Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.0% → 80.8%

Time: 16.6s

Alternatives: 22

Speedup: 2.9×

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: 66.0% 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: 80.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := {\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right)}^{2}\\ \mathbf{if}\;d \leq -1.75 \cdot 10^{-277}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot \left(-0.5 \cdot h\right), {\ell}^{-1}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right), \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{elif}\;d \leq 8.5 \cdot 10^{-182}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, t\_0, 1\right) \cdot d}{\sqrt{h}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right) \cdot h\right) \cdot {\ell}^{-1}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (* (* (/ D d) 0.5) M) 2.0)))
   (if (<= d -1.75e-277)
     (/
      (* (fma (* t_0 (* -0.5 h)) (pow l -1.0) 1.0) (/ (- d) (sqrt (- h))))
      (sqrt (- l)))
     (if (<= d -1e-310)
       (/
        (fma
         (* -0.125 (* (* D D) (/ (* M M) d)))
         (sqrt (pow (/ h l) 3.0))
         (* (sqrt (/ h l)) d))
        h)
       (if (<= d 8.5e-182)
         (/ (/ (* (fma (* (/ h l) -0.5) t_0 1.0) d) (sqrt h)) (sqrt l))
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (/ (sqrt d) (sqrt l)))
          (-
           1.0
           (*
            (* (* 0.5 (* D 0.5)) (/ M d))
            (* (* (* (/ 0.5 d) (* M D)) h) (pow l -1.0))))))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = pow((((D / d) * 0.5) * M), 2.0);
	double tmp;
	if (d <= -1.75e-277) {
		tmp = (fma((t_0 * (-0.5 * h)), pow(l, -1.0), 1.0) * (-d / sqrt(-h))) / sqrt(-l);
	} else if (d <= -1e-310) {
		tmp = fma((-0.125 * ((D * D) * ((M * M) / d))), sqrt(pow((h / l), 3.0)), (sqrt((h / l)) * d)) / h;
	} else if (d <= 8.5e-182) {
		tmp = ((fma(((h / l) * -0.5), t_0, 1.0) * d) / sqrt(h)) / sqrt(l);
	} else {
		tmp = (pow((d / h), pow(2.0, -1.0)) * (sqrt(d) / sqrt(l))) * (1.0 - (((0.5 * (D * 0.5)) * (M / d)) * ((((0.5 / d) * (M * D)) * h) * pow(l, -1.0))));
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(D / d) * 0.5) * M) ^ 2.0
	tmp = 0.0
	if (d <= -1.75e-277)
		tmp = Float64(Float64(fma(Float64(t_0 * Float64(-0.5 * h)), (l ^ -1.0), 1.0) * Float64(Float64(-d) / sqrt(Float64(-h)))) / sqrt(Float64(-l)));
	elseif (d <= -1e-310)
		tmp = Float64(fma(Float64(-0.125 * Float64(Float64(D * D) * Float64(Float64(M * M) / d))), sqrt((Float64(h / l) ^ 3.0)), Float64(sqrt(Float64(h / l)) * d)) / h);
	elseif (d <= 8.5e-182)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(h / l) * -0.5), t_0, 1.0) * d) / sqrt(h)) / sqrt(l));
	else
		tmp = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * Float64(sqrt(d) / sqrt(l))) * Float64(1.0 - Float64(Float64(Float64(0.5 * Float64(D * 0.5)) * Float64(M / d)) * Float64(Float64(Float64(Float64(0.5 / d) * Float64(M * D)) * h) * (l ^ -1.0)))));
	end
	return tmp
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(N[(N[(D / d), $MachinePrecision] * 0.5), $MachinePrecision] * M), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[d, -1.75e-277], N[(N[(N[(N[(t$95$0 * N[(-0.5 * h), $MachinePrecision]), $MachinePrecision] * N[Power[l, -1.0], $MachinePrecision] + 1.0), $MachinePrecision] * N[((-d) / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-310], N[(N[(N[(-0.125 * N[(N[(D * D), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Power[N[(h / l), $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 8.5e-182], N[(N[(N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * d), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(0.5 * N[(D * 0.5), $MachinePrecision]), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.5 / d), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] * N[Power[l, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.74999999999999991e-277

   1. Initial program 79.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 75.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 75.3

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 84.2

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

   6. Applied rewrites 84.2%

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

   8. Applied rewrites 90.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. unpow-1 N/A

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

      6. lift-pow.f64 N/A

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

      7. lower-fma.f64 N/A

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

   10. Applied rewrites 91.7%

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

   if -1.74999999999999991e-277 < d < -9.999999999999969e-311

   1. Initial program 26.9%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 14.0%

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

   6. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right), \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   if -9.999999999999969e-311 < d < 8.5000000000000001e-182

   1. Initial program 21.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 29.3

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

   4. Applied rewrites 29.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 29.3

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 45.6

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

   6. Applied rewrites 45.6%

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

   8. Applied rewrites 65.8%

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

   if 8.5000000000000001e-182 < d

   1. Initial program 72.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 71.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 79.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 79.0

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. pow1/2 N/A

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

      8. pow1/2 N/A

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

      9. lower-/.f64 N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 91.0

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

   8. Applied rewrites 91.0%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.75 \cdot 10^{-277}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right)}^{2} \cdot \left(-0.5 \cdot h\right), {\ell}^{-1}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right), \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{elif}\;d \leq 8.5 \cdot 10^{-182}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right)}^{2}, 1\right) \cdot d}{\sqrt{h}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right) \cdot h\right) \cdot {\ell}^{-1}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 58.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-159}:\\ \;\;\;\;\frac{\left(-0.125 \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(d \cdot d\right) \cdot \ell}\right) \cdot \frac{d}{\sqrt{-h}}}{-\sqrt{-\ell}}\\ \mathbf{elif}\;t\_0 \leq 10^{-252} \lor \neg \left(t\_0 \leq 5 \cdot 10^{+273}\right):\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 -5e-159)
     (/
      (*
       (* -0.125 (/ (* (* (* M M) h) (* D D)) (* (* d d) l)))
       (/ d (sqrt (- h))))
      (- (sqrt (- l))))
     (if (or (<= t_0 1e-252) (not (<= t_0 5e+273)))
       (fabs (/ d (sqrt (* l h))))
       (* (sqrt (/ d l)) (sqrt (/ d h)))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -5e-159) {
		tmp = ((-0.125 * ((((M * M) * h) * (D * D)) / ((d * d) * l))) * (d / sqrt(-h))) / -sqrt(-l);
	} else if ((t_0 <= 1e-252) || !(t_0 <= 5e+273)) {
		tmp = fabs((d / sqrt((l * h))));
	} else {
		tmp = sqrt((d / l)) * sqrt((d / h));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_0 <= (-5d-159)) then
        tmp = (((-0.125d0) * ((((m * m) * h) * (d_1 * d_1)) / ((d * d) * l))) * (d / sqrt(-h))) / -sqrt(-l)
    else if ((t_0 <= 1d-252) .or. (.not. (t_0 <= 5d+273))) then
        tmp = abs((d / sqrt((l * h))))
    else
        tmp = sqrt((d / l)) * sqrt((d / h))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -5e-159) {
		tmp = ((-0.125 * ((((M * M) * h) * (D * D)) / ((d * d) * l))) * (d / Math.sqrt(-h))) / -Math.sqrt(-l);
	} else if ((t_0 <= 1e-252) || !(t_0 <= 5e+273)) {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	} else {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= -5e-159:
		tmp = ((-0.125 * ((((M * M) * h) * (D * D)) / ((d * d) * l))) * (d / math.sqrt(-h))) / -math.sqrt(-l)
	elif (t_0 <= 1e-252) or not (t_0 <= 5e+273):
		tmp = math.fabs((d / math.sqrt((l * h))))
	else:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= -5e-159)
		tmp = Float64(Float64(Float64(-0.125 * Float64(Float64(Float64(Float64(M * M) * h) * Float64(D * D)) / Float64(Float64(d * d) * l))) * Float64(d / sqrt(Float64(-h)))) / Float64(-sqrt(Float64(-l))));
	elseif ((t_0 <= 1e-252) || !(t_0 <= 5e+273))
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= -5e-159)
		tmp = ((-0.125 * ((((M * M) * h) * (D * D)) / ((d * d) * l))) * (d / sqrt(-h))) / -sqrt(-l);
	elseif ((t_0 <= 1e-252) || ~((t_0 <= 5e+273)))
		tmp = abs((d / sqrt((l * h))));
	else
		tmp = sqrt((d / l)) * sqrt((d / h));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $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]}, If[LessEqual[t$95$0, -5e-159], N[(N[(N[(-0.125 * N[(N[(N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-N[Sqrt[(-l)], $MachinePrecision])), $MachinePrecision], If[Or[LessEqual[t$95$0, 1e-252], N[Not[LessEqual[t$95$0, 5e+273]], $MachinePrecision]], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $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)))) < -5.00000000000000032e-159

   1. Initial program 87.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 82.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 82.4

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 47.7

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

   6. Applied rewrites 47.7%

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

   8. Applied rewrites 50.2%

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

   9. Taylor expanded in d around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 24.2

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

   11. Applied rewrites 24.2%

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

   if -5.00000000000000032e-159 < (*.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)))) < 9.99999999999999943e-253 or 4.99999999999999961e273 < (*.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 24.4%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 30.1%

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

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 34.0

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

   8. Applied rewrites 34.0%

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

   10. Applied rewrites 34.1%

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

   12. Applied rewrites 51.0%

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

   if 9.99999999999999943e-253 < (*.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)))) < 4.99999999999999961e273

   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. Taylor expanded in d around inf

