Henrywood and Agarwal, Equation (12)

Percentage Accurate: 65.6% → 76.4%

Time: 22.3s

Alternatives: 19

Speedup: 4.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.6% 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: 76.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\left(\left(\sqrt{-d} \cdot \sqrt{\frac{-1}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+62}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 - h \cdot \frac{\left(M\_m \cdot D\_m\right) \cdot \left(\frac{M\_m \cdot D\_m}{d} \cdot 0.25\right)}{d \cdot \left(\ell \cdot 2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-D\_m, \frac{\left(h \cdot 0.125\right) \cdot \left(D\_m \cdot \left(M\_m \cdot M\_m\right)\right)}{d \cdot \left(\ell \cdot d\right)}, 1\right) \cdot \frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 -2e-311)
   (*
    (* (* (sqrt (- d)) (sqrt (/ -1.0 h))) (sqrt (/ d l)))
    (+ 1.0 (* (/ h l) (* (pow (/ (* M_m D_m) (* d 2.0)) 2.0) (/ -1.0 2.0)))))
   (if (<= l 1.75e+62)
     (*
      (/ d (sqrt (* l h)))
      (-
       1.0
       (* h (/ (* (* M_m D_m) (* (/ (* M_m D_m) d) 0.25)) (* d (* l 2.0))))))
     (/
      (*
       (fma (- D_m) (/ (* (* h 0.125) (* D_m (* M_m M_m))) (* d (* l d))) 1.0)
       (/ d (sqrt l)))
      (sqrt h)))))

C

D_m = fabs(D);
M_m = fabs(M);
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 <= -2e-311) {
		tmp = ((sqrt(-d) * sqrt((-1.0 / h))) * sqrt((d / l))) * (1.0 + ((h / l) * (pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	} else if (l <= 1.75e+62) {
		tmp = (d / sqrt((l * h))) * (1.0 - (h * (((M_m * D_m) * (((M_m * D_m) / d) * 0.25)) / (d * (l * 2.0)))));
	} else {
		tmp = (fma(-D_m, (((h * 0.125) * (D_m * (M_m * M_m))) / (d * (l * d))), 1.0) * (d / sqrt(l))) / sqrt(h);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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 <= -2e-311)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) * sqrt(Float64(-1.0 / h))) * sqrt(Float64(d / l))) * 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)))));
	elseif (l <= 1.75e+62)
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * Float64(1.0 - Float64(h * Float64(Float64(Float64(M_m * D_m) * Float64(Float64(Float64(M_m * D_m) / d) * 0.25)) / Float64(d * Float64(l * 2.0))))));
	else
		tmp = Float64(Float64(fma(Float64(-D_m), Float64(Float64(Float64(h * 0.125) * Float64(D_m * Float64(M_m * M_m))) / Float64(d * Float64(l * d))), 1.0) * Float64(d / sqrt(l))) / sqrt(h));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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, -2e-311], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] * N[Sqrt[N[(-1.0 / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $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[l, 1.75e+62], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(h * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / N[(d * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-D$95$m) * N[(N[(N[(h * 0.125), $MachinePrecision] * N[(D$95$m * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.9999999999999e-311

   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. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. sqrt-prod N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. associate-/r* N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 83.6

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

   4. Applied egg-rr 83.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 83.6

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

   6. Applied egg-rr 83.6%

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

   if -1.9999999999999e-311 < l < 1.74999999999999992e62

   1. Initial program 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. div-inv N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. times-frac N/A

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

   4. Applied egg-rr 79.1%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 79.1

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

   6. Applied egg-rr 79.1%

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

   8. Applied egg-rr 75.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   10. Applied egg-rr 91.7%

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

   if 1.74999999999999992e62 < l

   1. Initial program 56.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-/.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      20. lower-*.f64 43.5

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

   5. Simplified 43.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. sqrt-div N/A

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

      6. pow1/2 N/A

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

      7. metadata-eval N/A

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

      8. lift-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 53.8

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

   7. Applied egg-rr 53.8%

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

   9. Applied egg-rr 64.7%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\left(\left(\sqrt{-d} \cdot \sqrt{\frac{-1}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+62}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 - h \cdot \frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)}{d \cdot \left(\ell \cdot 2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-D, \frac{\left(h \cdot 0.125\right) \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot \left(\ell \cdot d\right)}, 1\right) \cdot \frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 48.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(0.125 \cdot \left(D\_m \cdot \left(D\_m \cdot \frac{M\_m \cdot M\_m}{d}\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+281}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (+
           1.0
           (* (/ h l) (* (pow (/ (* M_m D_m) (* d 2.0)) 2.0) (/ -1.0 2.0))))
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))))))
   (if (<= t_0 0.0)
     (* (sqrt (/ h (* l (* l l)))) (* 0.125 (* D_m (* D_m (/ (* M_m M_m) d)))))
     (if (<= t_0 4e+281)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (* (- d) (sqrt (/ 1.0 (* l h))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (1.0 + ((h / l) * (pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = sqrt((h / (l * (l * l)))) * (0.125 * (D_m * (D_m * ((M_m * M_m) / d))));
	} else if (t_0 <= 4e+281) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = -d * sqrt((1.0 / (l * h)));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + ((h / l) * ((((m_m * d_m) / (d * 2.0d0)) ** 2.0d0) * ((-1.0d0) / 2.0d0)))) * (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0)))
    if (t_0 <= 0.0d0) then
        tmp = sqrt((h / (l * (l * l)))) * (0.125d0 * (d_m * (d_m * ((m_m * m_m) / d))))
    else if (t_0 <= 4d+281) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = -d * sqrt((1.0d0 / (l * h)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (1.0 + ((h / l) * (Math.pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.sqrt((h / (l * (l * l)))) * (0.125 * (D_m * (D_m * ((M_m * M_m) / d))));
	} else if (t_0 <= 4e+281) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = -d * Math.sqrt((1.0 / (l * h)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (1.0 + ((h / l) * (math.pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.sqrt((h / (l * (l * l)))) * (0.125 * (D_m * (D_m * ((M_m * M_m) / d))))
	elif t_0 <= 4e+281:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = -d * math.sqrt((1.0 / (l * h)))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
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(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M_m * D_m) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))) * Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(0.125 * Float64(D_m * Float64(D_m * Float64(Float64(M_m * M_m) / d)))));
	elseif (t_0 <= 4e+281)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(l * h))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (1.0 + ((h / l) * ((((M_m * D_m) / (d * 2.0)) ^ 2.0) * (-1.0 / 2.0)))) * (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = sqrt((h / (l * (l * l)))) * (0.125 * (D_m * (D_m * ((M_m * M_m) / d))));
	elseif (t_0 <= 4e+281)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = -d * sqrt((1.0 / (l * h)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(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] * 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]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.125 * N[(D$95$m * N[(D$95$m * N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+281], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-d) * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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)))) < 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. lower-/.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      20. lower-*.f64 60.5

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

   5. Simplified 60.5%

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

   6. Taylor expanded in h around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. unpow2 N/A

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

      16. rem-square-sqrt N/A

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

      17. neg-mul-1 N/A

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

   8. Simplified 34.6%

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

   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)))) < 4.0000000000000001e281

   1. Initial program 98.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-/.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      20. lower-*.f64 63.9

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

   5. Simplified 63.9%

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

   6. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 35.5

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

   8. Simplified 35.5%

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

      1. lift-*.f64 N/A

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

      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. lift-sqrt.f64 N/A

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

      5. un-div-inv N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-prod N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. frac-times N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. sqrt-undiv N/A

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

      18. lift-/.f64 N/A

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

      19. unpow1/2 N/A

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

      20. metadata-eval N/A

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

      21. lift-/.f64 N/A

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

      22. lift-pow.f64 N/A

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

      23. lift-sqrt.f64 N/A

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

      24. lift-sqrt.f64 N/A

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

   10. Applied egg-rr 97.4%

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

   if 4.0000000000000001e281 < (*.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 17.3%

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

   3. 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. lower-/.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      20. lower-*.f64 22.3

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

   5. Simplified 22.3%

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

   6. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. rem-square-sqrt N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 32.4

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

   8. Simplified 32.4%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(0.125 \cdot \left(D \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right)\right)\\ \mathbf{elif}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 4 \cdot 10^{+281}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 + \frac{\frac{M\_m \cdot D\_m}{d \cdot 2}}{\ell} \cdot \frac{\frac{\left(M\_m \cdot D\_m\right) \cdot 0.5}{d \cdot 2}}{\frac{-1}{h}}\right)\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+62}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 - h \cdot \frac{\left(M\_m \cdot D\_m\right) \cdot \left(\frac{M\_m \cdot D\_m}{d} \cdot 0.25\right)}{d \cdot \left(\ell \cdot 2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-D\_m, \frac{\left(h \cdot 0.125\right) \cdot \left(D\_m \cdot \left(M\_m \cdot M\_m\right)\right)}{d \cdot \left(\ell \cdot d\right)}, 1\right) \cdot \frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 -2e-311)
   (*
    (* (sqrt (/ d h)) (/ (sqrt (- d)) (sqrt (- l))))
    (+
     1.0
     (*
      (/ (/ (* M_m D_m) (* d 2.0)) l)
      (/ (/ (* (* M_m D_m) 0.5) (* d 2.0)) (/ -1.0 h)))))
   (if (<= l 1.5e+62)
     (*
      (/ d (sqrt (* l h)))
      (-
       1.0
       (* h (/ (* (* M_m D_m) (* (/ (* M_m D_m) d) 0.25)) (* d (* l 2.0))))))
     (/
      (*
       (fma (- D_m) (/ (* (* h 0.125) (* D_m (* M_m M_m))) (* d (* l d))) 1.0)
       (/ d (sqrt l)))
      (sqrt h)))))

C

D_m = fabs(D);
M_m = fabs(M);
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 <= -2e-311) {
		tmp = (sqrt((d / h)) * (sqrt(-d) / sqrt(-l))) * (1.0 + ((((M_m * D_m) / (d * 2.0)) / l) * ((((M_m * D_m) * 0.5) / (d * 2.0)) / (-1.0 / h))));
	} else if (l <= 1.5e+62) {
		tmp = (d / sqrt((l * h))) * (1.0 - (h * (((M_m * D_m) * (((M_m * D_m) / d) * 0.25)) / (d * (l * 2.0)))));
	} else {
		tmp = (fma(-D_m, (((h * 0.125) * (D_m * (M_m * M_m))) / (d * (l * d))), 1.0) * (d / sqrt(l))) / sqrt(h);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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 <= -2e-311)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l)))) * Float64(1.0 + Float64(Float64(Float64(Float64(M_m * D_m) / Float64(d * 2.0)) / l) * Float64(Float64(Float64(Float64(M_m * D_m) * 0.5) / Float64(d * 2.0)) / Float64(-1.0 / h)))));
	elseif (l <= 1.5e+62)
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * Float64(1.0 - Float64(h * Float64(Float64(Float64(M_m * D_m) * Float64(Float64(Float64(M_m * D_m) / d) * 0.25)) / Float64(d * Float64(l * 2.0))))));
	else
		tmp = Float64(Float64(fma(Float64(-D_m), Float64(Float64(Float64(h * 0.125) * Float64(D_m * Float64(M_m * M_m))) / Float64(d * Float64(l * d))), 1.0) * Float64(d / sqrt(l))) / sqrt(h));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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, -2e-311], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] / N[(-1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.5e+62], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(h * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / N[(d * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-D$95$m) * N[(N[(N[(h * 0.125), $MachinePrecision] * N[(D$95$m * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.9999999999999e-311

   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. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. div-inv N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. times-frac N/A

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

   4. Applied egg-rr 76.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 76.6

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

   6. Applied egg-rr 76.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 76.6

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

   8. Applied egg-rr 76.6%

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

      1. frac-2neg N/A

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

      2. lift-neg.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 81.5

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

   10. Applied egg-rr 81.5%

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

   if -1.9999999999999e-311 < l < 1.5e62

   1. Initial program 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. div-inv N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. times-frac N/A

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

   4. Applied egg-rr 79.1%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 79.1

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

   6. Applied egg-rr 79.1%

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

   8. Applied egg-rr 75.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   10. Applied egg-rr 91.7%

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

   if 1.5e62 < l

   1. Initial program 56.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-/.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      20. lower-*.f64 43.5

