Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.2% → 82.4%

Time: 17.5s

Alternatives: 12

Speedup: 3.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.2% 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: 82.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{M\_m \cdot \left(D\_m \cdot \frac{0.5}{d}\right)}{\ell}, \left(\left(-0.25\right) \cdot D\_m\right) \cdot \left(\frac{M\_m}{d} \cdot h\right), 1\right)\\ \mathbf{if}\;d \leq -2.3 \cdot 10^{-144}:\\ \;\;\;\;t\_0 \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;d \leq -3.9 \cdot 10^{-301}:\\ \;\;\;\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right)\\ \mathbf{elif}\;d \leq 5.4 \cdot 10^{-164}:\\ \;\;\;\;\frac{\frac{\sqrt{d} \cdot \mathsf{fma}\left({\left(\frac{M\_m}{d} \cdot D\_m\right)}^{2} \cdot 0.25, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{\ell}} \cdot \sqrt{d}}{\sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{\sqrt{h}} \cdot \frac{-d}{\sqrt{\ell}}\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (fma
          (/ (* M_m (* D_m (/ 0.5 d))) l)
          (* (* (- 0.25) D_m) (* (/ M_m d) h))
          1.0)))
   (if (<= d -2.3e-144)
     (* t_0 (* (sqrt (/ d l)) (/ (sqrt (- d)) (sqrt (- h)))))
     (if (<= d -3.9e-301)
       (*
        (- 1.0 (* (/ h l) (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
        (* (sqrt (/ 1.0 (* l h))) (- d)))
       (if (<= d 5.4e-164)
         (/
          (*
           (/
            (*
             (sqrt d)
             (fma (* (pow (* (/ M_m d) D_m) 2.0) 0.25) (* -0.5 (/ h l)) 1.0))
            (sqrt l))
           (sqrt d))
          (sqrt h))
         (* (* (/ -1.0 (sqrt h)) (/ (- d) (sqrt l))) t_0))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = fma(((M_m * (D_m * (0.5 / d))) / l), ((-0.25 * D_m) * ((M_m / d) * h)), 1.0);
	double tmp;
	if (d <= -2.3e-144) {
		tmp = t_0 * (sqrt((d / l)) * (sqrt(-d) / sqrt(-h)));
	} else if (d <= -3.9e-301) {
		tmp = (1.0 - ((h / l) * (pow(((M_m * D_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (sqrt((1.0 / (l * h))) * -d);
	} else if (d <= 5.4e-164) {
		tmp = (((sqrt(d) * fma((pow(((M_m / d) * D_m), 2.0) * 0.25), (-0.5 * (h / l)), 1.0)) / sqrt(l)) * sqrt(d)) / sqrt(h);
	} else {
		tmp = ((-1.0 / sqrt(h)) * (-d / sqrt(l))) * t_0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = fma(Float64(Float64(M_m * Float64(D_m * Float64(0.5 / d))) / l), Float64(Float64(Float64(-0.25) * D_m) * Float64(Float64(M_m / d) * h)), 1.0)
	tmp = 0.0
	if (d <= -2.3e-144)
		tmp = Float64(t_0 * Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-h)))));
	elseif (d <= -3.9e-301)
		tmp = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d)));
	elseif (d <= 5.4e-164)
		tmp = Float64(Float64(Float64(Float64(sqrt(d) * fma(Float64((Float64(Float64(M_m / d) * D_m) ^ 2.0) * 0.25), Float64(-0.5 * Float64(h / l)), 1.0)) / sqrt(l)) * sqrt(d)) / sqrt(h));
	else
		tmp = Float64(Float64(Float64(-1.0 / sqrt(h)) * Float64(Float64(-d) / sqrt(l))) * t_0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[(M$95$m * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[((-0.25) * D$95$m), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[d, -2.3e-144], N[(t$95$0 * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -3.9e-301], N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.4e-164], N[(N[(N[(N[(N[Sqrt[d], $MachinePrecision] * N[(N[(N[Power[N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 0.25), $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[((-d) / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.3e-144

   1. Initial program 76.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-sqrt.f64 76.5

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

   4. Applied rewrites 76.5%

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

   6. Applied rewrites 78.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 78.4

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. pow1/2 N/A

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

      9. lower-/.f64 N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-neg.f64 87.8