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 36.3

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

   8. Applied rewrites 36.3%

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

   10. Applied rewrites 37.5%

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

   12. Applied rewrites 99.0%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-159}:\\ \;\;\;\;\frac{\left(-0.125 \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(d \cdot d\right) \cdot \ell}\right) \cdot \frac{d}{\sqrt{-h}}}{-\sqrt{-\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{-252} \lor \neg \left(\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+273}\right):\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{d}{\sqrt{\sqrt{\left|\ell\right|} \cdot \sqrt{\left|h\right| \cdot \left(\ell \cdot h\right)}}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+273}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 0.0)
     (/ d (sqrt (* (sqrt (fabs l)) (sqrt (* (fabs h) (* l h))))))
     (if (<= t_0 5e+273)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (fabs (/ d (sqrt (* l h))))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = d / sqrt((sqrt(fabs(l)) * sqrt((fabs(h) * (l * h)))));
	} else if (t_0 <= 5e+273) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_0 <= 0.0d0) then
        tmp = d / sqrt((sqrt(abs(l)) * sqrt((abs(h) * (l * h)))))
    else if (t_0 <= 5d+273) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = abs((d / sqrt((l * h))))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = d / Math.sqrt((Math.sqrt(Math.abs(l)) * Math.sqrt((Math.abs(h) * (l * h)))));
	} else if (t_0 <= 5e+273) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = d / math.sqrt((math.sqrt(math.fabs(l)) * math.sqrt((math.fabs(h) * (l * h)))))
	elif t_0 <= 5e+273:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = math.fabs((d / math.sqrt((l * h))))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(d / sqrt(Float64(sqrt(abs(l)) * sqrt(Float64(abs(h) * Float64(l * h))))));
	elseif (t_0 <= 5e+273)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = d / sqrt((sqrt(abs(l)) * sqrt((abs(h) * (l * h)))));
	elseif (t_0 <= 5e+273)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = abs((d / sqrt((l * h))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $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]}, If[LessEqual[t$95$0, 0.0], N[(d / N[Sqrt[N[(N[Sqrt[N[Abs[l], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[Abs[h], $MachinePrecision] * N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+273], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0

   1. Initial program 80.1%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 5.4%

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

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 19.2

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

   8. Applied rewrites 19.2%

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

   10. Applied rewrites 19.2%

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

   12. Applied rewrites 21.0%

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

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.99999999999999961e273

   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. Taylor expanded in d around inf

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 36.0

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

   8. Applied rewrites 36.0%

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

   10. Applied rewrites 37.1%

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

   12. Applied rewrites 99.0%

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

   if 4.99999999999999961e273 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 21.7%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 28.5%

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

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 28.3

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

   8. Applied rewrites 28.3%

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

   10. Applied rewrites 28.4%

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

   12. Applied rewrites 44.9%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\frac{d}{\sqrt{\sqrt{\left|\ell\right|} \cdot \sqrt{\left|h\right| \cdot \left(\ell \cdot h\right)}}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+273}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+273}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1\right|\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (/ d (sqrt (* l h)))))
   (if (<= t_0 0.0)
     t_1
     (if (<= t_0 5e+273) (* (sqrt (/ d l)) (sqrt (/ d h))) (fabs t_1)))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = d / sqrt((l * h));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+273) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = fabs(t_1);
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_1 = d / sqrt((l * h))
    if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 5d+273) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = abs(t_1)
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = d / Math.sqrt((l * h));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+273) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = Math.abs(t_1);
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = d / math.sqrt((l * h))
	tmp = 0
	if t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 5e+273:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = math.fabs(t_1)
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(d / sqrt(Float64(l * h)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e+273)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = abs(t_1);
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = d / sqrt((l * h));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e+273)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = abs(t_1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $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]}, Block[{t$95$1 = N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 5e+273], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[t$95$1], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0

   1. Initial program 80.1%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 5.4%

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

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 19.2

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

   8. Applied rewrites 19.2%

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

   10. Applied rewrites 19.2%

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

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.99999999999999961e273

   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. Taylor expanded in d around inf

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 36.0

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

   8. Applied rewrites 36.0%

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

   10. Applied rewrites 37.1%

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

   12. Applied rewrites 99.0%

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

   if 4.99999999999999961e273 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 21.7%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 28.5%

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

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 28.3

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

   8. Applied rewrites 28.3%

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

   10. Applied rewrites 28.4%

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

   12. Applied rewrites 44.9%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+273}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;t\_0 \leq 10^{-157}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 6 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1\right|\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (/ d (sqrt (* l h)))))
   (if (<= t_0 1e-157)
     t_1
     (if (<= t_0 6e+142) (sqrt (* (/ d l) (/ d h))) (fabs t_1)))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = d / sqrt((l * h));
	double tmp;
	if (t_0 <= 1e-157) {
		tmp = t_1;
	} else if (t_0 <= 6e+142) {
		tmp = sqrt(((d / l) * (d / h)));
	} else {
		tmp = fabs(t_1);
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_1 = d / sqrt((l * h))
    if (t_0 <= 1d-157) then
        tmp = t_1
    else if (t_0 <= 6d+142) then
        tmp = sqrt(((d / l) * (d / h)))
    else
        tmp = abs(t_1)
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = d / Math.sqrt((l * h));
	double tmp;
	if (t_0 <= 1e-157) {
		tmp = t_1;
	} else if (t_0 <= 6e+142) {
		tmp = Math.sqrt(((d / l) * (d / h)));
	} else {
		tmp = Math.abs(t_1);
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = d / math.sqrt((l * h))
	tmp = 0
	if t_0 <= 1e-157:
		tmp = t_1
	elif t_0 <= 6e+142:
		tmp = math.sqrt(((d / l) * (d / h)))
	else:
		tmp = math.fabs(t_1)
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(d / sqrt(Float64(l * h)))
	tmp = 0.0
	if (t_0 <= 1e-157)
		tmp = t_1;
	elseif (t_0 <= 6e+142)
		tmp = sqrt(Float64(Float64(d / l) * Float64(d / h)));
	else
		tmp = abs(t_1);
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = d / sqrt((l * h));
	tmp = 0.0;
	if (t_0 <= 1e-157)
		tmp = t_1;
	elseif (t_0 <= 6e+142)
		tmp = sqrt(((d / l) * (d / h)));
	else
		tmp = abs(t_1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $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]}, Block[{t$95$1 = N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-157], t$95$1, If[LessEqual[t$95$0, 6e+142], N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[t$95$1], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.99999999999999943e-158

   1. Initial program 82.0%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 14.5%

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

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 21.2

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

   8. Applied rewrites 21.2%

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

   10. Applied rewrites 21.2%

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

   if 9.99999999999999943e-158 < (*.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)))) < 5.99999999999999949e142

   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. Taylor expanded in d around inf

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 36.2

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

   8. Applied rewrites 36.2%

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

   10. Applied rewrites 38.0%

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

   12. Applied rewrites 99.1%

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

   if 5.99999999999999949e142 < (*.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 35.7%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 41.2%

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

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 29.1

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

   8. Applied rewrites 29.1%

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

   10. Applied rewrites 29.1%

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

   12. Applied rewrites 50.5%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{-157}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 6 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 45.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-159}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|t\_0\right|\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ d (sqrt (* l h)))))
   (if (<=
        (*
         (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
         (-
          1.0
          (* (* (pow 2.0 -1.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
        -5e-159)
     t_0
     (fabs t_0))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = d / sqrt((l * h));
	double tmp;
	if (((pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-159) {
		tmp = t_0;
	} else {
		tmp = fabs(t_0);
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d / sqrt((l * h))
    if (((((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= (-5d-159)) then
        tmp = t_0
    else
        tmp = abs(t_0)
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = d / Math.sqrt((l * h));
	double tmp;
	if (((Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-159) {
		tmp = t_0;
	} else {
		tmp = Math.abs(t_0);
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	t_0 = d / math.sqrt((l * h))
	tmp = 0
	if ((math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-159:
		tmp = t_0
	else:
		tmp = math.fabs(t_0)
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(d / sqrt(Float64(l * h)))
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -5e-159)
		tmp = t_0;
	else
		tmp = abs(t_0);
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = d / sqrt((l * h));
	tmp = 0.0;
	if (((((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= -5e-159)
		tmp = t_0;
	else
		tmp = abs(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $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], -5e-159], t$95$0, N[Abs[t$95$0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.00000000000000032e-159

   1. Initial program 87.6%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 0.9%

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

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 11.8

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

   8. Applied rewrites 11.8%

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

   10. Applied rewrites 11.8%

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

   if -5.00000000000000032e-159 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 60.1%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 62.9%

