Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.6% → 78.8%

Time: 16.6s

Alternatives: 21

Speedup: 3.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 78.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{\ell}}\\ t_1 := \frac{M\_m}{d} \cdot D\_m\\ \mathbf{if}\;d \leq -2 \cdot 10^{+88}:\\ \;\;\;\;\frac{-d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\_m\right) \cdot D\_m}{\ell} \cdot \frac{\left(0.5 \cdot \left(D\_m \cdot 0.5\right)\right) \cdot \frac{M\_m}{d}}{{h}^{-1}}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, \left(\left(0.25 \cdot \frac{h}{\ell}\right) \cdot t\_1\right) \cdot t\_1, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+172}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \sqrt{h}, t\_1 \cdot t\_1, \sqrt{{h}^{-1}} \cdot \ell\right)}{\ell} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.125 \cdot \frac{D\_m}{d}}{d}, \frac{\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot D\_m}{\ell}, 1\right)}{\sqrt{h}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ d (sqrt l))) (t_1 (* (/ M_m d) D_m)))
   (if (<= d -2e+88)
     (*
      (/ (- d) (sqrt (* h l)))
      (-
       1.0
       (*
        (/ (* (* (/ 0.5 d) M_m) D_m) l)
        (/ (* (* 0.5 (* D_m 0.5)) (/ M_m d)) (pow h -1.0)))))
     (if (<= d -5e-310)
       (*
        (* (fma -0.5 (* (* (* 0.25 (/ h l)) t_1) t_1) 1.0) (sqrt (/ d h)))
        (/ (sqrt (- d)) (sqrt (- l))))
       (if (<= d 1.85e+172)
         (*
          (/ (fma (* -0.125 (sqrt h)) (* t_1 t_1) (* (sqrt (pow h -1.0)) l)) l)
          t_0)
         (*
          (/
           (fma (/ (* -0.125 (/ D_m d)) d) (/ (* (* (* M_m M_m) h) D_m) l) 1.0)
           (sqrt h))
          t_0))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = d / sqrt(l);
	double t_1 = (M_m / d) * D_m;
	double tmp;
	if (d <= -2e+88) {
		tmp = (-d / sqrt((h * l))) * (1.0 - (((((0.5 / d) * M_m) * D_m) / l) * (((0.5 * (D_m * 0.5)) * (M_m / d)) / pow(h, -1.0))));
	} else if (d <= -5e-310) {
		tmp = (fma(-0.5, (((0.25 * (h / l)) * t_1) * t_1), 1.0) * sqrt((d / h))) * (sqrt(-d) / sqrt(-l));
	} else if (d <= 1.85e+172) {
		tmp = (fma((-0.125 * sqrt(h)), (t_1 * t_1), (sqrt(pow(h, -1.0)) * l)) / l) * t_0;
	} else {
		tmp = (fma(((-0.125 * (D_m / d)) / d), ((((M_m * M_m) * h) * D_m) / l), 1.0) / sqrt(h)) * t_0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(d / sqrt(l))
	t_1 = Float64(Float64(M_m / d) * D_m)
	tmp = 0.0
	if (d <= -2e+88)
		tmp = Float64(Float64(Float64(-d) / sqrt(Float64(h * l))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(0.5 / d) * M_m) * D_m) / l) * Float64(Float64(Float64(0.5 * Float64(D_m * 0.5)) * Float64(M_m / d)) / (h ^ -1.0)))));
	elseif (d <= -5e-310)
		tmp = Float64(Float64(fma(-0.5, Float64(Float64(Float64(0.25 * Float64(h / l)) * t_1) * t_1), 1.0) * sqrt(Float64(d / h))) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))));
	elseif (d <= 1.85e+172)
		tmp = Float64(Float64(fma(Float64(-0.125 * sqrt(h)), Float64(t_1 * t_1), Float64(sqrt((h ^ -1.0)) * l)) / l) * t_0);
	else
		tmp = Float64(Float64(fma(Float64(Float64(-0.125 * Float64(D_m / d)) / d), Float64(Float64(Float64(Float64(M_m * M_m) * h) * D_m) / l), 1.0) / sqrt(h)) * t_0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]}, If[LessEqual[d, -2e+88], N[(N[((-d) / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(0.5 * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] / N[Power[h, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(N[(-0.5 * N[(N[(N[(0.25 * N[(h / l), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.85e+172], N[(N[(N[(N[(-0.125 * N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision] + N[(N[Sqrt[N[Power[h, -1.0], $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(-0.125 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.99999999999999992e88

   1. Initial program 62.8%

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

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 69.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. unpow1/2 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-times N/A

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

      13. sqrt-div N/A

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

      14. sqr-neg N/A

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

      15. lift-neg.f64 N/A

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

      16. lift-neg.f64 N/A

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

      17. sqrt-unprod N/A

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

      18. rem-square-sqrt N/A

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

      19. lower-/.f64 N/A

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

      20. lower-sqrt.f64 N/A

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

      21. lower-*.f64 95.6

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

   6. Applied rewrites 95.6%

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

   if -1.99999999999999992e88 < d < -4.999999999999985e-310

   1. Initial program 78.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 0.0

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 77.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 77.2

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 77.2

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 77.2

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

   8. Applied rewrites 77.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. sqrt-div N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 87.2

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

   10. Applied rewrites 87.2%

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

   if -4.999999999999985e-310 < d < 1.84999999999999986e172

   1. Initial program 70.9%

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

   3. Taylor expanded in 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 34.7%

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

   7. Applied rewrites 34.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 34.2

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

      7. lift-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sqrt-prod N/A

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

      10. lift-/.f64 N/A

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

      11. sqrt-div N/A

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

      12. associate-*l/ N/A

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

      13. rem-square-sqrt N/A

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

   9. Applied rewrites 39.2%

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

   10. Taylor expanded in l around 0

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

       1. lower-/.f64 N/A

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

   12. Applied rewrites 81.3%

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

   if 1.84999999999999986e172 < d

   1. Initial program 61.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 38.4%

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

   7. Applied rewrites 60.9%

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

   8. Taylor expanded in d around 0

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

      1. div-sub N/A

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

      2. *-inverses N/A

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

      3. associate-*r/ N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r/ N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. times-frac N/A

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

   10. Applied rewrites 64.9%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/l* N/A

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

       4. *-commutative N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-/.f64 N/A

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

       8. associate-*l/ N/A

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

   12. Applied rewrites 79.8%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-0.125 \cdot \frac{D}{d}}{d}, \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot D}{\ell}, 1\right)}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{+88}:\\ \;\;\;\;\frac{-d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{{h}^{-1}}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, \left(\left(0.25 \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(\frac{M}{d} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+172}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \sqrt{h}, \left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right), \sqrt{{h}^{-1}} \cdot \ell\right)}{\ell} \cdot \frac{d}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.125 \cdot \frac{D}{d}}{d}, \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot D}{\ell}, 1\right)}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m}{d} \cdot D\_m\\ t_1 := \frac{d}{\sqrt{\ell}}\\ \mathbf{if}\;d \leq -8.8 \cdot 10^{+88}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot \left(0.25 \cdot {\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}\right), 1\right) \cdot \frac{-d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, \left(\left(0.25 \cdot \frac{h}{\ell}\right) \cdot t\_0\right) \cdot t\_0, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+172}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \sqrt{h}, t\_0 \cdot t\_0, \sqrt{{h}^{-1}} \cdot \ell\right)}{\ell} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.125 \cdot \frac{D\_m}{d}}{d}, \frac{\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot D\_m}{\ell}, 1\right)}{\sqrt{h}} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ M_m d) D_m)) (t_1 (/ d (sqrt l))))
   (if (<= d -8.8e+88)
     (*
      (fma -0.5 (* (/ h l) (* 0.25 (pow (* D_m (/ M_m d)) 2.0))) 1.0)
      (/ (- d) (sqrt (* h l))))
     (if (<= d -5e-310)
       (*
        (* (fma -0.5 (* (* (* 0.25 (/ h l)) t_0) t_0) 1.0) (sqrt (/ d h)))
        (/ (sqrt (- d)) (sqrt (- l))))
       (if (<= d 1.85e+172)
         (*
          (/ (fma (* -0.125 (sqrt h)) (* t_0 t_0) (* (sqrt (pow h -1.0)) l)) l)
          t_1)
         (*
          (/
           (fma (/ (* -0.125 (/ D_m d)) d) (/ (* (* (* M_m M_m) h) D_m) l) 1.0)
           (sqrt h))
          t_1))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (M_m / d) * D_m;
	double t_1 = d / sqrt(l);
	double tmp;
	if (d <= -8.8e+88) {
		tmp = fma(-0.5, ((h / l) * (0.25 * pow((D_m * (M_m / d)), 2.0))), 1.0) * (-d / sqrt((h * l)));
	} else if (d <= -5e-310) {
		tmp = (fma(-0.5, (((0.25 * (h / l)) * t_0) * t_0), 1.0) * sqrt((d / h))) * (sqrt(-d) / sqrt(-l));
	} else if (d <= 1.85e+172) {
		tmp = (fma((-0.125 * sqrt(h)), (t_0 * t_0), (sqrt(pow(h, -1.0)) * l)) / l) * t_1;
	} else {
		tmp = (fma(((-0.125 * (D_m / d)) / d), ((((M_m * M_m) * h) * D_m) / l), 1.0) / sqrt(h)) * t_1;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(M_m / d) * D_m)
	t_1 = Float64(d / sqrt(l))
	tmp = 0.0
	if (d <= -8.8e+88)
		tmp = Float64(fma(-0.5, Float64(Float64(h / l) * Float64(0.25 * (Float64(D_m * Float64(M_m / d)) ^ 2.0))), 1.0) * Float64(Float64(-d) / sqrt(Float64(h * l))));
	elseif (d <= -5e-310)
		tmp = Float64(Float64(fma(-0.5, Float64(Float64(Float64(0.25 * Float64(h / l)) * t_0) * t_0), 1.0) * sqrt(Float64(d / h))) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))));
	elseif (d <= 1.85e+172)
		tmp = Float64(Float64(fma(Float64(-0.125 * sqrt(h)), Float64(t_0 * t_0), Float64(sqrt((h ^ -1.0)) * l)) / l) * t_1);
	else
		tmp = Float64(Float64(fma(Float64(Float64(-0.125 * Float64(D_m / d)) / d), Float64(Float64(Float64(Float64(M_m * M_m) * h) * D_m) / l), 1.0) / sqrt(h)) * t_1);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -8.8e+88], N[(N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[(0.25 * N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[((-d) / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(N[(-0.5 * N[(N[(N[(0.25 * N[(h / l), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.85e+172], N[(N[(N[(N[(-0.125 * N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision] + N[(N[Sqrt[N[Power[h, -1.0], $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(N[(N[(-0.125 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -8.80000000000000035e88

   1. Initial program 62.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 0.0

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 87.5%

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

   if -8.80000000000000035e88 < d < -4.999999999999985e-310

   1. Initial program 78.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 0.0

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 77.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 77.2

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 77.2

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 77.2

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

   8. Applied rewrites 77.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. sqrt-div N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 87.2

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

   10. Applied rewrites 87.2%

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

   if -4.999999999999985e-310 < d < 1.84999999999999986e172

   1. Initial program 70.9%

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

   3. Taylor expanded in 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 34.7%

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

   7. Applied rewrites 34.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 34.2

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

      7. lift-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sqrt-prod N/A

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

      10. lift-/.f64 N/A

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

      11. sqrt-div N/A

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

      12. associate-*l/ N/A

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

      13. rem-square-sqrt N/A

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

   9. Applied rewrites 39.2%

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

   10. Taylor expanded in l around 0

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

       1. lower-/.f64 N/A

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

   12. Applied rewrites 81.3%

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

   if 1.84999999999999986e172 < d

   1. Initial program 61.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 38.4%

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

   7. Applied rewrites 60.9%

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

   8. Taylor expanded in d around 0

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

      1. div-sub N/A

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

      2. *-inverses N/A

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

      3. associate-*r/ N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r/ N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. times-frac N/A

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

   10. Applied rewrites 64.9%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/l* N/A

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

       4. *-commutative N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-/.f64 N/A

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

       8. associate-*l/ N/A

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

   12. Applied rewrites 79.8%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-0.125 \cdot \frac{D}{d}}{d}, \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot D}{\ell}, 1\right)}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.8 \cdot 10^{+88}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot \left(0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right), 1\right) \cdot \frac{-d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, \left(\left(0.25 \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(\frac{M}{d} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+172}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \sqrt{h}, \left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right), \sqrt{{h}^{-1}} \cdot \ell\right)}{\ell} \cdot \frac{d}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.125 \cdot \frac{D}{d}}{d}, \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot D}{\ell}, 1\right)}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m}{d} \cdot D\_m\\ \mathbf{if}\;\ell \leq -2.9 \cdot 10^{+129}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;\ell \leq 1.72 \cdot 10^{-153}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\frac{h \cdot 0.25}{d} \cdot \frac{M\_m}{\ell}\right) \cdot \left(D\_m \cdot \frac{M\_m}{d}\right), D\_m \cdot -0.5, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 1.22 \cdot 10^{+244}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \sqrt{h}, t\_0 \cdot t\_0, \sqrt{{h}^{-1}} \cdot \ell\right)}{\ell} \cdot \frac{d}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ M_m d) D_m)))
   (if (<= l -2.9e+129)
     (* (- d) (sqrt (pow (* l h) -1.0)))
     (if (<= l 1.72e-153)
       (*
        (*
         (fma
          (* (* (/ (* h 0.25) d) (/ M_m l)) (* D_m (/ M_m d)))
          (* D_m -0.5)
          1.0)
         (sqrt (/ d h)))
        (sqrt (/ d l)))
       (if (<= l 1.22e+244)
         (*
          (/ (fma (* -0.125 (sqrt h)) (* t_0 t_0) (* (sqrt (pow h -1.0)) l)) l)
          (/ d (sqrt l)))
         (/ d (sqrt (* l h))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (M_m / d) * D_m;
	double tmp;
	if (l <= -2.9e+129) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else if (l <= 1.72e-153) {
		tmp = (fma(((((h * 0.25) / d) * (M_m / l)) * (D_m * (M_m / d))), (D_m * -0.5), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else if (l <= 1.22e+244) {
		tmp = (fma((-0.125 * sqrt(h)), (t_0 * t_0), (sqrt(pow(h, -1.0)) * l)) / l) * (d / sqrt(l));
	} else {
		tmp = d / sqrt((l * h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(M_m / d) * D_m)
	tmp = 0.0
	if (l <= -2.9e+129)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	elseif (l <= 1.72e-153)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(h * 0.25) / d) * Float64(M_m / l)) * Float64(D_m * Float64(M_m / d))), Float64(D_m * -0.5), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	elseif (l <= 1.22e+244)
		tmp = Float64(Float64(fma(Float64(-0.125 * sqrt(h)), Float64(t_0 * t_0), Float64(sqrt((h ^ -1.0)) * l)) / l) * Float64(d / sqrt(l)));
	else
		tmp = Float64(d / sqrt(Float64(l * h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]}, If[LessEqual[l, -2.9e+129], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.72e-153], N[(N[(N[(N[(N[(N[(N[(h * 0.25), $MachinePrecision] / d), $MachinePrecision] * N[(M$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.22e+244], N[(N[(N[(N[(-0.125 * N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision] + N[(N[Sqrt[N[Power[h, -1.0], $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.90000000000000003e129

