Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.9% → 82.2%

Time: 17.8s

Alternatives: 22

Speedup: 2.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M\_m}{d}\\ t_1 := {\left(\frac{\ell}{h}\right)}^{-0.75}\\ t_2 := \mathsf{fma}\left(\frac{D\_m}{2} \cdot \frac{M\_m}{d}, \frac{\left(\frac{M\_m}{d} \cdot D\_m\right) \cdot \left(-0.5 \cdot h\right)}{2 \cdot \ell}, 1\right)\\ \mathbf{if}\;d \leq -4.2 \cdot 10^{-215}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{0.5 \cdot t\_0}{2}, \frac{\left(t\_0 \cdot h\right) \cdot -0.5}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 7.6 \cdot 10^{-211}:\\ \;\;\;\;\frac{\left(\left(D\_m \cdot D\_m\right) \cdot -0.125\right) \cdot \left(t\_1 \cdot \left(t\_1 \cdot \left(\frac{M\_m}{d} \cdot M\_m\right)\right)\right)}{h}\\ \mathbf{elif}\;d \leq 4.1 \cdot 10^{+195}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* D_m (/ M_m d)))
        (t_1 (pow (/ l h) -0.75))
        (t_2
         (fma
          (* (/ D_m 2.0) (/ M_m d))
          (/ (* (* (/ M_m d) D_m) (* -0.5 h)) (* 2.0 l))
          1.0)))
   (if (<= d -4.2e-215)
     (/
      (*
       (*
        (fma (/ (* 0.5 t_0) 2.0) (/ (* (* t_0 h) -0.5) l) 1.0)
        (sqrt (/ d l)))
       (sqrt (- d)))
      (sqrt (- h)))
     (if (<= d 7.6e-211)
       (/ (* (* (* D_m D_m) -0.125) (* t_1 (* t_1 (* (/ M_m d) M_m)))) h)
       (if (<= d 4.1e+195)
         (* (* (pow (/ d h) (pow 2.0 -1.0)) (/ (sqrt d) (sqrt l))) t_2)
         (* (* (/ (sqrt d) (sqrt h)) (pow (/ d l) (pow 2.0 -1.0))) 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 = D_m * (M_m / d);
	double t_1 = pow((l / h), -0.75);
	double t_2 = fma(((D_m / 2.0) * (M_m / d)), ((((M_m / d) * D_m) * (-0.5 * h)) / (2.0 * l)), 1.0);
	double tmp;
	if (d <= -4.2e-215) {
		tmp = ((fma(((0.5 * t_0) / 2.0), (((t_0 * h) * -0.5) / l), 1.0) * sqrt((d / l))) * sqrt(-d)) / sqrt(-h);
	} else if (d <= 7.6e-211) {
		tmp = (((D_m * D_m) * -0.125) * (t_1 * (t_1 * ((M_m / d) * M_m)))) / h;
	} else if (d <= 4.1e+195) {
		tmp = (pow((d / h), pow(2.0, -1.0)) * (sqrt(d) / sqrt(l))) * t_2;
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * pow((d / l), pow(2.0, -1.0))) * 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(D_m * Float64(M_m / d))
	t_1 = Float64(l / h) ^ -0.75
	t_2 = fma(Float64(Float64(D_m / 2.0) * Float64(M_m / d)), Float64(Float64(Float64(Float64(M_m / d) * D_m) * Float64(-0.5 * h)) / Float64(2.0 * l)), 1.0)
	tmp = 0.0
	if (d <= -4.2e-215)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(0.5 * t_0) / 2.0), Float64(Float64(Float64(t_0 * h) * -0.5) / l), 1.0) * sqrt(Float64(d / l))) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (d <= 7.6e-211)
		tmp = Float64(Float64(Float64(Float64(D_m * D_m) * -0.125) * Float64(t_1 * Float64(t_1 * Float64(Float64(M_m / d) * M_m)))) / h);
	elseif (d <= 4.1e+195)
		tmp = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * Float64(sqrt(d) / sqrt(l))) * t_2);
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * 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[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(l / h), $MachinePrecision], -0.75], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(D$95$m / 2.0), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(-0.5 * h), $MachinePrecision]), $MachinePrecision] / N[(2.0 * l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[d, -4.2e-215], N[(N[(N[(N[(N[(N[(0.5 * t$95$0), $MachinePrecision] / 2.0), $MachinePrecision] * N[(N[(N[(t$95$0 * h), $MachinePrecision] * -0.5), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.6e-211], N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(t$95$1 * N[(t$95$1 * N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 4.1e+195], N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -4.2e-215

   1. Initial program 70.5%

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

   4. Applied rewrites 81.4%

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

   6. Applied rewrites 88.7%

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

   if -4.2e-215 < d < 7.60000000000000023e-211

   1. Initial program 40.2%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 59.9%

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

   7. Applied rewrites 61.9%

      \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(M \cdot M\right), \frac{D \cdot D}{d} \cdot -0.125, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]

   8. Taylor expanded in d around 0

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

   10. Applied rewrites 56.8%

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

   12. Applied rewrites 66.8%

       \[\leadsto \frac{\left(\left(D \cdot D\right) \cdot -0.125\right) \cdot \left({\left(\frac{\ell}{h}\right)}^{-0.75} \cdot \left({\left(\frac{\ell}{h}\right)}^{-0.75} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right)}{h} \]

   if 7.60000000000000023e-211 < d < 4.1e195

   1. Initial program 72.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 71.9%

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

   6. Applied rewrites 74.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 74.6

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

   8. Applied rewrites 74.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. pow1/2 N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-sqrt.f64 85.4

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

   10. Applied rewrites 85.4%

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

   if 4.1e195 < d

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 72.2%

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

   6. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 72.2

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

   8. Applied rewrites 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. pow1/2 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow1/2 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. metadata-eval N/A