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

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 35.1

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

   8. Applied rewrites 35.1%

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

   10. Applied rewrites 35.7%

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

   12. Applied rewrites 61.5%

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

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-159}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := {\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right)}^{2}\\ \mathbf{if}\;d \leq -1.75 \cdot 10^{-277}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot \left(-0.5 \cdot h\right), {\ell}^{-1}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right), \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, t\_0, 1\right) \cdot d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (* (* (/ D d) 0.5) M) 2.0)))
   (if (<= d -1.75e-277)
     (/
      (* (fma (* t_0 (* -0.5 h)) (pow l -1.0) 1.0) (/ (- d) (sqrt (- h))))
      (sqrt (- l)))
     (if (<= d -1e-310)
       (/
        (fma
         (* -0.125 (* (* D D) (/ (* M M) d)))
         (sqrt (pow (/ h l) 3.0))
         (* (sqrt (/ h l)) d))
        h)
       (/ (/ (* (fma (* (/ h l) -0.5) t_0 1.0) d) (sqrt h)) (sqrt l))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = pow((((D / d) * 0.5) * M), 2.0);
	double tmp;
	if (d <= -1.75e-277) {
		tmp = (fma((t_0 * (-0.5 * h)), pow(l, -1.0), 1.0) * (-d / sqrt(-h))) / sqrt(-l);
	} else if (d <= -1e-310) {
		tmp = fma((-0.125 * ((D * D) * ((M * M) / d))), sqrt(pow((h / l), 3.0)), (sqrt((h / l)) * d)) / h;
	} else {
		tmp = ((fma(((h / l) * -0.5), t_0, 1.0) * d) / sqrt(h)) / sqrt(l);
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(D / d) * 0.5) * M) ^ 2.0
	tmp = 0.0
	if (d <= -1.75e-277)
		tmp = Float64(Float64(fma(Float64(t_0 * Float64(-0.5 * h)), (l ^ -1.0), 1.0) * Float64(Float64(-d) / sqrt(Float64(-h)))) / sqrt(Float64(-l)));
	elseif (d <= -1e-310)
		tmp = Float64(fma(Float64(-0.125 * Float64(Float64(D * D) * Float64(Float64(M * M) / d))), sqrt((Float64(h / l) ^ 3.0)), Float64(sqrt(Float64(h / l)) * d)) / h);
	else
		tmp = Float64(Float64(Float64(fma(Float64(Float64(h / l) * -0.5), t_0, 1.0) * d) / sqrt(h)) / sqrt(l));
	end
	return tmp
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(N[(N[(D / d), $MachinePrecision] * 0.5), $MachinePrecision] * M), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[d, -1.75e-277], N[(N[(N[(N[(t$95$0 * N[(-0.5 * h), $MachinePrecision]), $MachinePrecision] * N[Power[l, -1.0], $MachinePrecision] + 1.0), $MachinePrecision] * N[((-d) / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-310], N[(N[(N[(-0.125 * N[(N[(D * D), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Power[N[(h / l), $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * d), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.74999999999999991e-277

   1. Initial program 79.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 75.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 75.3

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 84.2

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

   6. Applied rewrites 84.2%

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

   8. Applied rewrites 90.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. unpow-1 N/A

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

      6. lift-pow.f64 N/A

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

      7. lower-fma.f64 N/A

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

   10. Applied rewrites 91.7%

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

   if -1.74999999999999991e-277 < d < -9.999999999999969e-311

   1. Initial program 26.9%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 14.0%

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

   6. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right), \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   if -9.999999999999969e-311 < d

   1. Initial program 62.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 69.3

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

   4. Applied rewrites 69.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 69.3

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 76.1

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

   6. Applied rewrites 76.1%

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

   8. Applied rewrites 78.9%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.75 \cdot 10^{-277}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right)}^{2} \cdot \left(-0.5 \cdot h\right), {\ell}^{-1}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right), \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right)}^{2}, 1\right) \cdot d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \left(\frac{D}{d} \cdot 0.5\right) \cdot M\\ \mathbf{if}\;d \leq -2.25 \cdot 10^{-277}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot \left(t\_0 \cdot -0.5\right), \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right), \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {t\_0}^{2}, 1\right) \cdot d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* (/ D d) 0.5) M)))
   (if (<= d -2.25e-277)
     (/
      (* (fma (* t_0 (* t_0 -0.5)) (/ h l) 1.0) (/ (- d) (sqrt (- h))))
      (sqrt (- l)))
     (if (<= d -1e-310)
       (/
        (fma
         (* -0.125 (* (* D D) (/ (* M M) d)))
         (sqrt (pow (/ h l) 3.0))
         (* (sqrt (/ h l)) d))
        h)
       (/
        (/ (* (fma (* (/ h l) -0.5) (pow t_0 2.0) 1.0) d) (sqrt h))
        (sqrt l))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = ((D / d) * 0.5) * M;
	double tmp;
	if (d <= -2.25e-277) {
		tmp = (fma((t_0 * (t_0 * -0.5)), (h / l), 1.0) * (-d / sqrt(-h))) / sqrt(-l);
	} else if (d <= -1e-310) {
		tmp = fma((-0.125 * ((D * D) * ((M * M) / d))), sqrt(pow((h / l), 3.0)), (sqrt((h / l)) * d)) / h;
	} else {
		tmp = ((fma(((h / l) * -0.5), pow(t_0, 2.0), 1.0) * d) / sqrt(h)) / sqrt(l);
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(D / d) * 0.5) * M)
	tmp = 0.0
	if (d <= -2.25e-277)
		tmp = Float64(Float64(fma(Float64(t_0 * Float64(t_0 * -0.5)), Float64(h / l), 1.0) * Float64(Float64(-d) / sqrt(Float64(-h)))) / sqrt(Float64(-l)));
	elseif (d <= -1e-310)
		tmp = Float64(fma(Float64(-0.125 * Float64(Float64(D * D) * Float64(Float64(M * M) / d))), sqrt((Float64(h / l) ^ 3.0)), Float64(sqrt(Float64(h / l)) * d)) / h);
	else
		tmp = Float64(Float64(Float64(fma(Float64(Float64(h / l) * -0.5), (t_0 ^ 2.0), 1.0) * d) / sqrt(h)) / sqrt(l));
	end
	return tmp
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(D / d), $MachinePrecision] * 0.5), $MachinePrecision] * M), $MachinePrecision]}, If[LessEqual[d, -2.25e-277], N[(N[(N[(N[(t$95$0 * N[(t$95$0 * -0.5), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[((-d) / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-310], N[(N[(N[(-0.125 * N[(N[(D * D), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Power[N[(h / l), $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision] + 1.0), $MachinePrecision] * d), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.24999999999999996e-277

   1. Initial program 79.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 75.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 75.3

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 84.2

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

   6. Applied rewrites 84.2%

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

   8. Applied rewrites 90.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. div-inv N/A

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

      11. lift-/.f64 N/A

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

      12. times-frac N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

      18. lower-*.f64 N/A

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

   10. Applied rewrites 90.8%

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

   if -2.24999999999999996e-277 < d < -9.999999999999969e-311

   1. Initial program 26.9%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 14.0%

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

   6. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right), \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   if -9.999999999999969e-311 < d

   1. Initial program 62.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 69.3

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

   4. Applied rewrites 69.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 69.3

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 76.1

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

   6. Applied rewrites 76.1%

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

   8. Applied rewrites 78.9%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.25 \cdot 10^{-277}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right) \cdot \left(\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right) \cdot -0.5\right), \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right), \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right)}^{2}, 1\right) \cdot d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := {\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right)}^{2}\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5 \cdot t\_0, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-\ell}}}{\sqrt{-h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, t\_0, 1\right) \cdot d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (* (* (/ D d) 0.5) M) 2.0)))
   (if (<= l -4e-310)
     (/ (* (fma (* -0.5 t_0) (/ h l) 1.0) (/ (- d) (sqrt (- l)))) (sqrt (- h)))
     (/ (/ (* (fma (* (/ h l) -0.5) t_0 1.0) d) (sqrt h)) (sqrt l)))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = pow((((D / d) * 0.5) * M), 2.0);
	double tmp;
	if (l <= -4e-310) {
		tmp = (fma((-0.5 * t_0), (h / l), 1.0) * (-d / sqrt(-l))) / sqrt(-h);
	} else {
		tmp = ((fma(((h / l) * -0.5), t_0, 1.0) * d) / sqrt(h)) / sqrt(l);
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(D / d) * 0.5) * M) ^ 2.0
	tmp = 0.0
	if (l <= -4e-310)
		tmp = Float64(Float64(fma(Float64(-0.5 * t_0), Float64(h / l), 1.0) * Float64(Float64(-d) / sqrt(Float64(-l)))) / sqrt(Float64(-h)));
	else
		tmp = Float64(Float64(Float64(fma(Float64(Float64(h / l) * -0.5), t_0, 1.0) * d) / sqrt(h)) / sqrt(l));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < -3.999999999999988e-310

   1. Initial program 76.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 71.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 71.6

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 79.9

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

   6. Applied rewrites 79.9%

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

   8. Applied rewrites 87.8%

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

   if -3.999999999999988e-310 < l

   1. Initial program 62.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 69.3

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

   4. Applied rewrites 69.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 69.3

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 76.1

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

   6. Applied rewrites 76.1%

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

   8. Applied rewrites 78.9%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5 \cdot {\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-\ell}}}{\sqrt{-h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right)}^{2}, 1\right) \cdot d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \left(\frac{D}{d} \cdot 0.5\right) \cdot M\\ \mathbf{if}\;d \leq -7.4 \cdot 10^{-302}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot \left(t\_0 \cdot -0.5\right), \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {t\_0}^{2}, 1\right) \cdot d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* (/ D d) 0.5) M)))
   (if (<= d -7.4e-302)
     (/
      (* (fma (* t_0 (* t_0 -0.5)) (/ h l) 1.0) (/ (- d) (sqrt (- h))))
      (sqrt (- l)))
     (/
      (/ (* (fma (* (/ h l) -0.5) (pow t_0 2.0) 1.0) d) (sqrt h))
      (sqrt l)))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = ((D / d) * 0.5) * M;
	double tmp;
	if (d <= -7.4e-302) {
		tmp = (fma((t_0 * (t_0 * -0.5)), (h / l), 1.0) * (-d / sqrt(-h))) / sqrt(-l);
	} else {
		tmp = ((fma(((h / l) * -0.5), pow(t_0, 2.0), 1.0) * d) / sqrt(h)) / sqrt(l);
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(D / d) * 0.5) * M)
	tmp = 0.0
	if (d <= -7.4e-302)
		tmp = Float64(Float64(fma(Float64(t_0 * Float64(t_0 * -0.5)), Float64(h / l), 1.0) * Float64(Float64(-d) / sqrt(Float64(-h)))) / sqrt(Float64(-l)));
	else
		tmp = Float64(Float64(Float64(fma(Float64(Float64(h / l) * -0.5), (t_0 ^ 2.0), 1.0) * d) / sqrt(h)) / sqrt(l));
	end
	return tmp
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(D / d), $MachinePrecision] * 0.5), $MachinePrecision] * M), $MachinePrecision]}, If[LessEqual[d, -7.4e-302], N[(N[(N[(N[(t$95$0 * N[(t$95$0 * -0.5), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[((-d) / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision] + 1.0), $MachinePrecision] * d), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -7.4e-302