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

   5. Simplified 43.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. sqrt-div N/A

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

      6. pow1/2 N/A

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

      7. metadata-eval N/A

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

      8. lift-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 53.8

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

   7. Applied egg-rr 53.8%

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

   9. Applied egg-rr 64.7%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 + \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\left(M \cdot D\right) \cdot 0.5}{d \cdot 2}}{\frac{-1}{h}}\right)\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+62}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 - h \cdot \frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)}{d \cdot \left(\ell \cdot 2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-D, \frac{\left(h \cdot 0.125\right) \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot \left(\ell \cdot d\right)}, 1\right) \cdot \frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := h \cdot \left(\frac{M\_m \cdot D\_m}{d} \cdot 0.25\right)\\ t_1 := \left(M\_m \cdot D\_m\right) \cdot 0.5\\ \mathbf{if}\;h \leq -1.4 \cdot 10^{-161}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - t\_0 \cdot \frac{M\_m \cdot D\_m}{d \cdot \left(\ell \cdot 2\right)}\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + \frac{D\_m \cdot \left(D\_m \cdot \left(0.125 \cdot \left(h \cdot \left(M\_m \cdot M\_m\right)\right)\right)\right)}{d \cdot \left(\ell \cdot d\right)}\right)\\ \mathbf{elif}\;h \leq 6.6 \cdot 10^{+277}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{t\_1}{d} \cdot \frac{t\_0}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}} \cdot \left(1 - h \cdot \frac{\left(M\_m \cdot D\_m\right) \cdot t\_1}{\left(d \cdot 2\right) \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}\right)}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* h (* (/ (* M_m D_m) d) 0.25))) (t_1 (* (* M_m D_m) 0.5)))
   (if (<= h -1.4e-161)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (* t_0 (/ (* M_m D_m) (* d (* l 2.0))))))
     (if (<= h -5e-310)
       (*
        (* d (sqrt (/ 1.0 (* l h))))
        (+ -1.0 (/ (* D_m (* D_m (* 0.125 (* h (* M_m M_m))))) (* d (* l d)))))
       (if (<= h 6.6e+277)
         (* (/ d (sqrt (* l h))) (- 1.0 (* (/ t_1 d) (/ t_0 l))))
         (/
          (*
           (/ d (sqrt h))
           (- 1.0 (* h (/ (* (* M_m D_m) t_1) (* (* d 2.0) (* l (* d 2.0)))))))
          (sqrt l)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = h * (((M_m * D_m) / d) * 0.25);
	double t_1 = (M_m * D_m) * 0.5;
	double tmp;
	if (h <= -1.4e-161) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (t_0 * ((M_m * D_m) / (d * (l * 2.0)))));
	} else if (h <= -5e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + ((D_m * (D_m * (0.125 * (h * (M_m * M_m))))) / (d * (l * d))));
	} else if (h <= 6.6e+277) {
		tmp = (d / sqrt((l * h))) * (1.0 - ((t_1 / d) * (t_0 / l)));
	} else {
		tmp = ((d / sqrt(h)) * (1.0 - (h * (((M_m * D_m) * t_1) / ((d * 2.0) * (l * (d * 2.0))))))) / sqrt(l);
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
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 = h * (((m_m * d_m) / d) * 0.25d0)
    t_1 = (m_m * d_m) * 0.5d0
    if (h <= (-1.4d-161)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (t_0 * ((m_m * d_m) / (d * (l * 2.0d0)))))
    else if (h <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((-1.0d0) + ((d_m * (d_m * (0.125d0 * (h * (m_m * m_m))))) / (d * (l * d))))
    else if (h <= 6.6d+277) then
        tmp = (d / sqrt((l * h))) * (1.0d0 - ((t_1 / d) * (t_0 / l)))
    else
        tmp = ((d / sqrt(h)) * (1.0d0 - (h * (((m_m * d_m) * t_1) / ((d * 2.0d0) * (l * (d * 2.0d0))))))) / sqrt(l)
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = h * (((M_m * D_m) / d) * 0.25);
	double t_1 = (M_m * D_m) * 0.5;
	double tmp;
	if (h <= -1.4e-161) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (t_0 * ((M_m * D_m) / (d * (l * 2.0)))));
	} else if (h <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (-1.0 + ((D_m * (D_m * (0.125 * (h * (M_m * M_m))))) / (d * (l * d))));
	} else if (h <= 6.6e+277) {
		tmp = (d / Math.sqrt((l * h))) * (1.0 - ((t_1 / d) * (t_0 / l)));
	} else {
		tmp = ((d / Math.sqrt(h)) * (1.0 - (h * (((M_m * D_m) * t_1) / ((d * 2.0) * (l * (d * 2.0))))))) / Math.sqrt(l);
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = h * (((M_m * D_m) / d) * 0.25)
	t_1 = (M_m * D_m) * 0.5
	tmp = 0
	if h <= -1.4e-161:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (t_0 * ((M_m * D_m) / (d * (l * 2.0)))))
	elif h <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (-1.0 + ((D_m * (D_m * (0.125 * (h * (M_m * M_m))))) / (d * (l * d))))
	elif h <= 6.6e+277:
		tmp = (d / math.sqrt((l * h))) * (1.0 - ((t_1 / d) * (t_0 / l)))
	else:
		tmp = ((d / math.sqrt(h)) * (1.0 - (h * (((M_m * D_m) * t_1) / ((d * 2.0) * (l * (d * 2.0))))))) / math.sqrt(l)
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
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(h * Float64(Float64(Float64(M_m * D_m) / d) * 0.25))
	t_1 = Float64(Float64(M_m * D_m) * 0.5)
	tmp = 0.0
	if (h <= -1.4e-161)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(t_0 * Float64(Float64(M_m * D_m) / Float64(d * Float64(l * 2.0))))));
	elseif (h <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + Float64(Float64(D_m * Float64(D_m * Float64(0.125 * Float64(h * Float64(M_m * M_m))))) / Float64(d * Float64(l * d)))));
	elseif (h <= 6.6e+277)
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * Float64(1.0 - Float64(Float64(t_1 / d) * Float64(t_0 / l))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(h)) * Float64(1.0 - Float64(h * Float64(Float64(Float64(M_m * D_m) * t_1) / Float64(Float64(d * 2.0) * Float64(l * Float64(d * 2.0))))))) / sqrt(l));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = h * (((M_m * D_m) / d) * 0.25);
	t_1 = (M_m * D_m) * 0.5;
	tmp = 0.0;
	if (h <= -1.4e-161)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (t_0 * ((M_m * D_m) / (d * (l * 2.0)))));
	elseif (h <= -5e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + ((D_m * (D_m * (0.125 * (h * (M_m * M_m))))) / (d * (l * d))));
	elseif (h <= 6.6e+277)
		tmp = (d / sqrt((l * h))) * (1.0 - ((t_1 / d) * (t_0 / l)));
	else
		tmp = ((d / sqrt(h)) * (1.0 - (h * (((M_m * D_m) * t_1) / ((d * 2.0) * (l * (d * 2.0))))))) / sqrt(l);
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(h * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(M$95$m * D$95$m), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[h, -1.4e-161], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(t$95$0 * N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(N[(D$95$m * N[(D$95$m * N[(0.125 * N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 6.6e+277], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(t$95$1 / d), $MachinePrecision] * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(h * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[(d * 2.0), $MachinePrecision] * N[(l * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -1.39999999999999996e-161

   1. Initial program 77.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. div-inv N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. times-frac N/A

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

   4. Applied egg-rr 81.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 81.4

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

   6. Applied egg-rr 81.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 81.4

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

   8. Applied egg-rr 81.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-2neg N/A

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

      12. lift-neg.f64 N/A

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

      13. frac-2neg N/A

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

      14. lift-/.f64 N/A

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

      15. frac-2neg N/A

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

      16. lift-/.f64 N/A

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

   10. Applied egg-rr 79.4%

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

   if -1.39999999999999996e-161 < h < -4.999999999999985e-310

   1. Initial program 59.8%

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

   3. 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. lower-/.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      20. lower-*.f64 43.6

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

   5. Simplified 43.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. sqrt-div N/A

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

      6. pow1/2 N/A

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

      7. metadata-eval N/A

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

      8. lift-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 0.0

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

   7. Applied egg-rr 0.0%

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

   8. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 68.1

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

   10. Simplified 68.1%

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

   if -4.999999999999985e-310 < h < 6.6000000000000003e277

   1. Initial program 69.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. div-inv N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. times-frac N/A

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

   4. Applied egg-rr 75.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 75.2

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

   6. Applied egg-rr 75.2%

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

   8. Applied egg-rr 72.2%

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

   10. Applied egg-rr 88.1%

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

   if 6.6000000000000003e277 < h

   1. Initial program 36.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. div-inv N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. times-frac N/A

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

   4. Applied egg-rr 29.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 29.0

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

   6. Applied egg-rr 29.0%

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

   8. Applied egg-rr 72.4%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.4 \cdot 10^{-161}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)\right) \cdot \frac{M \cdot D}{d \cdot \left(\ell \cdot 2\right)}\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(\ell \cdot d\right)}\right)\\ \mathbf{elif}\;h \leq 6.6 \cdot 10^{+277}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot 0.5}{d} \cdot \frac{h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}} \cdot \left(1 - h \cdot \frac{\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot 0.5\right)}{\left(d \cdot 2\right) \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}\right)}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left(M\_m \cdot D\_m\right) \cdot 0.5\\ t_1 := 1 - h \cdot \frac{\left(M\_m \cdot D\_m\right) \cdot t\_0}{\left(d \cdot 2\right) \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}\\ \mathbf{if}\;h \leq -2.9 \cdot 10^{-34}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot t\_1\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + \frac{D\_m \cdot \left(D\_m \cdot \left(0.125 \cdot \left(h \cdot \left(M\_m \cdot M\_m\right)\right)\right)\right)}{d \cdot \left(\ell \cdot d\right)}\right)\\ \mathbf{elif}\;h \leq 10^{+278}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{t\_0}{d} \cdot \frac{h \cdot \left(\frac{M\_m \cdot D\_m}{d} \cdot 0.25\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}} \cdot t\_1}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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) 0.5))
        (t_1
         (- 1.0 (* h (/ (* (* M_m D_m) t_0) (* (* d 2.0) (* l (* d 2.0))))))))
   (if (<= h -2.9e-34)
     (* (sqrt (/ d l)) (* (sqrt (/ d h)) t_1))
     (if (<= h -5e-310)
       (*
        (* d (sqrt (/ 1.0 (* l h))))
        (+ -1.0 (/ (* D_m (* D_m (* 0.125 (* h (* M_m M_m))))) (* d (* l d)))))
       (if (<= h 1e+278)
         (*
          (/ d (sqrt (* l h)))
          (- 1.0 (* (/ t_0 d) (/ (* h (* (/ (* M_m D_m) d) 0.25)) l))))
         (/ (* (/ d (sqrt h)) t_1) (sqrt l)))))))

C

D_m = fabs(D);
M_m = fabs(M);
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) * 0.5;
	double t_1 = 1.0 - (h * (((M_m * D_m) * t_0) / ((d * 2.0) * (l * (d * 2.0)))));
	double tmp;
	if (h <= -2.9e-34) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * t_1);
	} else if (h <= -5e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + ((D_m * (D_m * (0.125 * (h * (M_m * M_m))))) / (d * (l * d))));
	} else if (h <= 1e+278) {
		tmp = (d / sqrt((l * h))) * (1.0 - ((t_0 / d) * ((h * (((M_m * D_m) / d) * 0.25)) / l)));
	} else {
		tmp = ((d / sqrt(h)) * t_1) / sqrt(l);
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
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 = (m_m * d_m) * 0.5d0
    t_1 = 1.0d0 - (h * (((m_m * d_m) * t_0) / ((d * 2.0d0) * (l * (d * 2.0d0)))))
    if (h <= (-2.9d-34)) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * t_1)
    else if (h <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((-1.0d0) + ((d_m * (d_m * (0.125d0 * (h * (m_m * m_m))))) / (d * (l * d))))
    else if (h <= 1d+278) then
        tmp = (d / sqrt((l * h))) * (1.0d0 - ((t_0 / d) * ((h * (((m_m * d_m) / d) * 0.25d0)) / l)))
    else
        tmp = ((d / sqrt(h)) * t_1) / sqrt(l)
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
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 = (M_m * D_m) * 0.5;
	double t_1 = 1.0 - (h * (((M_m * D_m) * t_0) / ((d * 2.0) * (l * (d * 2.0)))));
	double tmp;
	if (h <= -2.9e-34) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * t_1);
	} else if (h <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (-1.0 + ((D_m * (D_m * (0.125 * (h * (M_m * M_m))))) / (d * (l * d))));
	} else if (h <= 1e+278) {
		tmp = (d / Math.sqrt((l * h))) * (1.0 - ((t_0 / d) * ((h * (((M_m * D_m) / d) * 0.25)) / l)));
	} else {
		tmp = ((d / Math.sqrt(h)) * t_1) / Math.sqrt(l);
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[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 = (M_m * D_m) * 0.5
	t_1 = 1.0 - (h * (((M_m * D_m) * t_0) / ((d * 2.0) * (l * (d * 2.0)))))
	tmp = 0
	if h <= -2.9e-34:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * t_1)
	elif h <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (-1.0 + ((D_m * (D_m * (0.125 * (h * (M_m * M_m))))) / (d * (l * d))))
	elif h <= 1e+278:
		tmp = (d / math.sqrt((l * h))) * (1.0 - ((t_0 / d) * ((h * (((M_m * D_m) / d) * 0.25)) / l)))
	else:
		tmp = ((d / math.sqrt(h)) * t_1) / math.sqrt(l)
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
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) * 0.5)
	t_1 = Float64(1.0 - Float64(h * Float64(Float64(Float64(M_m * D_m) * t_0) / Float64(Float64(d * 2.0) * Float64(l * Float64(d * 2.0))))))
	tmp = 0.0
	if (h <= -2.9e-34)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * t_1));
	elseif (h <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + Float64(Float64(D_m * Float64(D_m * Float64(0.125 * Float64(h * Float64(M_m * M_m))))) / Float64(d * Float64(l * d)))));
	elseif (h <= 1e+278)
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * Float64(1.0 - Float64(Float64(t_0 / d) * Float64(Float64(h * Float64(Float64(Float64(M_m * D_m) / d) * 0.25)) / l))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(h)) * t_1) / sqrt(l));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
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 = (M_m * D_m) * 0.5;
	t_1 = 1.0 - (h * (((M_m * D_m) * t_0) / ((d * 2.0) * (l * (d * 2.0)))));
	tmp = 0.0;
	if (h <= -2.9e-34)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * t_1);
	elseif (h <= -5e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + ((D_m * (D_m * (0.125 * (h * (M_m * M_m))))) / (d * (l * d))));
	elseif (h <= 1e+278)
		tmp = (d / sqrt((l * h))) * (1.0 - ((t_0 / d) * ((h * (((M_m * D_m) / d) * 0.25)) / l)));
	else
		tmp = ((d / sqrt(h)) * t_1) / sqrt(l);
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(h * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[(d * 2.0), $MachinePrecision] * N[(l * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -2.9e-34], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(N[(D$95$m * N[(D$95$m * N[(0.125 * N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 1e+278], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(t$95$0 / d), $MachinePrecision] * N[(N[(h * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -2.9000000000000002e-34

