}}}{\sqrt{h}}, 0\right) \]

      13. sqrt-lowering-sqrt.f64 47.4

          \[\leadsto \mathsf{fma}\left(d, \frac{\frac{1}{\sqrt{\ell}}}{\color{blue}{\sqrt{h}}}, 0\right) \]

   10. Applied egg-rr 47.4%

       \[\leadsto \mathsf{fma}\left(d, \color{blue}{\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{h}}}, 0\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.45 \cdot 10^{+264}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{-302}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 3.9 \cdot 10^{-149}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(d, \frac{\frac{1}{\sqrt{\ell}}}{\sqrt{h}}, 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 26.2% accurate, 8.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ t_1 := \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;d \leq -1.02 \cdot 10^{+24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{-293}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.82 \cdot 10^{+186}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* d (sqrt (/ 1.0 (* h l))))) (t_1 (/ d (sqrt (* h l)))))
   (if (<= d -1.02e+24)
     t_1
     (if (<= d 2.2e-293) t_0 (if (<= d 1.82e+186) t_1 t_0)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = d * sqrt((1.0 / (h * l)));
	double t_1 = d / sqrt((h * l));
	double tmp;
	if (d <= -1.02e+24) {
		tmp = t_1;
	} else if (d <= 2.2e-293) {
		tmp = t_0;
	} else if (d <= 1.82e+186) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = d * sqrt((1.0d0 / (h * l)))
    t_1 = d / sqrt((h * l))
    if (d <= (-1.02d+24)) then
        tmp = t_1
    else if (d <= 2.2d-293) then
        tmp = t_0
    else if (d <= 1.82d+186) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = d * Math.sqrt((1.0 / (h * l)));
	double t_1 = d / Math.sqrt((h * l));
	double tmp;
	if (d <= -1.02e+24) {
		tmp = t_1;
	} else if (d <= 2.2e-293) {
		tmp = t_0;
	} else if (d <= 1.82e+186) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = d * math.sqrt((1.0 / (h * l)))
	t_1 = d / math.sqrt((h * l))
	tmp = 0
	if d <= -1.02e+24:
		tmp = t_1
	elif d <= 2.2e-293:
		tmp = t_0
	elif d <= 1.82e+186:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(d * sqrt(Float64(1.0 / Float64(h * l))))
	t_1 = Float64(d / sqrt(Float64(h * l)))
	tmp = 0.0
	if (d <= -1.02e+24)
		tmp = t_1;
	elseif (d <= 2.2e-293)
		tmp = t_0;
	elseif (d <= 1.82e+186)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = d * sqrt((1.0 / (h * l)));
	t_1 = d / sqrt((h * l));
	tmp = 0.0;
	if (d <= -1.02e+24)
		tmp = t_1;
	elseif (d <= 2.2e-293)
		tmp = t_0;
	elseif (d <= 1.82e+186)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.02e+24], t$95$1, If[LessEqual[d, 2.2e-293], t$95$0, If[LessEqual[d, 1.82e+186], t$95$1, t$95$0]]]]]

Derivation

1. Split input into 2 regimes

2. if d < -1.02000000000000004e24 or 2.2e-293 < d < 1.8200000000000001e186

   1. Initial program 71.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. *-lowering-*.f64 57.7

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 57.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}} + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}, 0\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}, 0\right) \]

      5. *-lowering-*.f64 17.6

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}, 0\right) \]

   8. Simplified 17.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]
   9. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      3. metadata-eval N/A

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      7. *-lowering-*.f64 17.6

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   10. Applied egg-rr 17.6%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. frac-2neg N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)}} \]

       2. distribute-frac-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{d}{\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)}\right)} \]

       3. neg-lowering-neg.f64 N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{d}{\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)}\right)} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{d}{\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)}}\right) \]

       5. neg-sub0 N/A

          \[\leadsto \mathsf{neg}\left(\frac{d}{\color{blue}{0 - \sqrt{h \cdot \ell}}}\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{d}{\color{blue}{0 - \sqrt{h \cdot \ell}}}\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{d}{0 - \color{blue}{\sqrt{h \cdot \ell}}}\right) \]

       8. *-lowering-*.f64 28.9

          \[\leadsto -\frac{d}{0 - \sqrt{\color{blue}{h \cdot \ell}}} \]