   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. 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. frac-2neg N/A

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

      7. div-inv N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 52.8

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

   4. Applied rewrites 52.8%

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

   5. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 63.0

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

   7. Applied rewrites 63.0%

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

   if -2.90000000000000003e129 < l < 1.71999999999999997e-153

   1. Initial program 85.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 21.6

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

   4. Applied rewrites 21.6%

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

   6. Applied rewrites 84.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 84.4

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 84.4

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 84.4

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

   8. Applied rewrites 84.4%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   10. Applied rewrites 82.1%

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

   if 1.71999999999999997e-153 < l < 1.22000000000000005e244

   1. Initial program 68.7%

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

   3. Taylor expanded in 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 42.6%

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

   7. Applied rewrites 43.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 43.0

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

      7. lift-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sqrt-prod N/A

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

      10. lift-/.f64 N/A

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

      11. sqrt-div N/A

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

      12. associate-*l/ N/A

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

      13. rem-square-sqrt N/A

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

   9. Applied rewrites 48.0%

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

   10. Taylor expanded in l around 0

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

       1. lower-/.f64 N/A

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

   12. Applied rewrites 84.4%

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

   if 1.22000000000000005e244 < l

   1. Initial program 22.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 23.7%

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

   6. Taylor expanded in d around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 80.0

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

   8. Applied rewrites 80.0%

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

   10. Applied rewrites 80.0%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.9 \cdot 10^{+129}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;\ell \leq 1.72 \cdot 10^{-153}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\frac{h \cdot 0.25}{d} \cdot \frac{M}{\ell}\right) \cdot \left(D \cdot \frac{M}{d}\right), D \cdot -0.5, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 1.22 \cdot 10^{+244}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \sqrt{h}, \left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right), \sqrt{{h}^{-1}} \cdot \ell\right)}{\ell} \cdot \frac{d}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m}{d} \cdot D\_m\\ t_1 := \frac{d}{\sqrt{\ell}}\\ \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, \left(\left(0.25 \cdot \frac{h}{\ell}\right) \cdot t\_0\right) \cdot t\_0, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+172}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \sqrt{h}, t\_0 \cdot t\_0, \sqrt{{h}^{-1}} \cdot \ell\right)}{\ell} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.125 \cdot \frac{D\_m}{d}}{d}, \frac{\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot D\_m}{\ell}, 1\right)}{\sqrt{h}} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ M_m d) D_m)) (t_1 (/ d (sqrt l))))
   (if (<= d -5e-310)
     (*
      (* (fma -0.5 (* (* (* 0.25 (/ h l)) t_0) t_0) 1.0) (sqrt (/ d h)))
      (/ (sqrt (- d)) (sqrt (- l))))
     (if (<= d 1.85e+172)
       (*
        (/ (fma (* -0.125 (sqrt h)) (* t_0 t_0) (* (sqrt (pow h -1.0)) l)) l)
        t_1)
       (*
        (/
         (fma (/ (* -0.125 (/ D_m d)) d) (/ (* (* (* M_m M_m) h) D_m) l) 1.0)
         (sqrt h))
        t_1)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (M_m / d) * D_m;
	double t_1 = d / sqrt(l);
	double tmp;
	if (d <= -5e-310) {
		tmp = (fma(-0.5, (((0.25 * (h / l)) * t_0) * t_0), 1.0) * sqrt((d / h))) * (sqrt(-d) / sqrt(-l));
	} else if (d <= 1.85e+172) {
		tmp = (fma((-0.125 * sqrt(h)), (t_0 * t_0), (sqrt(pow(h, -1.0)) * l)) / l) * t_1;
	} else {
		tmp = (fma(((-0.125 * (D_m / d)) / d), ((((M_m * M_m) * h) * D_m) / l), 1.0) / sqrt(h)) * t_1;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(M_m / d) * D_m)
	t_1 = Float64(d / sqrt(l))
	tmp = 0.0
	if (d <= -5e-310)
		tmp = Float64(Float64(fma(-0.5, Float64(Float64(Float64(0.25 * Float64(h / l)) * t_0) * t_0), 1.0) * sqrt(Float64(d / h))) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))));
	elseif (d <= 1.85e+172)
		tmp = Float64(Float64(fma(Float64(-0.125 * sqrt(h)), Float64(t_0 * t_0), Float64(sqrt((h ^ -1.0)) * l)) / l) * t_1);
	else
		tmp = Float64(Float64(fma(Float64(Float64(-0.125 * Float64(D_m / d)) / d), Float64(Float64(Float64(Float64(M_m * M_m) * h) * D_m) / l), 1.0) / sqrt(h)) * t_1);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5e-310], N[(N[(N[(-0.5 * N[(N[(N[(0.25 * N[(h / l), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.85e+172], N[(N[(N[(N[(-0.125 * N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision] + N[(N[Sqrt[N[Power[h, -1.0], $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(N[(N[(-0.125 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.999999999999985e-310

   1. Initial program 72.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 0.0

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 73.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 73.4

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 73.4

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 73.4

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

   8. Applied rewrites 73.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. sqrt-div N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 81.6

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

   10. Applied rewrites 81.6%

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

   if -4.999999999999985e-310 < d < 1.84999999999999986e172

   1. Initial program 70.9%

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

   3. Taylor expanded in 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 34.7%

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

   7. Applied rewrites 34.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 34.2

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

      7. lift-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sqrt-prod N/A

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

      10. lift-/.f64 N/A

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

      11. sqrt-div N/A

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

      12. associate-*l/ N/A

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

      13. rem-square-sqrt N/A

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

   9. Applied rewrites 39.2%

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

   10. Taylor expanded in l around 0

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

       1. lower-/.f64 N/A

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

   12. Applied rewrites 81.3%

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

   if 1.84999999999999986e172 < d

   1. Initial program 61.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 38.4%

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

   7. Applied rewrites 60.9%

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

   8. Taylor expanded in d around 0

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

      1. div-sub N/A

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

      2. *-inverses N/A

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

      3. associate-*r/ N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r/ N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. times-frac N/A

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

   10. Applied rewrites 64.9%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/l* N/A

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

       4. *-commutative N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-/.f64 N/A

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

       8. associate-*l/ N/A

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

   12. Applied rewrites 79.8%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, \left(\left(0.25 \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(\frac{M}{d} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+172}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \sqrt{h}, \left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right), \sqrt{{h}^{-1}} \cdot \ell\right)}{\ell} \cdot \frac{d}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.125 \cdot \frac{D}{d}}{d}, \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot D}{\ell}, 1\right)}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \frac{-0.125}{\ell \cdot d}\\ t_2 := \left(\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot D\_m\right) \cdot \frac{D\_m}{d}\\ \mathbf{if}\;d \leq -8.6 \cdot 10^{+82}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;d \leq -1.75 \cdot 10^{+33}:\\ \;\;\;\;\left(\left(t\_1 \cdot t\_2\right) \cdot t\_0\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-304}:\\ \;\;\;\;\left(t\_0 \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{{\ell}^{-1}} \cdot d\right) \cdot \mathsf{fma}\left(t\_1, t\_2, 1\right)}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (/ -0.125 (* l d)))
        (t_2 (* (* (* (* M_m M_m) h) D_m) (/ D_m d))))
   (if (<= d -8.6e+82)
     (* (- d) (sqrt (pow (* l h) -1.0)))
     (if (<= d -1.75e+33)
       (* (* (* t_1 t_2) t_0) (sqrt (/ d l)))
       (if (<= d 1.15e-304)
         (* (* t_0 (* (sqrt (- d)) (sqrt (/ -1.0 l)))) 1.0)
         (/ (* (* (sqrt (pow l -1.0)) d) (fma t_1 t_2 1.0)) (sqrt h)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / h));
	double t_1 = -0.125 / (l * d);
	double t_2 = (((M_m * M_m) * h) * D_m) * (D_m / d);
	double tmp;
	if (d <= -8.6e+82) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else if (d <= -1.75e+33) {
		tmp = ((t_1 * t_2) * t_0) * sqrt((d / l));
	} else if (d <= 1.15e-304) {
		tmp = (t_0 * (sqrt(-d) * sqrt((-1.0 / l)))) * 1.0;
	} else {
		tmp = ((sqrt(pow(l, -1.0)) * d) * fma(t_1, t_2, 1.0)) / sqrt(h);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / h))
	t_1 = Float64(-0.125 / Float64(l * d))
	t_2 = Float64(Float64(Float64(Float64(M_m * M_m) * h) * D_m) * Float64(D_m / d))
	tmp = 0.0
	if (d <= -8.6e+82)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	elseif (d <= -1.75e+33)
		tmp = Float64(Float64(Float64(t_1 * t_2) * t_0) * sqrt(Float64(d / l)));
	elseif (d <= 1.15e-304)
		tmp = Float64(Float64(t_0 * Float64(sqrt(Float64(-d)) * sqrt(Float64(-1.0 / l)))) * 1.0);
	else
		tmp = Float64(Float64(Float64(sqrt((l ^ -1.0)) * d) * fma(t_1, t_2, 1.0)) / sqrt(h));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-0.125 / N[(l * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -8.6e+82], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.75e+33], N[(N[(N[(t$95$1 * t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.15e-304], N[(N[(t$95$0 * N[(N[Sqrt[(-d)], $MachinePrecision] * N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(N[Sqrt[N[Power[l, -1.0], $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] * N[(t$95$1 * t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -8.60000000000000029e82

   1. Initial program 64.2%

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

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

      7. div-inv N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 69.5

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

   4. Applied rewrites 69.5%

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

   5. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 79.6

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

   7. Applied rewrites 79.6%

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

   if -8.60000000000000029e82 < d < -1.75000000000000005e33

   1. Initial program 99.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 0.0

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 83.8

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

   9. Applied rewrites 83.8%

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

   if -1.75000000000000005e33 < d < 1.15e-304

   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. 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. frac-2neg N/A

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

      7. div-inv N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 81.7

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

   4. Applied rewrites 81.7%

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

   5. Taylor expanded in d around inf

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

   7. Applied rewrites 47.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 47.3

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

   9. Applied rewrites 47.3%

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

   if 1.15e-304 < d

   1. Initial program 68.8%

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

   3. Taylor expanded in 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 35.7%

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

   7. Applied rewrites 39.8%

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

   8. Taylor expanded in d around 0

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

      1. div-sub N/A

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

      2. *-inverses N/A

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

      3. associate-*r/ N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r/ N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. times-frac N/A

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

   10. Applied rewrites 57.8%

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

   11. Taylor expanded in d around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-/.f64 71.4

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

   13. Applied rewrites 71.4%

       \[\leadsto \frac{\color{blue}{\left(\sqrt{\frac{1}{\ell}} \cdot d\right)} \cdot \mathsf{fma}\left(\frac{-0.125}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.6 \cdot 10^{+82}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;d \leq -1.75 \cdot 10^{+33}:\\ \;\;\;\;\left(\left(\frac{-0.125}{\ell \cdot d} \cdot \left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-304}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{{\ell}^{-1}} \cdot d\right) \cdot \mathsf{fma}\left(\frac{-0.125}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.9 \cdot 10^{+129}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{-152}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\frac{h \cdot 0.25}{d} \cdot \frac{M\_m}{\ell}\right) \cdot \left(D\_m \cdot \frac{M\_m}{d}\right), D\_m \cdot -0.5, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{{\ell}^{-1}} \cdot d\right) \cdot \mathsf{fma}\left(\frac{-0.125}{\ell \cdot d}, \left(\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot D\_m\right) \cdot \frac{D\_m}{d}, 1\right)}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -2.9e+129)
   (* (- d) (sqrt (pow (* l h) -1.0)))
   (if (<= l 2.8e-152)
     (*
      (*
       (fma
        (* (* (/ (* h 0.25) d) (/ M_m l)) (* D_m (/ M_m d)))
        (* D_m -0.5)
        1.0)
       (sqrt (/ d h)))
      (sqrt (/ d l)))
     (/
      (*
       (* (sqrt (pow l -1.0)) d)
       (fma (/ -0.125 (* l d)) (* (* (* (* M_m M_m) h) D_m) (/ D_m d)) 1.0))
      (sqrt h)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -2.9e+129) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else if (l <= 2.8e-152) {
		tmp = (fma(((((h * 0.25) / d) * (M_m / l)) * (D_m * (M_m / d))), (D_m * -0.5), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else {
		tmp = ((sqrt(pow(l, -1.0)) * d) * fma((-0.125 / (l * d)), ((((M_m * M_m) * h) * D_m) * (D_m / d)), 1.0)) / sqrt(h);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -2.9e+129)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	elseif (l <= 2.8e-152)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(h * 0.25) / d) * Float64(M_m / l)) * Float64(D_m * Float64(M_m / d))), Float64(D_m * -0.5), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	else
		tmp = Float64(Float64(Float64(sqrt((l ^ -1.0)) * d) * fma(Float64(-0.125 / Float64(l * d)), Float64(Float64(Float64(Float64(M_m * M_m) * h) * D_m) * Float64(D_m / d)), 1.0)) / sqrt(h));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -2.9e+129], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.8e-152], N[(N[(N[(N[(N[(N[(N[(h * 0.25), $MachinePrecision] / d), $MachinePrecision] * N[(M$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[N[Power[l, -1.0], $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] * N[(N[(-0.125 / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.90000000000000003e129

   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. 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. frac-2neg N/A

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

      7. div-inv N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 52.8

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

   4. Applied rewrites 52.8%

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

   5. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 63.0

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

   7. Applied rewrites 63.0%

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

   if -2.90000000000000003e129 < l < 2.79999999999999984e-152

   1. Initial program 85.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 21.6

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

   4. Applied rewrites 21.6%

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

   6. Applied rewrites 84.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 84.4

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 84.4

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 84.4

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

   8. Applied rewrites 84.4%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   10. Applied rewrites 82.1%

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

   if 2.79999999999999984e-152 < l

   1. Initial program 63.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 40.6%

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

   7. Applied rewrites 46.0%

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

   8. Taylor expanded in d around 0

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

      1. div-sub N/A

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

      2. *-inverses N/A

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

      3. associate-*r/ N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r/ N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. times-frac N/A

