Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.6% → 79.2%

Time: 18.9s

Alternatives: 16

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.6% 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: 79.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M\_m}{d \cdot -2}\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 10^{+277}:\\ \;\;\;\;\left({\left(\frac{h}{d}\right)}^{-0.5} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - t\_0 \cdot \left(\frac{h}{\ell} \cdot \left(0.5 \cdot t\_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{h \cdot \left(D\_m \cdot \left(M\_m \cdot \left(M\_m \cdot D\_m\right)\right)\right)}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

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 (* D_m (/ M_m (* d -2.0)))))
   (if (<=
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (+
          1.0
          (* (/ h l) (* (pow (/ (* M_m D_m) (* d 2.0)) 2.0) (/ -1.0 2.0)))))
        1e+277)
     (*
      (* (pow (/ h d) -0.5) (sqrt (/ d l)))
      (- 1.0 (* t_0 (* (/ h l) (* 0.5 t_0)))))
     (*
      (fma
       -0.5
       (/ (* h (* D_m (* M_m (* M_m D_m)))) (* l (* d (* d 4.0))))
       1.0)
      (/ (fabs d) (sqrt (* h 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 = D_m * (M_m / (d * -2.0));
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))))) <= 1e+277) {
		tmp = (pow((h / d), -0.5) * sqrt((d / l))) * (1.0 - (t_0 * ((h / l) * (0.5 * t_0))));
	} else {
		tmp = fma(-0.5, ((h * (D_m * (M_m * (M_m * D_m)))) / (l * (d * (d * 4.0)))), 1.0) * (fabs(d) / sqrt((h * 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(D_m * Float64(M_m / Float64(d * -2.0)))
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M_m * D_m) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0))))) <= 1e+277)
		tmp = Float64(Float64((Float64(h / d) ^ -0.5) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(t_0 * Float64(Float64(h / l) * Float64(0.5 * t_0)))));
	else
		tmp = Float64(fma(-0.5, Float64(Float64(h * Float64(D_m * Float64(M_m * Float64(M_m * D_m)))) / Float64(l * Float64(d * Float64(d * 4.0)))), 1.0) * Float64(abs(d) / sqrt(Float64(h * l))));
	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$95$m * N[(M$95$m / N[(d * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[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], 1e+277], N[(N[(N[Power[N[(h / d), $MachinePrecision], -0.5], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(N[(h * N[(D$95$m * N[(M$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1e277

   1. Initial program 87.4%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. lift-/.f64 N/A

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

      4. pow-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 87.1

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

   4. Applied egg-rr 87.1%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 87.1

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

   6. Applied egg-rr 87.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-frac-neg N/A

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

      9. frac-2neg N/A

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

      10. distribute-frac-neg N/A

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

      11. sqr-neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. distribute-rgt-neg-in N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. distribute-rgt-neg-in N/A

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

      23. lower-*.f64 N/A

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

   8. Applied egg-rr 87.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

   10. Applied egg-rr 87.9%

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

   if 1e277 < (*.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 16.2%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. lift-/.f64 N/A

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

      4. pow-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 18.3

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

   4. Applied egg-rr 18.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 18.3

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

   6. Applied egg-rr 18.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-frac-neg N/A

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

      9. frac-2neg N/A

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

      10. distribute-frac-neg N/A

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

      11. sqr-neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. distribute-rgt-neg-in N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. distribute-rgt-neg-in N/A

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

      23. lower-*.f64 N/A

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

   8. Applied egg-rr 18.3%

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

   10. Applied egg-rr 68.7%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 10^{+277}:\\ \;\;\;\;\left({\left(\frac{h}{d}\right)}^{-0.5} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(D \cdot \frac{M}{d \cdot -2}\right) \cdot \left(\frac{h}{\ell} \cdot \left(0.5 \cdot \left(D \cdot \frac{M}{d \cdot -2}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{h \cdot \left(D \cdot \left(M \cdot \left(M \cdot D\right)\right)\right)}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 70.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-84}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \left(1 - \frac{\left(h \cdot 0.25\right) \cdot \left(M\_m \cdot \left(M\_m \cdot \left(D\_m \cdot D\_m\right)\right)\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot \ell\right)}\right)\right)\\ \mathbf{elif}\;t\_2 \leq 10^{-170}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;t\_2 \leq 10^{+277}:\\ \;\;\;\;t\_1 \cdot \left(t\_0 \cdot \mathsf{fma}\left(-D\_m, \frac{0.125 \cdot \left(D\_m \cdot \left(h \cdot \left(M\_m \cdot M\_m\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{h \cdot \left(D\_m \cdot \left(M\_m \cdot \left(M\_m \cdot D\_m\right)\right)\right)}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

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 (sqrt (/ d l)))
        (t_1 (sqrt (/ d h)))
        (t_2
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (+
           1.0
           (* (/ h l) (* (pow (/ (* M_m D_m) (* d 2.0)) 2.0) (/ -1.0 2.0)))))))
   (if (<= t_2 -1e-84)
     (*
      t_0
      (*
       t_1
       (-
        1.0
        (/ (* (* h 0.25) (* M_m (* M_m (* D_m D_m)))) (* (* d 2.0) (* d l))))))
     (if (<= t_2 1e-170)
       (* d (sqrt (/ 1.0 (* h l))))
       (if (<= t_2 2e+154)
         (sqrt (* (/ d h) (/ d l)))
         (if (<= t_2 1e+277)
           (*
            t_1
            (*
             t_0
             (fma
              (- D_m)
              (/ (* 0.125 (* D_m (* h (* M_m M_m)))) (* d (* d l)))
              1.0)))
           (*
            (fma
             -0.5
             (/ (* h (* D_m (* M_m (* M_m D_m)))) (* l (* d (* d 4.0))))
             1.0)
            (/ (fabs d) (sqrt (* h 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 = sqrt((d / l));
	double t_1 = sqrt((d / h));
	double t_2 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double tmp;
	if (t_2 <= -1e-84) {
		tmp = t_0 * (t_1 * (1.0 - (((h * 0.25) * (M_m * (M_m * (D_m * D_m)))) / ((d * 2.0) * (d * l)))));
	} else if (t_2 <= 1e-170) {
		tmp = d * sqrt((1.0 / (h * l)));
	} else if (t_2 <= 2e+154) {
		tmp = sqrt(((d / h) * (d / l)));
	} else if (t_2 <= 1e+277) {
		tmp = t_1 * (t_0 * fma(-D_m, ((0.125 * (D_m * (h * (M_m * M_m)))) / (d * (d * l))), 1.0));
	} else {
		tmp = fma(-0.5, ((h * (D_m * (M_m * (M_m * D_m)))) / (l * (d * (d * 4.0)))), 1.0) * (fabs(d) / sqrt((h * 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 = sqrt(Float64(d / l))
	t_1 = sqrt(Float64(d / h))
	t_2 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M_m * D_m) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))))
	tmp = 0.0
	if (t_2 <= -1e-84)
		tmp = Float64(t_0 * Float64(t_1 * Float64(1.0 - Float64(Float64(Float64(h * 0.25) * Float64(M_m * Float64(M_m * Float64(D_m * D_m)))) / Float64(Float64(d * 2.0) * Float64(d * l))))));
	elseif (t_2 <= 1e-170)
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (t_2 <= 2e+154)
		tmp = sqrt(Float64(Float64(d / h) * Float64(d / l)));
	elseif (t_2 <= 1e+277)
		tmp = Float64(t_1 * Float64(t_0 * fma(Float64(-D_m), Float64(Float64(0.125 * Float64(D_m * Float64(h * Float64(M_m * M_m)))) / Float64(d * Float64(d * l))), 1.0)));
	else
		tmp = Float64(fma(-0.5, Float64(Float64(h * Float64(D_m * Float64(M_m * Float64(M_m * D_m)))) / Float64(l * Float64(d * Float64(d * 4.0)))), 1.0) * Float64(abs(d) / sqrt(Float64(h * l))));
	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[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-84], N[(t$95$0 * N[(t$95$1 * N[(1.0 - N[(N[(N[(h * 0.25), $MachinePrecision] * N[(M$95$m * N[(M$95$m * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(d * 2.0), $MachinePrecision] * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e-170], N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+154], N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 1e+277], N[(t$95$1 * N[(t$95$0 * N[((-D$95$m) * N[(N[(0.125 * N[(D$95$m * N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(N[(h * N[(D$95$m * N[(M$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 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)))) < -1e-84

   1. Initial program 87.7%

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

      1. lift-/.f64 N/A

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

      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. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 47.8

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

   4. Applied egg-rr 47.8%

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

   6. Applied egg-rr 70.9%

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

   if -1e-84 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.99999999999999983e-171

   1. Initial program 49.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 22.0

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

   5. Simplified 22.0%

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

   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. *-commutative N/A

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

      5. lower-*.f64 54.5

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

   8. Simplified 54.5%

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

   if 9.99999999999999983e-171 < (*.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)))) < 2.00000000000000007e154