   12. Applied egg-rr 28.9%

       \[\leadsto \color{blue}{-\frac{d}{0 - \sqrt{h \cdot \ell}}} \]

   if -1.02000000000000004e24 < d < 2.2e-293 or 1.8200000000000001e186 < d

   1. Initial program 69.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. *-lowering-*.f64 44.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 44.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}} + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}, 0\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}, 0\right) \]

      5. *-lowering-*.f64 34.1

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}, 0\right) \]

   8. Simplified 34.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]
   9. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      6. *-lowering-*.f64 34.1

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot d \]

   10. Applied egg-rr 34.1%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.02 \cdot 10^{+24}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{-293}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq 1.82 \cdot 10^{+186}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 43.4% accurate, 8.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -6.6 \cdot 10^{-249}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq 1.08 \cdot 10^{+186}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -6.6e-249)
   (* (- 0.0 d) (sqrt (/ 1.0 (* h l))))
   (if (<= d 1.08e+186) (/ d (sqrt (* h l))) (/ d (* (sqrt l) (sqrt h))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -6.6e-249) {
		tmp = (0.0 - d) * sqrt((1.0 / (h * l)));
	} else if (d <= 1.08e+186) {
		tmp = d / sqrt((h * l));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-6.6d-249)) then
        tmp = (0.0d0 - d) * sqrt((1.0d0 / (h * l)))
    else if (d <= 1.08d+186) then
        tmp = d / sqrt((h * l))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -6.6e-249) {
		tmp = (0.0 - d) * Math.sqrt((1.0 / (h * l)));
	} else if (d <= 1.08e+186) {
		tmp = d / Math.sqrt((h * l));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -6.6e-249:
		tmp = (0.0 - d) * math.sqrt((1.0 / (h * l)))
	elif d <= 1.08e+186:
		tmp = d / math.sqrt((h * l))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -6.6e-249)
		tmp = Float64(Float64(0.0 - d) * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (d <= 1.08e+186)
		tmp = Float64(d / sqrt(Float64(h * l)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -6.6e-249)
		tmp = (0.0 - d) * sqrt((1.0 / (h * l)));
	elseif (d <= 1.08e+186)
		tmp = d / sqrt((h * l));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -6.6e-249], N[(N[(0.0 - d), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.08e+186], N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -6.6e-249

   1. Initial program 76.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      2. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\frac{\ell}{h}}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\frac{\ell}{h}}\right) \]

      6. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      7. times-frac N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\frac{M \cdot D}{2}}{d}}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\frac{M \cdot D}{2}}{d}}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      12. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{\frac{2}{M \cdot D}}}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      13. associate-/r/ N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

   4. Applied egg-rr 81.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      2. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      4. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      8. /-lowering-/.f64 80.5

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   6. Applied egg-rr 80.5%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   8. Applied egg-rr 69.0%

      \[\leadsto \color{blue}{\left(\left(1 - \frac{\left(0.5 \cdot \left(M \cdot D\right)\right) \cdot \left(0.25 \cdot \left(M \cdot D\right)\right)}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]

   9. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       2. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       3. rem-square-sqrt N/A

          \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

       9. mul-1-neg N/A

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

       10. neg-sub0 N/A

           \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(0 - d\right)} \]

       11. --lowering--.f64 45.1

           \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(0 - d\right)} \]

   11. Simplified 45.1%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(0 - d\right)} \]

   if -6.6e-249 < d < 1.08000000000000003e186

   1. Initial program 61.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. *-lowering-*.f64 49.7

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 49.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}} + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}, 0\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}, 0\right) \]

      5. *-lowering-*.f64 27.9

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}, 0\right) \]

   8. Simplified 27.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]
   9. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      3. metadata-eval N/A

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      7. *-lowering-*.f64 27.9

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   10. Applied egg-rr 27.9%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. frac-2neg N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)}} \]

       2. distribute-frac-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{d}{\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)}\right)} \]

       3. neg-lowering-neg.f64 N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{d}{\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)}\right)} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{d}{\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)}}\right) \]

       5. neg-sub0 N/A

          \[\leadsto \mathsf{neg}\left(\frac{d}{\color{blue}{0 - \sqrt{h \cdot \ell}}}\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{d}{\color{blue}{0 - \sqrt{h \cdot \ell}}}\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{d}{0 - \color{blue}{\sqrt{h \cdot \ell}}}\right) \]

       8. *-lowering-*.f64 36.1

          \[\leadsto -\frac{d}{0 - \sqrt{\color{blue}{h \cdot \ell}}} \]