   1. Initial program 76.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 72.1%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 72.1

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 80.5

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

   6. Applied rewrites 80.5%

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

   8. Applied rewrites 87.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. div-inv N/A

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

      11. lift-/.f64 N/A

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

      12. times-frac N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

      18. lower-*.f64 N/A

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

   10. Applied rewrites 87.6%

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

   if -7.4e-302 < d

   1. Initial program 62.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 68.7

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

   4. Applied rewrites 68.7%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 68.7

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 75.5

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

   6. Applied rewrites 75.5%

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

   8. Applied rewrites 78.3%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.4 \cdot 10^{-302}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right) \cdot \left(\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right) \cdot -0.5\right), \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right)}^{2}, 1\right) \cdot d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \left(\frac{D}{d} \cdot 0.5\right) \cdot M\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot \left(t\_0 \cdot -0.5\right), \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {t\_0}^{2}, 1\right) \cdot \frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* (/ D d) 0.5) M)))
   (if (<= l -4e-310)
     (/
      (* (fma (* t_0 (* t_0 -0.5)) (/ h l) 1.0) (/ (- d) (sqrt (- h))))
      (sqrt (- l)))
     (/
      (* (fma (* (/ h l) -0.5) (pow t_0 2.0) 1.0) (/ d (sqrt l)))
      (sqrt h)))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = ((D / d) * 0.5) * M;
	double tmp;
	if (l <= -4e-310) {
		tmp = (fma((t_0 * (t_0 * -0.5)), (h / l), 1.0) * (-d / sqrt(-h))) / sqrt(-l);
	} else {
		tmp = (fma(((h / l) * -0.5), pow(t_0, 2.0), 1.0) * (d / sqrt(l))) / sqrt(h);
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(D / d) * 0.5) * M)
	tmp = 0.0
	if (l <= -4e-310)
		tmp = Float64(Float64(fma(Float64(t_0 * Float64(t_0 * -0.5)), Float64(h / l), 1.0) * Float64(Float64(-d) / sqrt(Float64(-h)))) / sqrt(Float64(-l)));
	else
		tmp = Float64(Float64(fma(Float64(Float64(h / l) * -0.5), (t_0 ^ 2.0), 1.0) * Float64(d / sqrt(l))) / sqrt(h));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < -3.999999999999988e-310

   1. Initial program 76.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 71.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 71.6

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 79.9

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

   6. Applied rewrites 79.9%

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

   8. Applied rewrites 86.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. div-inv N/A

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

      11. lift-/.f64 N/A

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

      12. times-frac N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

      18. lower-*.f64 N/A

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

   10. Applied rewrites 86.9%

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

   if -3.999999999999988e-310 < l

   1. Initial program 62.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 69.3

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

   4. Applied rewrites 69.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 69.3

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 76.1

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

   6. Applied rewrites 76.1%

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

   8. Applied rewrites 77.2%

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

Alternative 12: 77.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \left(\frac{D}{d} \cdot 0.5\right) \cdot M\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot \left(t\_0 \cdot -0.5\right), \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {t\_0}^{2}, 1\right) \cdot \frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* (/ D d) 0.5) M)))
   (if (<= l -4e-310)
     (/
      (* (fma (* t_0 (* t_0 -0.5)) (/ h l) 1.0) (/ (- d) (sqrt (- h))))
      (sqrt (- l)))
     (/
      (* (fma (* (/ h l) -0.5) (pow t_0 2.0) 1.0) (/ d (sqrt h)))
      (sqrt l)))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = ((D / d) * 0.5) * M;
	double tmp;
	if (l <= -4e-310) {
		tmp = (fma((t_0 * (t_0 * -0.5)), (h / l), 1.0) * (-d / sqrt(-h))) / sqrt(-l);
	} else {
		tmp = (fma(((h / l) * -0.5), pow(t_0, 2.0), 1.0) * (d / sqrt(h))) / sqrt(l);
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(D / d) * 0.5) * M)
	tmp = 0.0
	if (l <= -4e-310)
		tmp = Float64(Float64(fma(Float64(t_0 * Float64(t_0 * -0.5)), Float64(h / l), 1.0) * Float64(Float64(-d) / sqrt(Float64(-h)))) / sqrt(Float64(-l)));
	else
		tmp = Float64(Float64(fma(Float64(Float64(h / l) * -0.5), (t_0 ^ 2.0), 1.0) * Float64(d / sqrt(h))) / sqrt(l));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < -3.999999999999988e-310

   1. Initial program 76.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 71.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 71.6

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 79.9

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

   6. Applied rewrites 79.9%

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

   8. Applied rewrites 86.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. div-inv N/A

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

      11. lift-/.f64 N/A

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

      12. times-frac N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

      18. lower-*.f64 N/A

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

   10. Applied rewrites 86.9%

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

   if -3.999999999999988e-310 < l

   1. Initial program 62.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 69.3

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

   4. Applied rewrites 69.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 69.3

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 76.1

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

   6. Applied rewrites 76.1%

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

   8. Applied rewrites 75.4%

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

Alternative 13: 74.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \left(\frac{D}{d} \cdot 0.5\right) \cdot M\\ \mathbf{if}\;h \leq -2.6 \cdot 10^{-301}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot \left(t\_0 \cdot -0.5\right), \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;h \leq 3.2 \cdot 10^{+186}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {t\_0}^{2}, 1\right) \cdot d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d}, -0.125, \ell\right)}{\ell}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* (/ D d) 0.5) M)))
   (if (<= h -2.6e-301)
     (/
      (* (fma (* t_0 (* t_0 -0.5)) (/ h l) 1.0) (/ (- d) (sqrt (- h))))
      (sqrt (- l)))
     (if (<= h 3.2e+186)
       (/ (* (fma (* (/ h l) -0.5) (pow t_0 2.0) 1.0) d) (sqrt (* l h)))
       (*
        (* (/ (sqrt d) (sqrt h)) (/ (sqrt d) (sqrt l)))
        (/ (fma (/ (* (* (* M M) h) (* D D)) (* d d)) -0.125 l) l))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = ((D / d) * 0.5) * M;
	double tmp;
	if (h <= -2.6e-301) {
		tmp = (fma((t_0 * (t_0 * -0.5)), (h / l), 1.0) * (-d / sqrt(-h))) / sqrt(-l);
	} else if (h <= 3.2e+186) {
		tmp = (fma(((h / l) * -0.5), pow(t_0, 2.0), 1.0) * d) / sqrt((l * h));
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * (sqrt(d) / sqrt(l))) * (fma(((((M * M) * h) * (D * D)) / (d * d)), -0.125, l) / l);
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(D / d) * 0.5) * M)
	tmp = 0.0
	if (h <= -2.6e-301)
		tmp = Float64(Float64(fma(Float64(t_0 * Float64(t_0 * -0.5)), Float64(h / l), 1.0) * Float64(Float64(-d) / sqrt(Float64(-h)))) / sqrt(Float64(-l)));
	elseif (h <= 3.2e+186)
		tmp = Float64(Float64(fma(Float64(Float64(h / l) * -0.5), (t_0 ^ 2.0), 1.0) * d) / sqrt(Float64(l * h)));
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * Float64(sqrt(d) / sqrt(l))) * Float64(fma(Float64(Float64(Float64(Float64(M * M) * h) * Float64(D * D)) / Float64(d * d)), -0.125, l) / l));
	end
	return tmp
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(D / d), $MachinePrecision] * 0.5), $MachinePrecision] * M), $MachinePrecision]}, If[LessEqual[h, -2.6e-301], N[(N[(N[(N[(t$95$0 * N[(t$95$0 * -0.5), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[((-d) / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 3.2e+186], N[(N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision] + 1.0), $MachinePrecision] * d), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * -0.125 + l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -2.5999999999999998e-301

   1. Initial program 76.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 72.1%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 72.1