   1. Initial program 99.3%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. lift-/.f64 N/A

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

      4. pow-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 99.4

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

   4. Applied egg-rr 99.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 99.4

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

   6. Applied egg-rr 99.4%

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

   8. Applied egg-rr 83.3%

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

   9. Taylor expanded in h around 0

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

   11. Simplified 99.2%

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

   if 2.00000000000000007e154 < (*.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)))) < 1e277

   1. Initial program 99.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 72.9

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

   5. Simplified 72.9%

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

   7. Applied egg-rr 72.9%

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

   if 1e277 < (*.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 16.2%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. lift-/.f64 N/A

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

      4. pow-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 18.3

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

   4. Applied egg-rr 18.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 18.3

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

   6. Applied egg-rr 18.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-frac-neg N/A

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

      9. frac-2neg N/A

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

      10. distribute-frac-neg N/A

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

      11. sqr-neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. distribute-rgt-neg-in N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. distribute-rgt-neg-in N/A

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

      23. lower-*.f64 N/A

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

   8. Applied egg-rr 18.3%

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

   10. Applied egg-rr 68.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{h \cdot \left(D \cdot \left(M \cdot \left(M \cdot D\right)\right)\right)}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq -1 \cdot 10^{-84}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - \frac{\left(h \cdot 0.25\right) \cdot \left(M \cdot \left(M \cdot \left(D \cdot D\right)\right)\right)}{\left(d \cdot 2\right) \cdot \left(d \cdot \ell\right)}\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 10^{-170}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 10^{+277}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-D, \frac{0.125 \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{h \cdot \left(D \cdot \left(M \cdot \left(M \cdot D\right)\right)\right)}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ t_3 := d \cdot \left(d \cdot \ell\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-84}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \mathsf{fma}\left(-D\_m, \frac{\left(M\_m \cdot \left(h \cdot M\_m\right)\right) \cdot \left(D\_m \cdot 0.125\right)}{t\_3}, 1\right)\right)\\ \mathbf{elif}\;t\_2 \leq 10^{-170}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;t\_2 \leq 10^{+277}:\\ \;\;\;\;t\_1 \cdot \left(t\_0 \cdot \mathsf{fma}\left(-D\_m, \frac{0.125 \cdot \left(D\_m \cdot \left(h \cdot \left(M\_m \cdot M\_m\right)\right)\right)}{t\_3}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{h \cdot \left(D\_m \cdot \left(M\_m \cdot \left(M\_m \cdot D\_m\right)\right)\right)}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

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 (sqrt (/ d l)))
        (t_1 (sqrt (/ d h)))
        (t_2
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (+
           1.0
           (* (/ h l) (* (pow (/ (* M_m D_m) (* d 2.0)) 2.0) (/ -1.0 2.0))))))
        (t_3 (* d (* d l))))
   (if (<= t_2 -1e-84)
     (*
      t_0
      (* t_1 (fma (- D_m) (/ (* (* M_m (* h M_m)) (* D_m 0.125)) t_3) 1.0)))
     (if (<= t_2 1e-170)
       (* d (sqrt (/ 1.0 (* h l))))
       (if (<= t_2 2e+154)
         (sqrt (* (/ d h) (/ d l)))
         (if (<= t_2 1e+277)
           (*
            t_1
            (*
             t_0
             (fma (- D_m) (/ (* 0.125 (* D_m (* h (* M_m M_m)))) t_3) 1.0)))
           (*
            (fma
             -0.5
             (/ (* h (* D_m (* M_m (* M_m D_m)))) (* l (* d (* d 4.0))))
             1.0)
            (/ (fabs d) (sqrt (* h 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 = sqrt((d / l));
	double t_1 = sqrt((d / h));
	double t_2 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double t_3 = d * (d * l);
	double tmp;
	if (t_2 <= -1e-84) {
		tmp = t_0 * (t_1 * fma(-D_m, (((M_m * (h * M_m)) * (D_m * 0.125)) / t_3), 1.0));
	} else if (t_2 <= 1e-170) {
		tmp = d * sqrt((1.0 / (h * l)));
	} else if (t_2 <= 2e+154) {
		tmp = sqrt(((d / h) * (d / l)));
	} else if (t_2 <= 1e+277) {
		tmp = t_1 * (t_0 * fma(-D_m, ((0.125 * (D_m * (h * (M_m * M_m)))) / t_3), 1.0));
	} else {
		tmp = fma(-0.5, ((h * (D_m * (M_m * (M_m * D_m)))) / (l * (d * (d * 4.0)))), 1.0) * (fabs(d) / sqrt((h * 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 = sqrt(Float64(d / l))
	t_1 = sqrt(Float64(d / h))
	t_2 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M_m * D_m) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))))
	t_3 = Float64(d * Float64(d * l))
	tmp = 0.0
	if (t_2 <= -1e-84)
		tmp = Float64(t_0 * Float64(t_1 * fma(Float64(-D_m), Float64(Float64(Float64(M_m * Float64(h * M_m)) * Float64(D_m * 0.125)) / t_3), 1.0)));
	elseif (t_2 <= 1e-170)
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (t_2 <= 2e+154)
		tmp = sqrt(Float64(Float64(d / h) * Float64(d / l)));
	elseif (t_2 <= 1e+277)
		tmp = Float64(t_1 * Float64(t_0 * fma(Float64(-D_m), Float64(Float64(0.125 * Float64(D_m * Float64(h * Float64(M_m * M_m)))) / t_3), 1.0)));
	else
		tmp = Float64(fma(-0.5, Float64(Float64(h * Float64(D_m * Float64(M_m * Float64(M_m * D_m)))) / Float64(l * Float64(d * Float64(d * 4.0)))), 1.0) * Float64(abs(d) / sqrt(Float64(h * l))));
	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[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-84], N[(t$95$0 * N[(t$95$1 * N[((-D$95$m) * N[(N[(N[(M$95$m * N[(h * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * 0.125), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e-170], N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+154], N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 1e+277], N[(t$95$1 * N[(t$95$0 * N[((-D$95$m) * N[(N[(0.125 * N[(D$95$m * N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(N[(h * N[(D$95$m * N[(M$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 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)))) < -1e-84

   1. Initial program 87.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 63.3

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

   5. Simplified 63.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 62.0

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

   7. Applied egg-rr 62.0%

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

   9. Applied egg-rr 67.0%

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

   if -1e-84 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.99999999999999983e-171

   1. Initial program 49.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 22.0

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

   5. Simplified 22.0%

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

   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. *-commutative N/A

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

      5. lower-*.f64 54.5

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

   8. Simplified 54.5%

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

   if 9.99999999999999983e-171 < (*.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)))) < 2.00000000000000007e154

   1. Initial program 99.3%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. lift-/.f64 N/A

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

      4. pow-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 99.4

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

   4. Applied egg-rr 99.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 99.4

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

   6. Applied egg-rr 99.4%

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

   8. Applied egg-rr 83.3%

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

   9. Taylor expanded in h around 0

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

   11. Simplified 99.2%

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

   if 2.00000000000000007e154 < (*.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)))) < 1e277

   1. Initial program 99.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 72.9

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

   5. Simplified 72.9%

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

   7. Applied egg-rr 72.9%

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

   if 1e277 < (*.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 16.2%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. lift-/.f64 N/A

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

      4. pow-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 18.3

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

   4. Applied egg-rr 18.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 18.3

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

   6. Applied egg-rr 18.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-frac-neg N/A

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

      9. frac-2neg N/A

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

      10. distribute-frac-neg N/A

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

      11. sqr-neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. distribute-rgt-neg-in N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. distribute-rgt-neg-in N/A

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

      23. lower-*.f64 N/A

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

   8. Applied egg-rr 18.3%

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

   10. Applied egg-rr 68.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{h \cdot \left(D \cdot \left(M \cdot \left(M \cdot D\right)\right)\right)}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq -1 \cdot 10^{-84}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left(-D, \frac{\left(M \cdot \left(h \cdot M\right)\right) \cdot \left(D \cdot 0.125\right)}{d \cdot \left(d \cdot \ell\right)}, 1\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 10^{-170}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 10^{+277}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-D, \frac{0.125 \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{h \cdot \left(D \cdot \left(M \cdot \left(M \cdot D\right)\right)\right)}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ t_3 := d \cdot \left(d \cdot \ell\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-84}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \mathsf{fma}\left(-D\_m, \frac{\left(M\_m \cdot \left(h \cdot M\_m\right)\right) \cdot \left(D\_m \cdot 0.125\right)}{t\_3}, 1\right)\right)\\ \mathbf{elif}\;t\_2 \leq 10^{-170}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;t\_2 \leq 10^{+277}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \mathsf{fma}\left(-D\_m, \frac{0.125 \cdot \left(D\_m \cdot \left(h \cdot \left(M\_m \cdot M\_m\right)\right)\right)}{t\_3}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{h \cdot \left(D\_m \cdot \left(M\_m \cdot \left(M\_m \cdot D\_m\right)\right)\right)}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