   1. Initial program 77.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. div-inv N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. times-frac N/A

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

   4. Applied egg-rr 82.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 82.2

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

   6. Applied egg-rr 82.2%

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

   8. Applied egg-rr 69.6%

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

   if -2.9000000000000002e-34 < h < -4.999999999999985e-310

   1. Initial program 66.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-/.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      20. lower-*.f64 42.9

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

   5. Simplified 42.9%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. sqrt-div N/A

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

      6. pow1/2 N/A

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

      7. metadata-eval N/A

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

      8. lift-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 0.0

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

   7. Applied egg-rr 0.0%

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

   8. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 64.0

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

   10. Simplified 64.0%

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

   if -4.999999999999985e-310 < h < 9.99999999999999964e277

   1. Initial program 69.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. div-inv N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. times-frac N/A

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

   4. Applied egg-rr 75.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 75.2

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

   6. Applied egg-rr 75.2%

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

   8. Applied egg-rr 72.2%

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

   10. Applied egg-rr 88.1%

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

   if 9.99999999999999964e277 < h

   1. Initial program 36.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. div-inv N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. times-frac N/A

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

   4. Applied egg-rr 29.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 29.0

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

   6. Applied egg-rr 29.0%

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

   8. Applied egg-rr 72.4%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.9 \cdot 10^{-34}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - h \cdot \frac{\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot 0.5\right)}{\left(d \cdot 2\right) \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(\ell \cdot d\right)}\right)\\ \mathbf{elif}\;h \leq 10^{+278}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot 0.5}{d} \cdot \frac{h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}} \cdot \left(1 - h \cdot \frac{\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot 0.5\right)}{\left(d \cdot 2\right) \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}\right)}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := h \cdot \left(\frac{M\_m \cdot D\_m}{d} \cdot 0.25\right)\\ t_1 := \left(M\_m \cdot D\_m\right) \cdot 0.5\\ \mathbf{if}\;h \leq 9.5 \cdot 10^{-284}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{t\_0}{\left(d \cdot 2\right) \cdot \frac{\ell}{M\_m \cdot D\_m}}\right)\\ \mathbf{elif}\;h \leq 9.5 \cdot 10^{+277}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{t\_1}{d} \cdot \frac{t\_0}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}} \cdot \left(1 - h \cdot \frac{\left(M\_m \cdot D\_m\right) \cdot t\_1}{\left(d \cdot 2\right) \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}\right)}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* h (* (/ (* M_m D_m) d) 0.25))) (t_1 (* (* M_m D_m) 0.5)))
   (if (<= h 9.5e-284)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (/ t_0 (* (* d 2.0) (/ l (* M_m D_m))))))
     (if (<= h 9.5e+277)
       (* (/ d (sqrt (* l h))) (- 1.0 (* (/ t_1 d) (/ t_0 l))))
       (/
        (*
         (/ d (sqrt h))
         (- 1.0 (* h (/ (* (* M_m D_m) t_1) (* (* d 2.0) (* l (* d 2.0)))))))
        (sqrt l))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = h * (((M_m * D_m) / d) * 0.25);
	double t_1 = (M_m * D_m) * 0.5;
	double tmp;
	if (h <= 9.5e-284) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (t_0 / ((d * 2.0) * (l / (M_m * D_m)))));
	} else if (h <= 9.5e+277) {
		tmp = (d / sqrt((l * h))) * (1.0 - ((t_1 / d) * (t_0 / l)));
	} else {
		tmp = ((d / sqrt(h)) * (1.0 - (h * (((M_m * D_m) * t_1) / ((d * 2.0) * (l * (d * 2.0))))))) / sqrt(l);
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
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 = h * (((m_m * d_m) / d) * 0.25d0)
    t_1 = (m_m * d_m) * 0.5d0
    if (h <= 9.5d-284) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (t_0 / ((d * 2.0d0) * (l / (m_m * d_m)))))
    else if (h <= 9.5d+277) then
        tmp = (d / sqrt((l * h))) * (1.0d0 - ((t_1 / d) * (t_0 / l)))
    else
        tmp = ((d / sqrt(h)) * (1.0d0 - (h * (((m_m * d_m) * t_1) / ((d * 2.0d0) * (l * (d * 2.0d0))))))) / sqrt(l)
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = h * (((M_m * D_m) / d) * 0.25);
	double t_1 = (M_m * D_m) * 0.5;
	double tmp;
	if (h <= 9.5e-284) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (t_0 / ((d * 2.0) * (l / (M_m * D_m)))));
	} else if (h <= 9.5e+277) {
		tmp = (d / Math.sqrt((l * h))) * (1.0 - ((t_1 / d) * (t_0 / l)));
	} else {
		tmp = ((d / Math.sqrt(h)) * (1.0 - (h * (((M_m * D_m) * t_1) / ((d * 2.0) * (l * (d * 2.0))))))) / Math.sqrt(l);
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = h * (((M_m * D_m) / d) * 0.25)
	t_1 = (M_m * D_m) * 0.5
	tmp = 0
	if h <= 9.5e-284:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (t_0 / ((d * 2.0) * (l / (M_m * D_m)))))
	elif h <= 9.5e+277:
		tmp = (d / math.sqrt((l * h))) * (1.0 - ((t_1 / d) * (t_0 / l)))
	else:
		tmp = ((d / math.sqrt(h)) * (1.0 - (h * (((M_m * D_m) * t_1) / ((d * 2.0) * (l * (d * 2.0))))))) / math.sqrt(l)
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
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(h * Float64(Float64(Float64(M_m * D_m) / d) * 0.25))
	t_1 = Float64(Float64(M_m * D_m) * 0.5)
	tmp = 0.0
	if (h <= 9.5e-284)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(t_0 / Float64(Float64(d * 2.0) * Float64(l / Float64(M_m * D_m))))));
	elseif (h <= 9.5e+277)
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * Float64(1.0 - Float64(Float64(t_1 / d) * Float64(t_0 / l))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(h)) * Float64(1.0 - Float64(h * Float64(Float64(Float64(M_m * D_m) * t_1) / Float64(Float64(d * 2.0) * Float64(l * Float64(d * 2.0))))))) / sqrt(l));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = h * (((M_m * D_m) / d) * 0.25);
	t_1 = (M_m * D_m) * 0.5;
	tmp = 0.0;
	if (h <= 9.5e-284)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (t_0 / ((d * 2.0) * (l / (M_m * D_m)))));
	elseif (h <= 9.5e+277)
		tmp = (d / sqrt((l * h))) * (1.0 - ((t_1 / d) * (t_0 / l)));
	else
		tmp = ((d / sqrt(h)) * (1.0 - (h * (((M_m * D_m) * t_1) / ((d * 2.0) * (l * (d * 2.0))))))) / sqrt(l);
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(h * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(M$95$m * D$95$m), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[h, 9.5e-284], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(t$95$0 / N[(N[(d * 2.0), $MachinePrecision] * N[(l / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 9.5e+277], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(t$95$1 / d), $MachinePrecision] * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(h * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[(d * 2.0), $MachinePrecision] * N[(l * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if h < 9.5000000000000003e-284

   1. Initial program 72.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. div-inv N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. times-frac N/A

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

   4. Applied egg-rr 76.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 76.4

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

   6. Applied egg-rr 76.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 76.4

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

   8. Applied egg-rr 76.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-2neg N/A

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

      12. lift-neg.f64 N/A

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

      13. frac-2neg N/A

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

      14. lift-/.f64 N/A

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

      15. frac-2neg N/A

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

      16. lift-/.f64 N/A

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

   10. Applied egg-rr 76.4%

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

   if 9.5000000000000003e-284 < h < 9.4999999999999997e277

   1. Initial program 69.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. div-inv N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. times-frac N/A

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

   4. Applied egg-rr 75.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 75.4

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

   6. Applied egg-rr 75.4%

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

   8. Applied egg-rr 74.0%

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

   10. Applied egg-rr 89.4%

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

   if 9.4999999999999997e277 < h

   1. Initial program 36.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. div-inv N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. times-frac N/A

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

   4. Applied egg-rr 29.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 29.0

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

   6. Applied egg-rr 29.0%

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

   8. Applied egg-rr 72.4%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 9.5 \cdot 10^{-284}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)}{\left(d \cdot 2\right) \cdot \frac{\ell}{M \cdot D}}\right)\\ \mathbf{elif}\;h \leq 9.5 \cdot 10^{+277}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot 0.5}{d} \cdot \frac{h \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}} \cdot \left(1 - h \cdot \frac{\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot 0.5\right)}{\left(d \cdot 2\right) \cdot \left(\ell \cdot \left(d \cdot 2\right)\right)}\right)}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := d \cdot \left(\ell \cdot d\right)\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + \frac{D\_m \cdot \left(D\_m \cdot \left(0.125 \cdot \left(h \cdot \left(M\_m \cdot M\_m\right)\right)\right)\right)}{t\_0}\right)\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+62}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 - h \cdot \frac{\left(M\_m \cdot D\_m\right) \cdot \left(\frac{M\_m \cdot D\_m}{d} \cdot 0.25\right)}{d \cdot \left(\ell \cdot 2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-D\_m, \frac{\left(h \cdot 0.125\right) \cdot \left(D\_m \cdot \left(M\_m \cdot M\_m\right)\right)}{t\_0}, 1\right) \cdot \frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (* l d))))
   (if (<= l -2e-311)
     (*
      (* d (sqrt (/ 1.0 (* l h))))
      (+ -1.0 (/ (* D_m (* D_m (* 0.125 (* h (* M_m M_m))))) t_0)))
     (if (<= l 1.8e+62)
       (*
        (/ d (sqrt (* l h)))
        (-
         1.0
         (* h (/ (* (* M_m D_m) (* (/ (* M_m D_m) d) 0.25)) (* d (* l 2.0))))))
       (/
        (*
         (fma (- D_m) (/ (* (* h 0.125) (* D_m (* M_m M_m))) t_0) 1.0)
         (/ d (sqrt l)))
        (sqrt h))))))

C

D_m = fabs(D);
M_m = fabs(M);
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 * (l * d);
	double tmp;
	if (l <= -2e-311) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + ((D_m * (D_m * (0.125 * (h * (M_m * M_m))))) / t_0));
	} else if (l <= 1.8e+62) {
		tmp = (d / sqrt((l * h))) * (1.0 - (h * (((M_m * D_m) * (((M_m * D_m) / d) * 0.25)) / (d * (l * 2.0)))));
	} else {
		tmp = (fma(-D_m, (((h * 0.125) * (D_m * (M_m * M_m))) / t_0), 1.0) * (d / sqrt(l))) / sqrt(h);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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 * Float64(l * d))
	tmp = 0.0
	if (l <= -2e-311)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + Float64(Float64(D_m * Float64(D_m * Float64(0.125 * Float64(h * Float64(M_m * M_m))))) / t_0)));
	elseif (l <= 1.8e+62)
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * Float64(1.0 - Float64(h * Float64(Float64(Float64(M_m * D_m) * Float64(Float64(Float64(M_m * D_m) / d) * 0.25)) / Float64(d * Float64(l * 2.0))))));
	else
		tmp = Float64(Float64(fma(Float64(-D_m), Float64(Float64(Float64(h * 0.125) * Float64(D_m * Float64(M_m * M_m))) / t_0), 1.0) * Float64(d / sqrt(l))) / sqrt(h));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(l * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2e-311], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(N[(D$95$m * N[(D$95$m * N[(0.125 * N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.8e+62], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(h * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / N[(d * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-D$95$m) * N[(N[(N[(h * 0.125), $MachinePrecision] * N[(D$95$m * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.9999999999999e-311

   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. 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. lower-/.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      20. lower-*.f64 50.5

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

   5. Simplified 50.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. sqrt-div N/A

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

      6. pow1/2 N/A

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

      7. metadata-eval N/A

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

      8. lift-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 0.0

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

   7. Applied egg-rr 0.0%

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

   8. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 60.6

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

   10. Simplified 60.6%

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

   if -1.9999999999999e-311 < l < 1.8e62

   1. Initial program 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. div-inv N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. times-frac N/A