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

   8. Applied rewrites 87.8%

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

   if -2.3e-144 < d < -3.9000000000000001e-301

   1. Initial program 38.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-sqrt.f64 38.1

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

   4. Applied rewrites 38.1%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 38.1

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 38.1

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

   6. Applied rewrites 38.1%

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

   7. Taylor expanded in h around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 67.9

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

   9. Applied rewrites 67.9%

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

   if -3.9000000000000001e-301 < d < 5.4000000000000003e-164

   1. Initial program 39.7%

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

   4. Applied rewrites 52.4%

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

   6. Applied rewrites 62.1%

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

   if 5.4000000000000003e-164 < d

   1. Initial program 78.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-sqrt.f64 78.7

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

   4. Applied rewrites 78.7%

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

   6. Applied rewrites 79.9%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 79.9

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 79.9

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

   8. Applied rewrites 79.9%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. sqrt-div N/A

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

      11. sqrt-unprod N/A

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

      12. rem-square-sqrt N/A

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

      13. lift-*.f64 N/A

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

      14. sqrt-prod N/A

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

      15. *-rgt-identity N/A

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

      16. times-frac N/A

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

      17. frac-2neg N/A

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

      18. times-frac N/A

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

      19. distribute-lft-neg-in N/A

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

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

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

      21. metadata-eval N/A

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

      22. times-frac N/A

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

      23. lower-*.f64 N/A

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

   10. Applied rewrites 92.7%

       \[\leadsto \color{blue}{\left(\frac{d}{-\sqrt{\ell}} \cdot \frac{-1}{\sqrt{h}}\right)} \cdot \mathsf{fma}\left(\frac{\left(\frac{0.5}{d} \cdot D\right) \cdot M}{\ell}, -\left(\frac{M}{d} \cdot h\right) \cdot \left(0.25 \cdot D\right), 1\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.3 \cdot 10^{-144}:\\ \;\;\;\;\mathsf{fma}\left(\frac{M \cdot \left(D \cdot \frac{0.5}{d}\right)}{\ell}, \left(\left(-0.25\right) \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right), 1\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;d \leq -3.9 \cdot 10^{-301}:\\ \;\;\;\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right)\\ \mathbf{elif}\;d \leq 5.4 \cdot 10^{-164}:\\ \;\;\;\;\frac{\frac{\sqrt{d} \cdot \mathsf{fma}\left({\left(\frac{M}{d} \cdot D\right)}^{2} \cdot 0.25, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{\ell}} \cdot \sqrt{d}}{\sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{\sqrt{h}} \cdot \frac{-d}{\sqrt{\ell}}\right) \cdot \mathsf{fma}\left(\frac{M \cdot \left(D \cdot \frac{0.5}{d}\right)}{\ell}, \left(\left(-0.25\right) \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 48.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-132}:\\ \;\;\;\;\frac{\left(\frac{D\_m \cdot D\_m}{d} \cdot \sqrt{\ell \cdot h}\right) \cdot \left(-0.125 \cdot \left(M\_m \cdot M\_m\right)\right)}{\ell \cdot \ell}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))
          (-
           1.0
           (* (/ h l) (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ 1.0 2.0)))))))
   (if (<= t_0 -5e-132)
     (/
      (* (* (/ (* D_m D_m) d) (sqrt (* l h))) (* -0.125 (* M_m M_m)))
      (* l l))
     (if (<= t_0 0.0)
       (* (sqrt (/ 1.0 (* l h))) (- d))
       (/ (sqrt (/ d l)) (sqrt (/ h d)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0))) * (1.0 - ((h / l) * (pow(((M_m * D_m) / (2.0 * d)), 2.0) * (1.0 / 2.0))));
	double tmp;
	if (t_0 <= -5e-132) {
		tmp = ((((D_m * D_m) / d) * sqrt((l * h))) * (-0.125 * (M_m * M_m))) / (l * l);
	} else if (t_0 <= 0.0) {
		tmp = sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = sqrt((d / l)) / sqrt((h / d));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0))) * (1.0d0 - ((h / l) * ((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0))))
    if (t_0 <= (-5d-132)) then
        tmp = ((((d_m * d_m) / d) * sqrt((l * h))) * ((-0.125d0) * (m_m * m_m))) / (l * l)
    else if (t_0 <= 0.0d0) then
        tmp = sqrt((1.0d0 / (l * h))) * -d
    else
        tmp = sqrt((d / l)) / sqrt((h / d))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (Math.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0))) * (1.0 - ((h / l) * (Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (1.0 / 2.0))));
	double tmp;
	if (t_0 <= -5e-132) {
		tmp = ((((D_m * D_m) / d) * Math.sqrt((l * h))) * (-0.125 * (M_m * M_m))) / (l * l);
	} else if (t_0 <= 0.0) {
		tmp = Math.sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = Math.sqrt((d / l)) / Math.sqrt((h / d));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0))) * (1.0 - ((h / l) * (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (1.0 / 2.0))))
	tmp = 0
	if t_0 <= -5e-132:
		tmp = ((((D_m * D_m) / d) * math.sqrt((l * h))) * (-0.125 * (M_m * M_m))) / (l * l)
	elif t_0 <= 0.0:
		tmp = math.sqrt((1.0 / (l * h))) * -d
	else:
		tmp = math.sqrt((d / l)) / math.sqrt((h / d))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))))
	tmp = 0.0
	if (t_0 <= -5e-132)
		tmp = Float64(Float64(Float64(Float64(Float64(D_m * D_m) / d) * sqrt(Float64(l * h))) * Float64(-0.125 * Float64(M_m * M_m))) / Float64(l * l));
	elseif (t_0 <= 0.0)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	else
		tmp = Float64(sqrt(Float64(d / l)) / sqrt(Float64(h / d)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0))) * (1.0 - ((h / l) * ((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0))));
	tmp = 0.0;
	if (t_0 <= -5e-132)
		tmp = ((((D_m * D_m) / d) * sqrt((l * h))) * (-0.125 * (M_m * M_m))) / (l * l);
	elseif (t_0 <= 0.0)
		tmp = sqrt((1.0 / (l * h))) * -d;
	else
		tmp = sqrt((d / l)) / sqrt((h / d));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-132], N[(N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] * N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.125 * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.9999999999999999e-132

   1. Initial program 86.4%

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

   3. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 29.2%

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

   6. Taylor expanded in h around inf

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

   8. Applied rewrites 34.8%

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

   if -4.9999999999999999e-132 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 30.4%

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

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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 58.7

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

   5. Applied rewrites 58.7%

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

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

   1. Initial program 61.4%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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 37.6