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 (sqrt (/ d h)))
        (t_1 (sqrt (/ d l)))
        (t_2
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (+
           1.0
           (* (/ h l) (* (pow (/ (* M_m D_m) (* d 2.0)) 2.0) (/ -1.0 2.0))))))
        (t_3 (* d (* d l))))
   (if (<= t_2 -1e-84)
     (*
      t_0
      (* t_1 (fma (- D_m) (/ (* (* M_m (* h M_m)) (* D_m 0.125)) t_3) 1.0)))
     (if (<= t_2 1e-170)
       (* d (sqrt (/ 1.0 (* h l))))
       (if (<= t_2 2e+154)
         (sqrt (* (/ d h) (/ d l)))
         (if (<= t_2 1e+277)
           (*
            t_0
            (*
             t_1
             (fma (- D_m) (/ (* 0.125 (* D_m (* h (* M_m M_m)))) t_3) 1.0)))
           (*
            (fma
             -0.5
             (/ (* h (* D_m (* M_m (* M_m D_m)))) (* l (* d (* d 4.0))))
             1.0)
            (/ (fabs d) (sqrt (* h 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 = sqrt((d / h));
	double t_1 = sqrt((d / l));
	double t_2 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double t_3 = d * (d * l);
	double tmp;
	if (t_2 <= -1e-84) {
		tmp = t_0 * (t_1 * fma(-D_m, (((M_m * (h * M_m)) * (D_m * 0.125)) / t_3), 1.0));
	} else if (t_2 <= 1e-170) {
		tmp = d * sqrt((1.0 / (h * l)));
	} else if (t_2 <= 2e+154) {
		tmp = sqrt(((d / h) * (d / l)));
	} else if (t_2 <= 1e+277) {
		tmp = t_0 * (t_1 * fma(-D_m, ((0.125 * (D_m * (h * (M_m * M_m)))) / t_3), 1.0));
	} else {
		tmp = fma(-0.5, ((h * (D_m * (M_m * (M_m * D_m)))) / (l * (d * (d * 4.0)))), 1.0) * (fabs(d) / sqrt((h * 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 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(d / l))
	t_2 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M_m * D_m) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))))
	t_3 = Float64(d * Float64(d * l))
	tmp = 0.0
	if (t_2 <= -1e-84)
		tmp = Float64(t_0 * Float64(t_1 * fma(Float64(-D_m), Float64(Float64(Float64(M_m * Float64(h * M_m)) * Float64(D_m * 0.125)) / t_3), 1.0)));
	elseif (t_2 <= 1e-170)
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (t_2 <= 2e+154)
		tmp = sqrt(Float64(Float64(d / h) * Float64(d / l)));
	elseif (t_2 <= 1e+277)
		tmp = Float64(t_0 * Float64(t_1 * fma(Float64(-D_m), Float64(Float64(0.125 * Float64(D_m * Float64(h * Float64(M_m * M_m)))) / t_3), 1.0)));
	else
		tmp = Float64(fma(-0.5, Float64(Float64(h * Float64(D_m * Float64(M_m * Float64(M_m * D_m)))) / Float64(l * Float64(d * Float64(d * 4.0)))), 1.0) * Float64(abs(d) / sqrt(Float64(h * l))));
	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[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-84], N[(t$95$0 * N[(t$95$1 * N[((-D$95$m) * N[(N[(N[(M$95$m * N[(h * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * 0.125), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e-170], N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+154], N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 1e+277], N[(t$95$0 * N[(t$95$1 * N[((-D$95$m) * N[(N[(0.125 * N[(D$95$m * N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(N[(h * N[(D$95$m * N[(M$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 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)))) < -1e-84

   1. Initial program 87.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 63.3

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

   5. Simplified 63.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 62.0

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

   7. Applied egg-rr 62.0%

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

   9. Applied egg-rr 67.0%

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

   if -1e-84 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.99999999999999983e-171

   1. Initial program 49.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 22.0

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

   5. Simplified 22.0%

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

   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. *-commutative N/A

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

      5. lower-*.f64 54.5

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

   8. Simplified 54.5%

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

   if 9.99999999999999983e-171 < (*.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)))) < 2.00000000000000007e154

   1. Initial program 99.3%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. lift-/.f64 N/A

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

      4. pow-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 99.4

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

   4. Applied egg-rr 99.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 99.4

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

   6. Applied egg-rr 99.4%

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

   8. Applied egg-rr 83.3%

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

   9. Taylor expanded in h around 0

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

   11. Simplified 99.2%

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

   if 2.00000000000000007e154 < (*.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)))) < 1e277

   1. Initial program 99.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 72.9

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

   5. Simplified 72.9%

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

   7. Applied egg-rr 72.9%

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

   if 1e277 < (*.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 16.2%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. lift-/.f64 N/A

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

      4. pow-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 18.3

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

   4. Applied egg-rr 18.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 18.3

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

   6. Applied egg-rr 18.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-frac-neg N/A

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

      9. frac-2neg N/A

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

      10. distribute-frac-neg N/A

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

      11. sqr-neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. distribute-rgt-neg-in N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. distribute-rgt-neg-in N/A

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

      23. lower-*.f64 N/A

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

   8. Applied egg-rr 18.3%

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

   10. Applied egg-rr 68.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{h \cdot \left(D \cdot \left(M \cdot \left(M \cdot D\right)\right)\right)}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq -1 \cdot 10^{-84}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-D, \frac{\left(M \cdot \left(h \cdot M\right)\right) \cdot \left(D \cdot 0.125\right)}{d \cdot \left(d \cdot \ell\right)}, 1\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 10^{-170}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 10^{+277}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-D, \frac{0.125 \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{h \cdot \left(D \cdot \left(M \cdot \left(M \cdot D\right)\right)\right)}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-D\_m, \frac{0.125 \cdot \left(D\_m \cdot \left(h \cdot \left(M\_m \cdot M\_m\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}, 1\right)\right)\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-34}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-170}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;t\_1 \leq 10^{+277}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{h \cdot \left(D\_m \cdot \left(M\_m \cdot \left(M\_m \cdot D\_m\right)\right)\right)}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

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
         (*
          (sqrt (/ d h))
          (*
           (sqrt (/ d l))
           (fma
            (- D_m)
            (/ (* 0.125 (* D_m (* h (* M_m M_m)))) (* d (* d l)))
            1.0))))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (+
           1.0
           (* (/ h l) (* (pow (/ (* M_m D_m) (* d 2.0)) 2.0) (/ -1.0 2.0)))))))
   (if (<= t_1 -1e-34)
     t_0
     (if (<= t_1 1e-170)
       (* d (sqrt (/ 1.0 (* h l))))
       (if (<= t_1 2e+154)
         (sqrt (* (/ d h) (/ d l)))
         (if (<= t_1 1e+277)
           t_0
           (*
            (fma
             -0.5
             (/ (* h (* D_m (* M_m (* M_m D_m)))) (* l (* d (* d 4.0))))
             1.0)
            (/ (fabs d) (sqrt (* h 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 = sqrt((d / h)) * (sqrt((d / l)) * fma(-D_m, ((0.125 * (D_m * (h * (M_m * M_m)))) / (d * (d * l))), 1.0));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double tmp;
	if (t_1 <= -1e-34) {
		tmp = t_0;
	} else if (t_1 <= 1e-170) {
		tmp = d * sqrt((1.0 / (h * l)));
	} else if (t_1 <= 2e+154) {
		tmp = sqrt(((d / h) * (d / l)));
	} else if (t_1 <= 1e+277) {
		tmp = t_0;
	} else {
		tmp = fma(-0.5, ((h * (D_m * (M_m * (M_m * D_m)))) / (l * (d * (d * 4.0)))), 1.0) * (fabs(d) / sqrt((h * 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(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * fma(Float64(-D_m), Float64(Float64(0.125 * Float64(D_m * Float64(h * Float64(M_m * M_m)))) / Float64(d * Float64(d * l))), 1.0)))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M_m * D_m) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))))
	tmp = 0.0
	if (t_1 <= -1e-34)
		tmp = t_0;
	elseif (t_1 <= 1e-170)
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (t_1 <= 2e+154)
		tmp = sqrt(Float64(Float64(d / h) * Float64(d / l)));
	elseif (t_1 <= 1e+277)
		tmp = t_0;
	else
		tmp = Float64(fma(-0.5, Float64(Float64(h * Float64(D_m * Float64(M_m * Float64(M_m * D_m)))) / Float64(l * Float64(d * Float64(d * 4.0)))), 1.0) * Float64(abs(d) / sqrt(Float64(h * l))));
	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[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[((-D$95$m) * N[(N[(0.125 * N[(D$95$m * N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-34], t$95$0, If[LessEqual[t$95$1, 1e-170], N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+154], N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 1e+277], t$95$0, N[(N[(-0.5 * N[(N[(h * N[(D$95$m * N[(M$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -9.99999999999999928e-35 or 2.00000000000000007e154 < (*.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)))) < 1e277

   1. Initial program 89.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 65.5

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

   5. Simplified 65.5%

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

   7. Applied egg-rr 66.6%

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

   if -9.99999999999999928e-35 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.99999999999999983e-171

   1. Initial program 51.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 21.3

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

   5. Simplified 21.3%

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

   6. Taylor expanded in d around inf

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

      1. 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. *-commutative N/A

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

      5. lower-*.f64 52.6

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

   8. Simplified 52.6%

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

   if 9.99999999999999983e-171 < (*.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)))) < 2.00000000000000007e154