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

   4. Applied egg-rr 79.1%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 79.1

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

   6. Applied egg-rr 79.1%

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

   8. Applied egg-rr 75.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   10. Applied egg-rr 91.7%

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

   if 1.8e62 < l

   1. Initial program 56.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-/.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      20. lower-*.f64 43.5

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

   5. Simplified 43.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. sqrt-div N/A

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

      6. pow1/2 N/A

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

      7. metadata-eval N/A

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

      8. lift-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 53.8

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

   7. Applied egg-rr 53.8%

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

   9. Applied egg-rr 64.7%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(\ell \cdot d\right)}\right)\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+62}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 - h \cdot \frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)}{d \cdot \left(\ell \cdot 2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-D, \frac{\left(h \cdot 0.125\right) \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot \left(\ell \cdot d\right)}, 1\right) \cdot \frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.85 \cdot 10^{-107}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(0.125 \cdot \left(D\_m \cdot \left(D\_m \cdot \frac{M\_m \cdot M\_m}{d}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 3.1 \cdot 10^{+144}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 - h \cdot \left(D\_m \cdot \left(M\_m \cdot \frac{\left(M\_m \cdot D\_m\right) \cdot 0.5}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 -1.85e-107)
   (* (- d) (sqrt (/ 1.0 (* l h))))
   (if (<= l -2e-311)
     (* (sqrt (/ h (* l (* l l)))) (* 0.125 (* D_m (* D_m (/ (* M_m M_m) d)))))
     (if (<= l 3.1e+144)
       (*
        (/ d (sqrt (* l h)))
        (-
         1.0
         (* h (* D_m (* M_m (/ (* (* M_m D_m) 0.5) (* l (* d (* d 4.0)))))))))
       (/ d (* (sqrt l) (sqrt h)))))))

C

D_m = fabs(D);
M_m = fabs(M);
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 <= -1.85e-107) {
		tmp = -d * sqrt((1.0 / (l * h)));
	} else if (l <= -2e-311) {
		tmp = sqrt((h / (l * (l * l)))) * (0.125 * (D_m * (D_m * ((M_m * M_m) / d))));
	} else if (l <= 3.1e+144) {
		tmp = (d / sqrt((l * h))) * (1.0 - (h * (D_m * (M_m * (((M_m * D_m) * 0.5) / (l * (d * (d * 4.0))))))));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-1.85d-107)) then
        tmp = -d * sqrt((1.0d0 / (l * h)))
    else if (l <= (-2d-311)) then
        tmp = sqrt((h / (l * (l * l)))) * (0.125d0 * (d_m * (d_m * ((m_m * m_m) / d))))
    else if (l <= 3.1d+144) then
        tmp = (d / sqrt((l * h))) * (1.0d0 - (h * (d_m * (m_m * (((m_m * d_m) * 0.5d0) / (l * (d * (d * 4.0d0))))))))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1.85e-107) {
		tmp = -d * Math.sqrt((1.0 / (l * h)));
	} else if (l <= -2e-311) {
		tmp = Math.sqrt((h / (l * (l * l)))) * (0.125 * (D_m * (D_m * ((M_m * M_m) / d))));
	} else if (l <= 3.1e+144) {
		tmp = (d / Math.sqrt((l * h))) * (1.0 - (h * (D_m * (M_m * (((M_m * D_m) * 0.5) / (l * (d * (d * 4.0))))))));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -1.85e-107:
		tmp = -d * math.sqrt((1.0 / (l * h)))
	elif l <= -2e-311:
		tmp = math.sqrt((h / (l * (l * l)))) * (0.125 * (D_m * (D_m * ((M_m * M_m) / d))))
	elif l <= 3.1e+144:
		tmp = (d / math.sqrt((l * h))) * (1.0 - (h * (D_m * (M_m * (((M_m * D_m) * 0.5) / (l * (d * (d * 4.0))))))))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
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 <= -1.85e-107)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(l * h))));
	elseif (l <= -2e-311)
		tmp = Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(0.125 * Float64(D_m * Float64(D_m * Float64(Float64(M_m * M_m) / d)))));
	elseif (l <= 3.1e+144)
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * Float64(1.0 - Float64(h * Float64(D_m * Float64(M_m * Float64(Float64(Float64(M_m * D_m) * 0.5) / Float64(l * Float64(d * Float64(d * 4.0)))))))));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -1.85e-107)
		tmp = -d * sqrt((1.0 / (l * h)));
	elseif (l <= -2e-311)
		tmp = sqrt((h / (l * (l * l)))) * (0.125 * (D_m * (D_m * ((M_m * M_m) / d))));
	elseif (l <= 3.1e+144)
		tmp = (d / sqrt((l * h))) * (1.0 - (h * (D_m * (M_m * (((M_m * D_m) * 0.5) / (l * (d * (d * 4.0))))))));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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, -1.85e-107], N[((-d) * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-311], N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.125 * N[(D$95$m * N[(D$95$m * N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.1e+144], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(h * N[(D$95$m * N[(M$95$m * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / N[(l * N[(d * N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.8500000000000001e-107

   1. Initial program 68.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-/.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      20. lower-*.f64 45.6

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

   5. Simplified 45.6%

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

   6. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. rem-square-sqrt N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 52.3

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

   8. Simplified 52.3%

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

   if -1.8500000000000001e-107 < l < -1.9999999999999e-311

   1. Initial program 81.3%

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

   3. 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. lower-/.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      20. lower-*.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 h around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. unpow2 N/A

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

      16. rem-square-sqrt N/A

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

      17. neg-mul-1 N/A

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

   8. Simplified 56.9%

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

   if -1.9999999999999e-311 < l < 3.1000000000000002e144

   1. Initial program 70.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. div-inv N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. times-frac N/A

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

   4. Applied egg-rr 76.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 76.5

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

   6. Applied egg-rr 76.5%

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

   8. Applied egg-rr 73.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 76.1

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

   10. Applied egg-rr 74.1%

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

   if 3.1000000000000002e144 < l

   1. Initial program 52.8%

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

   3. 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. lower-/.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      20. lower-*.f64 41.4

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

   5. Simplified 41.4%

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

   6. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 60.8

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

   8. Simplified 60.8%

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

      1. lift-*.f64 N/A

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

      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. lift-sqrt.f64 N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 60.9

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

   10. Applied egg-rr 60.9%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   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. pow1/2 N/A

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

       4. lower-*.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{{\ell}^{\frac{1}{2}} \cdot \sqrt{h}}} \]

       5. pow1/2 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell}} \cdot \sqrt{h}} \]

       6. lower-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell}} \cdot \sqrt{h}} \]

       7. lower-sqrt.f64 72.1

          \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

   12. Applied egg-rr 72.1%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.85 \cdot 10^{-107}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(0.125 \cdot \left(D \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 3.1 \cdot 10^{+144}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 - h \cdot \left(D \cdot \left(M \cdot \frac{\left(M \cdot D\right) \cdot 0.5}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.4% accurate, 4.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.7 \cdot 10^{-106}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(0.125 \cdot \left(D\_m \cdot \left(D\_m \cdot \frac{M\_m \cdot M\_m}{d}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2.02 \cdot 10^{+37}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{M\_m \cdot \left(h \cdot M\_m\right)}{d \cdot \left(\ell \cdot d\right)} \cdot \left(0.125 \cdot \left(D\_m \cdot D\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -2.7e-106)
   (* (- d) (sqrt (/ 1.0 (* l h))))
   (if (<= l -2e-311)
     (* (sqrt (/ h (* l (* l l)))) (* 0.125 (* D_m (* D_m (/ (* M_m M_m) d)))))
     (if (<= l 2.02e+37)
       (*
        (/ d (sqrt (* l h)))
        (- 1.0 (* (/ (* M_m (* h M_m)) (* d (* l d))) (* 0.125 (* D_m D_m)))))
       (/ d (* (sqrt l) (sqrt h)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -2.7e-106) {
		tmp = -d * sqrt((1.0 / (l * h)));
	} else if (l <= -2e-311) {
		tmp = sqrt((h / (l * (l * l)))) * (0.125 * (D_m * (D_m * ((M_m * M_m) / d))));
	} else if (l <= 2.02e+37) {
		tmp = (d / sqrt((l * h))) * (1.0 - (((M_m * (h * M_m)) / (d * (l * d))) * (0.125 * (D_m * D_m))));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-2.7d-106)) then
        tmp = -d * sqrt((1.0d0 / (l * h)))
    else if (l <= (-2d-311)) then
        tmp = sqrt((h / (l * (l * l)))) * (0.125d0 * (d_m * (d_m * ((m_m * m_m) / d))))
    else if (l <= 2.02d+37) then
        tmp = (d / sqrt((l * h))) * (1.0d0 - (((m_m * (h * m_m)) / (d * (l * d))) * (0.125d0 * (d_m * d_m))))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -2.7e-106) {
		tmp = -d * Math.sqrt((1.0 / (l * h)));
	} else if (l <= -2e-311) {
		tmp = Math.sqrt((h / (l * (l * l)))) * (0.125 * (D_m * (D_m * ((M_m * M_m) / d))));
	} else if (l <= 2.02e+37) {
		tmp = (d / Math.sqrt((l * h))) * (1.0 - (((M_m * (h * M_m)) / (d * (l * d))) * (0.125 * (D_m * D_m))));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -2.7e-106:
		tmp = -d * math.sqrt((1.0 / (l * h)))
	elif l <= -2e-311:
		tmp = math.sqrt((h / (l * (l * l)))) * (0.125 * (D_m * (D_m * ((M_m * M_m) / d))))
	elif l <= 2.02e+37:
		tmp = (d / math.sqrt((l * h))) * (1.0 - (((M_m * (h * M_m)) / (d * (l * d))) * (0.125 * (D_m * D_m))))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -2.7e-106)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(l * h))));
	elseif (l <= -2e-311)
		tmp = Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(0.125 * Float64(D_m * Float64(D_m * Float64(Float64(M_m * M_m) / d)))));
	elseif (l <= 2.02e+37)
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * Float64(1.0 - Float64(Float64(Float64(M_m * Float64(h * M_m)) / Float64(d * Float64(l * d))) * Float64(0.125 * Float64(D_m * D_m)))));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -2.7e-106)
		tmp = -d * sqrt((1.0 / (l * h)));
	elseif (l <= -2e-311)
		tmp = sqrt((h / (l * (l * l)))) * (0.125 * (D_m * (D_m * ((M_m * M_m) / d))));
	elseif (l <= 2.02e+37)
		tmp = (d / sqrt((l * h))) * (1.0 - (((M_m * (h * M_m)) / (d * (l * d))) * (0.125 * (D_m * D_m))));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -2.7e-106], N[((-d) * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-311], N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.125 * N[(D$95$m * N[(D$95$m * N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.02e+37], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(M$95$m * N[(h * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(d * N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.70000000000000022e-106

   1. Initial program 68.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      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. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 45.6

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 45.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      8. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. lower-neg.f64 52.3

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 52.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if -2.70000000000000022e-106 < l < -1.9999999999999e-311

   1. Initial program 81.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. 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. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      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. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.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 h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      6. cube-mult N/A

         \[\leadsto \sqrt{\frac{h}{\color{blue}{\ell \cdot \left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{{\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{h}{\color{blue}{\ell \cdot {\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      9. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \frac{-1}{8}\right)} \]

      12. associate-*r* N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot {\left(\sqrt{-1}\right)}^{2}}}{d} \cdot \frac{-1}{8}\right) \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \left({D}^{2} \cdot {M}^{2}\right)}}{d} \cdot \frac{-1}{8}\right) \]

      14. associate-/l* N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \cdot \frac{-1}{8}\right) \]

      15. unpow2 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \frac{-1}{8}\right) \]

      16. rem-square-sqrt N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(\color{blue}{-1} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \frac{-1}{8}\right) \]

      17. neg-mul-1 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right)} \cdot \frac{-1}{8}\right) \]

   8. Simplified 56.9%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right) \cdot 0.125\right)} \]

   if -1.9999999999999e-311 < l < 2.0199999999999999e37

   1. Initial program 72.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      7. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      8. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\ell \cdot \frac{1}{h}}\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      13. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      14. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\ell \cdot \frac{1}{h}}\right) \]

      15. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

   4. Applied egg-rr 79.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lower-sqrt.f64 79.7

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   6. Applied egg-rr 79.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   8. Applied egg-rr 77.9%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot \left(0.5 \cdot \left(M \cdot D\right)\right)}{\left(\left(d \cdot 2\right) \cdot \ell\right) \cdot \left(d \cdot 2\right)} \cdot h\right)} \]

   9. Taylor expanded in M around 0

      \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   10. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{1}{8} \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\frac{1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

       3. *-commutative N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell} \cdot \left(\frac{1}{8} \cdot {D}^{2}\right)}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell} \cdot \left(\frac{1}{8} \cdot {D}^{2}\right)}\right) \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}} \cdot \left(\frac{1}{8} \cdot {D}^{2}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2} \cdot \ell} \cdot \left(\frac{1}{8} \cdot {D}^{2}\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{{d}^{2} \cdot \ell} \cdot \left(\frac{1}{8} \cdot {D}^{2}\right)\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{{d}^{2} \cdot \ell} \cdot \left(\frac{1}{8} \cdot {D}^{2}\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{M \cdot \color{blue}{\left(M \cdot h\right)}}{{d}^{2} \cdot \ell} \cdot \left(\frac{1}{8} \cdot {D}^{2}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{M \cdot \left(M \cdot h\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell} \cdot \left(\frac{1}{8} \cdot {D}^{2}\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{M \cdot \left(M \cdot h\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}} \cdot \left(\frac{1}{8} \cdot {D}^{2}\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{M \cdot \left(M \cdot h\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}} \cdot \left(\frac{1}{8} \cdot {D}^{2}\right)\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{M \cdot \left(M \cdot h\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}} \cdot \left(\frac{1}{8} \cdot {D}^{2}\right)\right) \]

