Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.9% → 81.2%

Time: 17.9s

Alternatives: 12

Speedup: 3.8×

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: 66.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 81.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m}{d} \cdot D\_m\\ t_1 := \mathsf{fma}\left(\left(\frac{t\_0}{\ell} \cdot -0.25\right) \cdot h, t\_0 \cdot 0.5, 1\right)\\ \mathbf{if}\;d \leq -4 \cdot 10^{+150}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right) \cdot t\_1\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left(\left(\sqrt{\frac{-1}{h}} \cdot \sqrt{-d}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ M_m d) D_m))
        (t_1 (fma (* (* (/ t_0 l) -0.25) h) (* t_0 0.5) 1.0)))
   (if (<= d -4e+150)
     (* (* (sqrt (/ 1.0 (* l h))) (- d)) t_1)
     (if (<= d -5e-310)
       (*
        (- 1.0 (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
        (* (* (sqrt (/ -1.0 h)) (sqrt (- d))) (pow (/ d l) (/ 1.0 2.0))))
       (* t_1 (/ (/ 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 = (M_m / d) * D_m;
	double t_1 = fma((((t_0 / l) * -0.25) * h), (t_0 * 0.5), 1.0);
	double tmp;
	if (d <= -4e+150) {
		tmp = (sqrt((1.0 / (l * h))) * -d) * t_1;
	} else if (d <= -5e-310) {
		tmp = (1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * ((sqrt((-1.0 / h)) * sqrt(-d)) * pow((d / l), (1.0 / 2.0)));
	} else {
		tmp = t_1 * ((d / sqrt(l)) / sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(M_m / d) * D_m)
	t_1 = fma(Float64(Float64(Float64(t_0 / l) * -0.25) * h), Float64(t_0 * 0.5), 1.0)
	tmp = 0.0
	if (d <= -4e+150)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d)) * t_1);
	elseif (d <= -5e-310)
		tmp = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64(Float64(sqrt(Float64(-1.0 / h)) * sqrt(Float64(-d))) * (Float64(d / l) ^ Float64(1.0 / 2.0))));
	else
		tmp = Float64(t_1 * Float64(Float64(d / sqrt(l)) / sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(t$95$0 / l), $MachinePrecision] * -0.25), $MachinePrecision] * h), $MachinePrecision] * N[(t$95$0 * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[d, -4e+150], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[N[(-1.0 / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.99999999999999992e150

   1. Initial program 67.7%

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

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 67.6

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

   4. Applied rewrites 67.6%

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

   6. Applied rewrites 73.9%

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

   8. Applied rewrites 74.1%

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

   9. Taylor expanded in h around -inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. rem-square-sqrt N/A

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

       4. lower-*.f64 N/A

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

       5. mul-1-neg N/A

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

       6. lower-neg.f64 N/A

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

       7. lower-sqrt.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 93.7

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

   11. Applied rewrites 93.7%

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

   if -3.99999999999999992e150 < d < -4.999999999999985e-310

   1. Initial program 75.6%

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

      1. lift-pow.f64 N/A

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

      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. frac-2neg N/A

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

      7. div-inv N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 83.7

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

   4. Applied rewrites 83.7%

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

   if -4.999999999999985e-310 < d

   1. Initial program 68.5%

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

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 67.6

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

   4. Applied rewrites 67.6%

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

   6. Applied rewrites 70.1%

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

   8. Applied rewrites 70.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-pow.f64 N/A

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

      15. pow1/2 N/A

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

      16. lift-/.f64 N/A

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

      17. sqrt-div N/A

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

      18. lift-sqrt.f64 N/A

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

      19. frac-times N/A

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

      20. lift-sqrt.f64 N/A

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

      21. lift-sqrt.f64 N/A

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

      22. rem-square-sqrt N/A

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

   10. Applied rewrites 86.6%

       \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \cdot \mathsf{fma}\left(\left(-0.25 \cdot \frac{D \cdot \frac{M}{d}}{\ell}\right) \cdot h, 0.5 \cdot \left(D \cdot \frac{M}{d}\right), 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{+150}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right) \cdot \mathsf{fma}\left(\left(\frac{\frac{M}{d} \cdot D}{\ell} \cdot -0.25\right) \cdot h, \left(\frac{M}{d} \cdot D\right) \cdot 0.5, 1\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left(\left(\sqrt{\frac{-1}{h}} \cdot \sqrt{-d}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{\frac{M}{d} \cdot D}{\ell} \cdot -0.25\right) \cdot h, \left(\frac{M}{d} \cdot D\right) \cdot 0.5, 1\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.0% accurate, 1.2× speedup