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

      16. pow1/2 N/A

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

      17. lower-sqrt.f64 95.2

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

   10. Applied rewrites 95.2%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.2 \cdot 10^{-215}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot \frac{M}{d}\right)}{2}, \frac{\left(\left(D \cdot \frac{M}{d}\right) \cdot h\right) \cdot -0.5}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 7.6 \cdot 10^{-211}:\\ \;\;\;\;\frac{\left(\left(D \cdot D\right) \cdot -0.125\right) \cdot \left({\left(\frac{\ell}{h}\right)}^{-0.75} \cdot \left({\left(\frac{\ell}{h}\right)}^{-0.75} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right)}{h}\\ \mathbf{elif}\;d \leq 4.1 \cdot 10^{+195}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \mathsf{fma}\left(\frac{D}{2} \cdot \frac{M}{d}, \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(-0.5 \cdot h\right)}{2 \cdot \ell}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \mathsf{fma}\left(\frac{D}{2} \cdot \frac{M}{d}, \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(-0.5 \cdot h\right)}{2 \cdot \ell}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 58.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{h}{\ell}}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-191} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(M\_m \cdot M\_m\right) \cdot -0.125\right) \cdot \frac{D\_m \cdot D\_m}{d}, \frac{h}{\ell} \cdot t\_1, t\_1 \cdot d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (*
            (* (pow 2.0 -1.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0))
            (/ h l)))))
        (t_1 (sqrt (/ h l))))
   (if (or (<= t_0 2e-191) (not (<= t_0 INFINITY)))
     (/
      (fma
       (* (* (* M_m M_m) -0.125) (/ (* D_m D_m) d))
       (* (/ h l) t_1)
       (* t_1 d))
      h)
     (* (sqrt (/ d l)) (sqrt (/ d h))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = sqrt((h / l));
	double tmp;
	if ((t_0 <= 2e-191) || !(t_0 <= ((double) INFINITY))) {
		tmp = fma((((M_m * M_m) * -0.125) * ((D_m * D_m) / d)), ((h / l) * t_1), (t_1 * d)) / h;
	} else {
		tmp = sqrt((d / l)) * sqrt((d / h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = sqrt(Float64(h / l))
	tmp = 0.0
	if ((t_0 <= 2e-191) || !(t_0 <= Inf))
		tmp = Float64(fma(Float64(Float64(Float64(M_m * M_m) * -0.125) * Float64(Float64(D_m * D_m) / d)), Float64(Float64(h / l) * t_1), Float64(t_1 * d)) / h);
	else
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[t$95$0, 2e-191], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$1 * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2e-191 or +inf.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 53.1%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 44.5%

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

   7. Applied rewrites 50.7%

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

   if 2e-191 < (*.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)))) < +inf.0

   1. Initial program 84.7%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 44.4

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

   5. Applied rewrites 44.4%

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

   7. Applied rewrites 84.0%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{-191} \lor \neg \left(\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \frac{D \cdot D}{d}, \frac{h}{\ell} \cdot \sqrt{\frac{h}{\ell}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left(\left(M\_m \cdot M\_m\right) \cdot -0.125\right) \cdot \frac{D\_m \cdot D\_m}{d}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \sqrt{\frac{h}{\ell}}\\ t_3 := t\_2 \cdot d\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-191}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, \frac{\frac{h}{\ell}}{\sqrt{\frac{\ell}{h}}}, t\_3\right)}{h}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, \frac{h}{\ell} \cdot t\_2, t\_3\right)}{h}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (* (* M_m M_m) -0.125) (/ (* D_m D_m) d)))
        (t_1
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (*
            (* (pow 2.0 -1.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0))
            (/ h l)))))
        (t_2 (sqrt (/ h l)))
        (t_3 (* t_2 d)))
   (if (<= t_1 2e-191)
     (/ (fma t_0 (/ (/ h l) (sqrt (/ l h))) t_3) h)
     (if (<= t_1 INFINITY)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (/ (fma t_0 (* (/ h l) t_2) t_3) h)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = ((M_m * M_m) * -0.125) * ((D_m * D_m) / d);
	double t_1 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = sqrt((h / l));
	double t_3 = t_2 * d;
	double tmp;
	if (t_1 <= 2e-191) {
		tmp = fma(t_0, ((h / l) / sqrt((l / h))), t_3) / h;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = fma(t_0, ((h / l) * t_2), t_3) / h;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(Float64(M_m * M_m) * -0.125) * Float64(Float64(D_m * D_m) / d))
	t_1 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_2 = sqrt(Float64(h / l))
	t_3 = Float64(t_2 * d)
	tmp = 0.0
	if (t_1 <= 2e-191)
		tmp = Float64(fma(t_0, Float64(Float64(h / l) / sqrt(Float64(l / h))), t_3) / h);
	elseif (t_1 <= Inf)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(fma(t_0, Float64(Float64(h / l) * t_2), t_3) / h);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * d), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-191], N[(N[(t$95$0 * N[(N[(h / l), $MachinePrecision] / N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * t$95$2), $MachinePrecision] + t$95$3), $MachinePrecision] / h), $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)))) < 2e-191

   1. Initial program 78.4%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 50.5%

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

   7. Applied rewrites 58.8%

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

   if 2e-191 < (*.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)))) < +inf.0

   1. Initial program 84.7%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 44.4

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

   5. Applied rewrites 44.4%

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

   7. Applied rewrites 84.0%

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

   if +inf.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 0.0%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 31.8%

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

   7. Applied rewrites 33.8%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{-191}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \frac{D \cdot D}{d}, \frac{\frac{h}{\ell}}{\sqrt{\frac{\ell}{h}}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \frac{D \cdot D}{d}, \frac{h}{\ell} \cdot \sqrt{\frac{h}{\ell}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M\_m}{d}\\ t_1 := \mathsf{fma}\left(\frac{D\_m}{2} \cdot \frac{M\_m}{d}, \frac{\left(\frac{M\_m}{d} \cdot D\_m\right) \cdot \left(-0.5 \cdot h\right)}{2 \cdot \ell}, 1\right)\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{0.5 \cdot t\_0}{2}, \frac{\left(t\_0 \cdot h\right) \cdot -0.5}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{-134}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot t\_1\\ \end{array} \end{array} \]