       14. lower-*.f64 N/A

           \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{M \cdot \left(M \cdot h\right)}{d \cdot \left(d \cdot \ell\right)} \cdot \color{blue}{\left(\frac{1}{8} \cdot {D}^{2}\right)}\right) \]

       15. unpow2 N/A

           \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{M \cdot \left(M \cdot h\right)}{d \cdot \left(d \cdot \ell\right)} \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(D \cdot D\right)}\right)\right) \]

       16. lower-*.f64 66.0

           \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{M \cdot \left(M \cdot h\right)}{d \cdot \left(d \cdot \ell\right)} \cdot \left(0.125 \cdot \color{blue}{\left(D \cdot D\right)}\right)\right) \]

   11. Simplified 66.0%

       \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\frac{M \cdot \left(M \cdot h\right)}{d \cdot \left(d \cdot \ell\right)} \cdot \left(0.125 \cdot \left(D \cdot D\right)\right)}\right) \]

   if 2.0199999999999999e37 < l

   1. Initial program 58.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      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. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 41.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 41.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. lower-*.f64 52.8

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   8. Simplified 52.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      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. lift-sqrt.f64 N/A

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      6. lower-/.f64 52.9

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   10. Applied egg-rr 52.9%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   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. pow1/2 N/A

          \[\leadsto \frac{d}{\color{blue}{{\ell}^{\frac{1}{2}}} \cdot \sqrt{h}} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{{\ell}^{\frac{1}{2}} \cdot \sqrt{h}}} \]

       5. pow1/2 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell}} \cdot \sqrt{h}} \]

       6. lower-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell}} \cdot \sqrt{h}} \]

       7. lower-sqrt.f64 60.0

          \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

   12. Applied egg-rr 60.0%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.7 \cdot 10^{-106}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(0.125 \cdot \left(D \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2.02 \cdot 10^{+37}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{M \cdot \left(h \cdot M\right)}{d \cdot \left(\ell \cdot d\right)} \cdot \left(0.125 \cdot \left(D \cdot D\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.7% accurate, 4.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + \frac{D\_m \cdot \left(D\_m \cdot \left(0.125 \cdot \left(h \cdot \left(M\_m \cdot M\_m\right)\right)\right)\right)}{d \cdot \left(\ell \cdot d\right)}\right)\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+156}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 - h \cdot \left(D\_m \cdot \left(M\_m \cdot \frac{\left(M\_m \cdot D\_m\right) \cdot 0.5}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 -2e-311)
   (*
    (* d (sqrt (/ 1.0 (* l h))))
    (+ -1.0 (/ (* D_m (* D_m (* 0.125 (* h (* M_m M_m))))) (* d (* l d)))))
   (if (<= l 1.65e+156)
     (*
      (/ d (sqrt (* l h)))
      (-
       1.0
       (* h (* D_m (* M_m (/ (* (* M_m D_m) 0.5) (* l (* d (* d 4.0)))))))))
     (/ d (* (sqrt l) (sqrt h))))))

C

D_m = fabs(D);
M_m = fabs(M);
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 <= -2e-311) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + ((D_m * (D_m * (0.125 * (h * (M_m * M_m))))) / (d * (l * d))));
	} else if (l <= 1.65e+156) {
		tmp = (d / sqrt((l * h))) * (1.0 - (h * (D_m * (M_m * (((M_m * D_m) * 0.5) / (l * (d * (d * 4.0))))))));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-2d-311)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((-1.0d0) + ((d_m * (d_m * (0.125d0 * (h * (m_m * m_m))))) / (d * (l * d))))
    else if (l <= 1.65d+156) then
        tmp = (d / sqrt((l * h))) * (1.0d0 - (h * (d_m * (m_m * (((m_m * d_m) * 0.5d0) / (l * (d * (d * 4.0d0))))))))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -2e-311) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (-1.0 + ((D_m * (D_m * (0.125 * (h * (M_m * M_m))))) / (d * (l * d))));
	} else if (l <= 1.65e+156) {
		tmp = (d / Math.sqrt((l * h))) * (1.0 - (h * (D_m * (M_m * (((M_m * D_m) * 0.5) / (l * (d * (d * 4.0))))))));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -2e-311:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (-1.0 + ((D_m * (D_m * (0.125 * (h * (M_m * M_m))))) / (d * (l * d))))
	elif l <= 1.65e+156:
		tmp = (d / math.sqrt((l * h))) * (1.0 - (h * (D_m * (M_m * (((M_m * D_m) * 0.5) / (l * (d * (d * 4.0))))))))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
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 <= -2e-311)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + Float64(Float64(D_m * Float64(D_m * Float64(0.125 * Float64(h * Float64(M_m * M_m))))) / Float64(d * Float64(l * d)))));
	elseif (l <= 1.65e+156)
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * Float64(1.0 - Float64(h * Float64(D_m * Float64(M_m * Float64(Float64(Float64(M_m * D_m) * 0.5) / Float64(l * Float64(d * Float64(d * 4.0)))))))));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -2e-311)
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + ((D_m * (D_m * (0.125 * (h * (M_m * M_m))))) / (d * (l * d))));
	elseif (l <= 1.65e+156)
		tmp = (d / sqrt((l * h))) * (1.0 - (h * (D_m * (M_m * (((M_m * D_m) * 0.5) / (l * (d * (d * 4.0))))))));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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, -2e-311], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(N[(D$95$m * N[(D$95$m * N[(0.125 * N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.65e+156], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(h * N[(D$95$m * N[(M$95$m * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / N[(l * N[(d * N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 l < -1.9999999999999e-311

   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. 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. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      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. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 50.5

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 50.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      5. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      6. pow1/2 N/A

         \[\leadsto \left(\frac{\color{blue}{{d}^{\frac{1}{2}}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(\frac{{d}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\frac{{d}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{{d}^{\left(\frac{1}{2}\right)}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\frac{{d}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\frac{{d}^{\color{blue}{\frac{1}{2}}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      14. lower-sqrt.f64 0.0

          \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

   7. Applied egg-rr 0.0%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

   8. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{-1} \cdot d\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)\right)} \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      9. neg-mul-1 N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      10. lower-neg.f64 60.6

          \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

   10. Simplified 60.6%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\right)} \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

   if -1.9999999999999e-311 < l < 1.6499999999999999e156

   1. Initial program 70.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      7. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      8. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\ell \cdot \frac{1}{h}}\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      13. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      14. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\ell \cdot \frac{1}{h}}\right) \]

      15. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

   4. Applied egg-rr 76.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lower-sqrt.f64 76.5

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   6. Applied egg-rr 76.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   8. Applied egg-rr 73.2%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot \left(0.5 \cdot \left(M \cdot D\right)\right)}{\left(\left(d \cdot 2\right) \cdot \ell\right) \cdot \left(d \cdot 2\right)} \cdot h\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\color{blue}{\left(M \cdot D\right)} \cdot \left(\frac{1}{2} \cdot \left(M \cdot D\right)\right)}{\left(\left(d \cdot 2\right) \cdot \ell\right) \cdot \left(d \cdot 2\right)} \cdot h\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}\right)}{\left(\left(d \cdot 2\right) \cdot \ell\right) \cdot \left(d \cdot 2\right)} \cdot h\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(M \cdot D\right)\right)}}{\left(\left(d \cdot 2\right) \cdot \ell\right) \cdot \left(d \cdot 2\right)} \cdot h\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot \left(\frac{1}{2} \cdot \left(M \cdot D\right)\right)}{\left(\color{blue}{\left(d \cdot 2\right)} \cdot \ell\right) \cdot \left(d \cdot 2\right)} \cdot h\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot \left(\frac{1}{2} \cdot \left(M \cdot D\right)\right)}{\color{blue}{\left(\left(d \cdot 2\right) \cdot \ell\right)} \cdot \left(d \cdot 2\right)} \cdot h\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot \left(\frac{1}{2} \cdot \left(M \cdot D\right)\right)}{\left(\left(d \cdot 2\right) \cdot \ell\right) \cdot \color{blue}{\left(d \cdot 2\right)}} \cdot h\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot \left(\frac{1}{2} \cdot \left(M \cdot D\right)\right)}{\color{blue}{\left(\left(d \cdot 2\right) \cdot \ell\right) \cdot \left(d \cdot 2\right)}} \cdot h\right) \]

      8. associate-/l* N/A

         \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{\left(\left(d \cdot 2\right) \cdot \ell\right) \cdot \left(d \cdot 2\right)}\right)} \cdot h\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\color{blue}{\left(M \cdot D\right)} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{\left(\left(d \cdot 2\right) \cdot \ell\right) \cdot \left(d \cdot 2\right)}\right) \cdot h\right) \]

      10. *-commutative N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{\left(\left(d \cdot 2\right) \cdot \ell\right) \cdot \left(d \cdot 2\right)}\right) \cdot h\right) \]

      11. associate-*l* N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(D \cdot \left(M \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{\left(\left(d \cdot 2\right) \cdot \ell\right) \cdot \left(d \cdot 2\right)}\right)\right)} \cdot h\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(D \cdot \left(M \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{\left(\left(d \cdot 2\right) \cdot \ell\right) \cdot \left(d \cdot 2\right)}\right)\right)} \cdot h\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(D \cdot \color{blue}{\left(M \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{\left(\left(d \cdot 2\right) \cdot \ell\right) \cdot \left(d \cdot 2\right)}\right)}\right) \cdot h\right) \]

      14. lower-/.f64 76.1

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(D \cdot \left(M \cdot \color{blue}{\frac{0.5 \cdot \left(M \cdot D\right)}{\left(\left(d \cdot 2\right) \cdot \ell\right) \cdot \left(d \cdot 2\right)}}\right)\right) \cdot h\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(D \cdot \left(M \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{\color{blue}{\left(\left(d \cdot 2\right) \cdot \ell\right) \cdot \left(d \cdot 2\right)}}\right)\right) \cdot h\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(D \cdot \left(M \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{\color{blue}{\left(\left(d \cdot 2\right) \cdot \ell\right)} \cdot \left(d \cdot 2\right)}\right)\right) \cdot h\right) \]

      17. *-commutative N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(D \cdot \left(M \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{\color{blue}{\left(\ell \cdot \left(d \cdot 2\right)\right)} \cdot \left(d \cdot 2\right)}\right)\right) \cdot h\right) \]

   10. Applied egg-rr 74.1%

       \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(D \cdot \left(M \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{\ell \cdot \left(d \cdot \left(4 \cdot d\right)\right)}\right)\right)} \cdot h\right) \]

   if 1.6499999999999999e156 < l

   1. Initial program 52.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. 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. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      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. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 41.4

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 41.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. lower-*.f64 60.8

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   8. Simplified 60.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      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. lift-sqrt.f64 N/A

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      6. lower-/.f64 60.9

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   10. Applied egg-rr 60.9%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   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. pow1/2 N/A

          \[\leadsto \frac{d}{\color{blue}{{\ell}^{\frac{1}{2}}} \cdot \sqrt{h}} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{{\ell}^{\frac{1}{2}} \cdot \sqrt{h}}} \]

       5. pow1/2 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell}} \cdot \sqrt{h}} \]

       6. lower-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell}} \cdot \sqrt{h}} \]