   12. Applied egg-rr 36.1%

       \[\leadsto \color{blue}{-\frac{d}{0 - \sqrt{h \cdot \ell}}} \]

   if 1.08000000000000003e186 < d

   1. Initial program 74.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. *-lowering-*.f64 43.3

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 43.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}} + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}, 0\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}, 0\right) \]

      5. *-lowering-*.f64 73.4

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}, 0\right) \]

   8. Simplified 73.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]
   9. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      3. metadata-eval N/A

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      7. *-lowering-*.f64 73.2

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   10. Applied egg-rr 73.2%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell}} \cdot \sqrt{h}} \]

       5. sqrt-lowering-sqrt.f64 84.4

          \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

   12. Applied egg-rr 84.4%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.6 \cdot 10^{-249}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq 1.08 \cdot 10^{+186}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 42.3% accurate, 9.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;d \leq -6.6 \cdot 10^{-249}:\\ \;\;\;\;\left(0 - d\right) \cdot t\_0\\ \mathbf{elif}\;d \leq 1.56 \cdot 10^{+186}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot t\_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* h l)))))
   (if (<= d -6.6e-249)
     (* (- 0.0 d) t_0)
     (if (<= d 1.56e+186) (/ d (sqrt (* h l))) (* d t_0)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((1.0 / (h * l)));
	double tmp;
	if (d <= -6.6e-249) {
		tmp = (0.0 - d) * t_0;
	} else if (d <= 1.56e+186) {
		tmp = d / sqrt((h * l));
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / (h * l)))
    if (d <= (-6.6d-249)) then
        tmp = (0.0d0 - d) * t_0
    else if (d <= 1.56d+186) then
        tmp = d / sqrt((h * l))
    else
        tmp = d * t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((1.0 / (h * l)));
	double tmp;
	if (d <= -6.6e-249) {
		tmp = (0.0 - d) * t_0;
	} else if (d <= 1.56e+186) {
		tmp = d / Math.sqrt((h * l));
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((1.0 / (h * l)))
	tmp = 0
	if d <= -6.6e-249:
		tmp = (0.0 - d) * t_0
	elif d <= 1.56e+186:
		tmp = d / math.sqrt((h * l))
	else:
		tmp = d * t_0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(1.0 / Float64(h * l)))
	tmp = 0.0
	if (d <= -6.6e-249)
		tmp = Float64(Float64(0.0 - d) * t_0);
	elseif (d <= 1.56e+186)
		tmp = Float64(d / sqrt(Float64(h * l)));
	else
		tmp = Float64(d * t_0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((1.0 / (h * l)));
	tmp = 0.0;
	if (d <= -6.6e-249)
		tmp = (0.0 - d) * t_0;
	elseif (d <= 1.56e+186)
		tmp = d / sqrt((h * l));
	else
		tmp = d * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -6.6e-249], N[(N[(0.0 - d), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[d, 1.56e+186], N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -6.6e-249

   1. Initial program 76.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      2. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\frac{\ell}{h}}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\frac{\ell}{h}}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\frac{\ell}{h}}\right) \]

      6. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      7. times-frac N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      10. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\frac{M \cdot D}{2}}{d}}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\frac{M \cdot D}{2}}{d}}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      12. clear-num N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{\frac{2}{M \cdot D}}}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      13. associate-/r/ N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}\right) \]

   4. Applied egg-rr 81.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      2. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      3. clear-num N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      4. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

      8. /-lowering-/.f64 80.5

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   6. Applied egg-rr 80.5%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \frac{0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\frac{1}{h}}\right) \]

   8. Applied egg-rr 69.0%

      \[\leadsto \color{blue}{\left(\left(1 - \frac{\left(0.5 \cdot \left(M \cdot D\right)\right) \cdot \left(0.25 \cdot \left(M \cdot D\right)\right)}{d \cdot \left(d \cdot \ell\right)} \cdot h\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]

   9. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       2. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       3. rem-square-sqrt N/A

          \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

       9. mul-1-neg N/A

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

       10. neg-sub0 N/A

           \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(0 - d\right)} \]

       11. --lowering--.f64 45.1

           \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(0 - d\right)} \]

   11. Simplified 45.1%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(0 - d\right)} \]

   if -6.6e-249 < d < 1.5599999999999999e186

   1. Initial program 61.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. *-lowering-*.f64 49.7

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 49.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}} + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}, 0\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}, 0\right) \]