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 (* D_m (/ M_m d)))
        (t_1
         (fma
          (* (/ D_m 2.0) (/ M_m d))
          (/ (* (* (/ M_m d) D_m) (* -0.5 h)) (* 2.0 l))
          1.0)))
   (if (<= l -5e-310)
     (/
      (*
       (*
        (fma (/ (* 0.5 t_0) 2.0) (/ (* (* t_0 h) -0.5) l) 1.0)
        (sqrt (/ d l)))
       (sqrt (- d)))
      (sqrt (- h)))
     (if (<= l 2.9e-134)
       (* (* (pow (/ d h) (pow 2.0 -1.0)) (/ (sqrt d) (sqrt l))) t_1)
       (* (* (/ (sqrt d) (sqrt h)) (pow (/ d l) (pow 2.0 -1.0))) 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 = D_m * (M_m / d);
	double t_1 = fma(((D_m / 2.0) * (M_m / d)), ((((M_m / d) * D_m) * (-0.5 * h)) / (2.0 * l)), 1.0);
	double tmp;
	if (l <= -5e-310) {
		tmp = ((fma(((0.5 * t_0) / 2.0), (((t_0 * h) * -0.5) / l), 1.0) * sqrt((d / l))) * sqrt(-d)) / sqrt(-h);
	} else if (l <= 2.9e-134) {
		tmp = (pow((d / h), pow(2.0, -1.0)) * (sqrt(d) / sqrt(l))) * t_1;
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * pow((d / l), pow(2.0, -1.0))) * 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(D_m * Float64(M_m / d))
	t_1 = fma(Float64(Float64(D_m / 2.0) * Float64(M_m / d)), Float64(Float64(Float64(Float64(M_m / d) * D_m) * Float64(-0.5 * h)) / Float64(2.0 * l)), 1.0)
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(0.5 * t_0) / 2.0), Float64(Float64(Float64(t_0 * h) * -0.5) / l), 1.0) * sqrt(Float64(d / l))) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (l <= 2.9e-134)
		tmp = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * Float64(sqrt(d) / sqrt(l))) * t_1);
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * 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[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(D$95$m / 2.0), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(-0.5 * h), $MachinePrecision]), $MachinePrecision] / N[(2.0 * l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[l, -5e-310], N[(N[(N[(N[(N[(N[(0.5 * t$95$0), $MachinePrecision] / 2.0), $MachinePrecision] * N[(N[(N[(t$95$0 * h), $MachinePrecision] * -0.5), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.9e-134], N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 63.3%

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

   4. Applied rewrites 76.4%

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

   6. Applied rewrites 82.3%

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

   if -4.999999999999985e-310 < l < 2.89999999999999993e-134

   1. Initial program 60.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 65.4%

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

   6. Applied rewrites 57.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 61.9

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

   8. Applied rewrites 61.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. pow1/2 N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-sqrt.f64 80.7

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

   10. Applied rewrites 80.7%

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

   if 2.89999999999999993e-134 < l

   1. Initial program 69.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 68.0%

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

   6. Applied rewrites 71.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 71.2

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

   8. Applied rewrites 71.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. pow1/2 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow1/2 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. metadata-eval N/A

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

      16. pow1/2 N/A

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

      17. lower-sqrt.f64 84.4

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

   10. Applied rewrites 84.4%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot \frac{M}{d}\right)}{2}, \frac{\left(\left(D \cdot \frac{M}{d}\right) \cdot h\right) \cdot -0.5}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{-134}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \mathsf{fma}\left(\frac{D}{2} \cdot \frac{M}{d}, \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(-0.5 \cdot h\right)}{2 \cdot \ell}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \mathsf{fma}\left(\frac{D}{2} \cdot \frac{M}{d}, \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(-0.5 \cdot h\right)}{2 \cdot \ell}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M\_m}{d}\\ \mathbf{if}\;d \leq -3.8 \cdot 10^{-215}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{0.5 \cdot t\_0}{2}, \frac{\left(t\_0 \cdot h\right) \cdot -0.5}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 6 \cdot 10^{-210}:\\ \;\;\;\;\frac{\left(\frac{M\_m}{d} \cdot M\_m\right) \cdot \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\left(D\_m \cdot D\_m\right) \cdot -0.125\right)\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \mathsf{fma}\left(\frac{D\_m}{2} \cdot \frac{M\_m}{d}, \frac{\left(\frac{M\_m}{d} \cdot D\_m\right) \cdot \left(-0.5 \cdot h\right)}{2 \cdot \ell}, 1\right)\\ \end{array} \end{array} \]

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 (* D_m (/ M_m d))))
   (if (<= d -3.8e-215)
     (/
      (*
       (*
        (fma (/ (* 0.5 t_0) 2.0) (/ (* (* t_0 h) -0.5) l) 1.0)
        (sqrt (/ d l)))
       (sqrt (- d)))
      (sqrt (- h)))
     (if (<= d 6e-210)
       (/ (* (* (/ M_m d) M_m) (* (pow (/ h l) 1.5) (* (* D_m D_m) -0.125))) h)
       (*
        (* (/ (sqrt d) (sqrt h)) (pow (/ d l) (pow 2.0 -1.0)))
        (fma
         (* (/ D_m 2.0) (/ M_m d))
         (/ (* (* (/ M_m d) D_m) (* -0.5 h)) (* 2.0 l))
         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 = D_m * (M_m / d);
	double tmp;
	if (d <= -3.8e-215) {
		tmp = ((fma(((0.5 * t_0) / 2.0), (((t_0 * h) * -0.5) / l), 1.0) * sqrt((d / l))) * sqrt(-d)) / sqrt(-h);
	} else if (d <= 6e-210) {
		tmp = (((M_m / d) * M_m) * (pow((h / l), 1.5) * ((D_m * D_m) * -0.125))) / h;
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * pow((d / l), pow(2.0, -1.0))) * fma(((D_m / 2.0) * (M_m / d)), ((((M_m / d) * D_m) * (-0.5 * h)) / (2.0 * l)), 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 = Float64(D_m * Float64(M_m / d))
	tmp = 0.0
	if (d <= -3.8e-215)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(0.5 * t_0) / 2.0), Float64(Float64(Float64(t_0 * h) * -0.5) / l), 1.0) * sqrt(Float64(d / l))) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (d <= 6e-210)
		tmp = Float64(Float64(Float64(Float64(M_m / d) * M_m) * Float64((Float64(h / l) ^ 1.5) * Float64(Float64(D_m * D_m) * -0.125))) / h);
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * fma(Float64(Float64(D_m / 2.0) * Float64(M_m / d)), Float64(Float64(Float64(Float64(M_m / d) * D_m) * Float64(-0.5 * h)) / Float64(2.0 * l)), 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[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.8e-215], N[(N[(N[(N[(N[(N[(0.5 * t$95$0), $MachinePrecision] / 2.0), $MachinePrecision] * N[(N[(N[(t$95$0 * h), $MachinePrecision] * -0.5), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6e-210], N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D$95$m / 2.0), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(-0.5 * h), $MachinePrecision]), $MachinePrecision] / N[(2.0 * l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.79999999999999977e-215