Math

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

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ M_m d) D_m)))
   (if (<= h -1e-310)
     (*
      (-
       1.0
       (*
        (/ (* (/ M_m d) (* (* D_m 0.5) 0.5)) (pow h -1.0))
        (/ (* D_m (* M_m (/ 0.5 d))) l)))
      (* (pow (/ d l) (/ 1.0 2.0)) (/ (sqrt (- d)) (sqrt (- h)))))
     (*
      (fma (* (* (/ t_0 l) -0.25) h) (* t_0 0.5) 1.0)
      (/ (/ d (sqrt l)) (sqrt h))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (M_m / d) * D_m;
	double tmp;
	if (h <= -1e-310) {
		tmp = (1.0 - ((((M_m / d) * ((D_m * 0.5) * 0.5)) / pow(h, -1.0)) * ((D_m * (M_m * (0.5 / d))) / l))) * (pow((d / l), (1.0 / 2.0)) * (sqrt(-d) / sqrt(-h)));
	} else {
		tmp = fma((((t_0 / l) * -0.25) * h), (t_0 * 0.5), 1.0) * ((d / sqrt(l)) / sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(M_m / d) * D_m)
	tmp = 0.0
	if (h <= -1e-310)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(M_m / d) * Float64(Float64(D_m * 0.5) * 0.5)) / (h ^ -1.0)) * Float64(Float64(D_m * Float64(M_m * Float64(0.5 / d))) / l))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-h)))));
	else
		tmp = Float64(fma(Float64(Float64(Float64(t_0 / l) * -0.25) * h), Float64(t_0 * 0.5), 1.0) * Float64(Float64(d / sqrt(l)) / sqrt(h)));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if h < -9.999999999999969e-311

   1. Initial program 73.3%

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

      1. lift-*.f64 N/A

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

      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 73.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 73.0

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-neg.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 83.4

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

   6. Applied rewrites 83.4%

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

   if -9.999999999999969e-311 < h

   1. Initial program 68.5%

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

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 67.6

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

   4. Applied rewrites 67.6%

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

   6. Applied rewrites 70.1%

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

   8. Applied rewrites 70.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-pow.f64 N/A

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

      15. pow1/2 N/A

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

      16. lift-/.f64 N/A

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

      17. sqrt-div N/A

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

      18. lift-sqrt.f64 N/A

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

      19. frac-times N/A

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

      20. lift-sqrt.f64 N/A

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

      21. lift-sqrt.f64 N/A

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

      22. rem-square-sqrt N/A

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

   10. Applied rewrites 86.6%

       \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \cdot \mathsf{fma}\left(\left(-0.25 \cdot \frac{D \cdot \frac{M}{d}}{\ell}\right) \cdot h, 0.5 \cdot \left(D \cdot \frac{M}{d}\right), 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.2%

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

Alternative 3: 81.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m}{d} \cdot D\_m\\ t_1 := \mathsf{fma}\left(\left(\frac{t\_0}{\ell} \cdot -0.25\right) \cdot h, t\_0 \cdot 0.5, 1\right)\\ \mathbf{if}\;d \leq -4 \cdot 10^{+150}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right) \cdot t\_1\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ M_m d) D_m))
        (t_1 (fma (* (* (/ t_0 l) -0.25) h) (* t_0 0.5) 1.0)))
   (if (<= d -4e+150)
     (* (* (sqrt (/ 1.0 (* l h))) (- d)) t_1)
     (if (<= d -5e-310)
       (*
        (- 1.0 (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
        (* (pow (/ d l) (/ 1.0 2.0)) (/ (sqrt (- d)) (sqrt (- h)))))
       (* t_1 (/ (/ 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 = (M_m / d) * D_m;
	double t_1 = fma((((t_0 / l) * -0.25) * h), (t_0 * 0.5), 1.0);
	double tmp;
	if (d <= -4e+150) {
		tmp = (sqrt((1.0 / (l * h))) * -d) * t_1;
	} else if (d <= -5e-310) {
		tmp = (1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * (sqrt(-d) / sqrt(-h)));
	} else {
		tmp = t_1 * ((d / sqrt(l)) / sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(M_m / d) * D_m)
	t_1 = fma(Float64(Float64(Float64(t_0 / l) * -0.25) * h), Float64(t_0 * 0.5), 1.0)
	tmp = 0.0
	if (d <= -4e+150)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d)) * t_1);
	elseif (d <= -5e-310)
		tmp = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-h)))));
	else
		tmp = Float64(t_1 * Float64(Float64(d / sqrt(l)) / sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(t$95$0 / l), $MachinePrecision] * -0.25), $MachinePrecision] * h), $MachinePrecision] * N[(t$95$0 * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[d, -4e+150], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.99999999999999992e150