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

   5. Applied rewrites 37.6%

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

   7. Applied rewrites 64.6%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \leq -5 \cdot 10^{-132}:\\ \;\;\;\;\frac{\left(\frac{D \cdot D}{d} \cdot \sqrt{\ell \cdot h}\right) \cdot \left(-0.125 \cdot \left(M \cdot M\right)\right)}{\ell \cdot \ell}\\ \mathbf{elif}\;\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 48.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-132}:\\ \;\;\;\;\frac{\left(\frac{M\_m \cdot M\_m}{d} \cdot \sqrt{\ell \cdot h}\right) \cdot \left(-0.125 \cdot \left(D\_m \cdot D\_m\right)\right)}{\ell \cdot \ell}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))
          (-
           1.0
           (* (/ h l) (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ 1.0 2.0)))))))
   (if (<= t_0 -5e-132)
     (/
      (* (* (/ (* M_m M_m) d) (sqrt (* l h))) (* -0.125 (* D_m D_m)))
      (* l l))
     (if (<= t_0 0.0)
       (* (sqrt (/ 1.0 (* l h))) (- d))
       (/ (sqrt (/ d l)) (sqrt (/ h d)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0))) * (1.0 - ((h / l) * (pow(((M_m * D_m) / (2.0 * d)), 2.0) * (1.0 / 2.0))));
	double tmp;
	if (t_0 <= -5e-132) {
		tmp = ((((M_m * M_m) / d) * sqrt((l * h))) * (-0.125 * (D_m * D_m))) / (l * l);
	} else if (t_0 <= 0.0) {
		tmp = sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = sqrt((d / l)) / sqrt((h / d));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0))) * (1.0d0 - ((h / l) * ((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0))))
    if (t_0 <= (-5d-132)) then
        tmp = ((((m_m * m_m) / d) * sqrt((l * h))) * ((-0.125d0) * (d_m * d_m))) / (l * l)
    else if (t_0 <= 0.0d0) then
        tmp = sqrt((1.0d0 / (l * h))) * -d
    else
        tmp = sqrt((d / l)) / sqrt((h / d))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (Math.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0))) * (1.0 - ((h / l) * (Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (1.0 / 2.0))));
	double tmp;
	if (t_0 <= -5e-132) {
		tmp = ((((M_m * M_m) / d) * Math.sqrt((l * h))) * (-0.125 * (D_m * D_m))) / (l * l);
	} else if (t_0 <= 0.0) {
		tmp = Math.sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = Math.sqrt((d / l)) / Math.sqrt((h / d));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0))) * (1.0 - ((h / l) * (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (1.0 / 2.0))))
	tmp = 0
	if t_0 <= -5e-132:
		tmp = ((((M_m * M_m) / d) * math.sqrt((l * h))) * (-0.125 * (D_m * D_m))) / (l * l)
	elif t_0 <= 0.0:
		tmp = math.sqrt((1.0 / (l * h))) * -d
	else:
		tmp = math.sqrt((d / l)) / math.sqrt((h / d))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))))
	tmp = 0.0
	if (t_0 <= -5e-132)
		tmp = Float64(Float64(Float64(Float64(Float64(M_m * M_m) / d) * sqrt(Float64(l * h))) * Float64(-0.125 * Float64(D_m * D_m))) / Float64(l * l));
	elseif (t_0 <= 0.0)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	else
		tmp = Float64(sqrt(Float64(d / l)) / sqrt(Float64(h / d)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0))) * (1.0 - ((h / l) * ((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0))));
	tmp = 0.0;
	if (t_0 <= -5e-132)
		tmp = ((((M_m * M_m) / d) * sqrt((l * h))) * (-0.125 * (D_m * D_m))) / (l * l);
	elseif (t_0 <= 0.0)
		tmp = sqrt((1.0 / (l * h))) * -d;
	else
		tmp = sqrt((d / l)) / sqrt((h / d));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-132], N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision] * N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.9999999999999999e-132

   1. Initial program 86.4%

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

   3. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 29.2%

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

   6. Taylor expanded in h around 0

      \[\leadsto \frac{d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{\color{blue}{\ell} \cdot \ell} \]
   7. Step-by-step derivation

   8. Applied rewrites 2.1%

      \[\leadsto \frac{\sqrt{\frac{{\ell}^{3}}{h}} \cdot d}{\color{blue}{\ell} \cdot \ell} \]

   9. Taylor expanded in h around inf

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

   11. Applied rewrites 32.6%

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

   if -4.9999999999999999e-132 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 30.4%

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

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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 58.7

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

   5. Applied rewrites 58.7%

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

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

   1. Initial program 61.4%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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 37.6