       7. lower-sqrt.f64 72.1

          \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

   12. Applied egg-rr 72.1%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(\ell \cdot d\right)}\right)\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+156}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 - h \cdot \left(D \cdot \left(M \cdot \frac{\left(M \cdot D\right) \cdot 0.5}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.9% accurate, 4.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + \frac{D\_m \cdot \left(D\_m \cdot \left(0.125 \cdot \left(h \cdot \left(M\_m \cdot M\_m\right)\right)\right)\right)}{d \cdot \left(\ell \cdot d\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 - h \cdot \frac{\left(M\_m \cdot D\_m\right) \cdot \left(\frac{M\_m \cdot D\_m}{d} \cdot 0.25\right)}{d \cdot \left(\ell \cdot 2\right)}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= h -5e-310)
   (*
    (* d (sqrt (/ 1.0 (* l h))))
    (+ -1.0 (/ (* D_m (* D_m (* 0.125 (* h (* M_m M_m))))) (* d (* l d)))))
   (*
    (/ d (sqrt (* l h)))
    (-
     1.0
     (* h (/ (* (* M_m D_m) (* (/ (* M_m D_m) d) 0.25)) (* d (* l 2.0))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= -5e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + ((D_m * (D_m * (0.125 * (h * (M_m * M_m))))) / (d * (l * d))));
	} else {
		tmp = (d / sqrt((l * h))) * (1.0 - (h * (((M_m * D_m) * (((M_m * D_m) / d) * 0.25)) / (d * (l * 2.0)))));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (h <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((-1.0d0) + ((d_m * (d_m * (0.125d0 * (h * (m_m * m_m))))) / (d * (l * d))))
    else
        tmp = (d / sqrt((l * h))) * (1.0d0 - (h * (((m_m * d_m) * (((m_m * d_m) / d) * 0.25d0)) / (d * (l * 2.0d0)))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (-1.0 + ((D_m * (D_m * (0.125 * (h * (M_m * M_m))))) / (d * (l * d))));
	} else {
		tmp = (d / Math.sqrt((l * h))) * (1.0 - (h * (((M_m * D_m) * (((M_m * D_m) / d) * 0.25)) / (d * (l * 2.0)))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if h <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (-1.0 + ((D_m * (D_m * (0.125 * (h * (M_m * M_m))))) / (d * (l * d))))
	else:
		tmp = (d / math.sqrt((l * h))) * (1.0 - (h * (((M_m * D_m) * (((M_m * D_m) / d) * 0.25)) / (d * (l * 2.0)))))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (h <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + Float64(Float64(D_m * Float64(D_m * Float64(0.125 * Float64(h * Float64(M_m * M_m))))) / Float64(d * Float64(l * d)))));
	else
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * Float64(1.0 - Float64(h * Float64(Float64(Float64(M_m * D_m) * Float64(Float64(Float64(M_m * D_m) / d) * 0.25)) / Float64(d * Float64(l * 2.0))))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (h <= -5e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + ((D_m * (D_m * (0.125 * (h * (M_m * M_m))))) / (d * (l * d))));
	else
		tmp = (d / sqrt((l * h))) * (1.0 - (h * (((M_m * D_m) * (((M_m * D_m) / d) * 0.25)) / (d * (l * 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[h, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(N[(D$95$m * N[(D$95$m * N[(0.125 * N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(h * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / N[(d * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -4.999999999999985e-310

   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. 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. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      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. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 50.5

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 50.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      5. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      6. pow1/2 N/A

         \[\leadsto \left(\frac{\color{blue}{{d}^{\frac{1}{2}}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(\frac{{d}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\frac{{d}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{{d}^{\left(\frac{1}{2}\right)}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\frac{{d}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\frac{{d}^{\color{blue}{\frac{1}{2}}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      14. lower-sqrt.f64 0.0

          \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

   7. Applied egg-rr 0.0%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

   8. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{-1} \cdot d\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)\right)} \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right)\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      9. neg-mul-1 N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      10. lower-neg.f64 60.6

          \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

   10. Simplified 60.6%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\right)} \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

   if -4.999999999999985e-310 < h

   1. Initial program 67.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      7. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      8. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\ell \cdot \frac{1}{h}}\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      13. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      14. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\ell \cdot \frac{1}{h}}\right) \]

      15. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

   4. Applied egg-rr 71.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lower-sqrt.f64 71.2

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   6. Applied egg-rr 71.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   8. Applied egg-rr 69.2%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot \left(0.5 \cdot \left(M \cdot D\right)\right)}{\left(\left(d \cdot 2\right) \cdot \ell\right) \cdot \left(d \cdot 2\right)} \cdot h\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\color{blue}{\left(M \cdot D\right)} \cdot \left(\frac{1}{2} \cdot \left(M \cdot D\right)\right)}{\left(\left(d \cdot 2\right) \cdot \ell\right) \cdot \left(d \cdot 2\right)} \cdot h\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}\right)}{\left(\left(d \cdot 2\right) \cdot \ell\right) \cdot \left(d \cdot 2\right)} \cdot h\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(M \cdot D\right)\right)}}{\left(\left(d \cdot 2\right) \cdot \ell\right) \cdot \left(d \cdot 2\right)} \cdot h\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot \left(\frac{1}{2} \cdot \left(M \cdot D\right)\right)}{\left(\color{blue}{\left(d \cdot 2\right)} \cdot \ell\right) \cdot \left(d \cdot 2\right)} \cdot h\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot \left(\frac{1}{2} \cdot \left(M \cdot D\right)\right)}{\color{blue}{\left(\left(d \cdot 2\right) \cdot \ell\right)} \cdot \left(d \cdot 2\right)} \cdot h\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot \left(\frac{1}{2} \cdot \left(M \cdot D\right)\right)}{\left(\left(d \cdot 2\right) \cdot \ell\right) \cdot \color{blue}{\left(d \cdot 2\right)}} \cdot h\right) \]

      7. times-frac N/A

         \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\frac{M \cdot D}{\left(d \cdot 2\right) \cdot \ell} \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}\right)} \cdot h\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{M \cdot D}{\left(d \cdot 2\right) \cdot \ell} \cdot \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}\right) \cdot h\right) \]

      9. associate-*l/ N/A

         \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\frac{\left(M \cdot D\right) \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\left(d \cdot 2\right) \cdot \ell}} \cdot h\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\frac{\left(M \cdot D\right) \cdot \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\left(d \cdot 2\right) \cdot \ell}} \cdot h\right) \]

   10. Applied egg-rr 80.9%

       \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)}{d \cdot \left(2 \cdot \ell\right)}} \cdot h\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(\ell \cdot d\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 - h \cdot \frac{\left(M \cdot D\right) \cdot \left(\frac{M \cdot D}{d} \cdot 0.25\right)}{d \cdot \left(\ell \cdot 2\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.9 \cdot 10^{-105}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(0.125 \cdot \left(D\_m \cdot \left(D\_m \cdot \frac{M\_m \cdot M\_m}{d}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+125}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(h \cdot \left(\left(M\_m \cdot M\_m\right) \cdot -0.125\right), D\_m \cdot \frac{D\_m}{d \cdot \left(\ell \cdot d\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -2.9e-105)
   (* (- d) (sqrt (/ 1.0 (* l h))))
   (if (<= l -2e-311)
     (* (sqrt (/ h (* l (* l l)))) (* 0.125 (* D_m (* D_m (/ (* M_m M_m) d)))))
     (if (<= l 1.3e+125)
       (*
        (/ d (sqrt (* l h)))
        (fma (* h (* (* M_m M_m) -0.125)) (* D_m (/ D_m (* d (* l d)))) 1.0))
       (/ d (* (sqrt l) (sqrt h)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -2.9e-105) {
		tmp = -d * sqrt((1.0 / (l * h)));
	} else if (l <= -2e-311) {
		tmp = sqrt((h / (l * (l * l)))) * (0.125 * (D_m * (D_m * ((M_m * M_m) / d))));
	} else if (l <= 1.3e+125) {
		tmp = (d / sqrt((l * h))) * fma((h * ((M_m * M_m) * -0.125)), (D_m * (D_m / (d * (l * d)))), 1.0);
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -2.9e-105)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(l * h))));
	elseif (l <= -2e-311)
		tmp = Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(0.125 * Float64(D_m * Float64(D_m * Float64(Float64(M_m * M_m) / d)))));
	elseif (l <= 1.3e+125)
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * fma(Float64(h * Float64(Float64(M_m * M_m) * -0.125)), Float64(D_m * Float64(D_m / Float64(d * Float64(l * d)))), 1.0));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -2.9e-105], N[((-d) * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-311], N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.125 * N[(D$95$m * N[(D$95$m * N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.3e+125], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(N[(M$95$m * M$95$m), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * N[(D$95$m / N[(d * N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.90000000000000003e-105

   1. Initial program 68.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      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. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 45.6

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 45.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      8. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. lower-neg.f64 52.3

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 52.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if -2.90000000000000003e-105 < l < -1.9999999999999e-311

   1. Initial program 81.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. 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. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      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. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.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 h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      6. cube-mult N/A

         \[\leadsto \sqrt{\frac{h}{\color{blue}{\ell \cdot \left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{{\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{h}{\color{blue}{\ell \cdot {\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      9. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \frac{-1}{8}\right)} \]

      12. associate-*r* N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot {\left(\sqrt{-1}\right)}^{2}}}{d} \cdot \frac{-1}{8}\right) \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \left({D}^{2} \cdot {M}^{2}\right)}}{d} \cdot \frac{-1}{8}\right) \]

      14. associate-/l* N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \cdot \frac{-1}{8}\right) \]

      15. unpow2 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \frac{-1}{8}\right) \]

      16. rem-square-sqrt N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(\color{blue}{-1} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \frac{-1}{8}\right) \]

      17. neg-mul-1 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right)} \cdot \frac{-1}{8}\right) \]

   8. Simplified 56.9%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right) \cdot 0.125\right)} \]

   if -1.9999999999999e-311 < l < 1.30000000000000002e125

   1. Initial program 72.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      7. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}\right) \]

      8. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}}{\frac{\ell}{h}}\right) \]

      10. div-inv N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\color{blue}{\ell \cdot \frac{1}{h}}}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}}{\ell \cdot \frac{1}{h}}\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      13. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{1}{2}}{\ell \cdot \frac{1}{h}}\right) \]

      14. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}\right)}}{\ell \cdot \frac{1}{h}}\right) \]

      15. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{2}}{\frac{1}{h}}}\right) \]

   4. Applied egg-rr 77.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

      4. lower-sqrt.f64 77.6

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   6. Applied egg-rr 77.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{d \cdot 2}}{\ell} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{1}{h}}\right) \]

   8. Applied egg-rr 73.9%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot \left(0.5 \cdot \left(M \cdot D\right)\right)}{\left(\left(d \cdot 2\right) \cdot \ell\right) \cdot \left(d \cdot 2\right)} \cdot h\right)} \]

   9. Taylor expanded in M around 0

      \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} + 1\right)} \]

       2. associate-*r/ N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(\color{blue}{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}} + 1\right) \]

       3. *-commutative N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(\frac{\frac{-1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell} + 1\right) \]

       4. associate-*r* N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(\frac{\color{blue}{\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot {D}^{2}}}{{d}^{2} \cdot \ell} + 1\right) \]

       5. associate-/l* N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(\color{blue}{\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot \frac{{D}^{2}}{{d}^{2} \cdot \ell}} + 1\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right), \frac{{D}^{2}}{{d}^{2} \cdot \ell}, 1\right)} \]

   11. Simplified 60.6%

       \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(h \cdot \left(\left(M \cdot M\right) \cdot -0.125\right), D \cdot \frac{D}{d \cdot \left(d \cdot \ell\right)}, 1\right)} \]

   if 1.30000000000000002e125 < l

   1. Initial program 52.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      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. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 43.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 43.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. lower-*.f64 58.6

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   8. Simplified 58.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      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. lift-sqrt.f64 N/A

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      6. lower-/.f64 58.7

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   10. Applied egg-rr 58.7%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   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. pow1/2 N/A

          \[\leadsto \frac{d}{\color{blue}{{\ell}^{\frac{1}{2}}} \cdot \sqrt{h}} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{{\ell}^{\frac{1}{2}} \cdot \sqrt{h}}} \]

       5. pow1/2 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell}} \cdot \sqrt{h}} \]

       6. lower-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell}} \cdot \sqrt{h}} \]

       7. lower-sqrt.f64 69.9

          \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

   12. Applied egg-rr 69.9%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.9 \cdot 10^{-105}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(0.125 \cdot \left(D \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+125}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(h \cdot \left(\left(M \cdot M\right) \cdot -0.125\right), D \cdot \frac{D}{d \cdot \left(\ell \cdot d\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 54.5% accurate, 4.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.7 \cdot 10^{-105}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(0.125 \cdot \left(D\_m \cdot \left(D\_m \cdot \frac{M\_m \cdot M\_m}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(-D\_m, \frac{\left(h \cdot 0.125\right) \cdot \left(D\_m \cdot \left(M\_m \cdot M\_m\right)\right)}{d \cdot \left(\ell \cdot d\right)}, 1\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 -1.7e-105)
   (* (- d) (sqrt (/ 1.0 (* l h))))
   (if (<= l -2e-311)
     (* (sqrt (/ h (* l (* l l)))) (* 0.125 (* D_m (* D_m (/ (* M_m M_m) d)))))
     (*
      (/ d (sqrt (* l h)))
      (fma
       (- D_m)
       (/ (* (* h 0.125) (* D_m (* M_m M_m))) (* d (* l d)))
       1.0)))))

C

D_m = fabs(D);
M_m = fabs(M);
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 <= -1.7e-105) {
		tmp = -d * sqrt((1.0 / (l * h)));
	} else if (l <= -2e-311) {
		tmp = sqrt((h / (l * (l * l)))) * (0.125 * (D_m * (D_m * ((M_m * M_m) / d))));
	} else {
		tmp = (d / sqrt((l * h))) * fma(-D_m, (((h * 0.125) * (D_m * (M_m * M_m))) / (d * (l * d))), 1.0);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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 <= -1.7e-105)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(l * h))));
	elseif (l <= -2e-311)
		tmp = Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(0.125 * Float64(D_m * Float64(D_m * Float64(Float64(M_m * M_m) / d)))));
	else
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * fma(Float64(-D_m), Float64(Float64(Float64(h * 0.125) * Float64(D_m * Float64(M_m * M_m))) / Float64(d * Float64(l * d))), 1.0));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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, -1.7e-105], N[((-d) * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-311], N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.125 * N[(D$95$m * N[(D$95$m * N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[((-D$95$m) * N[(N[(N[(h * 0.125), $MachinePrecision] * N[(D$95$m * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.69999999999999996e-105

   1. Initial program 68.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      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. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 45.6

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 45.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      8. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. lower-neg.f64 52.3