   1. Initial program 67.7%

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

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 67.6

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

   4. Applied rewrites 67.6%

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

   6. Applied rewrites 73.9%

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

   8. Applied rewrites 74.1%

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

   9. Taylor expanded in h around -inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. rem-square-sqrt N/A

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

       4. lower-*.f64 N/A

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

       5. mul-1-neg N/A

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

       6. lower-neg.f64 N/A

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

       7. lower-sqrt.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 93.7

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

   11. Applied rewrites 93.7%

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

   if -3.99999999999999992e150 < d < -4.999999999999985e-310

   1. Initial program 75.6%

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

      1. lift-pow.f64 N/A

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

      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. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 83.7

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

   4. Applied rewrites 83.7%

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

   if -4.999999999999985e-310 < d

   1. Initial program 68.5%

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

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 67.6

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

   4. Applied rewrites 67.6%

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

   6. Applied rewrites 70.1%

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

   8. Applied rewrites 70.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-pow.f64 N/A

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

      15. pow1/2 N/A

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

      16. lift-/.f64 N/A

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

      17. sqrt-div N/A

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

      18. lift-sqrt.f64 N/A

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

      19. frac-times N/A

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

      20. lift-sqrt.f64 N/A

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

      21. lift-sqrt.f64 N/A

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

      22. rem-square-sqrt N/A

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

   10. Applied rewrites 86.6%

       \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \cdot \mathsf{fma}\left(\left(-0.25 \cdot \frac{D \cdot \frac{M}{d}}{\ell}\right) \cdot h, 0.5 \cdot \left(D \cdot \frac{M}{d}\right), 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{+150}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right) \cdot \mathsf{fma}\left(\left(\frac{\frac{M}{d} \cdot D}{\ell} \cdot -0.25\right) \cdot h, \left(\frac{M}{d} \cdot D\right) \cdot 0.5, 1\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{\frac{M}{d} \cdot D}{\ell} \cdot -0.25\right) \cdot h, \left(\frac{M}{d} \cdot D\right) \cdot 0.5, 1\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m}{d} \cdot D\_m\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{+199}:\\ \;\;\;\;\left(1 - \left({\left(D\_m \cdot \frac{0.5}{d}\right)}^{2} \cdot M\_m\right) \cdot \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot M\_m\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \left(\left(\frac{D\_m \cdot M\_m}{d} \cdot 0.5\right) \cdot h\right) \cdot \frac{\left(0.25 \cdot D\_m\right) \cdot \frac{M\_m}{d}}{\ell}\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{t\_0}{\ell} \cdot -0.25\right) \cdot h, t\_0 \cdot 0.5, 1\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ M_m d) D_m)))
   (if (<= l -4e+199)
     (*
      (- 1.0 (* (* (pow (* D_m (/ 0.5 d)) 2.0) M_m) (* (* (/ h l) 0.5) M_m)))
      (* (pow (/ d l) (/ 1.0 2.0)) (/ (sqrt (- d)) (sqrt (- h)))))
     (if (<= l -2e-310)
       (*
        (-
         1.0
         (* (* (* (/ (* D_m M_m) d) 0.5) h) (/ (* (* 0.25 D_m) (/ M_m d)) l)))
        (* (sqrt (/ 1.0 (* l h))) (- d)))
       (*
        (fma (* (* (/ t_0 l) -0.25) h) (* t_0 0.5) 1.0)
        (/ (/ d (sqrt l)) (sqrt h)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (M_m / d) * D_m;
	double tmp;
	if (l <= -4e+199) {
		tmp = (1.0 - ((pow((D_m * (0.5 / d)), 2.0) * M_m) * (((h / l) * 0.5) * M_m))) * (pow((d / l), (1.0 / 2.0)) * (sqrt(-d) / sqrt(-h)));
	} else if (l <= -2e-310) {
		tmp = (1.0 - (((((D_m * M_m) / d) * 0.5) * h) * (((0.25 * D_m) * (M_m / d)) / l))) * (sqrt((1.0 / (l * h))) * -d);
	} else {
		tmp = fma((((t_0 / l) * -0.25) * h), (t_0 * 0.5), 1.0) * ((d / sqrt(l)) / sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(M_m / d) * D_m)
	tmp = 0.0
	if (l <= -4e+199)
		tmp = Float64(Float64(1.0 - Float64(Float64((Float64(D_m * Float64(0.5 / d)) ^ 2.0) * M_m) * Float64(Float64(Float64(h / l) * 0.5) * M_m))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-h)))));
	elseif (l <= -2e-310)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(D_m * M_m) / d) * 0.5) * h) * Float64(Float64(Float64(0.25 * D_m) * Float64(M_m / d)) / l))) * Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d)));
	else
		tmp = Float64(fma(Float64(Float64(Float64(t_0 / l) * -0.25) * h), Float64(t_0 * 0.5), 1.0) * Float64(Float64(d / sqrt(l)) / sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]}, If[LessEqual[l, -4e+199], N[(N[(1.0 - N[(N[(N[Power[N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * M$95$m), $MachinePrecision] * N[(N[(N[(h / l), $MachinePrecision] * 0.5), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(N[(1.0 - N[(N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] * N[(N[(N[(0.25 * D$95$m), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(t$95$0 / l), $MachinePrecision] * -0.25), $MachinePrecision] * h), $MachinePrecision] * N[(t$95$0 * 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.00000000000000039e199