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 80.5

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

   6. Applied rewrites 80.5%

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

   8. Applied rewrites 87.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. div-inv N/A

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

      11. lift-/.f64 N/A

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

      12. times-frac N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

      18. lower-*.f64 N/A

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

   10. Applied rewrites 87.6%

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

   if -2.5999999999999998e-301 < h < 3.1999999999999999e186

   1. Initial program 65.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 73.2

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

   4. Applied rewrites 73.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 73.2

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 77.7

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

   6. Applied rewrites 77.7%

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

   8. Applied rewrites 74.6%

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

   if 3.1999999999999999e186 < h

   1. Initial program 48.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 48.2

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

   4. Applied rewrites 48.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 48.2

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 65.4

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

   6. Applied rewrites 65.4%

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

   7. Taylor expanded in l around 0

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

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. lower-/.f64 N/A

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

   9. Applied rewrites 61.0%

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

Alternative 14: 71.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \left(\frac{D}{d} \cdot 0.5\right) \cdot M\\ \mathbf{if}\;d \leq -5.6 \cdot 10^{-300}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot \left(t\_0 \cdot -0.5\right), \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq 9 \cdot 10^{-190}:\\ \;\;\;\;\left(-0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{+70}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}, -0.125, d \cdot d\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{-0.5}}{\sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* (/ D d) 0.5) M)))
   (if (<= d -5.6e-300)
     (/
      (* (fma (* t_0 (* t_0 -0.5)) (/ h l) 1.0) (/ (- d) (sqrt (- h))))
      (sqrt (- l)))
     (if (<= d 9e-190)
       (* (* -0.125 (* (* D D) (/ (* M M) d))) (sqrt (/ h (pow l 3.0))))
       (if (<= d 2.8e+70)
         (*
          (* (/ (sqrt d) (sqrt h)) (/ (sqrt d) (sqrt l)))
          (/ (fma (* (* D D) (/ (* (* M M) h) l)) -0.125 (* d d)) (* d d)))
         (* (/ (pow l -0.5) (sqrt h)) d))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = ((D / d) * 0.5) * M;
	double tmp;
	if (d <= -5.6e-300) {
		tmp = (fma((t_0 * (t_0 * -0.5)), (h / l), 1.0) * (-d / sqrt(-h))) / sqrt(-l);
	} else if (d <= 9e-190) {
		tmp = (-0.125 * ((D * D) * ((M * M) / d))) * sqrt((h / pow(l, 3.0)));
	} else if (d <= 2.8e+70) {
		tmp = ((sqrt(d) / sqrt(h)) * (sqrt(d) / sqrt(l))) * (fma(((D * D) * (((M * M) * h) / l)), -0.125, (d * d)) / (d * d));
	} else {
		tmp = (pow(l, -0.5) / sqrt(h)) * d;
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(D / d) * 0.5) * M)
	tmp = 0.0
	if (d <= -5.6e-300)
		tmp = Float64(Float64(fma(Float64(t_0 * Float64(t_0 * -0.5)), Float64(h / l), 1.0) * Float64(Float64(-d) / sqrt(Float64(-h)))) / sqrt(Float64(-l)));
	elseif (d <= 9e-190)
		tmp = Float64(Float64(-0.125 * Float64(Float64(D * D) * Float64(Float64(M * M) / d))) * sqrt(Float64(h / (l ^ 3.0))));
	elseif (d <= 2.8e+70)
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * Float64(sqrt(d) / sqrt(l))) * Float64(fma(Float64(Float64(D * D) * Float64(Float64(Float64(M * M) * h) / l)), -0.125, Float64(d * d)) / Float64(d * d)));
	else
		tmp = Float64(Float64((l ^ -0.5) / sqrt(h)) * d);
	end
	return tmp
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(D / d), $MachinePrecision] * 0.5), $MachinePrecision] * M), $MachinePrecision]}, If[LessEqual[d, -5.6e-300], N[(N[(N[(N[(t$95$0 * N[(t$95$0 * -0.5), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[((-d) / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 9e-190], N[(N[(-0.125 * N[(N[(D * D), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.8e+70], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(D * D), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.125 + N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[l, -0.5], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -5.59999999999999988e-300

   1. Initial program 77.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 73.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 73.8

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 82.4

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

   6. Applied rewrites 82.4%

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

   8. Applied rewrites 88.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. div-inv N/A

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

      11. lift-/.f64 N/A

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

      12. times-frac N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

      18. lower-*.f64 N/A

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

   10. Applied rewrites 88.8%

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

   if -5.59999999999999988e-300 < d < 9.00000000000000042e-190

   1. Initial program 16.6%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 13.4%

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

   6. Taylor expanded in d around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{\color{blue}{M \cdot M}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \]

      13. lower-pow.f64 23.5

          \[\leadsto \left(-0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\color{blue}{{\ell}^{3}}}} \]

   8. Applied rewrites 23.5%

      \[\leadsto \color{blue}{\left(-0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

   if 9.00000000000000042e-190 < d < 2.7999999999999999e70

   1. Initial program 73.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 77.9

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 77.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. metadata-eval 77.9

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 81.1

         \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 81.1%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   7. Taylor expanded in d around 0

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \color{blue}{\frac{{d}^{2} - \frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}} \]
   8. Step-by-step derivation

      1. cancel-sub-sign-inv N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \frac{\color{blue}{{d}^{2} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}}{{d}^{2}} \]

      2. metadata-eval N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \frac{{d}^{2} + \color{blue}{\frac{-1}{8}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}} \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \frac{\color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell} + {d}^{2}}}{{d}^{2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \color{blue}{\frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell} + {d}^{2}}{{d}^{2}}} \]

   9. Applied rewrites 77.7%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}, -0.125, d \cdot d\right)}{d \cdot d}} \]

   if 2.7999999999999999e70 < 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 d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 58.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 66.1

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 66.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 77.0%

       \[\leadsto \frac{{\ell}^{-0.5}}{\sqrt{h}} \cdot d \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 15: 69.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \left(\frac{D}{d} \cdot 0.5\right) \cdot M\\ \mathbf{if}\;d \leq -5.6 \cdot 10^{-300}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot \left(t\_0 \cdot -0.5\right), \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-196}:\\ \;\;\;\;{\left(\frac{\sqrt{\ell \cdot h}}{d}\right)}^{-1}\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{+70}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}, -0.125, d \cdot d\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{-0.5}}{\sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* (/ D d) 0.5) M)))
   (if (<= d -5.6e-300)
     (/
      (* (fma (* t_0 (* t_0 -0.5)) (/ h l) 1.0) (/ (- d) (sqrt (- h))))
      (sqrt (- l)))
     (if (<= d 1.9e-196)
       (pow (/ (sqrt (* l h)) d) -1.0)
       (if (<= d 2.8e+70)
         (*
          (* (/ (sqrt d) (sqrt h)) (/ (sqrt d) (sqrt l)))
          (/ (fma (* (* D D) (/ (* (* M M) h) l)) -0.125 (* d d)) (* d d)))
         (* (/ (pow l -0.5) (sqrt h)) d))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = ((D / d) * 0.5) * M;
	double tmp;
	if (d <= -5.6e-300) {
		tmp = (fma((t_0 * (t_0 * -0.5)), (h / l), 1.0) * (-d / sqrt(-h))) / sqrt(-l);
	} else if (d <= 1.9e-196) {
		tmp = pow((sqrt((l * h)) / d), -1.0);
	} else if (d <= 2.8e+70) {
		tmp = ((sqrt(d) / sqrt(h)) * (sqrt(d) / sqrt(l))) * (fma(((D * D) * (((M * M) * h) / l)), -0.125, (d * d)) / (d * d));
	} else {
		tmp = (pow(l, -0.5) / sqrt(h)) * d;
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(D / d) * 0.5) * M)
	tmp = 0.0
	if (d <= -5.6e-300)
		tmp = Float64(Float64(fma(Float64(t_0 * Float64(t_0 * -0.5)), Float64(h / l), 1.0) * Float64(Float64(-d) / sqrt(Float64(-h)))) / sqrt(Float64(-l)));
	elseif (d <= 1.9e-196)
		tmp = Float64(sqrt(Float64(l * h)) / d) ^ -1.0;
	elseif (d <= 2.8e+70)
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * Float64(sqrt(d) / sqrt(l))) * Float64(fma(Float64(Float64(D * D) * Float64(Float64(Float64(M * M) * h) / l)), -0.125, Float64(d * d)) / Float64(d * d)));
	else
		tmp = Float64(Float64((l ^ -0.5) / sqrt(h)) * d);
	end
	return tmp
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(D / d), $MachinePrecision] * 0.5), $MachinePrecision] * M), $MachinePrecision]}, If[LessEqual[d, -5.6e-300], N[(N[(N[(N[(t$95$0 * N[(t$95$0 * -0.5), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[((-d) / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.9e-196], N[Power[N[(N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision] / d), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[d, 2.8e+70], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(D * D), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.125 + N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[l, -0.5], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -5.59999999999999988e-300

   1. Initial program 77.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

   4. Applied rewrites 73.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      2. metadata-eval 73.8

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. frac-2neg N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto \left(\frac{\sqrt{\color{blue}{-d}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\sqrt{-d}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. lower-neg.f64 82.4

          \[\leadsto \left(\frac{\sqrt{-d}}{\sqrt{\color{blue}{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

   6. Applied rewrites 82.4%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

   8. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5 \cdot {\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot {\left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot M\right)}^{2}}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot M\right)}^{2} \cdot \frac{-1}{2}}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot M\right)}^{2}} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot M\right)}}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\left(M \cdot \left(\frac{D}{d} \cdot \frac{1}{2}\right)\right)}}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\left(M \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{1}{2}\right)}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\left(M \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{D}{d}\right)}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      8. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\left(\left(M \cdot \frac{1}{2}\right) \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\left(\left(M \cdot \color{blue}{\frac{1}{2}}\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      10. div-inv N/A

          \[\leadsto \frac{\mathsf{fma}\left({\left(\color{blue}{\frac{M}{2}} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{M}{2} \cdot \color{blue}{\frac{D}{d}}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      12. times-frac N/A

          \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      15. lift-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      16. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      17. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{-1}{2}\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{-1}{2}\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

   10. Applied rewrites 88.8%

       \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right) \cdot \left(\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right) \cdot -0.5\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

   if -5.59999999999999988e-300 < d < 1.9000000000000001e-196

   1. Initial program 14.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 14.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 26.4

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 26.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 26.4%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 26.4%

       \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\ell \cdot h}}{d}}} \]

   if 1.9000000000000001e-196 < d < 2.7999999999999999e70

   1. Initial program 71.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 75.9

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 75.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. metadata-eval 75.9

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 80.5

         \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 80.5%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   7. Taylor expanded in d around 0

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \color{blue}{\frac{{d}^{2} - \frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}} \]
   8. Step-by-step derivation

      1. cancel-sub-sign-inv N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \frac{\color{blue}{{d}^{2} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}}{{d}^{2}} \]

      2. metadata-eval N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \frac{{d}^{2} + \color{blue}{\frac{-1}{8}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}} \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \frac{\color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell} + {d}^{2}}}{{d}^{2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \color{blue}{\frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell} + {d}^{2}}{{d}^{2}}} \]

   9. Applied rewrites 75.6%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}, -0.125, d \cdot d\right)}{d \cdot d}} \]

   if 2.7999999999999999e70 < 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 d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 58.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 66.1