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

   10. Applied rewrites 55.2%

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

   11. Taylor expanded in d around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-/.f64 70.4

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

   13. Applied rewrites 70.4%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.9 \cdot 10^{+129}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{-152}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\frac{h \cdot 0.25}{d} \cdot \frac{M}{\ell}\right) \cdot \left(D \cdot \frac{M}{d}\right), D \cdot -0.5, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{{\ell}^{-1}} \cdot d\right) \cdot \mathsf{fma}\left(\frac{-0.125}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m}{d} \cdot D\_m\\ \mathbf{if}\;\ell \leq -1.15 \cdot 10^{+145}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{-152}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, \left(\left(0.25 \cdot \frac{h}{\ell}\right) \cdot t\_0\right) \cdot t\_0, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{{\ell}^{-1}} \cdot d\right) \cdot \mathsf{fma}\left(\frac{-0.125}{\ell \cdot d}, \left(\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot D\_m\right) \cdot \frac{D\_m}{d}, 1\right)}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ M_m d) D_m)))
   (if (<= l -1.15e+145)
     (* (- d) (sqrt (pow (* l h) -1.0)))
     (if (<= l 2.8e-152)
       (*
        (* (fma -0.5 (* (* (* 0.25 (/ h l)) t_0) t_0) 1.0) (sqrt (/ d h)))
        (sqrt (/ d l)))
       (/
        (*
         (* (sqrt (pow l -1.0)) d)
         (fma (/ -0.125 (* l d)) (* (* (* (* M_m M_m) h) D_m) (/ D_m d)) 1.0))
        (sqrt h))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (M_m / d) * D_m;
	double tmp;
	if (l <= -1.15e+145) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else if (l <= 2.8e-152) {
		tmp = (fma(-0.5, (((0.25 * (h / l)) * t_0) * t_0), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else {
		tmp = ((sqrt(pow(l, -1.0)) * d) * fma((-0.125 / (l * d)), ((((M_m * M_m) * h) * D_m) * (D_m / d)), 1.0)) / sqrt(h);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(M_m / d) * D_m)
	tmp = 0.0
	if (l <= -1.15e+145)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	elseif (l <= 2.8e-152)
		tmp = Float64(Float64(fma(-0.5, Float64(Float64(Float64(0.25 * Float64(h / l)) * t_0) * t_0), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	else
		tmp = Float64(Float64(Float64(sqrt((l ^ -1.0)) * d) * fma(Float64(-0.125 / Float64(l * d)), Float64(Float64(Float64(Float64(M_m * M_m) * h) * D_m) * Float64(D_m / d)), 1.0)) / sqrt(h));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]}, If[LessEqual[l, -1.15e+145], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.8e-152], N[(N[(N[(-0.5 * N[(N[(N[(0.25 * N[(h / l), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[N[Power[l, -1.0], $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] * N[(N[(-0.125 / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.15e145

   1. Initial program 27.4%

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

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

      7. div-inv N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 46.8

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 61.4

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

   7. Applied rewrites 61.4%

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

   if -1.15e145 < l < 2.79999999999999984e-152

   1. Initial program 86.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 20.9

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

   4. Applied rewrites 20.9%

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

   6. Applied rewrites 84.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 84.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 84.9

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 84.9

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

   8. Applied rewrites 84.9%

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

   if 2.79999999999999984e-152 < l

   1. Initial program 63.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 40.6%

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

   7. Applied rewrites 46.0%

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

   8. Taylor expanded in d around 0

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

      1. div-sub N/A

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

      2. *-inverses N/A

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

      3. associate-*r/ N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r/ N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. times-frac N/A

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

   10. Applied rewrites 55.2%

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

   11. Taylor expanded in d around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-/.f64 70.4

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

   13. Applied rewrites 70.4%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.15 \cdot 10^{+145}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{-152}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, \left(\left(0.25 \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(\frac{M}{d} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{{\ell}^{-1}} \cdot d\right) \cdot \mathsf{fma}\left(\frac{-0.125}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 46.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{if}\;\ell \leq -3.9 \cdot 10^{-222}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_0 \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (pow (* l h) -1.0))))
   (if (<= l -3.9e-222)
     (* (- d) t_0)
     (if (<= l -5e-310) (* t_0 d) (/ d (* (sqrt l) (sqrt h)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt(pow((l * h), -1.0));
	double tmp;
	if (l <= -3.9e-222) {
		tmp = -d * t_0;
	} else if (l <= -5e-310) {
		tmp = t_0 * d;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((l * h) ** (-1.0d0)))
    if (l <= (-3.9d-222)) then
        tmp = -d * t_0
    else if (l <= (-5d-310)) then
        tmp = t_0 * d
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt(Math.pow((l * h), -1.0));
	double tmp;
	if (l <= -3.9e-222) {
		tmp = -d * t_0;
	} else if (l <= -5e-310) {
		tmp = t_0 * d;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt(math.pow((l * h), -1.0))
	tmp = 0
	if l <= -3.9e-222:
		tmp = -d * t_0
	elif l <= -5e-310:
		tmp = t_0 * d
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt((Float64(l * h) ^ -1.0))
	tmp = 0.0
	if (l <= -3.9e-222)
		tmp = Float64(Float64(-d) * t_0);
	elseif (l <= -5e-310)
		tmp = Float64(t_0 * d);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt(((l * h) ^ -1.0));
	tmp = 0.0;
	if (l <= -3.9e-222)
		tmp = -d * t_0;
	elseif (l <= -5e-310)
		tmp = t_0 * d;
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -3.9e-222], N[((-d) * t$95$0), $MachinePrecision], If[LessEqual[l, -5e-310], N[(t$95$0 * 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 < -3.9000000000000001e-222

   1. Initial program 70.5%

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

      1. lift-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. frac-2neg N/A

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

      7. div-inv N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 78.0

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

   4. Applied rewrites 78.0%

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

   5. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 61.4

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

   7. Applied rewrites 61.4%

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

   if -3.9000000000000001e-222 < l < -4.999999999999985e-310

   1. Initial program 80.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 16.6%

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

   6. Taylor expanded in d around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 33.9

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

   8. Applied rewrites 33.9%

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

   if -4.999999999999985e-310 < l

   1. Initial program 69.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 35.4%

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

   6. Taylor expanded in d around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 40.5

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

   8. Applied rewrites 40.5%

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

   10. Applied rewrites 40.5%

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

   12. Applied rewrites 46.3%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.9 \cdot 10^{-222}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot D\_m\\ t_2 := \frac{d}{\sqrt{\ell}}\\ \mathbf{if}\;d \leq -8.6 \cdot 10^{+82}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;d \leq -1.75 \cdot 10^{+33}:\\ \;\;\;\;\left(\left(\frac{-0.125}{\ell \cdot d} \cdot \left(t\_1 \cdot \frac{D\_m}{d}\right)\right) \cdot t\_0\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-304}:\\ \;\;\;\;\left(t\_0 \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{-116}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.125}{d \cdot \ell}, \left(h \cdot \frac{M\_m \cdot M\_m}{d}\right) \cdot \left(D\_m \cdot D\_m\right), 1\right)}{\sqrt{h}} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.125 \cdot \frac{D\_m}{d}}{d}, \frac{t\_1}{\ell}, 1\right)}{\sqrt{h}} \cdot t\_2\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (* (* (* M_m M_m) h) D_m))
        (t_2 (/ d (sqrt l))))
   (if (<= d -8.6e+82)
     (* (- d) (sqrt (pow (* l h) -1.0)))
     (if (<= d -1.75e+33)
       (* (* (* (/ -0.125 (* l d)) (* t_1 (/ D_m d))) t_0) (sqrt (/ d l)))
       (if (<= d 1.15e-304)
         (* (* t_0 (* (sqrt (- d)) (sqrt (/ -1.0 l)))) 1.0)
         (if (<= d 2.5e-116)
           (*
            (/
             (fma
              (/ -0.125 (* d l))
              (* (* h (/ (* M_m M_m) d)) (* D_m D_m))
              1.0)
             (sqrt h))
            t_2)
           (*
            (/ (fma (/ (* -0.125 (/ D_m d)) d) (/ t_1 l) 1.0) (sqrt h))
            t_2)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / h));
	double t_1 = ((M_m * M_m) * h) * D_m;
	double t_2 = d / sqrt(l);
	double tmp;
	if (d <= -8.6e+82) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else if (d <= -1.75e+33) {
		tmp = (((-0.125 / (l * d)) * (t_1 * (D_m / d))) * t_0) * sqrt((d / l));
	} else if (d <= 1.15e-304) {
		tmp = (t_0 * (sqrt(-d) * sqrt((-1.0 / l)))) * 1.0;
	} else if (d <= 2.5e-116) {
		tmp = (fma((-0.125 / (d * l)), ((h * ((M_m * M_m) / d)) * (D_m * D_m)), 1.0) / sqrt(h)) * t_2;
	} else {
		tmp = (fma(((-0.125 * (D_m / d)) / d), (t_1 / l), 1.0) / sqrt(h)) * t_2;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / h))
	t_1 = Float64(Float64(Float64(M_m * M_m) * h) * D_m)
	t_2 = Float64(d / sqrt(l))
	tmp = 0.0
	if (d <= -8.6e+82)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	elseif (d <= -1.75e+33)
		tmp = Float64(Float64(Float64(Float64(-0.125 / Float64(l * d)) * Float64(t_1 * Float64(D_m / d))) * t_0) * sqrt(Float64(d / l)));
	elseif (d <= 1.15e-304)
		tmp = Float64(Float64(t_0 * Float64(sqrt(Float64(-d)) * sqrt(Float64(-1.0 / l)))) * 1.0);
	elseif (d <= 2.5e-116)
		tmp = Float64(Float64(fma(Float64(-0.125 / Float64(d * l)), Float64(Float64(h * Float64(Float64(M_m * M_m) / d)) * Float64(D_m * D_m)), 1.0) / sqrt(h)) * t_2);
	else
		tmp = Float64(Float64(fma(Float64(Float64(-0.125 * Float64(D_m / d)) / d), Float64(t_1 / l), 1.0) / sqrt(h)) * t_2);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision]}, Block[{t$95$2 = N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -8.6e+82], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.75e+33], N[(N[(N[(N[(-0.125 / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.15e-304], N[(N[(t$95$0 * N[(N[Sqrt[(-d)], $MachinePrecision] * N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[d, 2.5e-116], N[(N[(N[(N[(-0.125 / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(N[(N[(N[(-0.125 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(t$95$1 / l), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -8.60000000000000029e82

   1. Initial program 64.2%

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

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

      7. div-inv N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 69.5

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

   4. Applied rewrites 69.5%

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

   5. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 79.6

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

   7. Applied rewrites 79.6%

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

   if -8.60000000000000029e82 < d < -1.75000000000000005e33

   1. Initial program 99.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 0.0

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 83.8

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

   9. Applied rewrites 83.8%

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

   if -1.75000000000000005e33 < d < 1.15e-304

   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. 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. frac-2neg N/A

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

      7. div-inv N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. neg-mul-1 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{1}{\color{blue}{-1 \cdot \ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{\frac{1}{-1}}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{\color{blue}{-1}}{\ell}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-/.f64 81.7

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{-1}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 81.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot \color{blue}{1} \]
   6. Step-by-step derivation

   7. Applied rewrites 47.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot \color{blue}{1} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

      4. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

      5. lift-sqrt.f64 47.3

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

   9. Applied rewrites 47.3%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

   if 1.15e-304 < d < 2.5000000000000001e-116

   1. Initial program 41.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 19.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   7. Applied rewrites 16.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot 1}{\sqrt{h}}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot 1}{\sqrt{h}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell} \cdot d} \cdot 1}}{\sqrt{h}} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot d} \cdot \frac{1}{\sqrt{h}}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{\sqrt{h}} \cdot \sqrt{\frac{d}{\ell} \cdot d}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\sqrt{h}} \cdot \sqrt{\frac{d}{\ell} \cdot d}} \]

      6. lower-/.f64 16.5

         \[\leadsto \color{blue}{\frac{1}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell} \cdot d} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell} \cdot d}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell} \cdot d}} \]

      9. sqrt-prod N/A

         \[\leadsto \frac{1}{\sqrt{h}} \cdot \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{d}\right)} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{1}{\sqrt{h}} \cdot \left(\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{d}\right) \]

      11. sqrt-div N/A

          \[\leadsto \frac{1}{\sqrt{h}} \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \sqrt{d}\right) \]

      12. associate-*l/ N/A

          \[\leadsto \frac{1}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell}}} \]

      13. rem-square-sqrt N/A

          \[\leadsto \frac{1}{\sqrt{h}} \cdot \frac{\color{blue}{d}}{\sqrt{\ell}} \]

   9. Applied rewrites 23.3%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}} \]

   10. Taylor expanded in d around 0

       \[\leadsto \frac{\color{blue}{\frac{{d}^{2} - \frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}}}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}} \]
   11. Step-by-step derivation

       1. div-sub N/A

          \[\leadsto \frac{\color{blue}{\frac{{d}^{2}}{{d}^{2}} - \frac{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}}}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}} \]

       2. *-inverses N/A

          \[\leadsto \frac{\color{blue}{1} - \frac{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}} \]

       3. associate-/l* N/A

          \[\leadsto \frac{1 - \color{blue}{\frac{1}{8} \cdot \frac{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}}}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}} \]

       4. associate-/l/ N/A

          \[\leadsto \frac{1 - \frac{1}{8} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}} \]

       5. cancel-sign-sub-inv N/A

          \[\leadsto \frac{\color{blue}{1 + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}} \]

       6. metadata-eval N/A

          \[\leadsto \frac{1 + \color{blue}{\frac{-1}{8}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}} \]

       7. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} + 1}}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}} \]

       8. associate-*r/ N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}} + 1}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}} \]

       9. *-commutative N/A

          \[\leadsto \frac{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\ell \cdot {d}^{2}}} + 1}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}} \]

       10. unpow2 N/A

           \[\leadsto \frac{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot \color{blue}{\left(d \cdot d\right)}} + 1}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}} \]

       11. associate-*r* N/A

           \[\leadsto \frac{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\left(\ell \cdot d\right) \cdot d}} + 1}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}} \]