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

   5. Applied rewrites 37.6%

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

   7. Applied rewrites 64.6%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \leq -5 \cdot 10^{-132}:\\ \;\;\;\;\frac{\left(\frac{M \cdot M}{d} \cdot \sqrt{\ell \cdot h}\right) \cdot \left(-0.125 \cdot \left(D \cdot D\right)\right)}{\ell \cdot \ell}\\ \mathbf{elif}\;\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{M\_m \cdot \left(D\_m \cdot \frac{0.5}{d}\right)}{\ell}, \left(\left(-0.25\right) \cdot D\_m\right) \cdot \left(\frac{M\_m}{d} \cdot h\right), 1\right)\\ \mathbf{if}\;d \leq -2.3 \cdot 10^{-144}:\\ \;\;\;\;t\_0 \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;d \leq -3.9 \cdot 10^{-301}:\\ \;\;\;\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{\sqrt{h}} \cdot \frac{-d}{\sqrt{\ell}}\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (fma
          (/ (* M_m (* D_m (/ 0.5 d))) l)
          (* (* (- 0.25) D_m) (* (/ M_m d) h))
          1.0)))
   (if (<= d -2.3e-144)
     (* t_0 (* (sqrt (/ d l)) (/ (sqrt (- d)) (sqrt (- h)))))
     (if (<= d -3.9e-301)
       (*
        (- 1.0 (* (/ h l) (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
        (* (sqrt (/ 1.0 (* l h))) (- d)))
       (* (* (/ -1.0 (sqrt h)) (/ (- d) (sqrt l))) t_0)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = fma(((M_m * (D_m * (0.5 / d))) / l), ((-0.25 * D_m) * ((M_m / d) * h)), 1.0);
	double tmp;
	if (d <= -2.3e-144) {
		tmp = t_0 * (sqrt((d / l)) * (sqrt(-d) / sqrt(-h)));
	} else if (d <= -3.9e-301) {
		tmp = (1.0 - ((h / l) * (pow(((M_m * D_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (sqrt((1.0 / (l * h))) * -d);
	} else {
		tmp = ((-1.0 / sqrt(h)) * (-d / sqrt(l))) * t_0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = fma(Float64(Float64(M_m * Float64(D_m * Float64(0.5 / d))) / l), Float64(Float64(Float64(-0.25) * D_m) * Float64(Float64(M_m / d) * h)), 1.0)
	tmp = 0.0
	if (d <= -2.3e-144)
		tmp = Float64(t_0 * Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-h)))));
	elseif (d <= -3.9e-301)
		tmp = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d)));
	else
		tmp = Float64(Float64(Float64(-1.0 / sqrt(h)) * Float64(Float64(-d) / sqrt(l))) * t_0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[(M$95$m * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[((-0.25) * D$95$m), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[d, -2.3e-144], N[(t$95$0 * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -3.9e-301], N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[((-d) / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.3e-144

   1. Initial program 76.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-sqrt.f64 76.5

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

   4. Applied rewrites 76.5%

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

   6. Applied rewrites 78.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 78.4

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. pow1/2 N/A

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

      9. lower-/.f64 N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-neg.f64 87.8

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

   8. Applied rewrites 87.8%

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

   if -2.3e-144 < d < -3.9000000000000001e-301

   1. Initial program 38.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-sqrt.f64 38.1

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

   4. Applied rewrites 38.1%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 38.1

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 38.1

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

   6. Applied rewrites 38.1%

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

   7. Taylor expanded in h around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 67.9

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

   9. Applied rewrites 67.9%

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

   if -3.9000000000000001e-301 < d

   1. Initial program 70.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 70.3

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

   4. Applied rewrites 70.3%

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

   6. Applied rewrites 71.1%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 71.1

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 71.1

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

   8. Applied rewrites 71.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. sqrt-div N/A

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

      11. sqrt-unprod N/A

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

      12. rem-square-sqrt N/A

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

      13. lift-*.f64 N/A

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

      14. sqrt-prod N/A

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

      15. *-rgt-identity N/A

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

      16. times-frac N/A

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

      17. frac-2neg N/A

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

      18. times-frac N/A

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

      19. distribute-lft-neg-in N/A

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

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

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

      21. metadata-eval N/A

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

      22. times-frac N/A

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

      23. lower-*.f64 N/A

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

   10. Applied rewrites 83.9%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.3 \cdot 10^{-144}:\\ \;\;\;\;\mathsf{fma}\left(\frac{M \cdot \left(D \cdot \frac{0.5}{d}\right)}{\ell}, \left(\left(-0.25\right) \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right), 1\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;d \leq -3.9 \cdot 10^{-301}:\\ \;\;\;\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{\sqrt{h}} \cdot \frac{-d}{\sqrt{\ell}}\right) \cdot \mathsf{fma}\left(\frac{M \cdot \left(D \cdot \frac{0.5}{d}\right)}{\ell}, \left(\left(-0.25\right) \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{M\_m \cdot \left(D\_m \cdot \frac{0.5}{d}\right)}{\ell}, \left(\left(-0.25\right) \cdot D\_m\right) \cdot \left(\frac{M\_m}{d} \cdot h\right), 1\right)\\ \mathbf{if}\;d \leq -7.2 \cdot 10^{-55}:\\ \;\;\;\;t\_0 \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;d \leq -6.7 \cdot 10^{-307}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{\sqrt{h}} \cdot \frac{-d}{\sqrt{\ell}}\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (fma
          (/ (* M_m (* D_m (/ 0.5 d))) l)
          (* (* (- 0.25) D_m) (* (/ M_m d) h))
          1.0)))
   (if (<= d -7.2e-55)
     (* t_0 (* (sqrt (/ d l)) (/ (sqrt (- d)) (sqrt (- h)))))
     (if (<= d -6.7e-307)
       (* (* (sqrt (/ 1.0 (* l h))) (- d)) t_0)
       (* (* (/ -1.0 (sqrt h)) (/ (- d) (sqrt l))) t_0)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = fma(((M_m * (D_m * (0.5 / d))) / l), ((-0.25 * D_m) * ((M_m / d) * h)), 1.0);
	double tmp;
	if (d <= -7.2e-55) {
		tmp = t_0 * (sqrt((d / l)) * (sqrt(-d) / sqrt(-h)));
	} else if (d <= -6.7e-307) {
		tmp = (sqrt((1.0 / (l * h))) * -d) * t_0;
	} else {
		tmp = ((-1.0 / sqrt(h)) * (-d / sqrt(l))) * t_0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = fma(Float64(Float64(M_m * Float64(D_m * Float64(0.5 / d))) / l), Float64(Float64(Float64(-0.25) * D_m) * Float64(Float64(M_m / d) * h)), 1.0)
	tmp = 0.0
	if (d <= -7.2e-55)
		tmp = Float64(t_0 * Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-h)))));
	elseif (d <= -6.7e-307)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d)) * t_0);
	else
		tmp = Float64(Float64(Float64(-1.0 / sqrt(h)) * Float64(Float64(-d) / sqrt(l))) * t_0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[(M$95$m * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[((-0.25) * D$95$m), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[d, -7.2e-55], N[(t$95$0 * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -6.7e-307], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(-1.0 / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[((-d) / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -7.2000000000000001e-55