   1. Initial program 99.3%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. lift-/.f64 N/A

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

      4. pow-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 99.4

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

   4. Applied egg-rr 99.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 99.4

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

   6. Applied egg-rr 99.4%

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

   8. Applied egg-rr 83.3%

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

   9. Taylor expanded in h around 0

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

   11. Simplified 99.2%

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

   if 1e277 < (*.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 16.2%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. lift-/.f64 N/A

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

      4. pow-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 18.3

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

   4. Applied egg-rr 18.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 18.3

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

   6. Applied egg-rr 18.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-frac-neg N/A

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

      9. frac-2neg N/A

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

      10. distribute-frac-neg N/A

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

      11. sqr-neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. distribute-rgt-neg-in N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. distribute-rgt-neg-in N/A

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

      23. lower-*.f64 N/A

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

   8. Applied egg-rr 18.3%

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

   10. Applied egg-rr 68.7%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq -1 \cdot 10^{-34}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-D, \frac{0.125 \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}, 1\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 10^{-170}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 10^{+277}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-D, \frac{0.125 \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{h \cdot \left(D \cdot \left(M \cdot \left(M \cdot D\right)\right)\right)}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.5, \frac{h \cdot \left(D\_m \cdot \left(M\_m \cdot \left(M\_m \cdot D\_m\right)\right)\right)}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-34}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-170}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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
         (*
          (fma
           -0.5
           (/ (* h (* D_m (* M_m (* M_m D_m)))) (* l (* d (* d 4.0))))
           1.0)
          (/ (fabs d) (sqrt (* h l)))))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (+
           1.0
           (* (/ h l) (* (pow (/ (* M_m D_m) (* d 2.0)) 2.0) (/ -1.0 2.0)))))))
   (if (<= t_1 -1e-34)
     t_0
     (if (<= t_1 1e-170)
       (* d (sqrt (/ 1.0 (* h l))))
       (if (<= t_1 2e+154) (sqrt (* (/ d h) (/ d l))) t_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 t_0 = fma(-0.5, ((h * (D_m * (M_m * (M_m * D_m)))) / (l * (d * (d * 4.0)))), 1.0) * (fabs(d) / sqrt((h * l)));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double tmp;
	if (t_1 <= -1e-34) {
		tmp = t_0;
	} else if (t_1 <= 1e-170) {
		tmp = d * sqrt((1.0 / (h * l)));
	} else if (t_1 <= 2e+154) {
		tmp = sqrt(((d / h) * (d / l)));
	} else {
		tmp = t_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)
	t_0 = Float64(fma(-0.5, Float64(Float64(h * Float64(D_m * Float64(M_m * Float64(M_m * D_m)))) / Float64(l * Float64(d * Float64(d * 4.0)))), 1.0) * Float64(abs(d) / sqrt(Float64(h * l))))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M_m * D_m) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))))
	tmp = 0.0
	if (t_1 <= -1e-34)
		tmp = t_0;
	elseif (t_1 <= 1e-170)
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (t_1 <= 2e+154)
		tmp = sqrt(Float64(Float64(d / h) * Float64(d / l)));
	else
		tmp = t_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_] := Block[{t$95$0 = N[(N[(-0.5 * N[(N[(h * N[(D$95$m * N[(M$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-34], t$95$0, If[LessEqual[t$95$1, 1e-170], N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+154], N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -9.99999999999999928e-35 or 2.00000000000000007e154 < (*.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 56.3%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. lift-/.f64 N/A

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

      4. pow-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 57.3

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

   4. Applied egg-rr 57.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 57.3

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

   6. Applied egg-rr 57.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-frac-neg N/A

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

      9. frac-2neg N/A

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

      10. distribute-frac-neg N/A

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

      11. sqr-neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. distribute-rgt-neg-in N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. distribute-rgt-neg-in N/A

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

      23. lower-*.f64 N/A

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

   8. Applied egg-rr 57.3%

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

   10. Applied egg-rr 65.8%

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

   if -9.99999999999999928e-35 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.99999999999999983e-171

   1. Initial program 51.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 21.3

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

   5. Simplified 21.3%

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

   6. Taylor expanded in d around inf

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

      1. 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. *-commutative N/A

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

      5. lower-*.f64 52.6

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

   8. Simplified 52.6%

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

   if 9.99999999999999983e-171 < (*.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)))) < 2.00000000000000007e154

   1. Initial program 99.3%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. lift-/.f64 N/A

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

      4. pow-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 99.4

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

   4. Applied egg-rr 99.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 99.4

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

   6. Applied egg-rr 99.4%

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

   8. Applied egg-rr 83.3%

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

   9. Taylor expanded in h around 0

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

   11. Simplified 99.2%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq -1 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{h \cdot \left(D \cdot \left(M \cdot \left(M \cdot D\right)\right)\right)}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 10^{-170}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{h \cdot \left(D \cdot \left(M \cdot \left(M \cdot D\right)\right)\right)}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m \cdot D\_m}{d}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot \left(\left(t\_0 \cdot t\_0\right) \cdot 0.25\right), 1\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;t\_1 \leq 10^{+277}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-D\_m, \frac{0.125 \cdot \left(D\_m \cdot \left(h \cdot \left(M\_m \cdot M\_m\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{h \cdot \left(D\_m \cdot \left(M\_m \cdot \left(M\_m \cdot D\_m\right)\right)\right)}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

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) d))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (+
           1.0
           (* (/ h l) (* (pow (/ (* M_m D_m) (* d 2.0)) 2.0) (/ -1.0 2.0)))))))
   (if (<= t_1 2e+154)
     (*
      (fma (/ h l) (* -0.5 (* (* t_0 t_0) 0.25)) 1.0)
      (sqrt (* (/ d h) (/ d l))))
     (if (<= t_1 1e+277)
       (*
        (sqrt (/ d h))
        (*
         (sqrt (/ d l))
         (fma
          (- D_m)
          (/ (* 0.125 (* D_m (* h (* M_m M_m)))) (* d (* d l)))
          1.0)))
       (*
        (fma
         -0.5
         (/ (* h (* D_m (* M_m (* M_m D_m)))) (* l (* d (* d 4.0))))
         1.0)
        (/ (fabs d) (sqrt (* h 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) / d;
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double tmp;
	if (t_1 <= 2e+154) {
		tmp = fma((h / l), (-0.5 * ((t_0 * t_0) * 0.25)), 1.0) * sqrt(((d / h) * (d / l)));
	} else if (t_1 <= 1e+277) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * fma(-D_m, ((0.125 * (D_m * (h * (M_m * M_m)))) / (d * (d * l))), 1.0));
	} else {
		tmp = fma(-0.5, ((h * (D_m * (M_m * (M_m * D_m)))) / (l * (d * (d * 4.0)))), 1.0) * (fabs(d) / sqrt((h * 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) / d)
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M_m * D_m) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))))
	tmp = 0.0
	if (t_1 <= 2e+154)
		tmp = Float64(fma(Float64(h / l), Float64(-0.5 * Float64(Float64(t_0 * t_0) * 0.25)), 1.0) * sqrt(Float64(Float64(d / h) * Float64(d / l))));
	elseif (t_1 <= 1e+277)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * fma(Float64(-D_m), Float64(Float64(0.125 * Float64(D_m * Float64(h * Float64(M_m * M_m)))) / Float64(d * Float64(d * l))), 1.0)));
	else
		tmp = Float64(fma(-0.5, Float64(Float64(h * Float64(D_m * Float64(M_m * Float64(M_m * D_m)))) / Float64(l * Float64(d * Float64(d * 4.0)))), 1.0) * Float64(abs(d) / sqrt(Float64(h * l))));
	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[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+154], N[(N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+277], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[((-D$95$m) * N[(N[(0.125 * N[(D$95$m * N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(N[(h * N[(D$95$m * N[(M$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $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)))) < 2.00000000000000007e154

   1. Initial program 86.3%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. lift-/.f64 N/A

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

      4. pow-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 86.0

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

   4. Applied egg-rr 86.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 86.0

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

   6. Applied egg-rr 86.0%

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

   8. Applied egg-rr 57.3%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. swap-sqr N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. swap-sqr N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. frac-times N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. pow2 N/A

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

      17. lift-/.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-/r* N/A

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

      20. div-inv N/A

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

      21. metadata-eval N/A

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

      22. unpow-prod-down N/A

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

   10. Applied egg-rr 75.3%

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

   if 2.00000000000000007e154 < (*.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)))) < 1e277

   1. Initial program 99.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 72.9

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

   5. Simplified 72.9%

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

   7. Applied egg-rr 72.9%

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

   if 1e277 < (*.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 16.2%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. lift-/.f64 N/A

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

      4. pow-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 18.3

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

   4. Applied egg-rr 18.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 18.3

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

   6. Applied egg-rr 18.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-frac-neg N/A

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

      9. frac-2neg N/A

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

      10. distribute-frac-neg N/A

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

      11. sqr-neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. distribute-rgt-neg-in N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. distribute-rgt-neg-in N/A