   1. Initial program 69.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

   4. Applied rewrites 62.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 62.8

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-neg.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 79.3

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

   6. Applied rewrites 79.3%

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

   if -4.00000000000000039e199 < l < -1.999999999999994e-310

   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. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 73.8

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

   4. Applied rewrites 73.8%

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

   6. Applied rewrites 76.3%

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

   7. Taylor expanded in h around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 82.2

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

   9. Applied rewrites 82.2%

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

   if -1.999999999999994e-310 < l

   1. Initial program 68.5%

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

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 67.6

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

   4. Applied rewrites 67.6%

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

   6. Applied rewrites 70.1%

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

   8. Applied rewrites 70.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-pow.f64 N/A

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

      15. pow1/2 N/A

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

      16. lift-/.f64 N/A

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

      17. sqrt-div N/A

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

      18. lift-sqrt.f64 N/A

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

      19. frac-times N/A

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

      20. lift-sqrt.f64 N/A

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

      21. lift-sqrt.f64 N/A

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

      22. rem-square-sqrt N/A

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

   10. Applied rewrites 86.6%

       \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \cdot \mathsf{fma}\left(\left(-0.25 \cdot \frac{D \cdot \frac{M}{d}}{\ell}\right) \cdot h, 0.5 \cdot \left(D \cdot \frac{M}{d}\right), 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{+199}:\\ \;\;\;\;\left(1 - \left({\left(D \cdot \frac{0.5}{d}\right)}^{2} \cdot M\right) \cdot \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot M\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \left(\left(\frac{D \cdot M}{d} \cdot 0.5\right) \cdot h\right) \cdot \frac{\left(0.25 \cdot D\right) \cdot \frac{M}{d}}{\ell}\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{\frac{M}{d} \cdot D}{\ell} \cdot -0.25\right) \cdot h, \left(\frac{M}{d} \cdot D\right) \cdot 0.5, 1\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m}{d} \cdot D\_m\\ t_1 := \mathsf{fma}\left(\left(\frac{t\_0}{\ell} \cdot -0.25\right) \cdot h, t\_0 \cdot 0.5, 1\right)\\ \mathbf{if}\;d \leq -7.4 \cdot 10^{+149}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right) \cdot t\_1\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ M_m d) D_m))
        (t_1 (fma (* (* (/ t_0 l) -0.25) h) (* t_0 0.5) 1.0)))
   (if (<= d -7.4e+149)
     (* (* (sqrt (/ 1.0 (* l h))) (- d)) t_1)
     (if (<= d -5e-310)
       (* (* (sqrt (/ d h)) (sqrt (/ d l))) t_1)
       (* t_1 (/ (/ 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 = (M_m / d) * D_m;
	double t_1 = fma((((t_0 / l) * -0.25) * h), (t_0 * 0.5), 1.0);
	double tmp;
	if (d <= -7.4e+149) {
		tmp = (sqrt((1.0 / (l * h))) * -d) * t_1;
	} else if (d <= -5e-310) {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * t_1;
	} else {
		tmp = t_1 * ((d / sqrt(l)) / sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(M_m / d) * D_m)
	t_1 = fma(Float64(Float64(Float64(t_0 / l) * -0.25) * h), Float64(t_0 * 0.5), 1.0)
	tmp = 0.0
	if (d <= -7.4e+149)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d)) * t_1);
	elseif (d <= -5e-310)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * t_1);
	else
		tmp = Float64(t_1 * Float64(Float64(d / sqrt(l)) / sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(t$95$0 / l), $MachinePrecision] * -0.25), $MachinePrecision] * h), $MachinePrecision] * N[(t$95$0 * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[d, -7.4e+149], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(t$95$1 * N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -7.39999999999999957e149