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 66.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 77.0%

       \[\leadsto \frac{{\ell}^{-0.5}}{\sqrt{h}} \cdot d \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.6 \cdot 10^{-300}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right) \cdot \left(\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right) \cdot -0.5\right), \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-196}:\\ \;\;\;\;{\left(\frac{\sqrt{\ell \cdot h}}{d}\right)}^{-1}\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{+70}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}, -0.125, d \cdot d\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{-0.5}}{\sqrt{h}} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 69.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \left(\frac{D}{d} \cdot 0.5\right) \cdot M\\ \mathbf{if}\;d \leq -5.6 \cdot 10^{-300}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot \left(t\_0 \cdot -0.5\right), \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-196}:\\ \;\;\;\;{\left(\frac{\sqrt{\ell \cdot h}}{d}\right)}^{-1}\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{+70}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d}, -0.125, \ell\right)}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{-0.5}}{\sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* (/ D d) 0.5) M)))
   (if (<= d -5.6e-300)
     (/
      (* (fma (* t_0 (* t_0 -0.5)) (/ h l) 1.0) (/ (- d) (sqrt (- h))))
      (sqrt (- l)))
     (if (<= d 1.9e-196)
       (pow (/ (sqrt (* l h)) d) -1.0)
       (if (<= d 2.8e+70)
         (*
          (* (/ (sqrt d) (sqrt h)) (/ (sqrt d) (sqrt l)))
          (/ (fma (/ (* (* (* M M) h) (* D D)) (* d d)) -0.125 l) l))
         (* (/ (pow l -0.5) (sqrt h)) d))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = ((D / d) * 0.5) * M;
	double tmp;
	if (d <= -5.6e-300) {
		tmp = (fma((t_0 * (t_0 * -0.5)), (h / l), 1.0) * (-d / sqrt(-h))) / sqrt(-l);
	} else if (d <= 1.9e-196) {
		tmp = pow((sqrt((l * h)) / d), -1.0);
	} else if (d <= 2.8e+70) {
		tmp = ((sqrt(d) / sqrt(h)) * (sqrt(d) / sqrt(l))) * (fma(((((M * M) * h) * (D * D)) / (d * d)), -0.125, l) / l);
	} else {
		tmp = (pow(l, -0.5) / sqrt(h)) * d;
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(D / d) * 0.5) * M)
	tmp = 0.0
	if (d <= -5.6e-300)
		tmp = Float64(Float64(fma(Float64(t_0 * Float64(t_0 * -0.5)), Float64(h / l), 1.0) * Float64(Float64(-d) / sqrt(Float64(-h)))) / sqrt(Float64(-l)));
	elseif (d <= 1.9e-196)
		tmp = Float64(sqrt(Float64(l * h)) / d) ^ -1.0;
	elseif (d <= 2.8e+70)
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * Float64(sqrt(d) / sqrt(l))) * Float64(fma(Float64(Float64(Float64(Float64(M * M) * h) * Float64(D * D)) / Float64(d * d)), -0.125, l) / l));
	else
		tmp = Float64(Float64((l ^ -0.5) / sqrt(h)) * d);
	end
	return tmp
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(D / d), $MachinePrecision] * 0.5), $MachinePrecision] * M), $MachinePrecision]}, If[LessEqual[d, -5.6e-300], N[(N[(N[(N[(t$95$0 * N[(t$95$0 * -0.5), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[((-d) / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.9e-196], N[Power[N[(N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision] / d), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[d, 2.8e+70], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * -0.125 + l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[l, -0.5], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -5.59999999999999988e-300

   1. Initial program 77.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

   4. Applied rewrites 73.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      2. metadata-eval 73.8

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. frac-2neg N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto \left(\frac{\sqrt{\color{blue}{-d}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\sqrt{-d}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. lower-neg.f64 82.4

          \[\leadsto \left(\frac{\sqrt{-d}}{\sqrt{\color{blue}{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

   6. Applied rewrites 82.4%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

   8. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5 \cdot {\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot {\left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot M\right)}^{2}}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot M\right)}^{2} \cdot \frac{-1}{2}}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot M\right)}^{2}} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot M\right)}}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\left(M \cdot \left(\frac{D}{d} \cdot \frac{1}{2}\right)\right)}}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\left(M \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{1}{2}\right)}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\left(M \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{D}{d}\right)}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      8. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\left(\left(M \cdot \frac{1}{2}\right) \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\left(\left(M \cdot \color{blue}{\frac{1}{2}}\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      10. div-inv N/A

          \[\leadsto \frac{\mathsf{fma}\left({\left(\color{blue}{\frac{M}{2}} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{M}{2} \cdot \color{blue}{\frac{D}{d}}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      12. times-frac N/A

          \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      15. lift-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      16. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      17. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{-1}{2}\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{-1}{2}\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

   10. Applied rewrites 88.8%

       \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right) \cdot \left(\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right) \cdot -0.5\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

   if -5.59999999999999988e-300 < d < 1.9000000000000001e-196

   1. Initial program 14.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 14.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 26.4

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 26.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 26.4%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 26.4%

       \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\ell \cdot h}}{d}}} \]

   if 1.9000000000000001e-196 < d < 2.7999999999999999e70

   1. Initial program 71.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 75.9

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 75.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. metadata-eval 75.9

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 80.5

         \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 80.5%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   7. Taylor expanded in l around 0

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \color{blue}{\frac{\ell - \frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\ell}} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \frac{\ell - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{8}\right)\right)} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\ell} \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \frac{\color{blue}{\ell + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}}{\ell} \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \color{blue}{\frac{\ell + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\ell}} \]

   9. Applied rewrites 71.0%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d}, -0.125, \ell\right)}{\ell}} \]

   if 2.7999999999999999e70 < 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 d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 58.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 66.1

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 66.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 77.0%

       \[\leadsto \frac{{\ell}^{-0.5}}{\sqrt{h}} \cdot d \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.6 \cdot 10^{-300}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right) \cdot \left(\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right) \cdot -0.5\right), \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-196}:\\ \;\;\;\;{\left(\frac{\sqrt{\ell \cdot h}}{d}\right)}^{-1}\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{+70}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d}, -0.125, \ell\right)}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{-0.5}}{\sqrt{h}} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 69.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \left(\frac{D}{d} \cdot 0.5\right) \cdot M\\ \mathbf{if}\;d \leq -5.6 \cdot 10^{-300}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot \left(t\_0 \cdot -0.5\right), \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-196}:\\ \;\;\;\;{\left(\frac{\sqrt{\ell \cdot h}}{d}\right)}^{-1}\\ \mathbf{elif}\;d \leq 3 \cdot 10^{+70}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d}, -0.125, \ell\right)}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* (/ D d) 0.5) M)))
   (if (<= d -5.6e-300)
     (/
      (* (fma (* t_0 (* t_0 -0.5)) (/ h l) 1.0) (/ (- d) (sqrt (- h))))
      (sqrt (- l)))
     (if (<= d 1.9e-196)
       (pow (/ (sqrt (* l h)) d) -1.0)
       (if (<= d 3e+70)
         (*
          (* (/ (sqrt d) (sqrt h)) (/ (sqrt d) (sqrt l)))
          (/ (fma (/ (* (* (* M M) h) (* D D)) (* d d)) -0.125 l) l))
         (/ d (* (sqrt l) (sqrt h))))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = ((D / d) * 0.5) * M;
	double tmp;
	if (d <= -5.6e-300) {
		tmp = (fma((t_0 * (t_0 * -0.5)), (h / l), 1.0) * (-d / sqrt(-h))) / sqrt(-l);
	} else if (d <= 1.9e-196) {
		tmp = pow((sqrt((l * h)) / d), -1.0);
	} else if (d <= 3e+70) {
		tmp = ((sqrt(d) / sqrt(h)) * (sqrt(d) / sqrt(l))) * (fma(((((M * M) * h) * (D * D)) / (d * d)), -0.125, l) / l);
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(D / d) * 0.5) * M)
	tmp = 0.0
	if (d <= -5.6e-300)
		tmp = Float64(Float64(fma(Float64(t_0 * Float64(t_0 * -0.5)), Float64(h / l), 1.0) * Float64(Float64(-d) / sqrt(Float64(-h)))) / sqrt(Float64(-l)));
	elseif (d <= 1.9e-196)
		tmp = Float64(sqrt(Float64(l * h)) / d) ^ -1.0;
	elseif (d <= 3e+70)
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * Float64(sqrt(d) / sqrt(l))) * Float64(fma(Float64(Float64(Float64(Float64(M * M) * h) * Float64(D * D)) / Float64(d * d)), -0.125, l) / l));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(D / d), $MachinePrecision] * 0.5), $MachinePrecision] * M), $MachinePrecision]}, If[LessEqual[d, -5.6e-300], N[(N[(N[(N[(t$95$0 * N[(t$95$0 * -0.5), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[((-d) / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.9e-196], N[Power[N[(N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision] / d), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[d, 3e+70], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * -0.125 + l), $MachinePrecision] / l), $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 < -5.59999999999999988e-300

   1. Initial program 77.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

   4. Applied rewrites 73.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      2. metadata-eval 73.8

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. frac-2neg N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto \left(\frac{\sqrt{\color{blue}{-d}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\sqrt{-d}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. lower-neg.f64 82.4

          \[\leadsto \left(\frac{\sqrt{-d}}{\sqrt{\color{blue}{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

   6. Applied rewrites 82.4%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

   8. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5 \cdot {\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot {\left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot M\right)}^{2}}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot M\right)}^{2} \cdot \frac{-1}{2}}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot M\right)}^{2}} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot M\right)}}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\left(M \cdot \left(\frac{D}{d} \cdot \frac{1}{2}\right)\right)}}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\left(M \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{1}{2}\right)}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\left(M \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{D}{d}\right)}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      8. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\left(\left(M \cdot \frac{1}{2}\right) \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\left(\left(M \cdot \color{blue}{\frac{1}{2}}\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      10. div-inv N/A

          \[\leadsto \frac{\mathsf{fma}\left({\left(\color{blue}{\frac{M}{2}} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{M}{2} \cdot \color{blue}{\frac{D}{d}}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      12. times-frac N/A