   1. Initial program 70.5%

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

   4. Applied rewrites 81.4%

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

   6. Applied rewrites 88.7%

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

   if -3.79999999999999977e-215 < d < 6.0000000000000003e-210

   1. Initial program 39.5%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 58.8%

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

   7. Applied rewrites 62.6%

      \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(M \cdot M\right), \frac{D \cdot D}{d} \cdot -0.125, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]

   8. Taylor expanded in d around 0

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

   10. Applied rewrites 55.8%

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

   12. Applied rewrites 63.8%

       \[\leadsto \frac{\left(\frac{M}{d} \cdot M\right) \cdot \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\left(D \cdot D\right) \cdot -0.125\right)\right)}{h} \]

   if 6.0000000000000003e-210 < d

   1. Initial program 72.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 72.6%

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

   6. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 74.7

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

   8. Applied rewrites 74.7%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. pow1/2 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow1/2 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. metadata-eval N/A

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

      16. pow1/2 N/A

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

      17. lower-sqrt.f64 83.3

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

   10. Applied rewrites 83.3%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.8 \cdot 10^{-215}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot \frac{M}{d}\right)}{2}, \frac{\left(\left(D \cdot \frac{M}{d}\right) \cdot h\right) \cdot -0.5}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 6 \cdot 10^{-210}:\\ \;\;\;\;\frac{\left(\frac{M}{d} \cdot M\right) \cdot \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\left(D \cdot D\right) \cdot -0.125\right)\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \mathsf{fma}\left(\frac{D}{2} \cdot \frac{M}{d}, \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(-0.5 \cdot h\right)}{2 \cdot \ell}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M\_m}{d}\\ t_1 := \mathsf{fma}\left({\left(\frac{M\_m}{d} \cdot D\_m\right)}^{2} \cdot 0.25, \frac{h}{\ell} \cdot -0.5, 1\right)\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -3.8 \cdot 10^{-215}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{0.5 \cdot t\_0}{2}, \frac{\left(t\_0 \cdot h\right) \cdot -0.5}{\ell}, 1\right) \cdot t\_2\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 1.02 \cdot 10^{-241}:\\ \;\;\;\;\frac{\left(\frac{M\_m}{d} \cdot M\_m\right) \cdot \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\left(D\_m \cdot D\_m\right) \cdot -0.125\right)\right)}{h}\\ \mathbf{elif}\;d \leq 5.3 \cdot 10^{-104}:\\ \;\;\;\;t\_1 \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq 6.6 \cdot 10^{+146}:\\ \;\;\;\;\left(t\_1 \cdot \sqrt{\frac{d}{h}}\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* D_m (/ M_m d)))
        (t_1 (fma (* (pow (* (/ M_m d) D_m) 2.0) 0.25) (* (/ h l) -0.5) 1.0))
        (t_2 (sqrt (/ d l))))
   (if (<= d -3.8e-215)
     (/
      (*
       (* (fma (/ (* 0.5 t_0) 2.0) (/ (* (* t_0 h) -0.5) l) 1.0) t_2)
       (sqrt (- d)))
      (sqrt (- h)))
     (if (<= d 1.02e-241)
       (/ (* (* (/ M_m d) M_m) (* (pow (/ h l) 1.5) (* (* D_m D_m) -0.125))) h)
       (if (<= d 5.3e-104)
         (* t_1 (/ d (sqrt (* l h))))
         (if (<= d 6.6e+146)
           (* (* t_1 (sqrt (/ d h))) t_2)
           (/ d (* (sqrt l) (sqrt h)))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = D_m * (M_m / d);
	double t_1 = fma((pow(((M_m / d) * D_m), 2.0) * 0.25), ((h / l) * -0.5), 1.0);
	double t_2 = sqrt((d / l));
	double tmp;
	if (d <= -3.8e-215) {
		tmp = ((fma(((0.5 * t_0) / 2.0), (((t_0 * h) * -0.5) / l), 1.0) * t_2) * sqrt(-d)) / sqrt(-h);
	} else if (d <= 1.02e-241) {
		tmp = (((M_m / d) * M_m) * (pow((h / l), 1.5) * ((D_m * D_m) * -0.125))) / h;
	} else if (d <= 5.3e-104) {
		tmp = t_1 * (d / sqrt((l * h)));
	} else if (d <= 6.6e+146) {
		tmp = (t_1 * sqrt((d / h))) * t_2;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(D_m * Float64(M_m / d))
	t_1 = fma(Float64((Float64(Float64(M_m / d) * D_m) ^ 2.0) * 0.25), Float64(Float64(h / l) * -0.5), 1.0)
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (d <= -3.8e-215)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(0.5 * t_0) / 2.0), Float64(Float64(Float64(t_0 * h) * -0.5) / l), 1.0) * t_2) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (d <= 1.02e-241)
		tmp = Float64(Float64(Float64(Float64(M_m / d) * M_m) * Float64((Float64(h / l) ^ 1.5) * Float64(Float64(D_m * D_m) * -0.125))) / h);
	elseif (d <= 5.3e-104)
		tmp = Float64(t_1 * Float64(d / sqrt(Float64(l * h))));
	elseif (d <= 6.6e+146)
		tmp = Float64(Float64(t_1 * sqrt(Float64(d / h))) * t_2);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 0.25), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -3.8e-215], N[(N[(N[(N[(N[(N[(0.5 * t$95$0), $MachinePrecision] / 2.0), $MachinePrecision] * N[(N[(N[(t$95$0 * h), $MachinePrecision] * -0.5), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.02e-241], N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 5.3e-104], N[(t$95$1 * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6.6e+146], N[(N[(t$95$1 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -3.79999999999999977e-215