   1. Initial program 67.7%

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

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 67.6

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

   4. Applied rewrites 67.6%

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

   6. Applied rewrites 73.9%

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

   8. Applied rewrites 74.1%

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

   9. Taylor expanded in h around -inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. rem-square-sqrt N/A

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

       4. lower-*.f64 N/A

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

       5. mul-1-neg N/A

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

       6. lower-neg.f64 N/A

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

       7. lower-sqrt.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 93.7

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

   11. Applied rewrites 93.7%

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

   if -7.39999999999999957e149 < d < -4.999999999999985e-310

   1. Initial program 75.6%

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

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 74.0

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

   4. Applied rewrites 74.0%

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

   6. Applied rewrites 72.1%

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

   8. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. sqrt-div N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. clear-num N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 0.0

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

      12. lift-/.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. sqrt-undiv N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-/.f64 75.9

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 75.9

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

      20. lift-pow.f64 N/A

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

      21. pow1/2 N/A

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

      22. lift-sqrt.f64 75.9

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

   10. Applied rewrites 75.9%

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

   if -4.999999999999985e-310 < d

   1. Initial program 68.5%

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

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 67.6

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

   4. Applied rewrites 67.6%

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

   6. Applied rewrites 70.1%

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

   8. Applied rewrites 70.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-pow.f64 N/A

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

      15. pow1/2 N/A

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

      16. lift-/.f64 N/A

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

      17. sqrt-div N/A

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

      18. lift-sqrt.f64 N/A

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

      19. frac-times N/A

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

      20. lift-sqrt.f64 N/A

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

      21. lift-sqrt.f64 N/A

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

      22. rem-square-sqrt N/A

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

   10. Applied rewrites 86.6%

       \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \cdot \mathsf{fma}\left(\left(-0.25 \cdot \frac{D \cdot \frac{M}{d}}{\ell}\right) \cdot h, 0.5 \cdot \left(D \cdot \frac{M}{d}\right), 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.4 \cdot 10^{+149}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right) \cdot \mathsf{fma}\left(\left(\frac{\frac{M}{d} \cdot D}{\ell} \cdot -0.25\right) \cdot h, \left(\frac{M}{d} \cdot D\right) \cdot 0.5, 1\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\left(\frac{\frac{M}{d} \cdot D}{\ell} \cdot -0.25\right) \cdot h, \left(\frac{M}{d} \cdot D\right) \cdot 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{\frac{M}{d} \cdot D}{\ell} \cdot -0.25\right) \cdot h, \left(\frac{M}{d} \cdot D\right) \cdot 0.5, 1\right) \cdot \frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.3% accurate, 3.5× speedup

Math

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

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ M_m d) D_m))
        (t_1 (fma (* (* (/ t_0 l) -0.25) h) (* t_0 0.5) 1.0)))
   (if (<= l -2e-310)
     (* (* (sqrt (/ 1.0 (* l h))) (- d)) t_1)
     (* t_1 (/ (/ 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 = (M_m / d) * D_m;
	double t_1 = fma((((t_0 / l) * -0.25) * h), (t_0 * 0.5), 1.0);
	double tmp;
	if (l <= -2e-310) {
		tmp = (sqrt((1.0 / (l * h))) * -d) * t_1;
	} else {
		tmp = t_1 * ((d / sqrt(l)) / sqrt(h));
	}
	return tmp;
}

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 73.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 72.2