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

      23. lower-*.f64 N/A

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

   8. Applied egg-rr 18.3%

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

   10. Applied egg-rr 68.7%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot \left(\left(\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d}\right) \cdot 0.25\right), 1\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 10^{+277}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-D, \frac{0.125 \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{h \cdot \left(D \cdot \left(M \cdot \left(M \cdot D\right)\right)\right)}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(0.125 \cdot \left(M\_m \cdot M\_m\right)\right) \cdot \left(h \cdot \left(D\_m \cdot D\_m\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right)\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-201}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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
         (*
          (/ (fabs d) (sqrt (* h l)))
          (-
           1.0
           (/ (* (* 0.125 (* M_m M_m)) (* h (* D_m D_m))) (* d (* d l))))))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (+
           1.0
           (* (/ h l) (* (pow (/ (* M_m D_m) (* d 2.0)) 2.0) (/ -1.0 2.0)))))))
   (if (<= t_1 5e-201)
     t_0
     (if (<= t_1 2e+154) (sqrt (* (/ d h) (/ d l))) t_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 t_0 = (fabs(d) / sqrt((h * l))) * (1.0 - (((0.125 * (M_m * M_m)) * (h * (D_m * D_m))) / (d * (d * l))));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double tmp;
	if (t_1 <= 5e-201) {
		tmp = t_0;
	} else if (t_1 <= 2e+154) {
		tmp = sqrt(((d / h) * (d / l)));
	} else {
		tmp = t_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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (abs(d) / sqrt((h * l))) * (1.0d0 - (((0.125d0 * (m_m * m_m)) * (h * (d_m * d_m))) / (d * (d * l))))
    t_1 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 + ((h / l) * ((((m_m * d_m) / (d * 2.0d0)) ** 2.0d0) * ((-1.0d0) / 2.0d0))))
    if (t_1 <= 5d-201) then
        tmp = t_0
    else if (t_1 <= 2d+154) then
        tmp = sqrt(((d / h) * (d / l)))
    else
        tmp = t_0
    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.abs(d) / Math.sqrt((h * l))) * (1.0 - (((0.125 * (M_m * M_m)) * (h * (D_m * D_m))) / (d * (d * l))));
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (Math.pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	double tmp;
	if (t_1 <= 5e-201) {
		tmp = t_0;
	} else if (t_1 <= 2e+154) {
		tmp = Math.sqrt(((d / h) * (d / l)));
	} else {
		tmp = t_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):
	t_0 = (math.fabs(d) / math.sqrt((h * l))) * (1.0 - (((0.125 * (M_m * M_m)) * (h * (D_m * D_m))) / (d * (d * l))))
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (math.pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))))
	tmp = 0
	if t_1 <= 5e-201:
		tmp = t_0
	elif t_1 <= 2e+154:
		tmp = math.sqrt(((d / h) * (d / l)))
	else:
		tmp = t_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)
	t_0 = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * Float64(1.0 - Float64(Float64(Float64(0.125 * Float64(M_m * M_m)) * Float64(h * Float64(D_m * D_m))) / Float64(d * Float64(d * l)))))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M_m * D_m) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))))
	tmp = 0.0
	if (t_1 <= 5e-201)
		tmp = t_0;
	elseif (t_1 <= 2e+154)
		tmp = sqrt(Float64(Float64(d / h) * Float64(d / l)));
	else
		tmp = t_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)
	t_0 = (abs(d) / sqrt((h * l))) * (1.0 - (((0.125 * (M_m * M_m)) * (h * (D_m * D_m))) / (d * (d * l))));
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 + ((h / l) * ((((M_m * D_m) / (d * 2.0)) ^ 2.0) * (-1.0 / 2.0))));
	tmp = 0.0;
	if (t_1 <= 5e-201)
		tmp = t_0;
	elseif (t_1 <= 2e+154)
		tmp = sqrt(((d / h) * (d / l)));
	else
		tmp = t_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_] := Block[{t$95$0 = N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(0.125 * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(h * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-201], t$95$0, If[LessEqual[t$95$1, 2e+154], N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.9999999999999999e-201 or 2.00000000000000007e154 < (*.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 55.2%

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

   4. Applied egg-rr 51.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. sqrt-div N/A

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

      7. pow1/2 N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow1/2 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-neg.f64 26.9

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

   6. Applied egg-rr 26.9%

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

   7. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 26.4

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

   9. Simplified 26.4%

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

   11. Applied egg-rr 58.4%

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

   if 4.9999999999999999e-201 < (*.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)))) < 2.00000000000000007e154

   1. Initial program 98.6%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. lift-/.f64 N/A

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

      4. pow-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 97.9

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

   4. Applied egg-rr 97.9%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 97.9

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

   6. Applied egg-rr 97.9%

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

   8. Applied egg-rr 79.6%

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

   9. Taylor expanded in h around 0

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

   11. Simplified 94.9%

       \[\leadsto \color{blue}{1} \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(0.125 \cdot \left(M \cdot M\right)\right) \cdot \left(h \cdot \left(D \cdot D\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(0.125 \cdot \left(M \cdot M\right)\right) \cdot \left(h \cdot \left(D \cdot D\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m \cdot D\_m}{d \cdot -2}\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D\_m}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 10^{+277}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left(\left(t\_0 \cdot t\_0\right) \cdot \frac{-1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{h \cdot \left(D\_m \cdot \left(M\_m \cdot \left(M\_m \cdot D\_m\right)\right)\right)}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

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) (* d -2.0))))
   (if (<=
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (+
          1.0
          (* (/ h l) (* (pow (/ (* M_m D_m) (* d 2.0)) 2.0) (/ -1.0 2.0)))))
        1e+277)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (+ 1.0 (* (/ h l) (* (* t_0 t_0) (/ -1.0 2.0)))))
     (*
      (fma
       -0.5
       (/ (* h (* D_m (* M_m (* M_m D_m)))) (* l (* d (* d 4.0))))
       1.0)
      (/ (fabs d) (sqrt (* h 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) / (d * -2.0);
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 + ((h / l) * (pow(((M_m * D_m) / (d * 2.0)), 2.0) * (-1.0 / 2.0))))) <= 1e+277) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 + ((h / l) * ((t_0 * t_0) * (-1.0 / 2.0))));
	} else {
		tmp = fma(-0.5, ((h * (D_m * (M_m * (M_m * D_m)))) / (l * (d * (d * 4.0)))), 1.0) * (fabs(d) / sqrt((h * 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) / Float64(d * -2.0))
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M_m * D_m) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0))))) <= 1e+277)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 + Float64(Float64(h / l) * Float64(Float64(t_0 * t_0) * Float64(-1.0 / 2.0)))));
	else
		tmp = Float64(fma(-0.5, Float64(Float64(h * Float64(D_m * Float64(M_m * Float64(M_m * D_m)))) / Float64(l * Float64(d * Float64(d * 4.0)))), 1.0) * Float64(abs(d) / sqrt(Float64(h * l))));
	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[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[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], 1e+277], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(N[(h * N[(D$95$m * N[(M$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1e277

   1. Initial program 87.4%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. lift-/.f64 N/A

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

      4. pow-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 87.1

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

   4. Applied egg-rr 87.1%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 87.1

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

   6. Applied egg-rr 87.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-frac-neg N/A

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

      9. frac-2neg N/A

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

      10. distribute-frac-neg N/A

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

      11. sqr-neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. distribute-rgt-neg-in N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. distribute-rgt-neg-in N/A

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

      23. lower-*.f64 N/A

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

   8. Applied egg-rr 87.1%

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

      1. clear-num N/A

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

      2. lift-/.f64 N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. unpow1/2 N/A

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

      7. lower-sqrt.f64 87.4

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

   10. Applied egg-rr 87.4%

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

   if 1e277 < (*.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 16.2%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. lift-/.f64 N/A

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

      4. pow-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 18.3

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

   4. Applied egg-rr 18.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. unpow1/2 N/A

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

      4. lower-sqrt.f64 18.3

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

   6. Applied egg-rr 18.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-frac-neg N/A

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

      9. frac-2neg N/A

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

      10. distribute-frac-neg N/A

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

      11. sqr-neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. distribute-rgt-neg-in N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. distribute-rgt-neg-in N/A