   1. Initial program 70.5%

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

   4. Applied rewrites 81.4%

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

   6. Applied rewrites 88.7%

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

   if -3.79999999999999977e-215 < d < 1.01999999999999994e-241

   1. Initial program 40.2%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 59.0%

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

   7. Applied rewrites 61.3%

      \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(M \cdot M\right), \frac{D \cdot D}{d} \cdot -0.125, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]

   8. Taylor expanded in d around 0

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

   10. Applied rewrites 56.9%

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

   12. Applied rewrites 63.9%

       \[\leadsto \frac{\left(\frac{M}{d} \cdot M\right) \cdot \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\left(D \cdot D\right) \cdot -0.125\right)\right)}{h} \]

   if 1.01999999999999994e-241 < d < 5.30000000000000018e-104

   1. Initial program 49.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 45.6%

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

   8. Applied rewrites 78.1%

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

   if 5.30000000000000018e-104 < d < 6.60000000000000032e146

   1. Initial program 81.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 80.1%

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

   6. Applied rewrites 86.4%

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

   8. Applied rewrites 80.1%

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

   if 6.60000000000000032e146 < d

   1. Initial program 68.9%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 79.7

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

   5. Applied rewrites 79.7%

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

   7. Applied rewrites 79.6%

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

   9. Applied rewrites 90.8%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 7: 80.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M\_m}{d}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -3.8 \cdot 10^{-215}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{0.5 \cdot t\_0}{2}, \frac{\left(t\_0 \cdot h\right) \cdot -0.5}{\ell}, 1\right) \cdot t\_1\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 6 \cdot 10^{-210}:\\ \;\;\;\;\frac{\left(\frac{M\_m}{d} \cdot M\_m\right) \cdot \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\left(D\_m \cdot D\_m\right) \cdot -0.125\right)\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \left(\mathsf{fma}\left({\left(\frac{M\_m}{d} \cdot D\_m\right)}^{2} \cdot 0.25, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot t\_1\right)}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* D_m (/ M_m d))) (t_1 (sqrt (/ d l))))
   (if (<= d -3.8e-215)
     (/
      (*
       (* (fma (/ (* 0.5 t_0) 2.0) (/ (* (* t_0 h) -0.5) l) 1.0) t_1)
       (sqrt (- d)))
      (sqrt (- h)))
     (if (<= d 6e-210)
       (/ (* (* (/ M_m d) M_m) (* (pow (/ h l) 1.5) (* (* D_m D_m) -0.125))) h)
       (/
        (*
         (sqrt d)
         (*
          (fma (* (pow (* (/ M_m d) D_m) 2.0) 0.25) (* (/ h l) -0.5) 1.0)
          t_1))
        (sqrt h))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = D_m * (M_m / d);
	double t_1 = sqrt((d / l));
	double tmp;
	if (d <= -3.8e-215) {
		tmp = ((fma(((0.5 * t_0) / 2.0), (((t_0 * h) * -0.5) / l), 1.0) * t_1) * sqrt(-d)) / sqrt(-h);
	} else if (d <= 6e-210) {
		tmp = (((M_m / d) * M_m) * (pow((h / l), 1.5) * ((D_m * D_m) * -0.125))) / h;
	} else {
		tmp = (sqrt(d) * (fma((pow(((M_m / d) * D_m), 2.0) * 0.25), ((h / l) * -0.5), 1.0) * t_1)) / sqrt(h);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(D_m * Float64(M_m / d))
	t_1 = sqrt(Float64(d / l))
	tmp = 0.0
	if (d <= -3.8e-215)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(0.5 * t_0) / 2.0), Float64(Float64(Float64(t_0 * h) * -0.5) / l), 1.0) * t_1) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (d <= 6e-210)
		tmp = Float64(Float64(Float64(Float64(M_m / d) * M_m) * Float64((Float64(h / l) ^ 1.5) * Float64(Float64(D_m * D_m) * -0.125))) / h);
	else
		tmp = Float64(Float64(sqrt(d) * Float64(fma(Float64((Float64(Float64(M_m / d) * D_m) ^ 2.0) * 0.25), Float64(Float64(h / l) * -0.5), 1.0) * t_1)) / sqrt(h));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -3.8e-215], N[(N[(N[(N[(N[(N[(0.5 * t$95$0), $MachinePrecision] / 2.0), $MachinePrecision] * N[(N[(N[(t$95$0 * h), $MachinePrecision] * -0.5), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6e-210], N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] * N[(N[(N[(N[Power[N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 0.25), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.79999999999999977e-215

   1. Initial program 70.5%

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

   4. Applied rewrites 81.4%

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

   6. Applied rewrites 88.7%

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

   if -3.79999999999999977e-215 < d < 6.0000000000000003e-210

   1. Initial program 39.5%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 58.8%

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

   7. Applied rewrites 62.6%

      \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(M \cdot M\right), \frac{D \cdot D}{d} \cdot -0.125, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]

   8. Taylor expanded in d around 0

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

   10. Applied rewrites 55.8%

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

   12. Applied rewrites 63.8%

       \[\leadsto \frac{\left(\frac{M}{d} \cdot M\right) \cdot \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\left(D \cdot D\right) \cdot -0.125\right)\right)}{h} \]