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

   4. Applied rewrites 72.2%

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

   6. Applied rewrites 72.6%

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

   8. Applied rewrites 74.2%

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

   9. Taylor expanded in h around -inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. rem-square-sqrt N/A

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

       4. lower-*.f64 N/A

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

       5. mul-1-neg N/A

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

       6. lower-neg.f64 N/A

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

       7. lower-sqrt.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 76.5

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

   11. Applied rewrites 76.5%

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

   if -1.999999999999994e-310 < l

   1. Initial program 68.5%

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

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 67.6

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

   4. Applied rewrites 67.6%

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

   6. Applied rewrites 70.1%

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

   8. Applied rewrites 70.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-pow.f64 N/A

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

      15. pow1/2 N/A

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

      16. lift-/.f64 N/A

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

      17. sqrt-div N/A

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

      18. lift-sqrt.f64 N/A

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

      19. frac-times N/A

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

      20. lift-sqrt.f64 N/A

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

      21. lift-sqrt.f64 N/A

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

      22. rem-square-sqrt N/A

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

   10. Applied rewrites 86.6%

       \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \cdot \mathsf{fma}\left(\left(-0.25 \cdot \frac{D \cdot \frac{M}{d}}{\ell}\right) \cdot h, 0.5 \cdot \left(D \cdot \frac{M}{d}\right), 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.0%

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

Alternative 7: 78.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m}{d} \cdot D\_m\\ t_1 := \mathsf{fma}\left(\left(\frac{t\_0}{\ell} \cdot -0.25\right) \cdot h, t\_0 \cdot 0.5, 1\right)\\ \mathbf{if}\;\ell \leq -2.1 \cdot 10^{-308}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right) \cdot t\_1\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+92}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ M_m d) D_m))
        (t_1 (fma (* (* (/ t_0 l) -0.25) h) (* t_0 0.5) 1.0)))
   (if (<= l -2.1e-308)
     (* (* (sqrt (/ 1.0 (* l h))) (- d)) t_1)
     (if (<= l 1.55e+92)
       (* (/ d (sqrt (* l h))) t_1)
       (/ d (* (sqrt h) (sqrt l)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (M_m / d) * D_m;
	double t_1 = fma((((t_0 / l) * -0.25) * h), (t_0 * 0.5), 1.0);
	double tmp;
	if (l <= -2.1e-308) {
		tmp = (sqrt((1.0 / (l * h))) * -d) * t_1;
	} else if (l <= 1.55e+92) {
		tmp = (d / sqrt((l * h))) * t_1;
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(M_m / d) * D_m)
	t_1 = fma(Float64(Float64(Float64(t_0 / l) * -0.25) * h), Float64(t_0 * 0.5), 1.0)
	tmp = 0.0
	if (l <= -2.1e-308)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d)) * t_1);
	elseif (l <= 1.55e+92)
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * t_1);
	else
		tmp = Float64(d / Float64(sqrt(h) * 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_] := Block[{t$95$0 = N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(t$95$0 / l), $MachinePrecision] * -0.25), $MachinePrecision] * h), $MachinePrecision] * N[(t$95$0 * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[l, -2.1e-308], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[l, 1.55e+92], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.1e-308

   1. Initial program 73.1%

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

      1. 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. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 71.9

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

   4. Applied rewrites 71.9%

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

   6. Applied rewrites 73.2%

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

   8. Applied rewrites 74.8%

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

   9. Taylor expanded in h around -inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. rem-square-sqrt N/A

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

       4. lower-*.f64 N/A

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

       5. mul-1-neg N/A

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

       6. lower-neg.f64 N/A

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

       7. lower-sqrt.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 77.1

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

   11. Applied rewrites 77.1%

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

   if -2.1e-308 < l < 1.5500000000000001e92

   1. Initial program 78.8%

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

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 78.1

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

   4. Applied rewrites 78.1%

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

   6. Applied rewrites 79.6%

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

   8. Applied rewrites 79.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-undiv N/A

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

      10. un-div-inv N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. lift-/.f64 N/A

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

      14. frac-times N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. sqrt-div N/A