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

      23. lower-*.f64 N/A

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

   8. Applied egg-rr 18.3%

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

   10. Applied egg-rr 68.7%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \leq 10^{+277}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left(\left(\frac{M \cdot D}{d \cdot -2} \cdot \frac{M \cdot D}{d \cdot -2}\right) \cdot \frac{-1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{h \cdot \left(D \cdot \left(M \cdot \left(M \cdot D\right)\right)\right)}{\ell \cdot \left(d \cdot \left(d \cdot 4\right)\right)}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}\\ \mathbf{if}\;d \leq -3.2 \cdot 10^{-40}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-269}:\\ \;\;\;\;\frac{0.125}{d} \cdot \left(\left(D\_m \cdot \left(M\_m \cdot \left(M\_m \cdot D\_m\right)\right)\right) \cdot t\_0\right)\\ \mathbf{elif}\;d \leq 4.3 \cdot 10^{-36}:\\ \;\;\;\;\left(D\_m \cdot D\_m\right) \cdot \frac{t\_0 \cdot \left(\left(M\_m \cdot M\_m\right) \cdot -0.125\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 (<= d -3.2e-40)
     (* (sqrt (/ 1.0 (* h l))) (- d))
     (if (<= d 8e-269)
       (* (/ 0.125 d) (* (* D_m (* M_m (* M_m D_m))) t_0))
       (if (<= d 4.3e-36)
         (* (* 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 (d <= -3.2e-40) {
		tmp = sqrt((1.0 / (h * l))) * -d;
	} else if (d <= 8e-269) {
		tmp = (0.125 / d) * ((D_m * (M_m * (M_m * D_m))) * t_0);
	} else if (d <= 4.3e-36) {
		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 (d <= (-3.2d-40)) then
        tmp = sqrt((1.0d0 / (h * l))) * -d
    else if (d <= 8d-269) then
        tmp = (0.125d0 / d) * ((d_m * (m_m * (m_m * d_m))) * t_0)
    else if (d <= 4.3d-36) 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 (d <= -3.2e-40) {
		tmp = Math.sqrt((1.0 / (h * l))) * -d;
	} else if (d <= 8e-269) {
		tmp = (0.125 / d) * ((D_m * (M_m * (M_m * D_m))) * t_0);
	} else if (d <= 4.3e-36) {
		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 d <= -3.2e-40:
		tmp = math.sqrt((1.0 / (h * l))) * -d
	elif d <= 8e-269:
		tmp = (0.125 / d) * ((D_m * (M_m * (M_m * D_m))) * t_0)
	elif d <= 4.3e-36:
		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 (d <= -3.2e-40)
		tmp = Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(-d));
	elseif (d <= 8e-269)
		tmp = Float64(Float64(0.125 / d) * Float64(Float64(D_m * Float64(M_m * Float64(M_m * D_m))) * t_0));
	elseif (d <= 4.3e-36)
		tmp = Float64(Float64(D_m * D_m) * Float64(Float64(t_0 * 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 (d <= -3.2e-40)
		tmp = sqrt((1.0 / (h * l))) * -d;
	elseif (d <= 8e-269)
		tmp = (0.125 / d) * ((D_m * (M_m * (M_m * D_m))) * t_0);
	elseif (d <= 4.3e-36)
		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[d, -3.2e-40], N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], If[LessEqual[d, 8e-269], N[(N[(0.125 / d), $MachinePrecision] * N[(N[(D$95$m * N[(M$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.3e-36], N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(t$95$0 * N[(N[(M$95$m * M$95$m), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.20000000000000002e-40

   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. 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 57.4

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

   5. Simplified 57.4%

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

   6. Taylor expanded in 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. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. rem-square-sqrt N/A

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

      7. neg-mul-1 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 66.8

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

   8. Simplified 66.8%

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

   if -3.20000000000000002e-40 < d < 7.9999999999999997e-269

   1. Initial program 54.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 34.5

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

   5. Simplified 34.5%

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

   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. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. rem-square-sqrt N/A

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

      8. associate-/l* N/A

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

      9. mul-1-neg N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. associate-*r* N/A

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

      12. associate-/l* N/A

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

      13. distribute-lft-neg-in N/A

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

      14. metadata-eval N/A

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

   8. Simplified 42.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

   10. Applied egg-rr 48.9%

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

   if 7.9999999999999997e-269 < d < 4.3000000000000002e-36

   1. Initial program 65.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 40.8

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

   5. Simplified 40.8%

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

   6. Taylor expanded in d around 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. *-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. associate-*r/ N/A

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

      12. associate-*r/ N/A

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

      13. lower-/.f64 N/A

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

   8. Simplified 41.0%

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

   if 4.3000000000000002e-36 < d

   1. Initial program 73.6%

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

   3. 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 54.1

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

   5. Simplified 54.1%

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

   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. *-commutative N/A

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

      5. lower-*.f64 59.1

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

   8. Simplified 59.1%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 59.2

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 59.2

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

   10. Applied egg-rr 59.2%

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

       1. lift-*.f64 N/A

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

       2. pow1/2 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. metadata-eval N/A

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

       6. lift-/.f64 N/A

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

       7. unpow-prod-down N/A

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

       8. lower-*.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. metadata-eval N/A

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

       11. pow1/2 N/A

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

       12. lower-sqrt.f64 N/A

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

       13. lift-/.f64 N/A

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

       14. metadata-eval N/A

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

       15. pow1/2 N/A

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

       16. lower-sqrt.f64 75.0

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

   12. Applied egg-rr 75.0%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.2 \cdot 10^{-40}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq 8 \cdot 10^{-269}:\\ \;\;\;\;\frac{0.125}{d} \cdot \left(\left(D \cdot \left(M \cdot \left(M \cdot D\right)\right)\right) \cdot \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}\right)\\ \mathbf{elif}\;d \leq 4.3 \cdot 10^{-36}:\\ \;\;\;\;\left(D \cdot D\right) \cdot \frac{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(M \cdot M\right) \cdot -0.125\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}\\ \mathbf{if}\;d \leq -2.05 \cdot 10^{-41}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-310}:\\ \;\;\;\;t\_0 \cdot \left(\frac{0.125}{d} \cdot \left(D\_m \cdot \left(D\_m \cdot \left(M\_m \cdot M\_m\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 4.3 \cdot 10^{-36}:\\ \;\;\;\;\left(D\_m \cdot D\_m\right) \cdot \frac{t\_0 \cdot \left(\left(M\_m \cdot M\_m\right) \cdot -0.125\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 (<= d -2.05e-41)
     (* (sqrt (/ 1.0 (* h l))) (- d))
     (if (<= d -1e-310)
       (* t_0 (* (/ 0.125 d) (* D_m (* D_m (* M_m M_m)))))
       (if (<= d 4.3e-36)
         (* (* 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 (d <= -2.05e-41) {
		tmp = sqrt((1.0 / (h * l))) * -d;
	} else if (d <= -1e-310) {
		tmp = t_0 * ((0.125 / d) * (D_m * (D_m * (M_m * M_m))));
	} else if (d <= 4.3e-36) {
		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 (d <= (-2.05d-41)) then
        tmp = sqrt((1.0d0 / (h * l))) * -d
    else if (d <= (-1d-310)) then
        tmp = t_0 * ((0.125d0 / d) * (d_m * (d_m * (m_m * m_m))))
    else if (d <= 4.3d-36) 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 (d <= -2.05e-41) {
		tmp = Math.sqrt((1.0 / (h * l))) * -d;
	} else if (d <= -1e-310) {
		tmp = t_0 * ((0.125 / d) * (D_m * (D_m * (M_m * M_m))));
	} else if (d <= 4.3e-36) {
		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 d <= -2.05e-41:
		tmp = math.sqrt((1.0 / (h * l))) * -d
	elif d <= -1e-310:
		tmp = t_0 * ((0.125 / d) * (D_m * (D_m * (M_m * M_m))))
	elif d <= 4.3e-36:
		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 (d <= -2.05e-41)
		tmp = Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(-d));
	elseif (d <= -1e-310)
		tmp = Float64(t_0 * Float64(Float64(0.125 / d) * Float64(D_m * Float64(D_m * Float64(M_m * M_m)))));
	elseif (d <= 4.3e-36)
		tmp = Float64(Float64(D_m * D_m) * Float64(Float64(t_0 * 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 (d <= -2.05e-41)
		tmp = sqrt((1.0 / (h * l))) * -d;
	elseif (d <= -1e-310)
		tmp = t_0 * ((0.125 / d) * (D_m * (D_m * (M_m * M_m))));
	elseif (d <= 4.3e-36)
		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[d, -2.05e-41], N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], If[LessEqual[d, -1e-310], N[(t$95$0 * N[(N[(0.125 / d), $MachinePrecision] * N[(D$95$m * N[(D$95$m * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.3e-36], N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(t$95$0 * N[(N[(M$95$m * M$95$m), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.05000000000000007e-41

   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. 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 57.4

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

   5. Simplified 57.4%

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

   6. Taylor expanded in 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. associate-*l* N/A

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

      2. *-commutative N/A

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto d \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. unpow2 N/A

         \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      7. neg-mul-1 N/A

         \[\leadsto d \cdot \color{blue}{\left(\mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

      8. lower-neg.f64 N/A

         \[\leadsto d \cdot \color{blue}{\left(\mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \left(\mathsf{neg}\left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto d \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto d \cdot \left(\mathsf{neg}\left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right)\right) \]

      12. lower-*.f64 66.8

          \[\leadsto d \cdot \left(-\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \]

   8. Simplified 66.8%

      \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)} \]

   if -2.05000000000000007e-41 < d < -9.999999999999969e-311

   1. Initial program 56.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 36.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 36.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in 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. associate-/l* N/A

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot {M}^{2}}}{d}\right) \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot {M}^{2}}{d}\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{\color{blue}{-1} \cdot {M}^{2}}{d}\right) \]