   1. Initial program 78.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 78.0

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

   4. Applied rewrites 78.0%

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

   6. Applied rewrites 79.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 79.4

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. pow1/2 N/A

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

      9. lower-/.f64 N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-neg.f64 91.0

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

   8. Applied rewrites 91.0%

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

   if -7.2000000000000001e-55 < d < -6.6999999999999999e-307

   1. Initial program 47.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-sqrt.f64 47.3

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

   4. Applied rewrites 47.3%

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

   6. Applied rewrites 47.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 47.0

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 47.0

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

   8. Applied rewrites 47.0%

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

   9. Taylor expanded in h around -inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. rem-square-sqrt N/A

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

       4. lower-*.f64 N/A

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

       5. mul-1-neg N/A

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

       6. lower-neg.f64 N/A

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

       7. lower-sqrt.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 67.1

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

   11. Applied rewrites 67.1%

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

   if -6.6999999999999999e-307 < d

   1. Initial program 70.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-sqrt.f64 70.8

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

   4. Applied rewrites 70.8%

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

   6. Applied rewrites 71.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 71.6

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 71.6

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

   8. Applied rewrites 71.6%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. sqrt-div N/A

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

      11. sqrt-unprod N/A

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

      12. rem-square-sqrt N/A

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

      13. lift-*.f64 N/A

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

      14. sqrt-prod N/A

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

      15. *-rgt-identity N/A

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

      16. times-frac N/A

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

      17. frac-2neg N/A

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

      18. times-frac N/A

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

      19. distribute-lft-neg-in N/A

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

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

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

      21. metadata-eval N/A

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

      22. times-frac N/A

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

      23. lower-*.f64 N/A

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

   10. Applied rewrites 84.5%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(\frac{M \cdot \left(D \cdot \frac{0.5}{d}\right)}{\ell}, \left(\left(-0.25\right) \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right), 1\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;d \leq -6.7 \cdot 10^{-307}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right) \cdot \mathsf{fma}\left(\frac{M \cdot \left(D \cdot \frac{0.5}{d}\right)}{\ell}, \left(\left(-0.25\right) \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{\sqrt{h}} \cdot \frac{-d}{\sqrt{\ell}}\right) \cdot \mathsf{fma}\left(\frac{M \cdot \left(D \cdot \frac{0.5}{d}\right)}{\ell}, \left(\left(-0.25\right) \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m \cdot \left(D\_m \cdot \frac{0.5}{d}\right)}{\ell}\\ t_1 := \mathsf{fma}\left(t\_0, \left(\left(-0.25\right) \cdot D\_m\right) \cdot \left(\frac{M\_m}{d} \cdot h\right), 1\right)\\ t_2 := \mathsf{fma}\left(t\_0, \frac{\left(-0.25 \cdot D\_m\right) \cdot \left(M\_m \cdot h\right)}{d}, 1\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{if}\;h \leq -7.6 \cdot 10^{+151}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right) \cdot t\_1\\ \mathbf{elif}\;h \leq 1.5 \cdot 10^{+190}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ (* M_m (* D_m (/ 0.5 d))) l))
        (t_1 (fma t_0 (* (* (- 0.25) D_m) (* (/ M_m d) h)) 1.0))
        (t_2
         (*
          (fma t_0 (/ (* (* -0.25 D_m) (* M_m h)) d) 1.0)
          (* (sqrt (/ d h)) (sqrt (/ d l))))))
   (if (<= h -7.6e+151)
     t_2
     (if (<= h -4e-310)
       (* (* (sqrt (/ 1.0 (* l h))) (- d)) t_1)
       (if (<= h 1.5e+190) (* (/ d (sqrt (* l h))) t_1) t_2)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (M_m * (D_m * (0.5 / d))) / l;
	double t_1 = fma(t_0, ((-0.25 * D_m) * ((M_m / d) * h)), 1.0);
	double t_2 = fma(t_0, (((-0.25 * D_m) * (M_m * h)) / d), 1.0) * (sqrt((d / h)) * sqrt((d / l)));
	double tmp;
	if (h <= -7.6e+151) {
		tmp = t_2;
	} else if (h <= -4e-310) {
		tmp = (sqrt((1.0 / (l * h))) * -d) * t_1;
	} else if (h <= 1.5e+190) {
		tmp = (d / sqrt((l * h))) * t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(M_m * Float64(D_m * Float64(0.5 / d))) / l)
	t_1 = fma(t_0, Float64(Float64(Float64(-0.25) * D_m) * Float64(Float64(M_m / d) * h)), 1.0)
	t_2 = Float64(fma(t_0, Float64(Float64(Float64(-0.25 * D_m) * Float64(M_m * h)) / d), 1.0) * Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))))
	tmp = 0.0
	if (h <= -7.6e+151)
		tmp = t_2;
	elseif (h <= -4e-310)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d)) * t_1);
	elseif (h <= 1.5e+190)
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * t_1);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(M$95$m * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[((-0.25) * D$95$m), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * N[(N[(N[(-0.25 * D$95$m), $MachinePrecision] * N[(M$95$m * h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -7.6e+151], t$95$2, If[LessEqual[h, -4e-310], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[h, 1.5e+190], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -7.6000000000000001e151 or 1.49999999999999991e190 < h