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

      18. sqrt-unprod N/A

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

      19. rem-square-sqrt N/A

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

      20. lift-sqrt.f64 N/A

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

      21. lift-/.f64 87.4

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

   10. Applied rewrites 87.4%

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

   if 1.5500000000000001e92 < l

   1. Initial program 51.2%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 49.9%

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

   6. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 51.7

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

   8. Applied rewrites 51.7%

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

   10. Applied rewrites 51.6%

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

   12. Applied rewrites 72.0%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{-308}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right) \cdot \mathsf{fma}\left(\left(\frac{\frac{M}{d} \cdot D}{\ell} \cdot -0.25\right) \cdot h, \left(\frac{M}{d} \cdot D\right) \cdot 0.5, 1\right)\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+92}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\left(\frac{\frac{M}{d} \cdot D}{\ell} \cdot -0.25\right) \cdot h, \left(\frac{M}{d} \cdot D\right) \cdot 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m}{d} \cdot D\_m\\ \mathbf{if}\;\ell \leq -2.1 \cdot 10^{-308}:\\ \;\;\;\;1 \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+92}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\left(\frac{t\_0}{\ell} \cdot -0.25\right) \cdot h, t\_0 \cdot 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ M_m d) D_m)))
   (if (<= l -2.1e-308)
     (* 1.0 (* (/ (sqrt (- d)) (sqrt (- l))) (sqrt (/ d h))))
     (if (<= l 1.55e+92)
       (* (/ d (sqrt (* l h))) (fma (* (* (/ t_0 l) -0.25) h) (* t_0 0.5) 1.0))
       (/ d (* (sqrt h) (sqrt l)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (M_m / d) * D_m;
	double tmp;
	if (l <= -2.1e-308) {
		tmp = 1.0 * ((sqrt(-d) / sqrt(-l)) * sqrt((d / h)));
	} else if (l <= 1.55e+92) {
		tmp = (d / sqrt((l * h))) * fma((((t_0 / l) * -0.25) * h), (t_0 * 0.5), 1.0);
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(M_m / d) * D_m)
	tmp = 0.0
	if (l <= -2.1e-308)
		tmp = Float64(1.0 * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * sqrt(Float64(d / h))));
	elseif (l <= 1.55e+92)
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * fma(Float64(Float64(Float64(t_0 / l) * -0.25) * h), Float64(t_0 * 0.5), 1.0));
	else
		tmp = Float64(d / Float64(sqrt(h) * 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_] := Block[{t$95$0 = N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]}, If[LessEqual[l, -2.1e-308], N[(1.0 * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.55e+92], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(t$95$0 / l), $MachinePrecision] * -0.25), $MachinePrecision] * h), $MachinePrecision] * N[(t$95$0 * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.1e-308

   1. Initial program 73.1%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 43.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 43.5

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-neg.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 48.9

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

   7. Applied rewrites 48.9%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 48.9

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 48.9

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

   9. Applied rewrites 48.9%

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

   if -2.1e-308 < l < 1.5500000000000001e92

   1. Initial program 78.8%

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

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 78.1

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

   4. Applied rewrites 78.1%

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

   6. Applied rewrites 79.6%

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

   8. Applied rewrites 79.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-undiv N/A

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

      10. un-div-inv N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. lift-/.f64 N/A

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

      14. frac-times N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. sqrt-div N/A

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

      18. sqrt-unprod N/A

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

      19. rem-square-sqrt N/A

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

      20. lift-sqrt.f64 N/A

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

      21. lift-/.f64 87.4

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

   10. Applied rewrites 87.4%

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

   if 1.5500000000000001e92 < l

   1. Initial program 51.2%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 49.9%

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

   6. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 51.7

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

   8. Applied rewrites 51.7%

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

   10. Applied rewrites 51.6%

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

   12. Applied rewrites 72.0%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{-308}:\\ \;\;\;\;1 \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+92}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\left(\frac{\frac{M}{d} \cdot D}{\ell} \cdot -0.25\right) \cdot h, \left(\frac{M}{d} \cdot D\right) \cdot 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.72 \cdot 10^{-230}:\\ \;\;\;\;1 \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{-198}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