      8. associate-/l* N/A

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \color{blue}{\left(-1 \cdot \frac{{M}^{2}}{d}\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{{M}^{2}}{d}\right)\right)}\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2}}{d}\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{8} \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)}\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\mathsf{neg}\left(\frac{-1}{8} \cdot \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d}}\right)\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{8}\right)\right) \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\color{blue}{\frac{1}{8}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \]

   8. Simplified 45.5%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{0.125}{d}\right)} \]

   if -9.999999999999969e-311 < d < 4.3000000000000002e-36

   1. Initial program 62.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 38.0

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 38.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around 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. *-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. associate-*r/ N/A

          \[\leadsto \left(D \cdot D\right) \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\frac{\frac{-1}{8} \cdot {M}^{2}}{d}}\right) \]

      12. associate-*r/ N/A

          \[\leadsto \left(D \cdot D\right) \cdot \color{blue}{\frac{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot {M}^{2}\right)}{d}} \]

      13. lower-/.f64 N/A

          \[\leadsto \left(D \cdot D\right) \cdot \color{blue}{\frac{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot {M}^{2}\right)}{d}} \]

   8. Simplified 38.3%

      \[\leadsto \color{blue}{\left(D \cdot D\right) \cdot \frac{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(-0.125 \cdot \left(M \cdot M\right)\right)}{d}} \]

   if 4.3000000000000002e-36 < d

   1. Initial program 73.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. 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 54.1

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 54.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   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. *-commutative N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      5. lower-*.f64 59.1

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   8. Simplified 59.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \]

      3. metadata-eval N/A

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      6. lower-sqrt.f64 59.2

         \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      9. lower-*.f64 59.2

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   10. Applied egg-rr 59.2%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

       2. pow1/2 N/A

          \[\leadsto \frac{d}{\color{blue}{{\left(h \cdot \ell\right)}^{\frac{1}{2}}}} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{d}{{\color{blue}{\left(h \cdot \ell\right)}}^{\frac{1}{2}}} \]

       4. *-commutative N/A

          \[\leadsto \frac{d}{{\color{blue}{\left(\ell \cdot h\right)}}^{\frac{1}{2}}} \]

       5. metadata-eval N/A

          \[\leadsto \frac{d}{{\left(\ell \cdot h\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}} \]

       6. lift-/.f64 N/A

          \[\leadsto \frac{d}{{\left(\ell \cdot h\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}} \]

       7. unpow-prod-down N/A

          \[\leadsto \frac{d}{\color{blue}{{\ell}^{\left(\frac{1}{2}\right)} \cdot {h}^{\left(\frac{1}{2}\right)}}} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{{\ell}^{\left(\frac{1}{2}\right)} \cdot {h}^{\left(\frac{1}{2}\right)}}} \]

       9. lift-/.f64 N/A

          \[\leadsto \frac{d}{{\ell}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {h}^{\left(\frac{1}{2}\right)}} \]

       10. metadata-eval N/A

           \[\leadsto \frac{d}{{\ell}^{\color{blue}{\frac{1}{2}}} \cdot {h}^{\left(\frac{1}{2}\right)}} \]

       11. pow1/2 N/A

           \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell}} \cdot {h}^{\left(\frac{1}{2}\right)}} \]

       12. lower-sqrt.f64 N/A

           \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell}} \cdot {h}^{\left(\frac{1}{2}\right)}} \]

       13. lift-/.f64 N/A

           \[\leadsto \frac{d}{\sqrt{\ell} \cdot {h}^{\color{blue}{\left(\frac{1}{2}\right)}}} \]

       14. metadata-eval N/A

           \[\leadsto \frac{d}{\sqrt{\ell} \cdot {h}^{\color{blue}{\frac{1}{2}}}} \]

       15. pow1/2 N/A

           \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

       16. lower-sqrt.f64 75.0

           \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

   12. Applied egg-rr 75.0%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.05 \cdot 10^{-41}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\frac{0.125}{d} \cdot \left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 4.3 \cdot 10^{-36}:\\ \;\;\;\;\left(D \cdot D\right) \cdot \frac{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(M \cdot M\right) \cdot -0.125\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 49.3% accurate, 5.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2.05 \cdot 10^{-41}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\frac{0.125}{d} \cdot \left(D\_m \cdot \left(D\_m \cdot \left(M\_m \cdot M\_m\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 (<= d -2.05e-41)
   (* (sqrt (/ 1.0 (* h l))) (- d))
   (if (<= d -1e-310)
     (* (sqrt (/ h (* l (* l l)))) (* (/ 0.125 d) (* D_m (* D_m (* M_m M_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 (d <= -2.05e-41) {
		tmp = sqrt((1.0 / (h * l))) * -d;
	} else if (d <= -1e-310) {
		tmp = sqrt((h / (l * (l * l)))) * ((0.125 / d) * (D_m * (D_m * (M_m * M_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 (d <= (-2.05d-41)) then
        tmp = sqrt((1.0d0 / (h * l))) * -d
    else if (d <= (-1d-310)) then
        tmp = sqrt((h / (l * (l * l)))) * ((0.125d0 / d) * (d_m * (d_m * (m_m * m_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 (d <= -2.05e-41) {
		tmp = Math.sqrt((1.0 / (h * l))) * -d;
	} else if (d <= -1e-310) {
		tmp = Math.sqrt((h / (l * (l * l)))) * ((0.125 / d) * (D_m * (D_m * (M_m * M_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 d <= -2.05e-41:
		tmp = math.sqrt((1.0 / (h * l))) * -d
	elif d <= -1e-310:
		tmp = math.sqrt((h / (l * (l * l)))) * ((0.125 / d) * (D_m * (D_m * (M_m * M_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 (d <= -2.05e-41)
		tmp = Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(-d));
	elseif (d <= -1e-310)
		tmp = Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(Float64(0.125 / d) * Float64(D_m * Float64(D_m * Float64(M_m * M_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 (d <= -2.05e-41)
		tmp = sqrt((1.0 / (h * l))) * -d;
	elseif (d <= -1e-310)
		tmp = sqrt((h / (l * (l * l)))) * ((0.125 / d) * (D_m * (D_m * (M_m * M_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[d, -2.05e-41], N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], If[LessEqual[d, -1e-310], N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(0.125 / d), $MachinePrecision] * N[(D$95$m * N[(D$95$m * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.05000000000000007e-41

   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. 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 57.4

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 57.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in 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. associate-*l* N/A

         \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. *-commutative N/A

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto d \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. unpow2 N/A

         \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      7. neg-mul-1 N/A

         \[\leadsto d \cdot \color{blue}{\left(\mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

      8. lower-neg.f64 N/A

         \[\leadsto d \cdot \color{blue}{\left(\mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \left(\mathsf{neg}\left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto d \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto d \cdot \left(\mathsf{neg}\left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right)\right) \]

      12. lower-*.f64 66.8

          \[\leadsto d \cdot \left(-\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \]

   8. Simplified 66.8%

      \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)} \]

   if -2.05000000000000007e-41 < d < -9.999999999999969e-311

   1. Initial program 56.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 36.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 36.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in 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. associate-/l* N/A

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot {M}^{2}}}{d}\right) \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot {M}^{2}}{d}\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{\color{blue}{-1} \cdot {M}^{2}}{d}\right) \]

      8. associate-/l* N/A

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \color{blue}{\left(-1 \cdot \frac{{M}^{2}}{d}\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{{M}^{2}}{d}\right)\right)}\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2}}{d}\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{8} \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)}\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\mathsf{neg}\left(\frac{-1}{8} \cdot \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d}}\right)\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{8}\right)\right) \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\color{blue}{\frac{1}{8}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \]

   8. Simplified 45.5%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{0.125}{d}\right)} \]

   if -9.999999999999969e-311 < d

   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 47.1

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 47.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   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. *-commutative N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      5. lower-*.f64 43.6

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   8. Simplified 43.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \]

      3. metadata-eval N/A

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      6. lower-sqrt.f64 44.2

         \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      9. lower-*.f64 44.2

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   10. Applied egg-rr 44.2%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

       2. pow1/2 N/A

          \[\leadsto \frac{d}{\color{blue}{{\left(h \cdot \ell\right)}^{\frac{1}{2}}}} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{d}{{\color{blue}{\left(h \cdot \ell\right)}}^{\frac{1}{2}}} \]

       4. *-commutative N/A

          \[\leadsto \frac{d}{{\color{blue}{\left(\ell \cdot h\right)}}^{\frac{1}{2}}} \]

       5. metadata-eval N/A

          \[\leadsto \frac{d}{{\left(\ell \cdot h\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}} \]

       6. lift-/.f64 N/A

          \[\leadsto \frac{d}{{\left(\ell \cdot h\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}} \]

       7. unpow-prod-down N/A

          \[\leadsto \frac{d}{\color{blue}{{\ell}^{\left(\frac{1}{2}\right)} \cdot {h}^{\left(\frac{1}{2}\right)}}} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{{\ell}^{\left(\frac{1}{2}\right)} \cdot {h}^{\left(\frac{1}{2}\right)}}} \]

       9. lift-/.f64 N/A

          \[\leadsto \frac{d}{{\ell}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {h}^{\left(\frac{1}{2}\right)}} \]