   if 6.0000000000000003e-210 < d

   1. Initial program 72.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 72.6%

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

   6. Applied rewrites 73.5%

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

   8. Applied rewrites 79.4%

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

Alternative 8: 79.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M\_m}{d}\\ \mathbf{if}\;d \leq -3.8 \cdot 10^{-215}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{0.5 \cdot t\_0}{2}, \frac{\left(t\_0 \cdot h\right) \cdot -0.5}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 3.9 \cdot 10^{-224}:\\ \;\;\;\;\frac{\left(\frac{M\_m}{d} \cdot M\_m\right) \cdot \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\left(D\_m \cdot D\_m\right) \cdot -0.125\right)\right)}{h}\\ \mathbf{elif}\;d \leq 8.8 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}} \cdot \mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, 0.25 \cdot {\left(\frac{\frac{d}{D\_m}}{M\_m}\right)}^{-2}, 1\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* D_m (/ M_m d))))
   (if (<= d -3.8e-215)
     (/
      (*
       (*
        (fma (/ (* 0.5 t_0) 2.0) (/ (* (* t_0 h) -0.5) l) 1.0)
        (sqrt (/ d l)))
       (sqrt (- d)))
      (sqrt (- h)))
     (if (<= d 3.9e-224)
       (/ (* (* (/ M_m d) M_m) (* (pow (/ h l) 1.5) (* (* D_m D_m) -0.125))) h)
       (if (<= d 8.8e+126)
         (/
          (*
           (/ d (sqrt h))
           (fma (* -0.5 (/ h l)) (* 0.25 (pow (/ (/ d D_m) M_m) -2.0)) 1.0))
          (sqrt l))
         (/ d (* (sqrt l) (sqrt h))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = D_m * (M_m / d);
	double tmp;
	if (d <= -3.8e-215) {
		tmp = ((fma(((0.5 * t_0) / 2.0), (((t_0 * h) * -0.5) / l), 1.0) * sqrt((d / l))) * sqrt(-d)) / sqrt(-h);
	} else if (d <= 3.9e-224) {
		tmp = (((M_m / d) * M_m) * (pow((h / l), 1.5) * ((D_m * D_m) * -0.125))) / h;
	} else if (d <= 8.8e+126) {
		tmp = ((d / sqrt(h)) * fma((-0.5 * (h / l)), (0.25 * pow(((d / D_m) / M_m), -2.0)), 1.0)) / sqrt(l);
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(D_m * Float64(M_m / d))
	tmp = 0.0
	if (d <= -3.8e-215)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(0.5 * t_0) / 2.0), Float64(Float64(Float64(t_0 * h) * -0.5) / l), 1.0) * sqrt(Float64(d / l))) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (d <= 3.9e-224)
		tmp = Float64(Float64(Float64(Float64(M_m / d) * M_m) * Float64((Float64(h / l) ^ 1.5) * Float64(Float64(D_m * D_m) * -0.125))) / h);
	elseif (d <= 8.8e+126)
		tmp = Float64(Float64(Float64(d / sqrt(h)) * fma(Float64(-0.5 * Float64(h / l)), Float64(0.25 * (Float64(Float64(d / D_m) / M_m) ^ -2.0)), 1.0)) / sqrt(l));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.8e-215], N[(N[(N[(N[(N[(N[(0.5 * t$95$0), $MachinePrecision] / 2.0), $MachinePrecision] * N[(N[(N[(t$95$0 * h), $MachinePrecision] * -0.5), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.9e-224], N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 8.8e+126], N[(N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[(0.25 * N[Power[N[(N[(d / D$95$m), $MachinePrecision] / M$95$m), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.79999999999999977e-215

   1. Initial program 70.5%

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

   4. Applied rewrites 81.4%

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

   6. Applied rewrites 88.7%

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

   if -3.79999999999999977e-215 < d < 3.8999999999999998e-224

   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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 58.2%

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

   7. Applied rewrites 60.3%

      \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(M \cdot M\right), \frac{D \cdot D}{d} \cdot -0.125, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]

   8. Taylor expanded in d around 0

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

   10. Applied rewrites 55.0%

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

   12. Applied rewrites 63.5%

       \[\leadsto \frac{\left(\frac{M}{d} \cdot M\right) \cdot \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\left(D \cdot D\right) \cdot -0.125\right)\right)}{h} \]

   if 3.8999999999999998e-224 < d < 8.79999999999999994e126

   1. Initial program 72.9%

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 86.0%

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

   if 8.79999999999999994e126 < d

   1. Initial program 68.6%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 78.6

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

   5. Applied rewrites 78.6%

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

   7. Applied rewrites 78.5%

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

   9. Applied rewrites 88.9%

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

Alternative 9: 75.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M\_m}{d}\\ \mathbf{if}\;d \leq -3.8 \cdot 10^{-215}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{0.5 \cdot t\_0}{2}, \frac{\left(t\_0 \cdot h\right) \cdot -0.5}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 1.02 \cdot 10^{-241}:\\ \;\;\;\;\frac{\left(\frac{M\_m}{d} \cdot M\_m\right) \cdot \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\left(D\_m \cdot D\_m\right) \cdot -0.125\right)\right)}{h}\\ \mathbf{elif}\;d \leq 1.02 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{M\_m}{d} \cdot D\_m\right)}^{2} \cdot 0.25, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* D_m (/ M_m d))))
   (if (<= d -3.8e-215)
     (/
      (*
       (*
        (fma (/ (* 0.5 t_0) 2.0) (/ (* (* t_0 h) -0.5) l) 1.0)
        (sqrt (/ d l)))
       (sqrt (- d)))
      (sqrt (- h)))
     (if (<= d 1.02e-241)
       (/ (* (* (/ M_m d) M_m) (* (pow (/ h l) 1.5) (* (* D_m D_m) -0.125))) h)
       (if (<= d 1.02e+122)
         (*
          (fma (* (pow (* (/ M_m d) D_m) 2.0) 0.25) (* (/ h l) -0.5) 1.0)
          (/ d (sqrt (* l h))))
         (/ d (* (sqrt l) (sqrt h))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = D_m * (M_m / d);
	double tmp;
	if (d <= -3.8e-215) {
		tmp = ((fma(((0.5 * t_0) / 2.0), (((t_0 * h) * -0.5) / l), 1.0) * sqrt((d / l))) * sqrt(-d)) / sqrt(-h);
	} else if (d <= 1.02e-241) {
		tmp = (((M_m / d) * M_m) * (pow((h / l), 1.5) * ((D_m * D_m) * -0.125))) / h;
	} else if (d <= 1.02e+122) {
		tmp = fma((pow(((M_m / d) * D_m), 2.0) * 0.25), ((h / l) * -0.5), 1.0) * (d / sqrt((l * h)));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(D_m * Float64(M_m / d))
	tmp = 0.0
	if (d <= -3.8e-215)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(0.5 * t_0) / 2.0), Float64(Float64(Float64(t_0 * h) * -0.5) / l), 1.0) * sqrt(Float64(d / l))) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (d <= 1.02e-241)
		tmp = Float64(Float64(Float64(Float64(M_m / d) * M_m) * Float64((Float64(h / l) ^ 1.5) * Float64(Float64(D_m * D_m) * -0.125))) / h);
	elseif (d <= 1.02e+122)
		tmp = Float64(fma(Float64((Float64(Float64(M_m / d) * D_m) ^ 2.0) * 0.25), Float64(Float64(h / l) * -0.5), 1.0) * Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.8e-215], N[(N[(N[(N[(N[(N[(0.5 * t$95$0), $MachinePrecision] / 2.0), $MachinePrecision] * N[(N[(N[(t$95$0 * h), $MachinePrecision] * -0.5), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.02e-241], N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 1.02e+122], N[(N[(N[(N[Power[N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 0.25), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.79999999999999977e-215