       12. *-commutative N/A

           \[\leadsto \frac{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\left(d \cdot \ell\right)} \cdot d} + 1}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}} \]

       13. times-frac N/A

           \[\leadsto \frac{\color{blue}{\frac{\frac{-1}{8}}{d \cdot \ell} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d}} + 1}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}} \]

   12. Applied rewrites 49.5%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-0.125}{d \cdot \ell}, \left(h \cdot \frac{M \cdot M}{d}\right) \cdot \left(D \cdot D\right), 1\right)}}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}} \]

   if 2.5000000000000001e-116 < d

   1. Initial program 76.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 40.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   7. Applied rewrites 46.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot 1}{\sqrt{h}}} \]

   8. Taylor expanded in d around 0

      \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\frac{{d}^{2} - \frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}}}{\sqrt{h}} \]
   9. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\left(\frac{{d}^{2}}{{d}^{2}} - \frac{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}\right)}}{\sqrt{h}} \]

      2. *-inverses N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\color{blue}{1} - \frac{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}\right)}{\sqrt{h}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 - \frac{\color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell}}}{{d}^{2}}\right)}{\sqrt{h}} \]

      4. associate-/l/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right)}{\sqrt{h}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right)}{\sqrt{h}} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}}{\sqrt{h}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 + \color{blue}{\frac{-1}{8}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}{\sqrt{h}} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} + 1\right)}}{\sqrt{h}} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\color{blue}{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}} + 1\right)}{\sqrt{h}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\ell \cdot {d}^{2}}} + 1\right)}{\sqrt{h}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot \color{blue}{\left(d \cdot d\right)}} + 1\right)}{\sqrt{h}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\left(\ell \cdot d\right) \cdot d}} + 1\right)}{\sqrt{h}} \]

      13. times-frac N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\color{blue}{\frac{\frac{-1}{8}}{\ell \cdot d} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d}} + 1\right)}{\sqrt{h}} \]

   10. Applied rewrites 67.2%

       \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\mathsf{fma}\left(\frac{-0.125}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}}{\sqrt{h}} \]
   11. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}}} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell} \cdot d} \cdot \mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}}{\sqrt{h}} \]

       3. associate-/l* N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot d} \cdot \frac{\mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}}} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \cdot \sqrt{\frac{d}{\ell} \cdot d}} \]

       5. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell} \cdot d}} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell} \cdot d}} \]

       7. lift-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}} \cdot d} \]

       8. associate-*l/ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \cdot \sqrt{\color{blue}{\frac{d \cdot d}{\ell}}} \]

   12. Applied rewrites 76.2%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-0.125 \cdot \frac{D}{d}}{d}, \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot D}{\ell}, 1\right)}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.6 \cdot 10^{+82}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;d \leq -1.75 \cdot 10^{+33}:\\ \;\;\;\;\left(\left(\frac{-0.125}{\ell \cdot d} \cdot \left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-304}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{-116}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.125}{d \cdot \ell}, \left(h \cdot \frac{M \cdot M}{d}\right) \cdot \left(D \cdot D\right), 1\right)}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.125 \cdot \frac{D}{d}}{d}, \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot D}{\ell}, 1\right)}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.25 \cdot 10^{+91}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-228}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(-0.125 \cdot \left(\left(\frac{h}{\ell} \cdot D\_m\right) \cdot \frac{M\_m}{d}\right)\right) \cdot \frac{M\_m}{d}, D\_m, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.125 \cdot \frac{D\_m}{d}}{d}, \frac{\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot D\_m}{\ell}, 1\right)}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -1.25e+91)
   (* (- d) (sqrt (pow (* l h) -1.0)))
   (if (<= d 7e-228)
     (*
      (*
       (fma (* (* -0.125 (* (* (/ h l) D_m) (/ M_m d))) (/ M_m d)) D_m 1.0)
       (sqrt (/ d h)))
      (sqrt (/ d l)))
     (*
      (/
       (fma (/ (* -0.125 (/ D_m d)) d) (/ (* (* (* M_m M_m) h) D_m) l) 1.0)
       (sqrt h))
      (/ d (sqrt l))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1.25e+91) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else if (d <= 7e-228) {
		tmp = (fma(((-0.125 * (((h / l) * D_m) * (M_m / d))) * (M_m / d)), D_m, 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else {
		tmp = (fma(((-0.125 * (D_m / d)) / d), ((((M_m * M_m) * h) * D_m) / l), 1.0) / sqrt(h)) * (d / sqrt(l));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -1.25e+91)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	elseif (d <= 7e-228)
		tmp = Float64(Float64(fma(Float64(Float64(-0.125 * Float64(Float64(Float64(h / l) * D_m) * Float64(M_m / d))) * Float64(M_m / d)), D_m, 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	else
		tmp = Float64(Float64(fma(Float64(Float64(-0.125 * Float64(D_m / d)) / d), Float64(Float64(Float64(Float64(M_m * M_m) * h) * D_m) / l), 1.0) / sqrt(h)) * Float64(d / sqrt(l)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -1.25e+91], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7e-228], N[(N[(N[(N[(N[(-0.125 * N[(N[(N[(h / l), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * D$95$m + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.125 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.2500000000000001e91

   1. Initial program 62.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-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. frac-2neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. sqrt-prod N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-neg.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\color{blue}{-d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. neg-mul-1 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{1}{\color{blue}{-1 \cdot \ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{\frac{1}{-1}}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{\color{blue}{-1}}{\ell}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-/.f64 68.2

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{-1}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 68.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 78.8

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if -1.2500000000000001e91 < d < 6.9999999999999995e-228

   1. Initial program 76.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 7.8

         \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 7.8%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot \left(0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{4} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{4} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{4}\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\frac{h}{\ell} \cdot \frac{1}{4}\right) \cdot \color{blue}{{\left(D \cdot \frac{M}{d}\right)}^{2}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\frac{h}{\ell} \cdot \frac{1}{4}\right) \cdot \color{blue}{\left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{4}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(D \cdot \frac{M}{d}\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{4}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(D \cdot \frac{M}{d}\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{4}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)} \cdot \left(D \cdot \frac{M}{d}\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\color{blue}{\left(\frac{1}{4} \cdot \frac{h}{\ell}\right)} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(D \cdot \frac{M}{d}\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. lower-*.f64 73.1

          \[\leadsto \left(\mathsf{fma}\left(-0.5, \left(\color{blue}{\left(0.25 \cdot \frac{h}{\ell}\right)} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(D \cdot \frac{M}{d}\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\left(\frac{1}{4} \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}\right) \cdot \left(D \cdot \frac{M}{d}\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\left(\frac{1}{4} \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}\right) \cdot \left(D \cdot \frac{M}{d}\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. lower-*.f64 73.1

          \[\leadsto \left(\mathsf{fma}\left(-0.5, \left(\left(0.25 \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}\right) \cdot \left(D \cdot \frac{M}{d}\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\left(\frac{1}{4} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      15. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\left(\frac{1}{4} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      16. lower-*.f64 73.1

          \[\leadsto \left(\mathsf{fma}\left(-0.5, \left(\left(0.25 \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   8. Applied rewrites 73.1%

      \[\leadsto \left(\mathsf{fma}\left(-0.5, \color{blue}{\left(\left(0.25 \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(\frac{M}{d} \cdot D\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{2} \cdot \left(\left(\left(\frac{1}{4} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) + 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{2} \cdot \color{blue}{\left(\left(\left(\frac{1}{4} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(\frac{M}{d} \cdot D\right)\right)} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{-1}{2} \cdot \left(\left(\frac{1}{4} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right)\right) \cdot \left(\frac{M}{d} \cdot D\right)} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \left(\left(\frac{1}{4} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right)\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(\frac{-1}{2} \cdot \left(\left(\frac{1}{4} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right)\right) \cdot \frac{M}{d}\right) \cdot D} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot \left(\left(\frac{1}{4} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right)\right) \cdot \frac{M}{d}, D, 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   10. Applied rewrites 67.7%

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\left(-0.125 \cdot \left(\left(\frac{h}{\ell} \cdot D\right) \cdot \frac{M}{d}\right)\right) \cdot \frac{M}{d}, D, 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   if 6.9999999999999995e-228 < d

   1. Initial program 69.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. 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 36.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   7. Applied rewrites 41.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot 1}{\sqrt{h}}} \]

   8. Taylor expanded in d around 0

      \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\frac{{d}^{2} - \frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}}}{\sqrt{h}} \]
   9. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\left(\frac{{d}^{2}}{{d}^{2}} - \frac{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}\right)}}{\sqrt{h}} \]

      2. *-inverses N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\color{blue}{1} - \frac{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}\right)}{\sqrt{h}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 - \frac{\color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell}}}{{d}^{2}}\right)}{\sqrt{h}} \]

      4. associate-/l/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right)}{\sqrt{h}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right)}{\sqrt{h}} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}}{\sqrt{h}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 + \color{blue}{\frac{-1}{8}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}{\sqrt{h}} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} + 1\right)}}{\sqrt{h}} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\color{blue}{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}} + 1\right)}{\sqrt{h}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\ell \cdot {d}^{2}}} + 1\right)}{\sqrt{h}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot \color{blue}{\left(d \cdot d\right)}} + 1\right)}{\sqrt{h}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\left(\ell \cdot d\right) \cdot d}} + 1\right)}{\sqrt{h}} \]

      13. times-frac N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\color{blue}{\frac{\frac{-1}{8}}{\ell \cdot d} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d}} + 1\right)}{\sqrt{h}} \]

   10. Applied rewrites 60.5%

       \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\mathsf{fma}\left(\frac{-0.125}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}}{\sqrt{h}} \]
   11. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}}} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell} \cdot d} \cdot \mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}}{\sqrt{h}} \]

       3. associate-/l* N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot d} \cdot \frac{\mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}}} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \cdot \sqrt{\frac{d}{\ell} \cdot d}} \]

       5. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell} \cdot d}} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell} \cdot d}} \]

       7. lift-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}} \cdot d} \]

       8. associate-*l/ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \cdot \sqrt{\color{blue}{\frac{d \cdot d}{\ell}}} \]

   12. Applied rewrites 74.1%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-0.125 \cdot \frac{D}{d}}{d}, \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot D}{\ell}, 1\right)}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.25 \cdot 10^{+91}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-228}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(-0.125 \cdot \left(\left(\frac{h}{\ell} \cdot D\right) \cdot \frac{M}{d}\right)\right) \cdot \frac{M}{d}, D, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.125 \cdot \frac{D}{d}}{d}, \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot D}{\ell}, 1\right)}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.25 \cdot 10^{+91}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-228}:\\ \;\;\;\;\left(\mathsf{fma}\left(D\_m \cdot \frac{M\_m}{d}, -0.125 \cdot \left(\left(\frac{h}{\ell} \cdot D\_m\right) \cdot \frac{M\_m}{d}\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.125 \cdot \frac{D\_m}{d}}{d}, \frac{\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot D\_m}{\ell}, 1\right)}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -1.25e+91)
   (* (- d) (sqrt (pow (* l h) -1.0)))
   (if (<= d 7e-228)
     (*
      (*
       (fma (* D_m (/ M_m d)) (* -0.125 (* (* (/ h l) D_m) (/ M_m d))) 1.0)
       (sqrt (/ d h)))
      (sqrt (/ d l)))
     (*
      (/
       (fma (/ (* -0.125 (/ D_m d)) d) (/ (* (* (* M_m M_m) h) D_m) l) 1.0)
       (sqrt h))
      (/ d (sqrt l))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1.25e+91) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else if (d <= 7e-228) {
		tmp = (fma((D_m * (M_m / d)), (-0.125 * (((h / l) * D_m) * (M_m / d))), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else {
		tmp = (fma(((-0.125 * (D_m / d)) / d), ((((M_m * M_m) * h) * D_m) / l), 1.0) / sqrt(h)) * (d / sqrt(l));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -1.25e+91)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	elseif (d <= 7e-228)
		tmp = Float64(Float64(fma(Float64(D_m * Float64(M_m / d)), Float64(-0.125 * Float64(Float64(Float64(h / l) * D_m) * Float64(M_m / d))), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	else
		tmp = Float64(Float64(fma(Float64(Float64(-0.125 * Float64(D_m / d)) / d), Float64(Float64(Float64(Float64(M_m * M_m) * h) * D_m) / l), 1.0) / sqrt(h)) * Float64(d / sqrt(l)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -1.25e+91], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7e-228], N[(N[(N[(N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(-0.125 * N[(N[(N[(h / l), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.125 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.2500000000000001e91

   1. Initial program 62.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-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. frac-2neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. sqrt-prod N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-neg.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\color{blue}{-d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. neg-mul-1 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{1}{\color{blue}{-1 \cdot \ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{\frac{1}{-1}}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{\color{blue}{-1}}{\ell}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-/.f64 68.2

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{-1}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 68.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 78.8

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if -1.2500000000000001e91 < d < 6.9999999999999995e-228

   1. Initial program 76.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 7.8

         \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 7.8%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot \left(0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{4} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{4} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{4}\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\frac{h}{\ell} \cdot \frac{1}{4}\right) \cdot \color{blue}{{\left(D \cdot \frac{M}{d}\right)}^{2}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\frac{h}{\ell} \cdot \frac{1}{4}\right) \cdot \color{blue}{\left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{4}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(D \cdot \frac{M}{d}\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{4}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(D \cdot \frac{M}{d}\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{4}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)} \cdot \left(D \cdot \frac{M}{d}\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\color{blue}{\left(\frac{1}{4} \cdot \frac{h}{\ell}\right)} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(D \cdot \frac{M}{d}\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. lower-*.f64 73.1

          \[\leadsto \left(\mathsf{fma}\left(-0.5, \left(\color{blue}{\left(0.25 \cdot \frac{h}{\ell}\right)} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(D \cdot \frac{M}{d}\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\left(\frac{1}{4} \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}\right) \cdot \left(D \cdot \frac{M}{d}\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\left(\frac{1}{4} \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}\right) \cdot \left(D \cdot \frac{M}{d}\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. lower-*.f64 73.1

          \[\leadsto \left(\mathsf{fma}\left(-0.5, \left(\left(0.25 \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}\right) \cdot \left(D \cdot \frac{M}{d}\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\left(\frac{1}{4} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      15. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\left(\frac{1}{4} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      16. lower-*.f64 73.1

          \[\leadsto \left(\mathsf{fma}\left(-0.5, \left(\left(0.25 \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   8. Applied rewrites 73.1%

      \[\leadsto \left(\mathsf{fma}\left(-0.5, \color{blue}{\left(\left(0.25 \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(\frac{M}{d} \cdot D\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{2} \cdot \left(\left(\left(\frac{1}{4} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) + 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{2} \cdot \color{blue}{\left(\left(\left(\frac{1}{4} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(\frac{M}{d} \cdot D\right)\right)} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{-1}{2} \cdot \left(\left(\frac{1}{4} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right)\right) \cdot \left(\frac{M}{d} \cdot D\right)} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{-1}{2} \cdot \left(\left(\frac{1}{4} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right)\right)} + 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{M}{d} \cdot D, \frac{-1}{2} \cdot \left(\left(\frac{1}{4} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right), 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   10. Applied rewrites 69.8%