          \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      15. lift-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      16. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      17. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{-1}{2}\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{-1}{2}\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

   10. Applied rewrites 88.8%

       \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right) \cdot \left(\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right) \cdot -0.5\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

   if -5.59999999999999988e-300 < d < 1.9000000000000001e-196

   1. Initial program 14.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 14.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 26.4

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 26.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 26.4%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 26.4%

       \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\ell \cdot h}}{d}}} \]

   if 1.9000000000000001e-196 < d < 2.99999999999999976e70

   1. Initial program 71.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 75.9

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 75.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. metadata-eval 75.9

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 80.5

         \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 80.5%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   7. Taylor expanded in l around 0

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \color{blue}{\frac{\ell - \frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\ell}} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \frac{\ell - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{8}\right)\right)} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\ell} \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \frac{\color{blue}{\ell + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}}{\ell} \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \color{blue}{\frac{\ell + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\ell}} \]

   9. Applied rewrites 71.0%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d}, -0.125, \ell\right)}{\ell}} \]

   if 2.99999999999999976e70 < 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 d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 58.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 66.1

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 66.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 66.1%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 76.8%

       \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.6 \cdot 10^{-300}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right) \cdot \left(\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right) \cdot -0.5\right), \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-196}:\\ \;\;\;\;{\left(\frac{\sqrt{\ell \cdot h}}{d}\right)}^{-1}\\ \mathbf{elif}\;d \leq 3 \cdot 10^{+70}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d}, -0.125, \ell\right)}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 46.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -8.6 \cdot 10^{-188}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{-301}:\\ \;\;\;\;\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -8.6e-188)
   (fabs (/ d (sqrt (* l h))))
   (if (<= l 1.75e-301)
     (* (sqrt (pow (* l h) -1.0)) d)
     (/ d (* (sqrt l) (sqrt h))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -8.6e-188) {
		tmp = fabs((d / sqrt((l * h))));
	} else if (l <= 1.75e-301) {
		tmp = sqrt(pow((l * h), -1.0)) * d;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (l <= (-8.6d-188)) then
        tmp = abs((d / sqrt((l * h))))
    else if (l <= 1.75d-301) then
        tmp = sqrt(((l * h) ** (-1.0d0))) * d
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -8.6e-188) {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	} else if (l <= 1.75e-301) {
		tmp = Math.sqrt(Math.pow((l * h), -1.0)) * d;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -8.6e-188:
		tmp = math.fabs((d / math.sqrt((l * h))))
	elif l <= 1.75e-301:
		tmp = math.sqrt(math.pow((l * h), -1.0)) * d
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -8.6e-188)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	elseif (l <= 1.75e-301)
		tmp = Float64(sqrt((Float64(l * h) ^ -1.0)) * d);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -8.6e-188)
		tmp = abs((d / sqrt((l * h))));
	elseif (l <= 1.75e-301)
		tmp = sqrt(((l * h) ^ -1.0)) * d;
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -8.6e-188], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.75e-301], N[(N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -8.59999999999999975e-188

   1. Initial program 76.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 46.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 7.0

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 7.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 7.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 40.8%

       \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \]

   if -8.59999999999999975e-188 < l < 1.74999999999999996e-301

   1. Initial program 74.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 11.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 39.4

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 39.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   if 1.74999999999999996e-301 < l

   1. Initial program 62.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 41.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 43.5

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 43.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 44.3%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 52.7%

       \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.6 \cdot 10^{-188}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{-301}:\\ \;\;\;\;\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 64.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \left(\frac{D}{d} \cdot 0.5\right) \cdot M\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot \left(t\_0 \cdot -0.5\right), \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* (/ D d) 0.5) M)))
   (if (<= l -4e-310)
     (/
      (* (fma (* t_0 (* t_0 -0.5)) (/ h l) 1.0) (/ (- d) (sqrt (- h))))
      (sqrt (- l)))
     (/ d (* (sqrt l) (sqrt h))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = ((D / d) * 0.5) * M;
	double tmp;
	if (l <= -4e-310) {
		tmp = (fma((t_0 * (t_0 * -0.5)), (h / l), 1.0) * (-d / sqrt(-h))) / sqrt(-l);
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(D / d) * 0.5) * M)
	tmp = 0.0
	if (l <= -4e-310)
		tmp = Float64(Float64(fma(Float64(t_0 * Float64(t_0 * -0.5)), Float64(h / l), 1.0) * Float64(Float64(-d) / sqrt(Float64(-h)))) / sqrt(Float64(-l)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(D / d), $MachinePrecision] * 0.5), $MachinePrecision] * M), $MachinePrecision]}, If[LessEqual[l, -4e-310], N[(N[(N[(N[(t$95$0 * N[(t$95$0 * -0.5), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[((-d) / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -3.999999999999988e-310

   1. Initial program 76.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

   4. Applied rewrites 71.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      2. metadata-eval 71.6

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. frac-2neg N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto \left(\frac{\sqrt{\color{blue}{-d}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\sqrt{-d}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. lower-neg.f64 79.9

          \[\leadsto \left(\frac{\sqrt{-d}}{\sqrt{\color{blue}{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

   6. Applied rewrites 79.9%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

   8. Applied rewrites 86.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5 \cdot {\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot {\left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot M\right)}^{2}}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot M\right)}^{2} \cdot \frac{-1}{2}}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot M\right)}^{2}} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot M\right)}}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\left(M \cdot \left(\frac{D}{d} \cdot \frac{1}{2}\right)\right)}}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\left(M \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{1}{2}\right)}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\left(M \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{D}{d}\right)}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      8. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\left(\left(M \cdot \frac{1}{2}\right) \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\left(\left(M \cdot \color{blue}{\frac{1}{2}}\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      10. div-inv N/A

          \[\leadsto \frac{\mathsf{fma}\left({\left(\color{blue}{\frac{M}{2}} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{M}{2} \cdot \color{blue}{\frac{D}{d}}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      12. times-frac N/A

          \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      15. lift-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      16. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{-1}{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      17. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{-1}{2}\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{-1}{2}\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

   10. Applied rewrites 86.9%

       \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right) \cdot \left(\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right) \cdot -0.5\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

   if -3.999999999999988e-310 < l

   1. Initial program 62.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 41.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 43.6

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 43.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 44.4%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 52.6%

       \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 56.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{-h}\\ t_1 := \sqrt{-\ell}\\ \mathbf{if}\;d \leq -1.2 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{-d}{t\_1} \cdot 1}{t\_0}\\ \mathbf{elif}\;d \leq -7.4 \cdot 10^{-302}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot -0.125, \frac{h}{\ell}, 1\right) \cdot \frac{d}{t\_0}}{-t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (- h))) (t_1 (sqrt (- l))))
   (if (<= d -1.2e+142)
     (/ (* (/ (- d) t_1) 1.0) t_0)
     (if (<= d -7.4e-302)
       (/
        (*
         (fma (* (/ (* (* M M) (* D D)) (* d d)) -0.125) (/ h l) 1.0)
         (/ d t_0))
        (- t_1))
       (/ d (* (sqrt l) (sqrt h)))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(-h);
	double t_1 = sqrt(-l);
	double tmp;
	if (d <= -1.2e+142) {
		tmp = ((-d / t_1) * 1.0) / t_0;
	} else if (d <= -7.4e-302) {
		tmp = (fma(((((M * M) * (D * D)) / (d * d)) * -0.125), (h / l), 1.0) * (d / t_0)) / -t_1;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(-h))
	t_1 = sqrt(Float64(-l))
	tmp = 0.0
	if (d <= -1.2e+142)
		tmp = Float64(Float64(Float64(Float64(-d) / t_1) * 1.0) / t_0);
	elseif (d <= -7.4e-302)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(M * M) * Float64(D * D)) / Float64(d * d)) * -0.125), Float64(h / l), 1.0) * Float64(d / t_0)) / Float64(-t_1));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[(-h)], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-l)], $MachinePrecision]}, If[LessEqual[d, -1.2e+142], N[(N[(N[((-d) / t$95$1), $MachinePrecision] * 1.0), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[d, -7.4e-302], N[(N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / t$95$0), $MachinePrecision]), $MachinePrecision] / (-t$95$1)), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.2e142

   1. Initial program 91.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 64.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot 1 \]

      2. metadata-eval 64.9

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot 1 \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot 1 \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot 1 \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot 1 \]

      6. frac-2neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot 1 \]

      7. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}}\right) \cdot 1 \]

      8. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}}\right) \cdot 1 \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}}\right) \cdot 1 \]

      10. lower-neg.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{\color{blue}{-d}}}{\sqrt{\mathsf{neg}\left(\ell\right)}}\right) \cdot 1 \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{-d}}{\color{blue}{\sqrt{\mathsf{neg}\left(\ell\right)}}}\right) \cdot 1 \]

      12. lower-neg.f64 67.7

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{\color{blue}{-\ell}}}\right) \cdot 1 \]

   7. Applied rewrites 67.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}}\right) \cdot 1 \]

   9. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\frac{\frac{-d}{\sqrt{-\ell}} \cdot 1}{\sqrt{-h}}} \]

   if -1.2e142 < d < -7.4e-302

   1. Initial program 71.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

   4. Applied rewrites 68.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      2. metadata-eval 68.2

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. frac-2neg N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto \left(\frac{\sqrt{\color{blue}{-d}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\sqrt{-d}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot \left(D \cdot \frac{1}{2}\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. lower-neg.f64 77.6

          \[\leadsto \left(\frac{\sqrt{-d}}{\sqrt{\color{blue}{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

   6. Applied rewrites 77.6%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}\right) \cdot \left(\left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right) \]

   8. Applied rewrites 84.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5 \cdot {\left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}} \]

   9. Taylor expanded in d around 0

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{-1}{8}}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{-1}{8}}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