   1. Initial program 70.5%

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

   4. Applied rewrites 81.4%

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

   6. Applied rewrites 88.7%

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

   if -3.79999999999999977e-215 < d < 1.01999999999999994e-241

   1. Initial program 40.2%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 59.0%

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

   7. Applied rewrites 61.3%

      \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(M \cdot M\right), \frac{D \cdot D}{d} \cdot -0.125, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]

   8. Taylor expanded in d around 0

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

   10. Applied rewrites 56.9%

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

   12. Applied rewrites 63.9%

       \[\leadsto \frac{\left(\frac{M}{d} \cdot M\right) \cdot \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\left(D \cdot D\right) \cdot -0.125\right)\right)}{h} \]

   if 1.01999999999999994e-241 < d < 1.0200000000000001e122

   1. Initial program 70.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 68.9%

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

   6. Applied rewrites 71.9%

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

   8. Applied rewrites 73.2%

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

   if 1.0200000000000001e122 < d

   1. Initial program 69.4%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 79.2

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

   5. Applied rewrites 79.2%

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

   7. Applied rewrites 79.1%

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

   9. Applied rewrites 89.1%

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

Alternative 10: 74.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M\_m}{d}\\ t_1 := \frac{d}{\sqrt{\ell \cdot h}}\\ t_2 := D\_m \cdot \frac{0.5}{d}\\ \mathbf{if}\;d \leq -3.8 \cdot 10^{-215}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{0.5 \cdot t\_0}{2}, \frac{\left(t\_0 \cdot h\right) \cdot -0.5}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 1.32 \cdot 10^{-207}:\\ \;\;\;\;\frac{\left(\frac{M\_m}{d} \cdot M\_m\right) \cdot \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\left(D\_m \cdot D\_m\right) \cdot -0.125\right)\right)}{h}\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(t\_1 \cdot \left(\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot M\_m\right) \cdot t\_2\right), t\_2 \cdot M\_m, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* D_m (/ M_m d)))
        (t_1 (/ d (sqrt (* l h))))
        (t_2 (* D_m (/ 0.5 d))))
   (if (<= d -3.8e-215)
     (/
      (*
       (*
        (fma (/ (* 0.5 t_0) 2.0) (/ (* (* t_0 h) -0.5) l) 1.0)
        (sqrt (/ d l)))
       (sqrt (- d)))
      (sqrt (- h)))
     (if (<= d 1.32e-207)
       (/ (* (* (/ M_m d) M_m) (* (pow (/ h l) 1.5) (* (* D_m D_m) -0.125))) h)
       (if (<= d 1.6e+115)
         (fma (* t_1 (* (* (* (/ h l) -0.5) M_m) t_2)) (* t_2 M_m) t_1)
         (/ d (* (sqrt l) (sqrt h))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = D_m * (M_m / d);
	double t_1 = d / sqrt((l * h));
	double t_2 = D_m * (0.5 / d);
	double tmp;
	if (d <= -3.8e-215) {
		tmp = ((fma(((0.5 * t_0) / 2.0), (((t_0 * h) * -0.5) / l), 1.0) * sqrt((d / l))) * sqrt(-d)) / sqrt(-h);
	} else if (d <= 1.32e-207) {
		tmp = (((M_m / d) * M_m) * (pow((h / l), 1.5) * ((D_m * D_m) * -0.125))) / h;
	} else if (d <= 1.6e+115) {
		tmp = fma((t_1 * ((((h / l) * -0.5) * M_m) * t_2)), (t_2 * M_m), t_1);
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(D_m * Float64(M_m / d))
	t_1 = Float64(d / sqrt(Float64(l * h)))
	t_2 = Float64(D_m * Float64(0.5 / d))
	tmp = 0.0
	if (d <= -3.8e-215)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(0.5 * t_0) / 2.0), Float64(Float64(Float64(t_0 * h) * -0.5) / l), 1.0) * sqrt(Float64(d / l))) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (d <= 1.32e-207)
		tmp = Float64(Float64(Float64(Float64(M_m / d) * M_m) * Float64((Float64(h / l) ^ 1.5) * Float64(Float64(D_m * D_m) * -0.125))) / h);
	elseif (d <= 1.6e+115)
		tmp = fma(Float64(t_1 * Float64(Float64(Float64(Float64(h / l) * -0.5) * M_m) * t_2)), Float64(t_2 * M_m), t_1);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.8e-215], N[(N[(N[(N[(N[(N[(0.5 * t$95$0), $MachinePrecision] / 2.0), $MachinePrecision] * N[(N[(N[(t$95$0 * h), $MachinePrecision] * -0.5), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.32e-207], N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 1.6e+115], N[(N[(t$95$1 * N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * M$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * M$95$m), $MachinePrecision] + t$95$1), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.79999999999999977e-215

   1. Initial program 70.5%

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

   4. Applied rewrites 81.4%

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

   6. Applied rewrites 88.7%

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

   if -3.79999999999999977e-215 < d < 1.32000000000000003e-207

   1. Initial program 40.7%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 57.6%

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

   7. Applied rewrites 63.3%

      \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(M \cdot M\right), \frac{D \cdot D}{d} \cdot -0.125, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]