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(D \cdot \frac{M}{d}, -0.125 \cdot \left(\left(\frac{h}{\ell} \cdot D\right) \cdot \frac{M}{d}\right), 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   if 6.9999999999999995e-228 < d

   1. Initial program 69.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. 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 36.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   7. Applied rewrites 41.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot 1}{\sqrt{h}}} \]

   8. Taylor expanded in d around 0

      \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\frac{{d}^{2} - \frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}}}{\sqrt{h}} \]
   9. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\left(\frac{{d}^{2}}{{d}^{2}} - \frac{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}\right)}}{\sqrt{h}} \]

      2. *-inverses N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\color{blue}{1} - \frac{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}\right)}{\sqrt{h}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 - \frac{\color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell}}}{{d}^{2}}\right)}{\sqrt{h}} \]

      4. associate-/l/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right)}{\sqrt{h}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right)}{\sqrt{h}} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}}{\sqrt{h}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 + \color{blue}{\frac{-1}{8}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}{\sqrt{h}} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} + 1\right)}}{\sqrt{h}} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\color{blue}{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}} + 1\right)}{\sqrt{h}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\ell \cdot {d}^{2}}} + 1\right)}{\sqrt{h}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot \color{blue}{\left(d \cdot d\right)}} + 1\right)}{\sqrt{h}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\left(\ell \cdot d\right) \cdot d}} + 1\right)}{\sqrt{h}} \]

      13. times-frac N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\color{blue}{\frac{\frac{-1}{8}}{\ell \cdot d} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d}} + 1\right)}{\sqrt{h}} \]

   10. Applied rewrites 60.5%

       \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\mathsf{fma}\left(\frac{-0.125}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}}{\sqrt{h}} \]
   11. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}}} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell} \cdot d} \cdot \mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}}{\sqrt{h}} \]

       3. associate-/l* N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot d} \cdot \frac{\mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}}} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \cdot \sqrt{\frac{d}{\ell} \cdot d}} \]

       5. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell} \cdot d}} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell} \cdot d}} \]

       7. lift-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}} \cdot d} \]

       8. associate-*l/ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \cdot \sqrt{\color{blue}{\frac{d \cdot d}{\ell}}} \]

   12. Applied rewrites 74.1%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-0.125 \cdot \frac{D}{d}}{d}, \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot D}{\ell}, 1\right)}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.25 \cdot 10^{+91}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-228}:\\ \;\;\;\;\left(\mathsf{fma}\left(D \cdot \frac{M}{d}, -0.125 \cdot \left(\left(\frac{h}{\ell} \cdot D\right) \cdot \frac{M}{d}\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.125 \cdot \frac{D}{d}}{d}, \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot D}{\ell}, 1\right)}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+145}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-308}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, \frac{\left(h \cdot 0.25\right) \cdot \left(D\_m \cdot M\_m\right)}{d \cdot \ell} \cdot \left(\frac{M\_m}{d} \cdot D\_m\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.125 \cdot \frac{D\_m}{d}}{d}, \frac{\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot D\_m}{\ell}, 1\right)}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -1e+145)
   (* (- d) (sqrt (pow (* l h) -1.0)))
   (if (<= l 1.5e-308)
     (*
      (*
       (fma
        -0.5
        (* (/ (* (* h 0.25) (* D_m M_m)) (* d l)) (* (/ M_m d) D_m))
        1.0)
       (sqrt (/ d h)))
      (sqrt (/ d l)))
     (*
      (/
       (fma (/ (* -0.125 (/ D_m d)) d) (/ (* (* (* M_m M_m) h) D_m) l) 1.0)
       (sqrt h))
      (/ d (sqrt l))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1e+145) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else if (l <= 1.5e-308) {
		tmp = (fma(-0.5, ((((h * 0.25) * (D_m * M_m)) / (d * l)) * ((M_m / d) * D_m)), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else {
		tmp = (fma(((-0.125 * (D_m / d)) / d), ((((M_m * M_m) * h) * D_m) / l), 1.0) / sqrt(h)) * (d / sqrt(l));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -1e+145)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	elseif (l <= 1.5e-308)
		tmp = Float64(Float64(fma(-0.5, Float64(Float64(Float64(Float64(h * 0.25) * Float64(D_m * M_m)) / Float64(d * l)) * Float64(Float64(M_m / d) * D_m)), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	else
		tmp = Float64(Float64(fma(Float64(Float64(-0.125 * Float64(D_m / d)) / d), Float64(Float64(Float64(Float64(M_m * M_m) * h) * D_m) / l), 1.0) / sqrt(h)) * Float64(d / sqrt(l)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -1e+145], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.5e-308], N[(N[(N[(-0.5 * N[(N[(N[(N[(h * 0.25), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.125 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.9999999999999999e144

   1. Initial program 27.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. frac-2neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. sqrt-prod N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-neg.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\color{blue}{-d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. neg-mul-1 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{1}{\color{blue}{-1 \cdot \ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{\frac{1}{-1}}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{\color{blue}{-1}}{\ell}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-/.f64 46.8

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{-1}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 46.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 61.4

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Applied rewrites 61.4%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if -9.9999999999999999e144 < l < 1.4999999999999999e-308

   1. Initial program 86.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 0.0

         \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 0.0%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 86.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot \left(0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{4} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{4} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{4}\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\frac{h}{\ell} \cdot \frac{1}{4}\right) \cdot \color{blue}{{\left(D \cdot \frac{M}{d}\right)}^{2}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\frac{h}{\ell} \cdot \frac{1}{4}\right) \cdot \color{blue}{\left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{4}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(D \cdot \frac{M}{d}\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{4}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(D \cdot \frac{M}{d}\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{4}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)} \cdot \left(D \cdot \frac{M}{d}\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\color{blue}{\left(\frac{1}{4} \cdot \frac{h}{\ell}\right)} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(D \cdot \frac{M}{d}\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. lower-*.f64 86.9

          \[\leadsto \left(\mathsf{fma}\left(-0.5, \left(\color{blue}{\left(0.25 \cdot \frac{h}{\ell}\right)} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(D \cdot \frac{M}{d}\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\left(\frac{1}{4} \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}\right) \cdot \left(D \cdot \frac{M}{d}\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\left(\frac{1}{4} \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}\right) \cdot \left(D \cdot \frac{M}{d}\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. lower-*.f64 86.9

          \[\leadsto \left(\mathsf{fma}\left(-0.5, \left(\left(0.25 \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}\right) \cdot \left(D \cdot \frac{M}{d}\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\left(\frac{1}{4} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      15. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\left(\frac{1}{4} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      16. lower-*.f64 86.9

          \[\leadsto \left(\mathsf{fma}\left(-0.5, \left(\left(0.25 \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   8. Applied rewrites 86.9%

      \[\leadsto \left(\mathsf{fma}\left(-0.5, \color{blue}{\left(\left(0.25 \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(\frac{M}{d} \cdot D\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(\left(\frac{1}{4} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right)} \cdot \left(\frac{M}{d} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\color{blue}{\left(\frac{1}{4} \cdot \frac{h}{\ell}\right)} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(\frac{M}{d} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\left(\frac{1}{4} \cdot \color{blue}{\frac{h}{\ell}}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(\frac{M}{d} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. associate-*r/ N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\color{blue}{\frac{\frac{1}{4} \cdot h}{\ell}} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(\frac{M}{d} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\frac{\frac{1}{4} \cdot h}{\ell} \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}\right) \cdot \left(\frac{M}{d} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\frac{\frac{1}{4} \cdot h}{\ell} \cdot \left(\color{blue}{\frac{M}{d}} \cdot D\right)\right) \cdot \left(\frac{M}{d} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. associate-*l/ N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\frac{\frac{1}{4} \cdot h}{\ell} \cdot \color{blue}{\frac{M \cdot D}{d}}\right) \cdot \left(\frac{M}{d} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. frac-times N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{\left(\frac{1}{4} \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot d}} \cdot \left(\frac{M}{d} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \frac{\left(\frac{1}{4} \cdot h\right) \cdot \left(M \cdot D\right)}{\color{blue}{\ell \cdot d}} \cdot \left(\frac{M}{d} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{\left(\frac{1}{4} \cdot h\right) \cdot \left(M \cdot D\right)}{\ell \cdot d}} \cdot \left(\frac{M}{d} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\left(\frac{1}{4} \cdot h\right) \cdot \left(M \cdot D\right)}}{\ell \cdot d} \cdot \left(\frac{M}{d} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\left(h \cdot \frac{1}{4}\right)} \cdot \left(M \cdot D\right)}{\ell \cdot d} \cdot \left(\frac{M}{d} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\left(h \cdot \frac{1}{4}\right)} \cdot \left(M \cdot D\right)}{\ell \cdot d} \cdot \left(\frac{M}{d} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      14. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \frac{\left(h \cdot \frac{1}{4}\right) \cdot \color{blue}{\left(D \cdot M\right)}}{\ell \cdot d} \cdot \left(\frac{M}{d} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      15. lower-*.f64 83.9

          \[\leadsto \left(\mathsf{fma}\left(-0.5, \frac{\left(h \cdot 0.25\right) \cdot \color{blue}{\left(D \cdot M\right)}}{\ell \cdot d} \cdot \left(\frac{M}{d} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      16. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \frac{\left(h \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)}{\color{blue}{\ell \cdot d}} \cdot \left(\frac{M}{d} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      17. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \frac{\left(h \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)}{\color{blue}{d \cdot \ell}} \cdot \left(\frac{M}{d} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      18. lower-*.f64 83.9

          \[\leadsto \left(\mathsf{fma}\left(-0.5, \frac{\left(h \cdot 0.25\right) \cdot \left(D \cdot M\right)}{\color{blue}{d \cdot \ell}} \cdot \left(\frac{M}{d} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   10. Applied rewrites 83.9%

       \[\leadsto \left(\mathsf{fma}\left(-0.5, \color{blue}{\frac{\left(h \cdot 0.25\right) \cdot \left(D \cdot M\right)}{d \cdot \ell}} \cdot \left(\frac{M}{d} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   if 1.4999999999999999e-308 < l

   1. Initial program 69.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 35.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   7. Applied rewrites 39.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot 1}{\sqrt{h}}} \]

   8. Taylor expanded in d around 0

      \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\frac{{d}^{2} - \frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}}}{\sqrt{h}} \]
   9. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\left(\frac{{d}^{2}}{{d}^{2}} - \frac{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}\right)}}{\sqrt{h}} \]

      2. *-inverses N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\color{blue}{1} - \frac{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}\right)}{\sqrt{h}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 - \frac{\color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell}}}{{d}^{2}}\right)}{\sqrt{h}} \]

      4. associate-/l/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right)}{\sqrt{h}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right)}{\sqrt{h}} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}}{\sqrt{h}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 + \color{blue}{\frac{-1}{8}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}{\sqrt{h}} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} + 1\right)}}{\sqrt{h}} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\color{blue}{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}} + 1\right)}{\sqrt{h}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\ell \cdot {d}^{2}}} + 1\right)}{\sqrt{h}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot \color{blue}{\left(d \cdot d\right)}} + 1\right)}{\sqrt{h}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\left(\ell \cdot d\right) \cdot d}} + 1\right)}{\sqrt{h}} \]

      13. times-frac N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\color{blue}{\frac{\frac{-1}{8}}{\ell \cdot d} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d}} + 1\right)}{\sqrt{h}} \]

   10. Applied rewrites 57.3%

       \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\mathsf{fma}\left(\frac{-0.125}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}}{\sqrt{h}} \]
   11. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}}} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell} \cdot d} \cdot \mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}}{\sqrt{h}} \]

       3. associate-/l* N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot d} \cdot \frac{\mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}}} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \cdot \sqrt{\frac{d}{\ell} \cdot d}} \]

       5. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell} \cdot d}} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell} \cdot d}} \]

       7. lift-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}} \cdot d} \]

       8. associate-*l/ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \cdot \sqrt{\color{blue}{\frac{d \cdot d}{\ell}}} \]

   12. Applied rewrites 72.3%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-0.125 \cdot \frac{D}{d}}{d}, \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot D}{\ell}, 1\right)}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+145}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-308}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, \frac{\left(h \cdot 0.25\right) \cdot \left(D \cdot M\right)}{d \cdot \ell} \cdot \left(\frac{M}{d} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.125 \cdot \frac{D}{d}}{d}, \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot D}{\ell}, 1\right)}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \frac{-0.125}{\ell \cdot d}\\ t_2 := \left(\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot D\_m\right) \cdot \frac{D\_m}{d}\\ \mathbf{if}\;d \leq -8.6 \cdot 10^{+82}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;d \leq -1.75 \cdot 10^{+33}:\\ \;\;\;\;\left(\left(t\_1 \cdot t\_2\right) \cdot t\_0\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-304}:\\ \;\;\;\;\left(t\_0 \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}} \cdot \mathsf{fma}\left(t\_1, t\_2, 1\right)}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (/ -0.125 (* l d)))
        (t_2 (* (* (* (* M_m M_m) h) D_m) (/ D_m d))))
   (if (<= d -8.6e+82)
     (* (- d) (sqrt (pow (* l h) -1.0)))
     (if (<= d -1.75e+33)
       (* (* (* t_1 t_2) t_0) (sqrt (/ d l)))
       (if (<= d 1.15e-304)
         (* (* t_0 (* (sqrt (- d)) (sqrt (/ -1.0 l)))) 1.0)
         (/ (* (/ d (sqrt l)) (fma t_1 t_2 1.0)) (sqrt h)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / h));
	double t_1 = -0.125 / (l * d);
	double t_2 = (((M_m * M_m) * h) * D_m) * (D_m / d);
	double tmp;
	if (d <= -8.6e+82) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else if (d <= -1.75e+33) {
		tmp = ((t_1 * t_2) * t_0) * sqrt((d / l));
	} else if (d <= 1.15e-304) {
		tmp = (t_0 * (sqrt(-d) * sqrt((-1.0 / l)))) * 1.0;
	} else {
		tmp = ((d / sqrt(l)) * fma(t_1, t_2, 1.0)) / sqrt(h);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / h))
	t_1 = Float64(-0.125 / Float64(l * d))
	t_2 = Float64(Float64(Float64(Float64(M_m * M_m) * h) * D_m) * Float64(D_m / d))
	tmp = 0.0
	if (d <= -8.6e+82)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	elseif (d <= -1.75e+33)
		tmp = Float64(Float64(Float64(t_1 * t_2) * t_0) * sqrt(Float64(d / l)));
	elseif (d <= 1.15e-304)
		tmp = Float64(Float64(t_0 * Float64(sqrt(Float64(-d)) * sqrt(Float64(-1.0 / l)))) * 1.0);
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) * fma(t_1, t_2, 1.0)) / sqrt(h));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-0.125 / N[(l * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -8.6e+82], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.75e+33], N[(N[(N[(t$95$1 * t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.15e-304], N[(N[(t$95$0 * N[(N[Sqrt[(-d)], $MachinePrecision] * N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -8.60000000000000029e82