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 52.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if -1.69999999999999996e-105 < l < -1.9999999999999e-311

   1. Initial program 81.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. 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. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      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. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.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 h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      6. cube-mult N/A

         \[\leadsto \sqrt{\frac{h}{\color{blue}{\ell \cdot \left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{{\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{h}{\color{blue}{\ell \cdot {\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      9. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \frac{-1}{8}\right)} \]

      12. associate-*r* N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot {\left(\sqrt{-1}\right)}^{2}}}{d} \cdot \frac{-1}{8}\right) \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \left({D}^{2} \cdot {M}^{2}\right)}}{d} \cdot \frac{-1}{8}\right) \]

      14. associate-/l* N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \cdot \frac{-1}{8}\right) \]

      15. unpow2 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \frac{-1}{8}\right) \]

      16. rem-square-sqrt N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(\color{blue}{-1} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \frac{-1}{8}\right) \]

      17. neg-mul-1 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right)} \cdot \frac{-1}{8}\right) \]

   8. Simplified 56.9%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right) \cdot 0.125\right)} \]

   if -1.9999999999999e-311 < l

   1. Initial program 67.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      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. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 51.4

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 51.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      5. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      6. pow1/2 N/A

         \[\leadsto \left(\frac{\color{blue}{{d}^{\frac{1}{2}}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(\frac{{d}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\frac{{d}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{{d}^{\left(\frac{1}{2}\right)}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\frac{{d}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\frac{{d}^{\color{blue}{\frac{1}{2}}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      14. lower-sqrt.f64 58.5

          \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

   7. Applied egg-rr 58.5%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

   9. Applied egg-rr 64.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-D, \frac{\left(h \cdot 0.125\right) \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot \left(d \cdot \ell\right)}, 1\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.7 \cdot 10^{-105}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(0.125 \cdot \left(D \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(-D, \frac{\left(h \cdot 0.125\right) \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d \cdot \left(\ell \cdot d\right)}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 53.1% accurate, 4.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.4 \cdot 10^{-107}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(0.125 \cdot \left(D\_m \cdot \left(D\_m \cdot \frac{M\_m \cdot M\_m}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{D\_m \cdot \left(D\_m \cdot \left(0.125 \cdot \left(h \cdot \left(M\_m \cdot M\_m\right)\right)\right)\right)}{d \cdot \left(\ell \cdot d\right)}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 -6.4e-107)
   (* (- d) (sqrt (/ 1.0 (* l h))))
   (if (<= l -2e-311)
     (* (sqrt (/ h (* l (* l l)))) (* 0.125 (* D_m (* D_m (/ (* M_m M_m) d)))))
     (*
      (/ d (sqrt (* l h)))
      (- 1.0 (/ (* D_m (* D_m (* 0.125 (* h (* M_m M_m))))) (* d (* l d))))))))

C

D_m = fabs(D);
M_m = fabs(M);
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 <= -6.4e-107) {
		tmp = -d * sqrt((1.0 / (l * h)));
	} else if (l <= -2e-311) {
		tmp = sqrt((h / (l * (l * l)))) * (0.125 * (D_m * (D_m * ((M_m * M_m) / d))));
	} else {
		tmp = (d / sqrt((l * h))) * (1.0 - ((D_m * (D_m * (0.125 * (h * (M_m * M_m))))) / (d * (l * d))));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-6.4d-107)) then
        tmp = -d * sqrt((1.0d0 / (l * h)))
    else if (l <= (-2d-311)) then
        tmp = sqrt((h / (l * (l * l)))) * (0.125d0 * (d_m * (d_m * ((m_m * m_m) / d))))
    else
        tmp = (d / sqrt((l * h))) * (1.0d0 - ((d_m * (d_m * (0.125d0 * (h * (m_m * m_m))))) / (d * (l * d))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -6.4e-107) {
		tmp = -d * Math.sqrt((1.0 / (l * h)));
	} else if (l <= -2e-311) {
		tmp = Math.sqrt((h / (l * (l * l)))) * (0.125 * (D_m * (D_m * ((M_m * M_m) / d))));
	} else {
		tmp = (d / Math.sqrt((l * h))) * (1.0 - ((D_m * (D_m * (0.125 * (h * (M_m * M_m))))) / (d * (l * d))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -6.4e-107:
		tmp = -d * math.sqrt((1.0 / (l * h)))
	elif l <= -2e-311:
		tmp = math.sqrt((h / (l * (l * l)))) * (0.125 * (D_m * (D_m * ((M_m * M_m) / d))))
	else:
		tmp = (d / math.sqrt((l * h))) * (1.0 - ((D_m * (D_m * (0.125 * (h * (M_m * M_m))))) / (d * (l * d))))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
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 <= -6.4e-107)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(l * h))));
	elseif (l <= -2e-311)
		tmp = Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(0.125 * Float64(D_m * Float64(D_m * Float64(Float64(M_m * M_m) / d)))));
	else
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * Float64(1.0 - Float64(Float64(D_m * Float64(D_m * Float64(0.125 * Float64(h * Float64(M_m * M_m))))) / Float64(d * Float64(l * d)))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -6.4e-107)
		tmp = -d * sqrt((1.0 / (l * h)));
	elseif (l <= -2e-311)
		tmp = sqrt((h / (l * (l * l)))) * (0.125 * (D_m * (D_m * ((M_m * M_m) / d))));
	else
		tmp = (d / sqrt((l * h))) * (1.0 - ((D_m * (D_m * (0.125 * (h * (M_m * M_m))))) / (d * (l * d))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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, -6.4e-107], N[((-d) * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-311], N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.125 * N[(D$95$m * N[(D$95$m * N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(D$95$m * N[(D$95$m * N[(0.125 * N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.40000000000000025e-107

   1. Initial program 68.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      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. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 45.6

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 45.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      8. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. lower-neg.f64 52.3

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 52.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if -6.40000000000000025e-107 < l < -1.9999999999999e-311

   1. Initial program 81.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. 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. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      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. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.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 h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      6. cube-mult N/A

         \[\leadsto \sqrt{\frac{h}{\color{blue}{\ell \cdot \left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{{\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{h}{\color{blue}{\ell \cdot {\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      9. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \frac{-1}{8}\right)} \]

      12. associate-*r* N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot {\left(\sqrt{-1}\right)}^{2}}}{d} \cdot \frac{-1}{8}\right) \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \left({D}^{2} \cdot {M}^{2}\right)}}{d} \cdot \frac{-1}{8}\right) \]

      14. associate-/l* N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \cdot \frac{-1}{8}\right) \]

      15. unpow2 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \frac{-1}{8}\right) \]

      16. rem-square-sqrt N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(\color{blue}{-1} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \frac{-1}{8}\right) \]

      17. neg-mul-1 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right)} \cdot \frac{-1}{8}\right) \]

   8. Simplified 56.9%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right) \cdot 0.125\right)} \]

   if -1.9999999999999e-311 < l

   1. Initial program 67.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      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. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 51.4

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 51.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      5. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      6. pow1/2 N/A

         \[\leadsto \left(\frac{\color{blue}{{d}^{\frac{1}{2}}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(\frac{{d}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\frac{{d}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{{d}^{\left(\frac{1}{2}\right)}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\frac{{d}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\frac{{d}^{\color{blue}{\frac{1}{2}}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      14. lower-sqrt.f64 58.5

          \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

   7. Applied egg-rr 58.5%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      5. pow1/2 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      7. sqrt-div N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{\ell}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      10. frac-times N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}} \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d}} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{d} \cdot \color{blue}{\sqrt{d}}}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      13. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      14. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h}} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      15. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h} \cdot \color{blue}{\sqrt{\ell}}} \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      16. sqrt-prod N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      18. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      19. lower-/.f64 60.9

          \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

   9. Applied egg-rr 60.9%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.4 \cdot 10^{-107}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(0.125 \cdot \left(D \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(\ell \cdot d\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 46.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-108}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-311}:\\ \;\;\;\;t\_0 \cdot \left(0.125 \cdot \left(D\_m \cdot \left(D\_m \cdot \frac{M\_m \cdot M\_m}{d}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+37}:\\ \;\;\;\;\frac{-0.125 \cdot \left(t\_0 \cdot \left(D\_m \cdot \left(D\_m \cdot \left(M\_m \cdot M\_m\right)\right)\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (/ h (* l (* l l))))))
   (if (<= l -5e-108)
     (* (- d) (sqrt (/ 1.0 (* l h))))
     (if (<= l -2e-311)
       (* t_0 (* 0.125 (* D_m (* D_m (/ (* M_m M_m) d)))))
       (if (<= l 5e+37)
         (/ (* -0.125 (* t_0 (* D_m (* D_m (* M_m M_m))))) d)
         (/ d (* (sqrt l) (sqrt h))))))))

C

D_m = fabs(D);
M_m = fabs(M);
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((h / (l * (l * l))));
	double tmp;
	if (l <= -5e-108) {
		tmp = -d * sqrt((1.0 / (l * h)));
	} else if (l <= -2e-311) {
		tmp = t_0 * (0.125 * (D_m * (D_m * ((M_m * M_m) / d))));
	} else if (l <= 5e+37) {
		tmp = (-0.125 * (t_0 * (D_m * (D_m * (M_m * M_m))))) / d;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
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((h / (l * (l * l))))
    if (l <= (-5d-108)) then
        tmp = -d * sqrt((1.0d0 / (l * h)))
    else if (l <= (-2d-311)) then
        tmp = t_0 * (0.125d0 * (d_m * (d_m * ((m_m * m_m) / d))))
    else if (l <= 5d+37) then
        tmp = ((-0.125d0) * (t_0 * (d_m * (d_m * (m_m * m_m))))) / d
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
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((h / (l * (l * l))));
	double tmp;
	if (l <= -5e-108) {
		tmp = -d * Math.sqrt((1.0 / (l * h)));
	} else if (l <= -2e-311) {
		tmp = t_0 * (0.125 * (D_m * (D_m * ((M_m * M_m) / d))));
	} else if (l <= 5e+37) {
		tmp = (-0.125 * (t_0 * (D_m * (D_m * (M_m * M_m))))) / d;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[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((h / (l * (l * l))))
	tmp = 0
	if l <= -5e-108:
		tmp = -d * math.sqrt((1.0 / (l * h)))
	elif l <= -2e-311:
		tmp = t_0 * (0.125 * (D_m * (D_m * ((M_m * M_m) / d))))
	elif l <= 5e+37:
		tmp = (-0.125 * (t_0 * (D_m * (D_m * (M_m * M_m))))) / d
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
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(h / Float64(l * Float64(l * l))))
	tmp = 0.0
	if (l <= -5e-108)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(l * h))));
	elseif (l <= -2e-311)
		tmp = Float64(t_0 * Float64(0.125 * Float64(D_m * Float64(D_m * Float64(Float64(M_m * M_m) / d)))));
	elseif (l <= 5e+37)
		tmp = Float64(Float64(-0.125 * Float64(t_0 * Float64(D_m * Float64(D_m * Float64(M_m * M_m))))) / d);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
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((h / (l * (l * l))));
	tmp = 0.0;
	if (l <= -5e-108)
		tmp = -d * sqrt((1.0 / (l * h)));
	elseif (l <= -2e-311)
		tmp = t_0 * (0.125 * (D_m * (D_m * ((M_m * M_m) / d))));
	elseif (l <= 5e+37)
		tmp = (-0.125 * (t_0 * (D_m * (D_m * (M_m * M_m))))) / d;
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -5e-108], N[((-d) * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-311], N[(t$95$0 * N[(0.125 * N[(D$95$m * N[(D$95$m * N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5e+37], N[(N[(-0.125 * N[(t$95$0 * N[(D$95$m * N[(D$95$m * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -5e-108

   1. Initial program 68.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      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. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 45.6

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 45.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      8. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. lower-neg.f64 52.3

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 52.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if -5e-108 < l < -1.9999999999999e-311

   1. Initial program 81.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. 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. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      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. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.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 h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      6. cube-mult N/A

         \[\leadsto \sqrt{\frac{h}{\color{blue}{\ell \cdot \left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{{\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{h}{\color{blue}{\ell \cdot {\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      9. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \frac{-1}{8}\right)} \]

      12. associate-*r* N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot {\left(\sqrt{-1}\right)}^{2}}}{d} \cdot \frac{-1}{8}\right) \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \left({D}^{2} \cdot {M}^{2}\right)}}{d} \cdot \frac{-1}{8}\right) \]

      14. associate-/l* N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \cdot \frac{-1}{8}\right) \]

      15. unpow2 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \frac{-1}{8}\right) \]

      16. rem-square-sqrt N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(\color{blue}{-1} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \frac{-1}{8}\right) \]

      17. neg-mul-1 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right)} \cdot \frac{-1}{8}\right) \]

   8. Simplified 56.9%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right) \cdot 0.125\right)} \]

   if -1.9999999999999e-311 < l < 4.99999999999999989e37

   1. Initial program 72.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      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. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 56.5

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 56.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{-1}{8} \cdot \color{blue}{\frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}{d}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}{d}} \]

   8. Simplified 49.0%

      \[\leadsto \color{blue}{\frac{-0.125 \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}\right)}{d}} \]

   if 4.99999999999999989e37 < l

   1. Initial program 59.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      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. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 43.4

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 43.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. lower-*.f64 54.9