       3. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}} \cdot \frac{-1}{8}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

       4. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}} \cdot \frac{-1}{8}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}} \cdot \frac{-1}{8}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

       6. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{\left(M \cdot M\right)} \cdot {D}^{2}}{{d}^{2}} \cdot \frac{-1}{8}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{\left(M \cdot M\right)} \cdot {D}^{2}}{{d}^{2}} \cdot \frac{-1}{8}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

       8. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(M \cdot M\right) \cdot \color{blue}{\left(D \cdot D\right)}}{{d}^{2}} \cdot \frac{-1}{8}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

       9. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(M \cdot M\right) \cdot \color{blue}{\left(D \cdot D\right)}}{{d}^{2}} \cdot \frac{-1}{8}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

       10. unpow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{\color{blue}{d \cdot d}} \cdot \frac{-1}{8}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

       11. lower-*.f64 64.9

           \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{\color{blue}{d \cdot d}} \cdot -0.125, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

   11. Applied rewrites 64.9%

       \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot -0.125}, \frac{h}{\ell}, 1\right) \cdot \frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}} \]

   if -7.4e-302 < d

   1. Initial program 62.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 40.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 43.3

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 43.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 44.1%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 52.2%

       \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.2 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{-d}{\sqrt{-\ell}} \cdot 1}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq -7.4 \cdot 10^{-302}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot -0.125, \frac{h}{\ell}, 1\right) \cdot \frac{d}{\sqrt{-h}}}{-\sqrt{-\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 52.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.4 \cdot 10^{-160}:\\ \;\;\;\;\frac{-d}{\sqrt{-\ell} \cdot \sqrt{-h}} \cdot 1\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt{\sqrt{\left|h\right| \cdot \left(\ell \cdot \ell\right)} \cdot \sqrt{\left|h\right|}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -6.4e-160)
   (* (/ (- d) (* (sqrt (- l)) (sqrt (- h)))) 1.0)
   (if (<= l -4e-310)
     (/ d (sqrt (* (sqrt (* (fabs h) (* l l))) (sqrt (fabs h)))))
     (/ d (* (sqrt l) (sqrt h))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -6.4e-160) {
		tmp = (-d / (sqrt(-l) * sqrt(-h))) * 1.0;
	} else if (l <= -4e-310) {
		tmp = d / sqrt((sqrt((fabs(h) * (l * l))) * sqrt(fabs(h))));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (l <= (-6.4d-160)) then
        tmp = (-d / (sqrt(-l) * sqrt(-h))) * 1.0d0
    else if (l <= (-4d-310)) then
        tmp = d / sqrt((sqrt((abs(h) * (l * l))) * sqrt(abs(h))))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -6.4e-160) {
		tmp = (-d / (Math.sqrt(-l) * Math.sqrt(-h))) * 1.0;
	} else if (l <= -4e-310) {
		tmp = d / Math.sqrt((Math.sqrt((Math.abs(h) * (l * l))) * Math.sqrt(Math.abs(h))));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -6.4e-160:
		tmp = (-d / (math.sqrt(-l) * math.sqrt(-h))) * 1.0
	elif l <= -4e-310:
		tmp = d / math.sqrt((math.sqrt((math.fabs(h) * (l * l))) * math.sqrt(math.fabs(h))))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -6.4e-160)
		tmp = Float64(Float64(Float64(-d) / Float64(sqrt(Float64(-l)) * sqrt(Float64(-h)))) * 1.0);
	elseif (l <= -4e-310)
		tmp = Float64(d / sqrt(Float64(sqrt(Float64(abs(h) * Float64(l * l))) * sqrt(abs(h)))));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -6.4e-160)
		tmp = (-d / (sqrt(-l) * sqrt(-h))) * 1.0;
	elseif (l <= -4e-310)
		tmp = d / sqrt((sqrt((abs(h) * (l * l))) * sqrt(abs(h))));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -6.4e-160], N[(N[((-d) / N[(N[Sqrt[(-l)], $MachinePrecision] * N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[l, -4e-310], N[(d / N[Sqrt[N[(N[Sqrt[N[(N[Abs[h], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[h], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.40000000000000018e-160

   1. Initial program 78.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 48.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot 1 \]

      2. metadata-eval 48.7

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot 1 \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot 1 \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot 1 \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot 1 \]

      6. frac-2neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot 1 \]

      7. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}}\right) \cdot 1 \]

      8. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}}\right) \cdot 1 \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}}\right) \cdot 1 \]

      10. lower-neg.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{\color{blue}{-d}}}{\sqrt{\mathsf{neg}\left(\ell\right)}}\right) \cdot 1 \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{-d}}{\color{blue}{\sqrt{\mathsf{neg}\left(\ell\right)}}}\right) \cdot 1 \]

      12. lower-neg.f64 50.5

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{\color{blue}{-\ell}}}\right) \cdot 1 \]

   7. Applied rewrites 50.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}}\right) \cdot 1 \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)} \cdot 1 \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot 1 \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      4. frac-2neg N/A

         \[\leadsto \left(\color{blue}{\frac{\mathsf{neg}\left(\sqrt{-d}\right)}{\mathsf{neg}\left(\sqrt{-\ell}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\sqrt{-d}\right)}{\mathsf{neg}\left(\sqrt{-\ell}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot 1 \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\sqrt{-d}\right)}{\mathsf{neg}\left(\sqrt{-\ell}\right)} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot 1 \]

      7. lift-pow.f64 N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\sqrt{-d}\right)}{\mathsf{neg}\left(\sqrt{-\ell}\right)} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}}\right) \cdot 1 \]

      8. unpow1/2 N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\sqrt{-d}\right)}{\mathsf{neg}\left(\sqrt{-\ell}\right)} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot 1 \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\sqrt{-d}\right)}{\mathsf{neg}\left(\sqrt{-\ell}\right)} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot 1 \]

      10. frac-2neg N/A

          \[\leadsto \left(\frac{\mathsf{neg}\left(\sqrt{-d}\right)}{\mathsf{neg}\left(\sqrt{-\ell}\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(h\right)}}}\right) \cdot 1 \]

      11. lift-neg.f64 N/A

          \[\leadsto \left(\frac{\mathsf{neg}\left(\sqrt{-d}\right)}{\mathsf{neg}\left(\sqrt{-\ell}\right)} \cdot \sqrt{\frac{\color{blue}{-d}}{\mathsf{neg}\left(h\right)}}\right) \cdot 1 \]

      12. lift-neg.f64 N/A

          \[\leadsto \left(\frac{\mathsf{neg}\left(\sqrt{-d}\right)}{\mathsf{neg}\left(\sqrt{-\ell}\right)} \cdot \sqrt{\frac{-d}{\color{blue}{-h}}}\right) \cdot 1 \]

      13. sqrt-undiv N/A

          \[\leadsto \left(\frac{\mathsf{neg}\left(\sqrt{-d}\right)}{\mathsf{neg}\left(\sqrt{-\ell}\right)} \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}}\right) \cdot 1 \]

      14. lift-sqrt.f64 N/A

          \[\leadsto \left(\frac{\mathsf{neg}\left(\sqrt{-d}\right)}{\mathsf{neg}\left(\sqrt{-\ell}\right)} \cdot \frac{\color{blue}{\sqrt{-d}}}{\sqrt{-h}}\right) \cdot 1 \]

      15. lift-sqrt.f64 N/A

          \[\leadsto \left(\frac{\mathsf{neg}\left(\sqrt{-d}\right)}{\mathsf{neg}\left(\sqrt{-\ell}\right)} \cdot \frac{\sqrt{-d}}{\color{blue}{\sqrt{-h}}}\right) \cdot 1 \]

      16. frac-2neg N/A

          \[\leadsto \left(\frac{\mathsf{neg}\left(\sqrt{-d}\right)}{\mathsf{neg}\left(\sqrt{-\ell}\right)} \cdot \color{blue}{\frac{\mathsf{neg}\left(\sqrt{-d}\right)}{\mathsf{neg}\left(\sqrt{-h}\right)}}\right) \cdot 1 \]

      17. frac-times N/A

          \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(\sqrt{-d}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{-d}\right)\right)}{\left(\mathsf{neg}\left(\sqrt{-\ell}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{-h}\right)\right)}} \cdot 1 \]

   9. Applied rewrites 53.9%

      \[\leadsto \color{blue}{\frac{-d}{\left(-\sqrt{-\ell}\right) \cdot \left(-\sqrt{-h}\right)}} \cdot 1 \]

   if -6.40000000000000018e-160 < l < -3.999999999999988e-310

   1. Initial program 64.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 10.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 29.9

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 29.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 29.9%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 64.7%

       \[\leadsto \frac{d}{\sqrt{\sqrt{\left|h\right| \cdot \left(\ell \cdot \ell\right)} \cdot \sqrt{\left|h\right|}}} \]

   if -3.999999999999988e-310 < l

   1. Initial program 62.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 41.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 43.6

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 43.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 44.4%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 52.6%

       \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.4 \cdot 10^{-160}:\\ \;\;\;\;\frac{-d}{\sqrt{-\ell} \cdot \sqrt{-h}} \cdot 1\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt{\sqrt{\left|h\right| \cdot \left(\ell \cdot \ell\right)} \cdot \sqrt{\left|h\right|}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 26.5% accurate, 15.3× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D) :precision binary64 (/ d (sqrt (* l h))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	return d / sqrt((l * h));
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
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 / sqrt((l * h))
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	return d / Math.sqrt((l * h));
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	return d / math.sqrt((l * h))

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp = code(d, h, l, M, D)
	tmp = d / sqrt((l * h));
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.5%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing

3. Taylor expanded in d around inf

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 41.6%

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

6. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   6. lower-*.f64 27.1

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

8. Applied rewrites 27.1%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
9. Step-by-step derivation

10. Applied rewrites 27.5%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024305 
(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)))))