   1. Initial program 64.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. frac-2neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. sqrt-prod N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-neg.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\color{blue}{-d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. neg-mul-1 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{1}{\color{blue}{-1 \cdot \ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{\frac{1}{-1}}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{\color{blue}{-1}}{\ell}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-/.f64 69.5

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{-1}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 69.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 79.6

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Applied rewrites 79.6%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if -8.60000000000000029e82 < d < -1.75000000000000005e33

   1. Initial program 99.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 0.0

         \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 0.0%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot \left(0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]

   7. Taylor expanded in d around 0

      \[\leadsto \left(\color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\ell \cdot {d}^{2}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. unpow2 N/A

         \[\leadsto \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot \color{blue}{\left(d \cdot d\right)}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. associate-*r* N/A

         \[\leadsto \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\left(\ell \cdot d\right) \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. times-frac N/A

         \[\leadsto \left(\color{blue}{\left(\frac{\frac{-1}{8}}{\ell \cdot d} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d}\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\frac{\frac{-1}{8}}{\ell \cdot d} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d}\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\frac{-1}{8}}{\ell \cdot d}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d}\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{\frac{-1}{8}}{\color{blue}{\ell \cdot d}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d}\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. *-commutative N/A

         \[\leadsto \left(\left(\frac{\frac{-1}{8}}{\ell \cdot d} \cdot \frac{\color{blue}{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}}{d}\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. unpow2 N/A

          \[\leadsto \left(\left(\frac{\frac{-1}{8}}{\ell \cdot d} \cdot \frac{\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}}{d}\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. associate-*r* N/A

          \[\leadsto \left(\left(\frac{\frac{-1}{8}}{\ell \cdot d} \cdot \frac{\color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D}}{d}\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. associate-/l* N/A

          \[\leadsto \left(\left(\frac{\frac{-1}{8}}{\ell \cdot d} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot \frac{D}{d}\right)}\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{\frac{-1}{8}}{\ell \cdot d} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot \frac{D}{d}\right)}\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{\frac{-1}{8}}{\ell \cdot d} \cdot \left(\color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)} \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{\frac{-1}{8}}{\ell \cdot d} \cdot \left(\left(\color{blue}{\left({M}^{2} \cdot h\right)} \cdot D\right) \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      16. unpow2 N/A

          \[\leadsto \left(\left(\frac{\frac{-1}{8}}{\ell \cdot d} \cdot \left(\left(\left(\color{blue}{\left(M \cdot M\right)} \cdot h\right) \cdot D\right) \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{\frac{-1}{8}}{\ell \cdot d} \cdot \left(\left(\left(\color{blue}{\left(M \cdot M\right)} \cdot h\right) \cdot D\right) \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      18. lower-/.f64 83.8

          \[\leadsto \left(\left(\frac{-0.125}{\ell \cdot d} \cdot \left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \color{blue}{\frac{D}{d}}\right)\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   9. Applied rewrites 83.8%

      \[\leadsto \left(\color{blue}{\left(\frac{-0.125}{\ell \cdot d} \cdot \left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}\right)\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   if -1.75000000000000005e33 < d < 1.15e-304

   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. 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. frac-2neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. sqrt-prod N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-neg.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\color{blue}{-d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. neg-mul-1 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{1}{\color{blue}{-1 \cdot \ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{\frac{1}{-1}}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{\color{blue}{-1}}{\ell}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-/.f64 81.7

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{-1}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 81.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot \color{blue}{1} \]
   6. Step-by-step derivation

   7. Applied rewrites 47.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot \color{blue}{1} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

      4. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

      5. lift-sqrt.f64 47.3

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

   9. Applied rewrites 47.3%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

   if 1.15e-304 < d

   1. Initial program 68.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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 35.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   7. Applied rewrites 39.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot 1}{\sqrt{h}}} \]

   8. Taylor expanded in d around 0

      \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\frac{{d}^{2} - \frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}}}{\sqrt{h}} \]
   9. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\left(\frac{{d}^{2}}{{d}^{2}} - \frac{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}\right)}}{\sqrt{h}} \]

      2. *-inverses N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\color{blue}{1} - \frac{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}\right)}{\sqrt{h}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 - \frac{\color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell}}}{{d}^{2}}\right)}{\sqrt{h}} \]

      4. associate-/l/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right)}{\sqrt{h}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right)}{\sqrt{h}} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}}{\sqrt{h}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 + \color{blue}{\frac{-1}{8}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}{\sqrt{h}} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} + 1\right)}}{\sqrt{h}} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\color{blue}{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}} + 1\right)}{\sqrt{h}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\ell \cdot {d}^{2}}} + 1\right)}{\sqrt{h}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot \color{blue}{\left(d \cdot d\right)}} + 1\right)}{\sqrt{h}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\left(\ell \cdot d\right) \cdot d}} + 1\right)}{\sqrt{h}} \]

      13. times-frac N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\color{blue}{\frac{\frac{-1}{8}}{\ell \cdot d} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d}} + 1\right)}{\sqrt{h}} \]

   10. Applied rewrites 57.8%

       \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\mathsf{fma}\left(\frac{-0.125}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}}{\sqrt{h}} \]
   11. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell} \cdot d}} \cdot \mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{\ell} \cdot d}} \cdot \mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{\ell}} \cdot d} \cdot \mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \]

       4. associate-*l/ N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\frac{d \cdot d}{\ell}}} \cdot \mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \]

       5. sqrt-div N/A

          \[\leadsto \frac{\color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \]

       6. sqrt-prod N/A

          \[\leadsto \frac{\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \]

       7. rem-square-sqrt N/A

          \[\leadsto \frac{\frac{\color{blue}{d}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \]

       8. lift-sqrt.f64 N/A

          \[\leadsto \frac{\frac{d}{\color{blue}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \]

       9. lift-/.f64 71.4

          \[\leadsto \frac{\color{blue}{\frac{d}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(\frac{-0.125}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \]

   12. Applied rewrites 71.4%

       \[\leadsto \frac{\color{blue}{\frac{d}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(\frac{-0.125}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.6 \cdot 10^{+82}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;d \leq -1.75 \cdot 10^{+33}:\\ \;\;\;\;\left(\left(\frac{-0.125}{\ell \cdot d} \cdot \left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-304}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{-0.125}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -2.3 \cdot 10^{+82}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;d \leq -2.5 \cdot 10^{+33}:\\ \;\;\;\;\left(\left(\frac{-0.125}{d \cdot \ell} \cdot \left(\left(h \cdot \frac{M\_m \cdot M\_m}{d}\right) \cdot \left(D\_m \cdot D\_m\right)\right)\right) \cdot t\_0\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-304}:\\ \;\;\;\;\left(t\_0 \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{-0.125}{\ell \cdot d}, \left(\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot D\_m\right) \cdot \frac{D\_m}{d}, 1\right)}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h))))
   (if (<= d -2.3e+82)
     (* (- d) (sqrt (pow (* l h) -1.0)))
     (if (<= d -2.5e+33)
       (*
        (* (* (/ -0.125 (* d l)) (* (* h (/ (* M_m M_m) d)) (* D_m D_m))) t_0)
        (sqrt (/ d l)))
       (if (<= d 1.15e-304)
         (* (* t_0 (* (sqrt (- d)) (sqrt (/ -1.0 l)))) 1.0)
         (/
          (*
           (/ d (sqrt l))
           (fma
            (/ -0.125 (* l d))
            (* (* (* (* M_m M_m) h) D_m) (/ D_m d))
            1.0))
          (sqrt h)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / h));
	double tmp;
	if (d <= -2.3e+82) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else if (d <= -2.5e+33) {
		tmp = (((-0.125 / (d * l)) * ((h * ((M_m * M_m) / d)) * (D_m * D_m))) * t_0) * sqrt((d / l));
	} else if (d <= 1.15e-304) {
		tmp = (t_0 * (sqrt(-d) * sqrt((-1.0 / l)))) * 1.0;
	} else {
		tmp = ((d / sqrt(l)) * fma((-0.125 / (l * d)), ((((M_m * M_m) * h) * D_m) * (D_m / d)), 1.0)) / sqrt(h);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / h))
	tmp = 0.0
	if (d <= -2.3e+82)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	elseif (d <= -2.5e+33)
		tmp = Float64(Float64(Float64(Float64(-0.125 / Float64(d * l)) * Float64(Float64(h * Float64(Float64(M_m * M_m) / d)) * Float64(D_m * D_m))) * t_0) * sqrt(Float64(d / l)));
	elseif (d <= 1.15e-304)
		tmp = Float64(Float64(t_0 * Float64(sqrt(Float64(-d)) * sqrt(Float64(-1.0 / l)))) * 1.0);
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) * fma(Float64(-0.125 / Float64(l * d)), Float64(Float64(Float64(Float64(M_m * M_m) * h) * D_m) * Float64(D_m / d)), 1.0)) / sqrt(h));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -2.3e+82], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2.5e+33], N[(N[(N[(N[(-0.125 / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.15e-304], N[(N[(t$95$0 * N[(N[Sqrt[(-d)], $MachinePrecision] * N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.125 / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.29999999999999988e82

   1. Initial program 64.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. frac-2neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. sqrt-prod N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-neg.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\color{blue}{-d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. neg-mul-1 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{1}{\color{blue}{-1 \cdot \ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{\frac{1}{-1}}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{\color{blue}{-1}}{\ell}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-/.f64 69.5

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{-1}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 69.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 79.6

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Applied rewrites 79.6%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if -2.29999999999999988e82 < d < -2.49999999999999986e33

   1. Initial program 99.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 0.0

         \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 0.0%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot \left(0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{4} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{4} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{4}\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\frac{h}{\ell} \cdot \frac{1}{4}\right) \cdot \color{blue}{{\left(D \cdot \frac{M}{d}\right)}^{2}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\frac{h}{\ell} \cdot \frac{1}{4}\right) \cdot \color{blue}{\left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{4}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(D \cdot \frac{M}{d}\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{4}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(D \cdot \frac{M}{d}\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{4}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)} \cdot \left(D \cdot \frac{M}{d}\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\color{blue}{\left(\frac{1}{4} \cdot \frac{h}{\ell}\right)} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(D \cdot \frac{M}{d}\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. lower-*.f64 99.7

          \[\leadsto \left(\mathsf{fma}\left(-0.5, \left(\color{blue}{\left(0.25 \cdot \frac{h}{\ell}\right)} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \left(D \cdot \frac{M}{d}\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\left(\frac{1}{4} \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}\right) \cdot \left(D \cdot \frac{M}{d}\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\left(\frac{1}{4} \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}\right) \cdot \left(D \cdot \frac{M}{d}\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. lower-*.f64 99.7

          \[\leadsto \left(\mathsf{fma}\left(-0.5, \left(\left(0.25 \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}\right) \cdot \left(D \cdot \frac{M}{d}\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\left(\frac{1}{4} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      15. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \left(\left(\frac{1}{4} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      16. lower-*.f64 99.7

          \[\leadsto \left(\mathsf{fma}\left(-0.5, \left(\left(0.25 \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   8. Applied rewrites 99.7%

      \[\leadsto \left(\mathsf{fma}\left(-0.5, \color{blue}{\left(\left(0.25 \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(\frac{M}{d} \cdot D\right)}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   9. Taylor expanded in d around 0

      \[\leadsto \left(\color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \left(\color{blue}{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

       2. *-commutative N/A

          \[\leadsto \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\ell \cdot {d}^{2}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

       3. unpow2 N/A

          \[\leadsto \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot \color{blue}{\left(d \cdot d\right)}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

       4. associate-*r* N/A

          \[\leadsto \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\left(\ell \cdot d\right) \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

       5. *-commutative N/A

          \[\leadsto \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\left(d \cdot \ell\right)} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

       6. times-frac N/A

          \[\leadsto \left(\color{blue}{\left(\frac{\frac{-1}{8}}{d \cdot \ell} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d}\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

       7. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\frac{\frac{-1}{8}}{d \cdot \ell} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d}\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

       8. lower-/.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\frac{\frac{-1}{8}}{d \cdot \ell}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d}\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{\frac{-1}{8}}{\color{blue}{d \cdot \ell}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d}\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

       10. associate-/l* N/A

           \[\leadsto \left(\left(\frac{\frac{-1}{8}}{d \cdot \ell} \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{d}\right)}\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

       11. *-commutative N/A

           \[\leadsto \left(\left(\frac{\frac{-1}{8}}{d \cdot \ell} \cdot \color{blue}{\left(\frac{{M}^{2} \cdot h}{d} \cdot {D}^{2}\right)}\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

       12. lower-*.f64 N/A

           \[\leadsto \left(\left(\frac{\frac{-1}{8}}{d \cdot \ell} \cdot \color{blue}{\left(\frac{{M}^{2} \cdot h}{d} \cdot {D}^{2}\right)}\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

       13. *-commutative N/A

           \[\leadsto \left(\left(\frac{\frac{-1}{8}}{d \cdot \ell} \cdot \left(\frac{\color{blue}{h \cdot {M}^{2}}}{d} \cdot {D}^{2}\right)\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

       14. associate-/l* N/A

           \[\leadsto \left(\left(\frac{\frac{-1}{8}}{d \cdot \ell} \cdot \left(\color{blue}{\left(h \cdot \frac{{M}^{2}}{d}\right)} \cdot {D}^{2}\right)\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

       15. lower-*.f64 N/A

           \[\leadsto \left(\left(\frac{\frac{-1}{8}}{d \cdot \ell} \cdot \left(\color{blue}{\left(h \cdot \frac{{M}^{2}}{d}\right)} \cdot {D}^{2}\right)\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