   1. Initial program 56.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-sqrt.f64 56.8

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

   4. Applied rewrites 56.8%

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

   6. Applied rewrites 52.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 52.2

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 52.2

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

   8. Applied rewrites 52.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-lft-neg-in N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 49.4

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

   10. Applied rewrites 49.4%

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

   if -7.6000000000000001e151 < h < -3.999999999999988e-310

   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. lower-sqrt.f64 68.5

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

   4. Applied rewrites 68.5%

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

   6. Applied rewrites 70.9%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 70.9

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 70.9

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

   8. Applied rewrites 70.9%

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

   9. Taylor expanded in h around -inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. rem-square-sqrt N/A

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

       4. lower-*.f64 N/A

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

       5. mul-1-neg N/A

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

       6. lower-neg.f64 N/A

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

       7. lower-sqrt.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 84.0

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

   11. Applied rewrites 84.0%

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

   if -3.999999999999988e-310 < h < 1.49999999999999991e190

   1. Initial program 73.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 73.4

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

   4. Applied rewrites 73.4%

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

   6. Applied rewrites 75.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 75.3

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 75.3

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

   8. Applied rewrites 75.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. sqrt-div N/A

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

      11. sqrt-unprod N/A

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

      12. rem-square-sqrt N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lower-/.f64 85.4

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

   10. Applied rewrites 85.4%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -7.6 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(\frac{M \cdot \left(D \cdot \frac{0.5}{d}\right)}{\ell}, \frac{\left(-0.25 \cdot D\right) \cdot \left(M \cdot h\right)}{d}, 1\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right) \cdot \mathsf{fma}\left(\frac{M \cdot \left(D \cdot \frac{0.5}{d}\right)}{\ell}, \left(\left(-0.25\right) \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right), 1\right)\\ \mathbf{elif}\;h \leq 1.5 \cdot 10^{+190}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{M \cdot \left(D \cdot \frac{0.5}{d}\right)}{\ell}, \left(\left(-0.25\right) \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{M \cdot \left(D \cdot \frac{0.5}{d}\right)}{\ell}, \frac{\left(-0.25 \cdot D\right) \cdot \left(M \cdot h\right)}{d}, 1\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m \cdot \left(D\_m \cdot \frac{0.5}{d}\right)}{\ell}\\ t_1 := \mathsf{fma}\left(t\_0, \left(\left(-0.25\right) \cdot D\_m\right) \cdot \left(\frac{M\_m}{d} \cdot h\right), 1\right)\\ \mathbf{if}\;h \leq -7.6 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \frac{\left(-0.25 \cdot D\_m\right) \cdot \left(M\_m \cdot h\right)}{d}, 1\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{\sqrt{h}} \cdot \frac{-d}{\sqrt{\ell}}\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ (* M_m (* D_m (/ 0.5 d))) l))
        (t_1 (fma t_0 (* (* (- 0.25) D_m) (* (/ M_m d) h)) 1.0)))
   (if (<= h -7.6e+151)
     (*
      (fma t_0 (/ (* (* -0.25 D_m) (* M_m h)) d) 1.0)
      (* (sqrt (/ d h)) (sqrt (/ d l))))
     (if (<= h -4e-310)
       (* (* (sqrt (/ 1.0 (* l h))) (- d)) t_1)
       (* (* (/ -1.0 (sqrt h)) (/ (- d) (sqrt l))) t_1)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (M_m * (D_m * (0.5 / d))) / l;
	double t_1 = fma(t_0, ((-0.25 * D_m) * ((M_m / d) * h)), 1.0);
	double tmp;
	if (h <= -7.6e+151) {
		tmp = fma(t_0, (((-0.25 * D_m) * (M_m * h)) / d), 1.0) * (sqrt((d / h)) * sqrt((d / l)));
	} else if (h <= -4e-310) {
		tmp = (sqrt((1.0 / (l * h))) * -d) * t_1;
	} else {
		tmp = ((-1.0 / sqrt(h)) * (-d / sqrt(l))) * t_1;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(M_m * Float64(D_m * Float64(0.5 / d))) / l)
	t_1 = fma(t_0, Float64(Float64(Float64(-0.25) * D_m) * Float64(Float64(M_m / d) * h)), 1.0)
	tmp = 0.0
	if (h <= -7.6e+151)
		tmp = Float64(fma(t_0, Float64(Float64(Float64(-0.25 * D_m) * Float64(M_m * h)) / d), 1.0) * Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))));
	elseif (h <= -4e-310)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d)) * t_1);
	else
		tmp = Float64(Float64(Float64(-1.0 / sqrt(h)) * Float64(Float64(-d) / sqrt(l))) * t_1);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(M$95$m * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[((-0.25) * D$95$m), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[h, -7.6e+151], N[(N[(t$95$0 * N[(N[(N[(-0.25 * D$95$m), $MachinePrecision] * N[(M$95$m * h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -4e-310], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(-1.0 / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[((-d) / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -7.6000000000000001e151