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   8. Simplified 54.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      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. lift-sqrt.f64 N/A

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      6. lower-/.f64 54.9

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   10. Applied egg-rr 54.9%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   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. pow1/2 N/A

          \[\leadsto \frac{d}{\color{blue}{{\ell}^{\frac{1}{2}}} \cdot \sqrt{h}} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{{\ell}^{\frac{1}{2}} \cdot \sqrt{h}}} \]

       5. pow1/2 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell}} \cdot \sqrt{h}} \]

       6. lower-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell}} \cdot \sqrt{h}} \]

       7. lower-sqrt.f64 62.3

          \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

   12. Applied egg-rr 62.3%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-108}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(0.125 \cdot \left(D \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+37}:\\ \;\;\;\;\frac{-0.125 \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 46.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}\\ \mathbf{if}\;\ell \leq -1.42 \cdot 10^{-105}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-311}:\\ \;\;\;\;t\_0 \cdot \left(0.125 \cdot \left(D\_m \cdot \left(D\_m \cdot \frac{M\_m \cdot M\_m}{d}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{+29}:\\ \;\;\;\;\left(D\_m \cdot D\_m\right) \cdot \left(t\_0 \cdot \frac{\left(M\_m \cdot M\_m\right) \cdot -0.125}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (/ h (* l (* l l))))))
   (if (<= l -1.42e-105)
     (* (- d) (sqrt (/ 1.0 (* l h))))
     (if (<= l -2e-311)
       (* t_0 (* 0.125 (* D_m (* D_m (/ (* M_m M_m) d)))))
       (if (<= l 8.5e+29)
         (* (* D_m D_m) (* t_0 (/ (* (* M_m M_m) -0.125) d)))
         (/ d (* (sqrt l) (sqrt h))))))))

C

D_m = fabs(D);
M_m = fabs(M);
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((h / (l * (l * l))));
	double tmp;
	if (l <= -1.42e-105) {
		tmp = -d * sqrt((1.0 / (l * h)));
	} else if (l <= -2e-311) {
		tmp = t_0 * (0.125 * (D_m * (D_m * ((M_m * M_m) / d))));
	} else if (l <= 8.5e+29) {
		tmp = (D_m * D_m) * (t_0 * (((M_m * M_m) * -0.125) / d));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
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((h / (l * (l * l))))
    if (l <= (-1.42d-105)) then
        tmp = -d * sqrt((1.0d0 / (l * h)))
    else if (l <= (-2d-311)) then
        tmp = t_0 * (0.125d0 * (d_m * (d_m * ((m_m * m_m) / d))))
    else if (l <= 8.5d+29) then
        tmp = (d_m * d_m) * (t_0 * (((m_m * m_m) * (-0.125d0)) / d))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
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((h / (l * (l * l))));
	double tmp;
	if (l <= -1.42e-105) {
		tmp = -d * Math.sqrt((1.0 / (l * h)));
	} else if (l <= -2e-311) {
		tmp = t_0 * (0.125 * (D_m * (D_m * ((M_m * M_m) / d))));
	} else if (l <= 8.5e+29) {
		tmp = (D_m * D_m) * (t_0 * (((M_m * M_m) * -0.125) / d));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[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((h / (l * (l * l))))
	tmp = 0
	if l <= -1.42e-105:
		tmp = -d * math.sqrt((1.0 / (l * h)))
	elif l <= -2e-311:
		tmp = t_0 * (0.125 * (D_m * (D_m * ((M_m * M_m) / d))))
	elif l <= 8.5e+29:
		tmp = (D_m * D_m) * (t_0 * (((M_m * M_m) * -0.125) / d))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
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(h / Float64(l * Float64(l * l))))
	tmp = 0.0
	if (l <= -1.42e-105)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(l * h))));
	elseif (l <= -2e-311)
		tmp = Float64(t_0 * Float64(0.125 * Float64(D_m * Float64(D_m * Float64(Float64(M_m * M_m) / d)))));
	elseif (l <= 8.5e+29)
		tmp = Float64(Float64(D_m * D_m) * Float64(t_0 * Float64(Float64(Float64(M_m * M_m) * -0.125) / d)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
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((h / (l * (l * l))));
	tmp = 0.0;
	if (l <= -1.42e-105)
		tmp = -d * sqrt((1.0 / (l * h)));
	elseif (l <= -2e-311)
		tmp = t_0 * (0.125 * (D_m * (D_m * ((M_m * M_m) / d))));
	elseif (l <= 8.5e+29)
		tmp = (D_m * D_m) * (t_0 * (((M_m * M_m) * -0.125) / d));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.42e-105], N[((-d) * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-311], N[(t$95$0 * N[(0.125 * N[(D$95$m * N[(D$95$m * N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8.5e+29], N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(t$95$0 * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * -0.125), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.4199999999999999e-105

   1. Initial program 68.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      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. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 45.6

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 45.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      8. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. lower-neg.f64 52.3

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 52.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if -1.4199999999999999e-105 < l < -1.9999999999999e-311

   1. Initial program 81.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. 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. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      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. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.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 h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      6. cube-mult N/A

         \[\leadsto \sqrt{\frac{h}{\color{blue}{\ell \cdot \left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{{\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{h}{\color{blue}{\ell \cdot {\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      9. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \frac{-1}{8}\right)} \]

      12. associate-*r* N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot {\left(\sqrt{-1}\right)}^{2}}}{d} \cdot \frac{-1}{8}\right) \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \left({D}^{2} \cdot {M}^{2}\right)}}{d} \cdot \frac{-1}{8}\right) \]

      14. associate-/l* N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \cdot \frac{-1}{8}\right) \]

      15. unpow2 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \frac{-1}{8}\right) \]

      16. rem-square-sqrt N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(\color{blue}{-1} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \frac{-1}{8}\right) \]

      17. neg-mul-1 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{{D}^{2} \cdot {M}^{2}}{d}\right)\right)} \cdot \frac{-1}{8}\right) \]

   8. Simplified 56.9%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right) \cdot 0.125\right)} \]

   if -1.9999999999999e-311 < l < 8.5000000000000006e29

   1. Initial program 72.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      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. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 58.0

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 58.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      3. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      5. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      6. pow1/2 N/A

         \[\leadsto \left(\frac{\color{blue}{{d}^{\frac{1}{2}}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(\frac{{d}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\frac{{d}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{{d}^{\left(\frac{1}{2}\right)}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\frac{{d}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\frac{{d}^{\color{blue}{\frac{1}{2}}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      14. lower-sqrt.f64 62.7

          \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

   7. Applied egg-rr 62.7%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

   8. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   9. 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. lower-*.f64 N/A

         \[\leadsto \color{blue}{{D}^{2} \cdot \left(\frac{-1}{8} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]

      7. unpow2 N/A

         \[\leadsto \color{blue}{\left(D \cdot D\right)} \cdot \left(\frac{-1}{8} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(D \cdot D\right)} \cdot \left(\frac{-1}{8} \cdot \left(\frac{{M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \left(D \cdot D\right) \cdot \color{blue}{\left(\left(\frac{-1}{8} \cdot \frac{{M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]

      10. *-commutative N/A

          \[\leadsto \left(D \cdot D\right) \cdot \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{M}^{2}}{d}\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(D \cdot D\right) \cdot \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{M}^{2}}{d}\right)\right)} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(D \cdot D\right) \cdot \left(\color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{M}^{2}}{d}\right)\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \left(D \cdot D\right) \cdot \left(\sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{M}^{2}}{d}\right)\right) \]

      14. cube-mult N/A

          \[\leadsto \left(D \cdot D\right) \cdot \left(\sqrt{\frac{h}{\color{blue}{\ell \cdot \left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{M}^{2}}{d}\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \left(D \cdot D\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \color{blue}{{\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{M}^{2}}{d}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left(D \cdot D\right) \cdot \left(\sqrt{\frac{h}{\color{blue}{\ell \cdot {\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{M}^{2}}{d}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \left(D \cdot D\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{M}^{2}}{d}\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \left(D \cdot D\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{M}^{2}}{d}\right)\right) \]

   10. Simplified 49.3%

       \[\leadsto \color{blue}{\left(D \cdot D\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \frac{-0.125 \cdot \left(M \cdot M\right)}{d}\right)} \]

   if 8.5000000000000006e29 < l

   1. Initial program 58.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      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. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 41.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 41.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. lower-*.f64 52.8

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   8. Simplified 52.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      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. lift-sqrt.f64 N/A

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      6. lower-/.f64 52.9

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   10. Applied egg-rr 52.9%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   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. pow1/2 N/A

          \[\leadsto \frac{d}{\color{blue}{{\ell}^{\frac{1}{2}}} \cdot \sqrt{h}} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{{\ell}^{\frac{1}{2}} \cdot \sqrt{h}}} \]

       5. pow1/2 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell}} \cdot \sqrt{h}} \]

       6. lower-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell}} \cdot \sqrt{h}} \]

       7. lower-sqrt.f64 60.0

          \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

   12. Applied egg-rr 60.0%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.42 \cdot 10^{-105}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(0.125 \cdot \left(D \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{+29}:\\ \;\;\;\;\left(D \cdot D\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 45.3% accurate, 9.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.65 \cdot 10^{-276}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 1.65e-276)
   (* (- d) (sqrt (/ 1.0 (* l h))))
   (/ d (* (sqrt l) (sqrt h)))))

C

D_m = fabs(D);
M_m = fabs(M);
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 <= 1.65e-276) {
		tmp = -d * sqrt((1.0 / (l * h)));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= 1.65d-276) then
        tmp = -d * sqrt((1.0d0 / (l * h)))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 1.65e-276) {
		tmp = -d * Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= 1.65e-276:
		tmp = -d * math.sqrt((1.0 / (l * h)))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
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 <= 1.65e-276)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(l * h))));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= 1.65e-276)
		tmp = -d * sqrt((1.0 / (l * h)));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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, 1.65e-276], N[((-d) * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.64999999999999996e-276

   1. Initial program 72.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      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. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 51.7

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 51.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      8. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. lower-neg.f64 43.7

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 43.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if 1.64999999999999996e-276 < l

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      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. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 50.2

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 50.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. lower-*.f64 40.3

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   8. Simplified 40.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      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. lift-sqrt.f64 N/A

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      6. lower-/.f64 41.0

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   10. Applied egg-rr 41.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   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. pow1/2 N/A

          \[\leadsto \frac{d}{\color{blue}{{\ell}^{\frac{1}{2}}} \cdot \sqrt{h}} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{{\ell}^{\frac{1}{2}} \cdot \sqrt{h}}} \]

       5. pow1/2 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell}} \cdot \sqrt{h}} \]

       6. lower-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell}} \cdot \sqrt{h}} \]

       7. lower-sqrt.f64 45.7

          \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

   12. Applied egg-rr 45.7%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.65 \cdot 10^{-276}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 41.9% accurate, 10.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -4.8 \cdot 10^{-241}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 -4.8e-241) (* (- d) (sqrt (/ 1.0 (* l h)))) (/ d (sqrt (* l h)))))

C

D_m = fabs(D);
M_m = fabs(M);
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 <= -4.8e-241) {
		tmp = -d * sqrt((1.0 / (l * h)));
	} else {
		tmp = d / sqrt((l * h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
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 <= (-4.8d-241)) then
        tmp = -d * sqrt((1.0d0 / (l * h)))
    else
        tmp = d / sqrt((l * h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
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 <= -4.8e-241) {
		tmp = -d * Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = d / Math.sqrt((l * h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[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 <= -4.8e-241:
		tmp = -d * math.sqrt((1.0 / (l * h)))
	else:
		tmp = d / math.sqrt((l * h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
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 <= -4.8e-241)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(l * h))));
	else
		tmp = Float64(d / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
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 <= -4.8e-241)
		tmp = -d * sqrt((1.0 / (l * h)));
	else
		tmp = d / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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, -4.8e-241], N[((-d) * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4.8e-241

   1. Initial program 72.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      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. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 50.2

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 50.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      8. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. lower-neg.f64 47.5

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 47.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if -4.8e-241 < d

   1. Initial program 67.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      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. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      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. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. lower-*.f64 51.6

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 51.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. lower-*.f64 38.5

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   8. Simplified 38.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      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. lift-sqrt.f64 N/A

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      6. lower-/.f64 39.1

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   10. Applied egg-rr 39.1%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.8 \cdot 10^{-241}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 26.8% accurate, 15.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (* l h))))

C

D_m = fabs(D);
M_m = fabs(M);
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((l * h));
}

Fortran

D_m = abs(d)
M_m = abs(m)
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((l * h))
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
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((l * h));
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[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((l * h))

Julia

D_m = abs(D)
M_m = abs(M)
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(l * h)))
end

MATLAB

D_m = abs(D);
M_m = abs(M);
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((l * h));
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.7%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
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. lower-/.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

   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. lower-*.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

   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. lower-*.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

   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. lower-*.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

   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. lower-*.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

   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. lower-*.f64 N/A

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   20. lower-*.f64 51.0

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

5. Simplified 51.0%

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

6. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

   3. lower-/.f64 N/A

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

   4. lower-*.f64 25.1

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

8. Simplified 25.1%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   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. lift-sqrt.f64 N/A

      \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   5. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   6. lower-/.f64 25.5

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

10. Applied egg-rr 25.5%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

11. Final simplification 25.5%

    \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(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)))))