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 -1.72e-230)
   (* 1.0 (* (/ (sqrt (- d)) (sqrt (- l))) (sqrt (/ d h))))
   (if (<= l 6.5e-198)
     (/ (* (sqrt (/ h l)) (- d)) h)
     (/ d (* (sqrt h) (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 <= -1.72e-230) {
		tmp = 1.0 * ((sqrt(-d) / sqrt(-l)) * sqrt((d / h)));
	} else if (l <= 6.5e-198) {
		tmp = (sqrt((h / l)) * -d) / h;
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-1.72d-230)) then
        tmp = 1.0d0 * ((sqrt(-d) / sqrt(-l)) * sqrt((d / h)))
    else if (l <= 6.5d-198) then
        tmp = (sqrt((h / l)) * -d) / h
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1.72e-230) {
		tmp = 1.0 * ((Math.sqrt(-d) / Math.sqrt(-l)) * Math.sqrt((d / h)));
	} else if (l <= 6.5e-198) {
		tmp = (Math.sqrt((h / l)) * -d) / h;
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -1.72e-230:
		tmp = 1.0 * ((math.sqrt(-d) / math.sqrt(-l)) * math.sqrt((d / h)))
	elif l <= 6.5e-198:
		tmp = (math.sqrt((h / l)) * -d) / h
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -1.72e-230)
		tmp = Float64(1.0 * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * sqrt(Float64(d / h))));
	elseif (l <= 6.5e-198)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * Float64(-d)) / h);
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -1.72e-230)
		tmp = 1.0 * ((sqrt(-d) / sqrt(-l)) * sqrt((d / h)));
	elseif (l <= 6.5e-198)
		tmp = (sqrt((h / l)) * -d) / h;
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -1.72e-230], N[(1.0 * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.5e-198], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision] / h), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.71999999999999999e-230

   1. Initial program 71.8%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 46.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 46.4

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-neg.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 52.6

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

   7. Applied rewrites 52.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 52.6

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 52.6

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

   9. Applied rewrites 52.6%

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

   if -1.71999999999999999e-230 < l < 6.5000000000000004e-198

   1. Initial program 83.3%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 14.6%

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

   6. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 38.9%

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

   9. Taylor expanded in l around -inf

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

   11. Applied rewrites 38.1%

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

   if 6.5000000000000004e-198 < l

   1. Initial program 66.0%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 49.3%

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

   6. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 52.3

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

   8. Applied rewrites 52.3%

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

   10. Applied rewrites 52.8%

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

   12. Applied rewrites 64.0%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.72 \cdot 10^{-230}:\\ \;\;\;\;1 \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{-198}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 45.3% accurate, 9.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.4 \cdot 10^{-198}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

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 5.4e-198)
   (* (sqrt (/ 1.0 (* l h))) (- d))
   (/ d (* (sqrt h) (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 <= 5.4e-198) {
		tmp = sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= 5.4d-198) then
        tmp = sqrt((1.0d0 / (l * h))) * -d
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 5.4e-198) {
		tmp = Math.sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= 5.4e-198:
		tmp = math.sqrt((1.0 / (l * h))) * -d
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= 5.4e-198)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= 5.4e-198)
		tmp = sqrt((1.0 / (l * h))) * -d;
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, 5.4e-198], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.4000000000000003e-198

   1. Initial program 74.9%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 38.0%

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

   6. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      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 44.8

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

   8. Applied rewrites 44.8%

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

   if 5.4000000000000003e-198 < l

   1. Initial program 66.0%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 49.3%

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

   6. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 52.3

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

   8. Applied rewrites 52.3%

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

   10. Applied rewrites 52.8%

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

   12. Applied rewrites 64.0%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.4 \cdot 10^{-198}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.0% accurate, 10.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.4 \cdot 10^{-198}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l 5.4e-198) (* (sqrt (/ 1.0 (* l h))) (- d)) (/ 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 (l <= 5.4e-198) {
		tmp = sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / sqrt((l * h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= 5.4d-198) then
        tmp = sqrt((1.0d0 / (l * h))) * -d
    else
        tmp = d / sqrt((l * h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 5.4e-198) {
		tmp = Math.sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / Math.sqrt((l * h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= 5.4e-198:
		tmp = math.sqrt((1.0 / (l * h))) * -d
	else:
		tmp = d / math.sqrt((l * h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= 5.4e-198)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	else
		tmp = Float64(d / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= 5.4e-198)
		tmp = sqrt((1.0 / (l * h))) * -d;
	else
		tmp = d / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, 5.4e-198], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.4000000000000003e-198

   1. Initial program 74.9%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 38.0%

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

   6. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      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 44.8

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

   8. Applied rewrites 44.8%

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

   if 5.4000000000000003e-198 < l

   1. Initial program 66.0%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 49.3%

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

   6. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 52.3

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

   8. Applied rewrites 52.3%

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

   10. Applied rewrites 52.8%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.4 \cdot 10^{-198}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 25.6% 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 h around 0

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

5. Applied rewrites 43.3%

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

6. Taylor expanded in h around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 31.0

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

8. Applied rewrites 31.0%

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

10. Applied rewrites 31.2%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024250 
(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)))))