   1. Initial program 51.9%

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

      1. lift-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. lower-sqrt.f64 51.9

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

   4. Applied rewrites 51.9%

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

   6. Applied rewrites 47.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 47.0

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 47.0

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

   8. Applied rewrites 47.0%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-lft-neg-in N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 41.3

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

   10. Applied rewrites 41.3%

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

   if -7.6000000000000001e151 < h < -3.999999999999988e-310

   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. lower-sqrt.f64 68.5

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

   4. Applied rewrites 68.5%

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

   6. Applied rewrites 70.9%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 70.9

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 70.9

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

   8. Applied rewrites 70.9%

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

   9. Taylor expanded in h around -inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. rem-square-sqrt N/A

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

       4. lower-*.f64 N/A

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

       5. mul-1-neg N/A

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

       6. lower-neg.f64 N/A

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

       7. lower-sqrt.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 84.0

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

   11. Applied rewrites 84.0%

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

   if -3.999999999999988e-310 < h

   1. Initial program 71.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-sqrt.f64 71.3

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

   4. Applied rewrites 71.3%

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

   6. Applied rewrites 72.1%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 72.1

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 72.1

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

   8. Applied rewrites 72.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. sqrt-div N/A

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

      11. sqrt-unprod N/A

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

      12. rem-square-sqrt N/A

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

      13. lift-*.f64 N/A

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

      14. sqrt-prod N/A

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

      15. *-rgt-identity N/A

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

      16. times-frac N/A

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

      17. frac-2neg N/A

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

      18. times-frac N/A

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

      19. distribute-lft-neg-in N/A

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

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

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

      21. metadata-eval N/A

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

      22. times-frac N/A

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

      23. lower-*.f64 N/A

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

   10. Applied rewrites 85.1%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -7.6 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(\frac{M \cdot \left(D \cdot \frac{0.5}{d}\right)}{\ell}, \frac{\left(-0.25 \cdot D\right) \cdot \left(M \cdot h\right)}{d}, 1\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\right) \cdot \mathsf{fma}\left(\frac{M \cdot \left(D \cdot \frac{0.5}{d}\right)}{\ell}, \left(\left(-0.25\right) \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{\sqrt{h}} \cdot \frac{-d}{\sqrt{\ell}}\right) \cdot \mathsf{fma}\left(\frac{M \cdot \left(D \cdot \frac{0.5}{d}\right)}{\ell}, \left(\left(-0.25\right) \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := \frac{\sqrt{\frac{h}{\ell}}}{\left|\ell\right|} \cdot \left(\frac{M\_m \cdot M\_m}{d} \cdot \left(\left(D\_m \cdot D\_m\right) \cdot 0.125\right)\right)\\ \mathbf{if}\;d \leq -1.45 \cdot 10^{+65}:\\ \;\;\;\;\sqrt{\frac{\frac{-d}{\ell}}{h}} \cdot t\_0\\ \mathbf{elif}\;d \leq -2.8 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -3.8 \cdot 10^{-75}:\\ \;\;\;\;\frac{\frac{t\_0}{\sqrt{-\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-311}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{M\_m \cdot \left(D\_m \cdot \frac{0.5}{d}\right)}{\ell}, \left(\left(-0.25\right) \cdot D\_m\right) \cdot \left(\frac{M\_m}{d} \cdot h\right), 1\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (- d)))
        (t_1
         (*
          (/ (sqrt (/ h l)) (fabs l))
          (* (/ (* M_m M_m) d) (* (* D_m D_m) 0.125)))))
   (if (<= d -1.45e+65)
     (* (sqrt (/ (/ (- d) l) h)) t_0)
     (if (<= d -2.8e+23)
       t_1
       (if (<= d -3.8e-75)
         (/ (/ t_0 (sqrt (- l))) (sqrt (/ h d)))
         (if (<= d -5e-311)
           t_1
           (*
            (/ d (sqrt (* l h)))
            (fma
             (/ (* M_m (* D_m (/ 0.5 d))) l)
             (* (* (- 0.25) D_m) (* (/ M_m d) h))
             1.0))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt(-d);
	double t_1 = (sqrt((h / l)) / fabs(l)) * (((M_m * M_m) / d) * ((D_m * D_m) * 0.125));
	double tmp;
	if (d <= -1.45e+65) {
		tmp = sqrt(((-d / l) / h)) * t_0;
	} else if (d <= -2.8e+23) {
		tmp = t_1;
	} else if (d <= -3.8e-75) {
		tmp = (t_0 / sqrt(-l)) / sqrt((h / d));
	} else if (d <= -5e-311) {
		tmp = t_1;
	} else {
		tmp = (d / sqrt((l * h))) * fma(((M_m * (D_m * (0.5 / d))) / l), ((-0.25 * D_m) * ((M_m / d) * h)), 1.0);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(-d))
	t_1 = Float64(Float64(sqrt(Float64(h / l)) / abs(l)) * Float64(Float64(Float64(M_m * M_m) / d) * Float64(Float64(D_m * D_m) * 0.125)))
	tmp = 0.0
	if (d <= -1.45e+65)
		tmp = Float64(sqrt(Float64(Float64(Float64(-d) / l) / h)) * t_0);
	elseif (d <= -2.8e+23)
		tmp = t_1;
	elseif (d <= -3.8e-75)
		tmp = Float64(Float64(t_0 / sqrt(Float64(-l))) / sqrt(Float64(h / d)));
	elseif (d <= -5e-311)
		tmp = t_1;
	else
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * fma(Float64(Float64(M_m * Float64(D_m * Float64(0.5 / d))) / l), Float64(Float64(Float64(-0.25) * D_m) * Float64(Float64(M_m / d) * h)), 1.0));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.45e+65], N[(N[Sqrt[N[(N[((-d) / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[d, -2.8e+23], t$95$1, If[LessEqual[d, -3.8e-75], N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-311], t$95$1, N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[((-0.25) * D$95$m), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.45e65