       10. metadata-eval N/A

           \[\leadsto \frac{d}{{\ell}^{\color{blue}{\frac{1}{2}}} \cdot {h}^{\left(\frac{1}{2}\right)}} \]

       11. pow1/2 N/A

           \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell}} \cdot {h}^{\left(\frac{1}{2}\right)}} \]

       12. lower-sqrt.f64 N/A

           \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell}} \cdot {h}^{\left(\frac{1}{2}\right)}} \]

       13. lift-/.f64 N/A

           \[\leadsto \frac{d}{\sqrt{\ell} \cdot {h}^{\color{blue}{\left(\frac{1}{2}\right)}}} \]

       14. metadata-eval N/A

           \[\leadsto \frac{d}{\sqrt{\ell} \cdot {h}^{\color{blue}{\frac{1}{2}}}} \]

       15. pow1/2 N/A

           \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

       16. lower-sqrt.f64 56.1

           \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

   12. Applied egg-rr 56.1%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.05 \cdot 10^{-41}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\frac{0.125}{d} \cdot \left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.1% accurate, 9.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.25 \cdot 10^{-226}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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
 (if (<= l 1.25e-226)
   (* (sqrt (/ 1.0 (* h l))) (- 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 tmp;
	if (l <= 1.25e-226) {
		tmp = sqrt((1.0 / (h * l))) * -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) :: tmp
    if (l <= 1.25d-226) then
        tmp = sqrt((1.0d0 / (h * l))) * -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 tmp;
	if (l <= 1.25e-226) {
		tmp = Math.sqrt((1.0 / (h * l))) * -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):
	tmp = 0
	if l <= 1.25e-226:
		tmp = math.sqrt((1.0 / (h * l))) * -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)
	tmp = 0.0
	if (l <= 1.25e-226)
		tmp = Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(-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)
	tmp = 0.0;
	if (l <= 1.25e-226)
		tmp = sqrt((1.0 / (h * l))) * -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_] := If[LessEqual[l, 1.25e-226], N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.2499999999999999e-226

   1. Initial program 64.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 46.5

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 46.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in 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. associate-*l* N/A

         \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. *-commutative N/A

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto d \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. unpow2 N/A

         \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      7. neg-mul-1 N/A

         \[\leadsto d \cdot \color{blue}{\left(\mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

      8. lower-neg.f64 N/A

         \[\leadsto d \cdot \color{blue}{\left(\mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \left(\mathsf{neg}\left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto d \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto d \cdot \left(\mathsf{neg}\left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right)\right) \]

      12. lower-*.f64 43.5

          \[\leadsto d \cdot \left(-\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \]

   8. Simplified 43.5%

      \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)} \]

   if 1.2499999999999999e-226 < l

   1. Initial program 68.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 47.7

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 47.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. 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. *-commutative N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      5. lower-*.f64 48.0

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   8. Simplified 48.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \]

      3. metadata-eval N/A

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      6. lower-sqrt.f64 48.6

         \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      9. lower-*.f64 48.6

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   10. Applied egg-rr 48.6%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

       2. pow1/2 N/A

          \[\leadsto \frac{d}{\color{blue}{{\left(h \cdot \ell\right)}^{\frac{1}{2}}}} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{d}{{\color{blue}{\left(h \cdot \ell\right)}}^{\frac{1}{2}}} \]

       4. *-commutative N/A

          \[\leadsto \frac{d}{{\color{blue}{\left(\ell \cdot h\right)}}^{\frac{1}{2}}} \]

       5. metadata-eval N/A

          \[\leadsto \frac{d}{{\left(\ell \cdot h\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}} \]

       6. lift-/.f64 N/A

          \[\leadsto \frac{d}{{\left(\ell \cdot h\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}} \]

       7. unpow-prod-down N/A

          \[\leadsto \frac{d}{\color{blue}{{\ell}^{\left(\frac{1}{2}\right)} \cdot {h}^{\left(\frac{1}{2}\right)}}} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{{\ell}^{\left(\frac{1}{2}\right)} \cdot {h}^{\left(\frac{1}{2}\right)}}} \]

       9. lift-/.f64 N/A

          \[\leadsto \frac{d}{{\ell}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {h}^{\left(\frac{1}{2}\right)}} \]

       10. metadata-eval N/A

           \[\leadsto \frac{d}{{\ell}^{\color{blue}{\frac{1}{2}}} \cdot {h}^{\left(\frac{1}{2}\right)}} \]

       11. pow1/2 N/A

           \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell}} \cdot {h}^{\left(\frac{1}{2}\right)}} \]

       12. lower-sqrt.f64 N/A

           \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell}} \cdot {h}^{\left(\frac{1}{2}\right)}} \]

       13. lift-/.f64 N/A

           \[\leadsto \frac{d}{\sqrt{\ell} \cdot {h}^{\color{blue}{\left(\frac{1}{2}\right)}}} \]

       14. metadata-eval N/A

           \[\leadsto \frac{d}{\sqrt{\ell} \cdot {h}^{\color{blue}{\frac{1}{2}}}} \]

       15. pow1/2 N/A

           \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

       16. lower-sqrt.f64 61.0

           \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

   12. Applied egg-rr 61.0%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.25 \cdot 10^{-226}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 42.8% accurate, 10.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.35 \cdot 10^{-226}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

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
 (if (<= l 2.35e-226) (* (sqrt (/ 1.0 (* h l))) (- d)) (/ d (sqrt (* h 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 tmp;
	if (l <= 2.35e-226) {
		tmp = sqrt((1.0 / (h * l))) * -d;
	} else {
		tmp = d / sqrt((h * 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) :: tmp
    if (l <= 2.35d-226) then
        tmp = sqrt((1.0d0 / (h * l))) * -d
    else
        tmp = d / sqrt((h * 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 tmp;
	if (l <= 2.35e-226) {
		tmp = Math.sqrt((1.0 / (h * l))) * -d;
	} else {
		tmp = d / Math.sqrt((h * 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):
	tmp = 0
	if l <= 2.35e-226:
		tmp = math.sqrt((1.0 / (h * l))) * -d
	else:
		tmp = d / math.sqrt((h * 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)
	tmp = 0.0
	if (l <= 2.35e-226)
		tmp = Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(-d));
	else
		tmp = Float64(d / sqrt(Float64(h * 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)
	tmp = 0.0;
	if (l <= 2.35e-226)
		tmp = sqrt((1.0 / (h * l))) * -d;
	else
		tmp = d / sqrt((h * 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_] := If[LessEqual[l, 2.35e-226], N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.34999999999999999e-226

   1. Initial program 64.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 46.5

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 46.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in 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. associate-*l* N/A

         \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. *-commutative N/A

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto d \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. unpow2 N/A

         \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      7. neg-mul-1 N/A

         \[\leadsto d \cdot \color{blue}{\left(\mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

      8. lower-neg.f64 N/A

         \[\leadsto d \cdot \color{blue}{\left(\mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \left(\mathsf{neg}\left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto d \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto d \cdot \left(\mathsf{neg}\left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right)\right) \]

      12. lower-*.f64 43.5

          \[\leadsto d \cdot \left(-\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \]

   8. Simplified 43.5%

      \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)} \]

   if 2.34999999999999999e-226 < l

   1. Initial program 68.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 47.7

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

   5. Simplified 47.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. 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. *-commutative N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      5. lower-*.f64 48.0

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   8. Simplified 48.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \]

      3. metadata-eval N/A

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

      6. lower-sqrt.f64 48.6

         \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      9. lower-*.f64 48.6

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   10. Applied egg-rr 48.6%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.35 \cdot 10^{-226}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 27.1% accurate, 12.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ d \cdot \sqrt{\frac{1}{h \cdot \ell}} \end{array} \]

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 (/ 1.0 (* h 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) {
	return d * sqrt((1.0 / (h * l)));
}

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((1.0d0 / (h * l)))
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((1.0 / (h * l)));
}

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((1.0 / (h * l)))

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(1.0 / Float64(h * l))))
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((1.0 / (h * l)));
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[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.3%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing

3. 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 47.0

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

5. Simplified 47.0%

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

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. *-commutative N/A

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   5. lower-*.f64 26.5

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

8. Simplified 26.5%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

9. Final simplification 26.5%

   \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]
10. Add Preprocessing

Alternative 16: 26.9% accurate, 15.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \frac{d}{\sqrt{h \cdot \ell}} \end{array} \]

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 (* h 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) {
	return d / sqrt((h * l));
}

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((h * l))
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((h * l));
}

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((h * l))

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(h * l)))
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((h * l));
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[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.3%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing

3. 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 47.0

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right) \]

5. Simplified 47.0%

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\right) \]

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. *-commutative N/A

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   5. lower-*.f64 26.5

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

8. Simplified 26.5%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   2. sqrt-div N/A

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \]

   3. metadata-eval N/A

      \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}} \]

   4. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

   6. lower-sqrt.f64 26.4

      \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

   8. *-commutative N/A

      \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   9. lower-*.f64 26.4

      \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

10. Applied egg-rr 26.4%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024208 
(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)))))