       16. lower-/.f64 N/A

           \[\leadsto \left(\left(\frac{\frac{-1}{8}}{d \cdot \ell} \cdot \left(\left(h \cdot \color{blue}{\frac{{M}^{2}}{d}}\right) \cdot {D}^{2}\right)\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

       17. unpow2 N/A

           \[\leadsto \left(\left(\frac{\frac{-1}{8}}{d \cdot \ell} \cdot \left(\left(h \cdot \frac{\color{blue}{M \cdot M}}{d}\right) \cdot {D}^{2}\right)\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

       18. lower-*.f64 N/A

           \[\leadsto \left(\left(\frac{\frac{-1}{8}}{d \cdot \ell} \cdot \left(\left(h \cdot \frac{\color{blue}{M \cdot M}}{d}\right) \cdot {D}^{2}\right)\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

       19. unpow2 N/A

           \[\leadsto \left(\left(\frac{\frac{-1}{8}}{d \cdot \ell} \cdot \left(\left(h \cdot \frac{M \cdot M}{d}\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

       20. lower-*.f64 75.6

           \[\leadsto \left(\left(\frac{-0.125}{d \cdot \ell} \cdot \left(\left(h \cdot \frac{M \cdot M}{d}\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   11. Applied rewrites 75.6%

       \[\leadsto \left(\color{blue}{\left(\frac{-0.125}{d \cdot \ell} \cdot \left(\left(h \cdot \frac{M \cdot M}{d}\right) \cdot \left(D \cdot D\right)\right)\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   if -2.49999999999999986e33 < d < 1.15e-304

   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. 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. frac-2neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. sqrt-prod N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-neg.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\color{blue}{-d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. neg-mul-1 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{1}{\color{blue}{-1 \cdot \ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{\frac{1}{-1}}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{\color{blue}{-1}}{\ell}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-/.f64 81.7

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{-1}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 81.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot \color{blue}{1} \]
   6. Step-by-step derivation

   7. Applied rewrites 47.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot \color{blue}{1} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

      4. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

      5. lift-sqrt.f64 47.3

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

   9. Applied rewrites 47.3%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

   if 1.15e-304 < d

   1. Initial program 68.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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 35.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   7. Applied rewrites 39.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot 1}{\sqrt{h}}} \]

   8. Taylor expanded in d around 0

      \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\frac{{d}^{2} - \frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}}}{\sqrt{h}} \]
   9. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\left(\frac{{d}^{2}}{{d}^{2}} - \frac{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}\right)}}{\sqrt{h}} \]

      2. *-inverses N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\color{blue}{1} - \frac{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}\right)}{\sqrt{h}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 - \frac{\color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell}}}{{d}^{2}}\right)}{\sqrt{h}} \]

      4. associate-/l/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right)}{\sqrt{h}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right)}{\sqrt{h}} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}}{\sqrt{h}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 + \color{blue}{\frac{-1}{8}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}{\sqrt{h}} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} + 1\right)}}{\sqrt{h}} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\color{blue}{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}} + 1\right)}{\sqrt{h}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\ell \cdot {d}^{2}}} + 1\right)}{\sqrt{h}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot \color{blue}{\left(d \cdot d\right)}} + 1\right)}{\sqrt{h}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\left(\ell \cdot d\right) \cdot d}} + 1\right)}{\sqrt{h}} \]

      13. times-frac N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\color{blue}{\frac{\frac{-1}{8}}{\ell \cdot d} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d}} + 1\right)}{\sqrt{h}} \]

   10. Applied rewrites 57.8%

       \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\mathsf{fma}\left(\frac{-0.125}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}}{\sqrt{h}} \]
   11. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell} \cdot d}} \cdot \mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{\ell} \cdot d}} \cdot \mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{\ell}} \cdot d} \cdot \mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \]

       4. associate-*l/ N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\frac{d \cdot d}{\ell}}} \cdot \mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \]

       5. sqrt-div N/A

          \[\leadsto \frac{\color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \]

       6. sqrt-prod N/A

          \[\leadsto \frac{\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \]

       7. rem-square-sqrt N/A

          \[\leadsto \frac{\frac{\color{blue}{d}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \]

       8. lift-sqrt.f64 N/A

          \[\leadsto \frac{\frac{d}{\color{blue}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \]

       9. lift-/.f64 71.4

          \[\leadsto \frac{\color{blue}{\frac{d}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(\frac{-0.125}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \]

   12. Applied rewrites 71.4%

       \[\leadsto \frac{\color{blue}{\frac{d}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(\frac{-0.125}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.3 \cdot 10^{+82}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;d \leq -2.5 \cdot 10^{+33}:\\ \;\;\;\;\left(\left(\frac{-0.125}{d \cdot \ell} \cdot \left(\left(h \cdot \frac{M \cdot M}{d}\right) \cdot \left(D \cdot D\right)\right)\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-304}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{-0.125}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 59.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{+31}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-304}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{-0.125}{\ell \cdot d}, \left(\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot D\_m\right) \cdot \frac{D\_m}{d}, 1\right)}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -5e+31)
   (* (- d) (sqrt (pow (* l h) -1.0)))
   (if (<= d 1.15e-304)
     (* (* (sqrt (/ d h)) (* (sqrt (- d)) (sqrt (/ -1.0 l)))) 1.0)
     (/
      (*
       (/ d (sqrt l))
       (fma (/ -0.125 (* l d)) (* (* (* (* M_m M_m) h) D_m) (/ D_m d)) 1.0))
      (sqrt h)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -5e+31) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else if (d <= 1.15e-304) {
		tmp = (sqrt((d / h)) * (sqrt(-d) * sqrt((-1.0 / l)))) * 1.0;
	} else {
		tmp = ((d / sqrt(l)) * fma((-0.125 / (l * d)), ((((M_m * M_m) * h) * D_m) * (D_m / d)), 1.0)) / sqrt(h);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -5e+31)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	elseif (d <= 1.15e-304)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(-d)) * sqrt(Float64(-1.0 / l)))) * 1.0);
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) * fma(Float64(-0.125 / Float64(l * d)), Float64(Float64(Float64(Float64(M_m * M_m) * h) * D_m) * Float64(D_m / d)), 1.0)) / sqrt(h));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -5e+31], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.15e-304], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] * N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.125 / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.00000000000000027e31

   1. Initial program 70.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. frac-2neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. sqrt-prod N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-neg.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\color{blue}{-d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. neg-mul-1 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{1}{\color{blue}{-1 \cdot \ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{\frac{1}{-1}}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{\color{blue}{-1}}{\ell}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-/.f64 74.4

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{-1}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 74.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 65.6

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Applied rewrites 65.6%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if -5.00000000000000027e31 < d < 1.15e-304

   1. Initial program 75.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-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. frac-2neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. sqrt-prod N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-neg.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\color{blue}{-d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. neg-mul-1 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{1}{\color{blue}{-1 \cdot \ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{\frac{1}{-1}}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{\color{blue}{-1}}{\ell}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-/.f64 82.7

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{-1}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 82.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot \color{blue}{1} \]
   6. Step-by-step derivation

   7. Applied rewrites 47.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot \color{blue}{1} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

      4. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

      5. lift-sqrt.f64 47.3

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

   9. Applied rewrites 47.3%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

   if 1.15e-304 < d

   1. Initial program 68.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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 35.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   7. Applied rewrites 39.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot 1}{\sqrt{h}}} \]

   8. Taylor expanded in d around 0

      \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\frac{{d}^{2} - \frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}}}{\sqrt{h}} \]
   9. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\left(\frac{{d}^{2}}{{d}^{2}} - \frac{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}\right)}}{\sqrt{h}} \]

      2. *-inverses N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\color{blue}{1} - \frac{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}\right)}{\sqrt{h}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 - \frac{\color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell}}}{{d}^{2}}\right)}{\sqrt{h}} \]

      4. associate-/l/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right)}{\sqrt{h}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right)}{\sqrt{h}} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}}{\sqrt{h}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 + \color{blue}{\frac{-1}{8}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}{\sqrt{h}} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} + 1\right)}}{\sqrt{h}} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\color{blue}{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}} + 1\right)}{\sqrt{h}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\ell \cdot {d}^{2}}} + 1\right)}{\sqrt{h}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot \color{blue}{\left(d \cdot d\right)}} + 1\right)}{\sqrt{h}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\left(\ell \cdot d\right) \cdot d}} + 1\right)}{\sqrt{h}} \]

      13. times-frac N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\color{blue}{\frac{\frac{-1}{8}}{\ell \cdot d} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d}} + 1\right)}{\sqrt{h}} \]

   10. Applied rewrites 57.8%

       \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\mathsf{fma}\left(\frac{-0.125}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}}{\sqrt{h}} \]
   11. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell} \cdot d}} \cdot \mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{\ell} \cdot d}} \cdot \mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{\ell}} \cdot d} \cdot \mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \]

       4. associate-*l/ N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\frac{d \cdot d}{\ell}}} \cdot \mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \]

       5. sqrt-div N/A

          \[\leadsto \frac{\color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \]

       6. sqrt-prod N/A

          \[\leadsto \frac{\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \]

       7. rem-square-sqrt N/A

          \[\leadsto \frac{\frac{\color{blue}{d}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \]

       8. lift-sqrt.f64 N/A

          \[\leadsto \frac{\frac{d}{\color{blue}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(\frac{\frac{-1}{8}}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \]

       9. lift-/.f64 71.4

          \[\leadsto \frac{\color{blue}{\frac{d}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(\frac{-0.125}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \]

   12. Applied rewrites 71.4%

       \[\leadsto \frac{\color{blue}{\frac{d}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(\frac{-0.125}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{+31}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-304}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{-0.125}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 57.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{+31}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-304}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1\\ \mathbf{elif}\;d \leq 1.34 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.125}{d \cdot \ell}, \left(h \cdot \frac{M\_m \cdot M\_m}{d}\right) \cdot \left(D\_m \cdot D\_m\right), 1\right)}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.125 \cdot \frac{D\_m}{d}}{d}, \frac{\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot D\_m}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -5e+31)
   (* (- d) (sqrt (pow (* l h) -1.0)))
   (if (<= d 1.15e-304)
     (* (* (sqrt (/ d h)) (* (sqrt (- d)) (sqrt (/ -1.0 l)))) 1.0)
     (if (<= d 1.34e-20)
       (*
        (/
         (fma (/ -0.125 (* d l)) (* (* h (/ (* M_m M_m) d)) (* D_m D_m)) 1.0)
         (sqrt h))
        (/ d (sqrt l)))
       (*
        (fma (/ (* -0.125 (/ D_m d)) d) (/ (* (* (* M_m M_m) h) D_m) l) 1.0)
        (/ d (sqrt (* l h))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -5e+31) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else if (d <= 1.15e-304) {
		tmp = (sqrt((d / h)) * (sqrt(-d) * sqrt((-1.0 / l)))) * 1.0;
	} else if (d <= 1.34e-20) {
		tmp = (fma((-0.125 / (d * l)), ((h * ((M_m * M_m) / d)) * (D_m * D_m)), 1.0) / sqrt(h)) * (d / sqrt(l));
	} else {
		tmp = fma(((-0.125 * (D_m / d)) / d), ((((M_m * M_m) * h) * D_m) / l), 1.0) * (d / sqrt((l * h)));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -5e+31)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	elseif (d <= 1.15e-304)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(-d)) * sqrt(Float64(-1.0 / l)))) * 1.0);
	elseif (d <= 1.34e-20)
		tmp = Float64(Float64(fma(Float64(-0.125 / Float64(d * l)), Float64(Float64(h * Float64(Float64(M_m * M_m) / d)) * Float64(D_m * D_m)), 1.0) / sqrt(h)) * Float64(d / sqrt(l)));
	else
		tmp = Float64(fma(Float64(Float64(-0.125 * Float64(D_m / d)) / d), Float64(Float64(Float64(Float64(M_m * M_m) * h) * D_m) / l), 1.0) * Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -5e+31], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.15e-304], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] * N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[d, 1.34e-20], N[(N[(N[(N[(-0.125 / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.125 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -5.00000000000000027e31

   1. Initial program 70.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. frac-2neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. sqrt-prod N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-neg.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\color{blue}{-d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. neg-mul-1 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{1}{\color{blue}{-1 \cdot \ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{\frac{1}{-1}}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{\color{blue}{-1}}{\ell}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-/.f64 74.4

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{-1}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 74.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 65.6

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Applied rewrites 65.6%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if -5.00000000000000027e31 < d < 1.15e-304

   1. Initial program 75.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-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. frac-2neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. sqrt-prod N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-neg.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\color{blue}{-d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. neg-mul-1 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{1}{\color{blue}{-1 \cdot \ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{\frac{1}{-1}}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{\color{blue}{-1}}{\ell}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-/.f64 82.7

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{-1}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 82.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot \color{blue}{1} \]
   6. Step-by-step derivation

   7. Applied rewrites 47.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot \color{blue}{1} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

      4. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

      5. lift-sqrt.f64 47.3

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

   9. Applied rewrites 47.3%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

   if 1.15e-304 < d < 1.3399999999999999e-20

   1. Initial program 59.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 26.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   7. Applied rewrites 25.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot 1}{\sqrt{h}}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot 1}{\sqrt{h}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell} \cdot d} \cdot 1}}{\sqrt{h}} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot d} \cdot \frac{1}{\sqrt{h}}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{\sqrt{h}} \cdot \sqrt{\frac{d}{\ell} \cdot d}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\sqrt{h}} \cdot \sqrt{\frac{d}{\ell} \cdot d}} \]

      6. lower-/.f64 25.7

         \[\leadsto \color{blue}{\frac{1}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell} \cdot d} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell} \cdot d}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell} \cdot d}} \]

      9. sqrt-prod N/A

         \[\leadsto \frac{1}{\sqrt{h}} \cdot \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{d}\right)} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{1}{\sqrt{h}} \cdot \left(\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{d}\right) \]

      11. sqrt-div N/A

          \[\leadsto \frac{1}{\sqrt{h}} \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \sqrt{d}\right) \]

      12. associate-*l/ N/A

          \[\leadsto \frac{1}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell}}} \]

      13. rem-square-sqrt N/A

          \[\leadsto \frac{1}{\sqrt{h}} \cdot \frac{\color{blue}{d}}{\sqrt{\ell}} \]

   9. Applied rewrites 29.6%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}} \]

   10. Taylor expanded in d around 0

       \[\leadsto \frac{\color{blue}{\frac{{d}^{2} - \frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}}}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}} \]
   11. Step-by-step derivation