   1. Initial program 72.0%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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 1.9

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

   5. Applied rewrites 1.9%

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

   7. Applied rewrites 60.4%

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

   9. Applied rewrites 76.3%

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

   if -1.45e65 < d < -2.8e23 or -3.79999999999999994e-75 < d < -5.00000000000023e-311

   1. Initial program 51.4%

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

   3. Taylor expanded in h around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 37.4%

      \[\leadsto \color{blue}{\left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.3%

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

   if -2.8e23 < d < -3.79999999999999994e-75

   1. Initial program 86.5%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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 2.7

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

   5. Applied rewrites 2.7%

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

   7. Applied rewrites 61.1%

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

   9. Applied rewrites 64.3%

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

   if -5.00000000000023e-311 < d

   1. Initial program 71.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-sqrt.f64 71.3

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

   4. Applied rewrites 71.3%

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

   6. Applied rewrites 72.1%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 72.1

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 72.1

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

   8. Applied rewrites 72.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. sqrt-div N/A

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

      11. sqrt-unprod N/A

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

      12. rem-square-sqrt N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lower-/.f64 77.0

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

   10. Applied rewrites 77.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{\left(\frac{0.5}{d} \cdot D\right) \cdot M}{\ell}, -\left(\frac{M}{d} \cdot h\right) \cdot \left(0.25 \cdot D\right), 1\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.45 \cdot 10^{+65}:\\ \;\;\;\;\sqrt{\frac{\frac{-d}{\ell}}{h}} \cdot \sqrt{-d}\\ \mathbf{elif}\;d \leq -2.8 \cdot 10^{+23}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}}}{\left|\ell\right|} \cdot \left(\frac{M \cdot M}{d} \cdot \left(\left(D \cdot D\right) \cdot 0.125\right)\right)\\ \mathbf{elif}\;d \leq -3.8 \cdot 10^{-75}:\\ \;\;\;\;\frac{\frac{\sqrt{-d}}{\sqrt{-\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}}}{\left|\ell\right|} \cdot \left(\frac{M \cdot M}{d} \cdot \left(\left(D \cdot D\right) \cdot 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{M \cdot \left(D \cdot \frac{0.5}{d}\right)}{\ell}, \left(\left(-0.25\right) \cdot D\right) \cdot \left(\frac{M}{d} \cdot h\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.3% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < -3.999999999999988e-310

   1. Initial program 64.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 64.4

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

   4. Applied rewrites 64.4%

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

   6. Applied rewrites 65.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 65.0

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 65.0

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

   8. Applied rewrites 65.0%

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

   9. Taylor expanded in h around -inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. rem-square-sqrt N/A

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

       4. lower-*.f64 N/A

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

       5. mul-1-neg N/A

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

       6. lower-neg.f64 N/A

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

       7. lower-sqrt.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 72.4

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

   11. Applied rewrites 72.4%

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

   if -3.999999999999988e-310 < l

   1. Initial program 71.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-sqrt.f64 71.3

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

   4. Applied rewrites 71.3%

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

   6. Applied rewrites 72.1%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 72.1

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 72.1

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

   8. Applied rewrites 72.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. sqrt-div N/A

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

      11. sqrt-unprod N/A

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

      12. rem-square-sqrt N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lower-/.f64 77.0

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

   10. Applied rewrites 77.0%

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

4. Final simplification 75.0%

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

Alternative 10: 46.5% accurate, 9.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7 \cdot 10^{-295}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

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 7e-295)
   (* (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 <= 7e-295) {
		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 <= 7d-295) 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 <= 7e-295) {
		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 <= 7e-295:
		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 <= 7e-295)
		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 <= 7e-295)
		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, 7e-295], 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 < 6.99999999999999977e-295

   1. Initial program 65.3%

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

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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 44.2

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

   5. Applied rewrites 44.2%

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

   if 6.99999999999999977e-295 < l

   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

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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 44.1

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

   5. Applied rewrites 44.1%

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

   7. Applied rewrites 44.1%

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

   9. Applied rewrites 50.0%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7 \cdot 10^{-295}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.2% accurate, 10.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 10^{-188}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

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 1e-188) (* (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 <= 1e-188) {
		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 <= 1d-188) 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 <= 1e-188) {
		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 <= 1e-188:
		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 <= 1e-188)
		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 <= 1e-188)
		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, 1e-188], 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 < 9.9999999999999995e-189

   1. Initial program 67.5%

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

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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 41.6

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

   5. Applied rewrites 41.6%

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

   if 9.9999999999999995e-189 < l

   1. Initial program 69.2%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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 48.3

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

   5. Applied rewrites 48.3%

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

   7. Applied rewrites 48.3%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 10^{-188}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 26.2% 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 68.2%

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

3. Taylor expanded in h around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 27.5

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

5. Applied rewrites 27.5%

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

7. Applied rewrites 27.5%

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

Reproduce

herbie shell --seed 2024256 
(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)))))