      5. *-lowering-*.f64 27.9

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}, 0\right) \]

   8. Simplified 27.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]
   9. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      3. metadata-eval N/A

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      7. *-lowering-*.f64 27.9

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   10. Applied egg-rr 27.9%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. frac-2neg N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)}} \]

       2. distribute-frac-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{d}{\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)}\right)} \]

       3. neg-lowering-neg.f64 N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{d}{\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)}\right)} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{d}{\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)}}\right) \]

       5. neg-sub0 N/A

          \[\leadsto \mathsf{neg}\left(\frac{d}{\color{blue}{0 - \sqrt{h \cdot \ell}}}\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{d}{\color{blue}{0 - \sqrt{h \cdot \ell}}}\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{d}{0 - \color{blue}{\sqrt{h \cdot \ell}}}\right) \]

       8. *-lowering-*.f64 36.1

          \[\leadsto -\frac{d}{0 - \sqrt{\color{blue}{h \cdot \ell}}} \]

   12. Applied egg-rr 36.1%

       \[\leadsto \color{blue}{-\frac{d}{0 - \sqrt{h \cdot \ell}}} \]

   if 1.5599999999999999e186 < d

   1. Initial program 74.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. *-lowering-*.f64 43.3

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 43.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}} + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}, 0\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}, 0\right) \]

      5. *-lowering-*.f64 73.4

         \[\leadsto \mathsf{fma}\left(d, \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}, 0\right) \]

   8. Simplified 73.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]
   9. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      6. *-lowering-*.f64 73.4

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot d \]

   10. Applied egg-rr 73.4%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.6 \cdot 10^{-249}:\\ \;\;\;\;\left(0 - d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq 1.56 \cdot 10^{+186}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 26.0% accurate, 15.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \frac{d}{\sqrt{h \cdot \ell}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (/ d (sqrt (* h l))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return d / sqrt((h * l));
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = d / sqrt((h * l))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return d / Math.sqrt((h * l));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return d / math.sqrt((h * l))

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(d / sqrt(Float64(h * l)))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = d / sqrt((h * l));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.2%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing

3. Taylor expanded in M around 0

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

   3. *-commutative N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

   4. unpow2 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

   5. associate-*r* N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

   6. associate-*r* N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

   7. *-commutative N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

   9. associate-*r* N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

   10. *-commutative N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

   13. *-commutative N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

   15. unpow2 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

   17. unpow2 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

   18. associate-*l* N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   19. *-lowering-*.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   20. *-lowering-*.f64 52.0

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

5. Simplified 52.0%

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

6. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
7. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}} + 0} \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]

   3. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(d, \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}, 0\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(d, \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}, 0\right) \]

   5. *-lowering-*.f64 25.0

      \[\leadsto \mathsf{fma}\left(d, \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}, 0\right) \]

8. Simplified 25.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{1}{h \cdot \ell}}, 0\right)} \]
9. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   2. sqrt-div N/A

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

   3. metadata-eval N/A

      \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

   4. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   7. *-lowering-*.f64 24.6

      \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

10. Applied egg-rr 24.6%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
11. Step-by-step derivation

    1. frac-2neg N/A

       \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)}} \]

    2. distribute-frac-neg N/A

       \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{d}{\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)}\right)} \]

    3. neg-lowering-neg.f64 N/A

       \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{d}{\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)}\right)} \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{d}{\mathsf{neg}\left(\sqrt{h \cdot \ell}\right)}}\right) \]

    5. neg-sub0 N/A

       \[\leadsto \mathsf{neg}\left(\frac{d}{\color{blue}{0 - \sqrt{h \cdot \ell}}}\right) \]

    6. --lowering--.f64 N/A

       \[\leadsto \mathsf{neg}\left(\frac{d}{\color{blue}{0 - \sqrt{h \cdot \ell}}}\right) \]

    7. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{neg}\left(\frac{d}{0 - \color{blue}{\sqrt{h \cdot \ell}}}\right) \]

    8. *-lowering-*.f64 26.5

       \[\leadsto -\frac{d}{0 - \sqrt{\color{blue}{h \cdot \ell}}} \]

12. Applied egg-rr 26.5%

    \[\leadsto \color{blue}{-\frac{d}{0 - \sqrt{h \cdot \ell}}} \]

13. Final simplification 24.6%

    \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(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)))))