       1. div-sub N/A

          \[\leadsto \frac{\color{blue}{\frac{{d}^{2}}{{d}^{2}} - \frac{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}}}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}} \]

       2. *-inverses N/A

          \[\leadsto \frac{\color{blue}{1} - \frac{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}} \]

       3. associate-/l* N/A

          \[\leadsto \frac{1 - \color{blue}{\frac{1}{8} \cdot \frac{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}}}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}} \]

       4. associate-/l/ N/A

          \[\leadsto \frac{1 - \frac{1}{8} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}} \]

       5. cancel-sign-sub-inv N/A

          \[\leadsto \frac{\color{blue}{1 + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}} \]

       6. metadata-eval N/A

          \[\leadsto \frac{1 + \color{blue}{\frac{-1}{8}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}} \]

       7. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} + 1}}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}} \]

       8. associate-*r/ N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}} + 1}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}} \]

       9. *-commutative N/A

          \[\leadsto \frac{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\ell \cdot {d}^{2}}} + 1}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}} \]

       10. unpow2 N/A

           \[\leadsto \frac{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot \color{blue}{\left(d \cdot d\right)}} + 1}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}} \]

       11. associate-*r* N/A

           \[\leadsto \frac{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\left(\ell \cdot d\right) \cdot d}} + 1}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}} \]

       12. *-commutative N/A

           \[\leadsto \frac{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\left(d \cdot \ell\right)} \cdot d} + 1}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}} \]

       13. times-frac N/A

           \[\leadsto \frac{\color{blue}{\frac{\frac{-1}{8}}{d \cdot \ell} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d}} + 1}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}} \]

   12. Applied rewrites 57.1%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-0.125}{d \cdot \ell}, \left(h \cdot \frac{M \cdot M}{d}\right) \cdot \left(D \cdot D\right), 1\right)}}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}} \]

   if 1.3399999999999999e-20 < d

   1. Initial program 77.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 43.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   7. Applied rewrites 52.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot 1}{\sqrt{h}}} \]

   8. Taylor expanded in d around 0

      \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\frac{{d}^{2} - \frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}}}{\sqrt{h}} \]
   9. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\left(\frac{{d}^{2}}{{d}^{2}} - \frac{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}\right)}}{\sqrt{h}} \]

      2. *-inverses N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\color{blue}{1} - \frac{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}\right)}{\sqrt{h}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 - \frac{\color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell}}}{{d}^{2}}\right)}{\sqrt{h}} \]

      4. associate-/l/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right)}{\sqrt{h}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right)}{\sqrt{h}} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}}{\sqrt{h}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 + \color{blue}{\frac{-1}{8}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}{\sqrt{h}} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} + 1\right)}}{\sqrt{h}} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\color{blue}{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}} + 1\right)}{\sqrt{h}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\ell \cdot {d}^{2}}} + 1\right)}{\sqrt{h}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot \color{blue}{\left(d \cdot d\right)}} + 1\right)}{\sqrt{h}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\left(\ell \cdot d\right) \cdot d}} + 1\right)}{\sqrt{h}} \]

      13. times-frac N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\color{blue}{\frac{\frac{-1}{8}}{\ell \cdot d} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d}} + 1\right)}{\sqrt{h}} \]

   10. Applied rewrites 67.0%

       \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\mathsf{fma}\left(\frac{-0.125}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}}{\sqrt{h}} \]

   12. Applied rewrites 69.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.125 \cdot \frac{D}{d}}{d}, \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot D}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{+31}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-304}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1\\ \mathbf{elif}\;d \leq 1.34 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.125}{d \cdot \ell}, \left(h \cdot \frac{M \cdot M}{d}\right) \cdot \left(D \cdot D\right), 1\right)}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.125 \cdot \frac{D}{d}}{d}, \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot D}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 54.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{+31}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.125 \cdot \frac{D\_m}{d}}{d}, \frac{\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot D\_m}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -5e+31)
   (* (- d) (sqrt (pow (* l h) -1.0)))
   (if (<= d -5e-310)
     (* (* (sqrt (/ d h)) (* (sqrt (- d)) (sqrt (/ -1.0 l)))) 1.0)
     (*
      (fma (/ (* -0.125 (/ D_m d)) d) (/ (* (* (* M_m M_m) h) D_m) l) 1.0)
      (/ d (sqrt (* l h)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -5e+31) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else if (d <= -5e-310) {
		tmp = (sqrt((d / h)) * (sqrt(-d) * sqrt((-1.0 / l)))) * 1.0;
	} else {
		tmp = fma(((-0.125 * (D_m / d)) / d), ((((M_m * M_m) * h) * D_m) / l), 1.0) * (d / sqrt((l * h)));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -5e+31)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	elseif (d <= -5e-310)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(-d)) * sqrt(Float64(-1.0 / l)))) * 1.0);
	else
		tmp = Float64(fma(Float64(Float64(-0.125 * Float64(D_m / d)) / d), Float64(Float64(Float64(Float64(M_m * M_m) * h) * D_m) / l), 1.0) * Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -5e+31], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] * N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(N[(-0.125 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.00000000000000027e31

   1. Initial program 70.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. frac-2neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. sqrt-prod N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-neg.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\color{blue}{-d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. neg-mul-1 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{1}{\color{blue}{-1 \cdot \ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{\frac{1}{-1}}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{\color{blue}{-1}}{\ell}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-/.f64 74.4

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{-1}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 74.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 65.6

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Applied rewrites 65.6%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if -5.00000000000000027e31 < d < -4.999999999999985e-310

   1. Initial program 74.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. frac-2neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. sqrt-prod N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-neg.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\color{blue}{-d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. neg-mul-1 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{1}{\color{blue}{-1 \cdot \ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{\frac{1}{-1}}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{\color{blue}{-1}}{\ell}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-/.f64 84.1

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{-1}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 84.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot \color{blue}{1} \]
   6. Step-by-step derivation

   7. Applied rewrites 48.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot \color{blue}{1} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

      4. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

      5. lift-sqrt.f64 48.0

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

   9. Applied rewrites 48.0%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

   if -4.999999999999985e-310 < d

   1. Initial program 69.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 35.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   7. Applied rewrites 39.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot 1}{\sqrt{h}}} \]

   8. Taylor expanded in d around 0

      \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\frac{{d}^{2} - \frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}}}{\sqrt{h}} \]
   9. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\left(\frac{{d}^{2}}{{d}^{2}} - \frac{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}\right)}}{\sqrt{h}} \]

      2. *-inverses N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\color{blue}{1} - \frac{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}{{d}^{2}}\right)}{\sqrt{h}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 - \frac{\color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell}}}{{d}^{2}}\right)}{\sqrt{h}} \]

      4. associate-/l/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right)}{\sqrt{h}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 - \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right)}{\sqrt{h}} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}}{\sqrt{h}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(1 + \color{blue}{\frac{-1}{8}} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}{\sqrt{h}} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} + 1\right)}}{\sqrt{h}} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\color{blue}{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}} + 1\right)}{\sqrt{h}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\ell \cdot {d}^{2}}} + 1\right)}{\sqrt{h}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot \color{blue}{\left(d \cdot d\right)}} + 1\right)}{\sqrt{h}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\left(\ell \cdot d\right) \cdot d}} + 1\right)}{\sqrt{h}} \]

      13. times-frac N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \left(\color{blue}{\frac{\frac{-1}{8}}{\ell \cdot d} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d}} + 1\right)}{\sqrt{h}} \]

   10. Applied rewrites 57.3%

       \[\leadsto \frac{\sqrt{\frac{d}{\ell} \cdot d} \cdot \color{blue}{\mathsf{fma}\left(\frac{-0.125}{\ell \cdot d}, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot \frac{D}{d}, 1\right)}}{\sqrt{h}} \]

   12. Applied rewrites 63.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.125 \cdot \frac{D}{d}}{d}, \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot D}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{+31}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.125 \cdot \frac{D}{d}}{d}, \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot D}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 47.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{+31}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -5e+31)
   (* (- d) (sqrt (pow (* l h) -1.0)))
   (if (<= d -5e-310)
     (* (* (sqrt (/ d h)) (* (sqrt (- d)) (sqrt (/ -1.0 l)))) 1.0)
     (/ d (* (sqrt l) (sqrt h))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -5e+31) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else if (d <= -5e-310) {
		tmp = (sqrt((d / h)) * (sqrt(-d) * sqrt((-1.0 / l)))) * 1.0;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-5d+31)) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else if (d <= (-5d-310)) then
        tmp = (sqrt((d / h)) * (sqrt(-d) * sqrt(((-1.0d0) / l)))) * 1.0d0
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -5e+31) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else if (d <= -5e-310) {
		tmp = (Math.sqrt((d / h)) * (Math.sqrt(-d) * Math.sqrt((-1.0 / l)))) * 1.0;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -5e+31:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	elif d <= -5e-310:
		tmp = (math.sqrt((d / h)) * (math.sqrt(-d) * math.sqrt((-1.0 / l)))) * 1.0
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -5e+31)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	elseif (d <= -5e-310)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(-d)) * sqrt(Float64(-1.0 / l)))) * 1.0);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -5e+31)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	elseif (d <= -5e-310)
		tmp = (sqrt((d / h)) * (sqrt(-d) * sqrt((-1.0 / l)))) * 1.0;
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -5e+31], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] * N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.00000000000000027e31

   1. Initial program 70.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. frac-2neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. sqrt-prod N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-neg.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\color{blue}{-d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. neg-mul-1 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{1}{\color{blue}{-1 \cdot \ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{\frac{1}{-1}}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{\color{blue}{-1}}{\ell}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-/.f64 74.4

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{-1}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 74.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 65.6

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Applied rewrites 65.6%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if -5.00000000000000027e31 < d < -4.999999999999985e-310

   1. Initial program 74.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. frac-2neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. sqrt-prod N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-neg.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\color{blue}{-d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. neg-mul-1 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{1}{\color{blue}{-1 \cdot \ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{\frac{1}{-1}}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{\color{blue}{-1}}{\ell}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-/.f64 84.1

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{-1}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 84.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot \color{blue}{1} \]
   6. Step-by-step derivation

   7. Applied rewrites 48.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot \color{blue}{1} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

      4. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

      5. lift-sqrt.f64 48.0

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

   9. Applied rewrites 48.0%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1 \]

   if -4.999999999999985e-310 < d

   1. Initial program 69.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 35.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot d}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. unpow2 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      5. rem-square-sqrt N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{-1} \cdot d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. remove-double-neg N/A

         \[\leadsto \color{blue}{d} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      13. lower-*.f64 40.5

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 40.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 40.5%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 46.3%

       \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{+31}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 42.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{if}\;\ell \leq -3.9 \cdot 10^{-222}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot d\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (pow (* l h) -1.0))))
   (if (<= l -3.9e-222) (* (- d) t_0) (* t_0 d))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt(pow((l * h), -1.0));
	double tmp;
	if (l <= -3.9e-222) {
		tmp = -d * t_0;
	} else {
		tmp = t_0 * d;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((l * h) ** (-1.0d0)))
    if (l <= (-3.9d-222)) then
        tmp = -d * t_0
    else
        tmp = t_0 * d
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt(Math.pow((l * h), -1.0));
	double tmp;
	if (l <= -3.9e-222) {
		tmp = -d * t_0;
	} else {
		tmp = t_0 * d;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt(math.pow((l * h), -1.0))
	tmp = 0
	if l <= -3.9e-222:
		tmp = -d * t_0
	else:
		tmp = t_0 * d
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt((Float64(l * h) ^ -1.0))
	tmp = 0.0
	if (l <= -3.9e-222)
		tmp = Float64(Float64(-d) * t_0);
	else
		tmp = Float64(t_0 * d);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt(((l * h) ^ -1.0));
	tmp = 0.0;
	if (l <= -3.9e-222)
		tmp = -d * t_0;
	else
		tmp = t_0 * d;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -3.9e-222], N[((-d) * t$95$0), $MachinePrecision], N[(t$95$0 * d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -3.9000000000000001e-222

   1. Initial program 70.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-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. frac-2neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. sqrt-prod N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-neg.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\color{blue}{-d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. neg-mul-1 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{1}{\color{blue}{-1 \cdot \ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. associate-/r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{\frac{1}{-1}}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{\color{blue}{-1}}{\ell}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-/.f64 78.0

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{-1}{\ell}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 78.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 61.4

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Applied rewrites 61.4%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if -3.9000000000000001e-222 < l

   1. Initial program 70.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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 32.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot d}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. unpow2 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      5. rem-square-sqrt N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{-1} \cdot d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. remove-double-neg N/A

         \[\leadsto \color{blue}{d} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      13. lower-*.f64 39.4

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 39.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.9 \cdot 10^{-222}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 26.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (* (sqrt (pow (* l h) -1.0)) d))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return sqrt(pow((l * h), -1.0)) * d;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = sqrt(((l * h) ** (-1.0d0))) * d
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return Math.sqrt(Math.pow((l * h), -1.0)) * d;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return math.sqrt(math.pow((l * h), -1.0)) * d

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(sqrt((Float64(l * h) ^ -1.0)) * d)
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = sqrt(((l * h) ^ -1.0)) * d;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]

Derivation

1. Initial program 70.7%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
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 39.6%

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

6. Taylor expanded in d around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
7. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   2. mul-1-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   3. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot d}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   4. unpow2 N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   5. rem-square-sqrt N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{-1} \cdot d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   6. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   7. remove-double-neg N/A

      \[\leadsto \color{blue}{d} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   8. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   9. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

   11. lower-/.f64 N/A

       \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   12. *-commutative N/A

       \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   13. lower-*.f64 26.2

       \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

8. Applied rewrites 26.2%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

9. Final simplification 26.2%

   \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]
10. Add Preprocessing

Alternative 21: 26.5% accurate, 15.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (/ d (sqrt (* l h))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return d / sqrt((l * h));
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = d / sqrt((l * h))
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return d / Math.sqrt((l * h));
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return d / math.sqrt((l * h))

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = d / sqrt((l * h));
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.7%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
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 39.6%

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

6. Taylor expanded in d around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
7. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   2. mul-1-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   3. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot d}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   4. unpow2 N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   5. rem-square-sqrt N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{-1} \cdot d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   6. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   7. remove-double-neg N/A

      \[\leadsto \color{blue}{d} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   8. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   9. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

   11. lower-/.f64 N/A

       \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   12. *-commutative N/A

       \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   13. lower-*.f64 26.2

       \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

8. Applied rewrites 26.2%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
9. Step-by-step derivation

10. Applied rewrites 25.5%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024304 
(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)))))