   8. Taylor expanded in d around 0

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

   10. Applied rewrites 54.7%

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

   12. Applied rewrites 64.5%

       \[\leadsto \frac{\left(\frac{M}{d} \cdot M\right) \cdot \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\left(D \cdot D\right) \cdot -0.125\right)\right)}{h} \]

   if 1.32000000000000003e-207 < d < 1.6e115

   1. Initial program 73.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 72.3%

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

   6. Applied rewrites 75.7%

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

   8. Applied rewrites 70.0%

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

   if 1.6e115 < d

   1. Initial program 69.4%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 79.2

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

   5. Applied rewrites 79.2%

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

   7. Applied rewrites 79.1%

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

   9. Applied rewrites 89.1%

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

Alternative 11: 74.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M\_m}{d}\\ t_1 := \frac{d}{\sqrt{\ell \cdot h}}\\ t_2 := D\_m \cdot \frac{0.5}{d}\\ \mathbf{if}\;d \leq -5.7 \cdot 10^{-214}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{0.5 \cdot t\_0}{2}, \frac{\left(t\_0 \cdot h\right) \cdot -0.5}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 1.32 \cdot 10^{-207}:\\ \;\;\;\;\frac{D\_m \cdot \left(D\_m \cdot \left(\left(-0.125 \cdot {\left(\frac{h}{\ell}\right)}^{1.5}\right) \cdot \left(\frac{M\_m}{d} \cdot M\_m\right)\right)\right)}{h}\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(t\_1 \cdot \left(\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot M\_m\right) \cdot t\_2\right), t\_2 \cdot M\_m, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* D_m (/ M_m d)))
        (t_1 (/ d (sqrt (* l h))))
        (t_2 (* D_m (/ 0.5 d))))
   (if (<= d -5.7e-214)
     (/
      (*
       (*
        (fma (/ (* 0.5 t_0) 2.0) (/ (* (* t_0 h) -0.5) l) 1.0)
        (sqrt (/ d l)))
       (sqrt (- d)))
      (sqrt (- h)))
     (if (<= d 1.32e-207)
       (/ (* D_m (* D_m (* (* -0.125 (pow (/ h l) 1.5)) (* (/ M_m d) M_m)))) h)
       (if (<= d 1.6e+115)
         (fma (* t_1 (* (* (* (/ h l) -0.5) M_m) t_2)) (* t_2 M_m) t_1)
         (/ d (* (sqrt l) (sqrt h))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = D_m * (M_m / d);
	double t_1 = d / sqrt((l * h));
	double t_2 = D_m * (0.5 / d);
	double tmp;
	if (d <= -5.7e-214) {
		tmp = ((fma(((0.5 * t_0) / 2.0), (((t_0 * h) * -0.5) / l), 1.0) * sqrt((d / l))) * sqrt(-d)) / sqrt(-h);
	} else if (d <= 1.32e-207) {
		tmp = (D_m * (D_m * ((-0.125 * pow((h / l), 1.5)) * ((M_m / d) * M_m)))) / h;
	} else if (d <= 1.6e+115) {
		tmp = fma((t_1 * ((((h / l) * -0.5) * M_m) * t_2)), (t_2 * M_m), t_1);
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(D_m * Float64(M_m / d))
	t_1 = Float64(d / sqrt(Float64(l * h)))
	t_2 = Float64(D_m * Float64(0.5 / d))
	tmp = 0.0
	if (d <= -5.7e-214)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(0.5 * t_0) / 2.0), Float64(Float64(Float64(t_0 * h) * -0.5) / l), 1.0) * sqrt(Float64(d / l))) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (d <= 1.32e-207)
		tmp = Float64(Float64(D_m * Float64(D_m * Float64(Float64(-0.125 * (Float64(h / l) ^ 1.5)) * Float64(Float64(M_m / d) * M_m)))) / h);
	elseif (d <= 1.6e+115)
		tmp = fma(Float64(t_1 * Float64(Float64(Float64(Float64(h / l) * -0.5) * M_m) * t_2)), Float64(t_2 * M_m), t_1);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5.7e-214], N[(N[(N[(N[(N[(N[(0.5 * t$95$0), $MachinePrecision] / 2.0), $MachinePrecision] * N[(N[(N[(t$95$0 * h), $MachinePrecision] * -0.5), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.32e-207], N[(N[(D$95$m * N[(D$95$m * N[(N[(-0.125 * N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 1.6e+115], N[(N[(t$95$1 * N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * M$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * M$95$m), $MachinePrecision] + t$95$1), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -5.6999999999999996e-214

   1. Initial program 71.9%

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

   4. Applied rewrites 83.0%

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

   6. Applied rewrites 89.5%

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

   if -5.6999999999999996e-214 < d < 1.32000000000000003e-207

   1. Initial program 39.2%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 55.5%

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

   7. Applied rewrites 60.9%

      \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(M \cdot M\right), \frac{D \cdot D}{d} \cdot -0.125, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]

   8. Taylor expanded in d around 0

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

   10. Applied rewrites 53.0%

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

   12. Applied rewrites 70.4%

       \[\leadsto \frac{D \cdot \left(D \cdot \left(\left(-0.125 \cdot {\left(\frac{h}{\ell}\right)}^{1.5}\right) \cdot \left(\frac{M}{d} \cdot M\right)\right)\right)}{h} \]

   if 1.32000000000000003e-207 < d < 1.6e115

   1. Initial program 73.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 72.3%

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

   6. Applied rewrites 75.7%

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

   8. Applied rewrites 70.0%

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

   if 1.6e115 < d

   1. Initial program 69.4%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 79.2

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

   5. Applied rewrites 79.2%

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

   7. Applied rewrites 79.1%

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

   9. Applied rewrites 89.1%

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

Alternative 12: 63.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{0.5}{d}\\ t_1 := \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;d \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;d \leq -1.5 \cdot 10^{-186}:\\ \;\;\;\;\frac{\left(\frac{\mathsf{fma}\left(-0.125 \cdot \left(D\_m \cdot D\_m\right), h \cdot \frac{M\_m \cdot M\_m}{\ell}, d \cdot d\right)}{d \cdot d} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{-206}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(M\_m \cdot M\_m\right) \cdot -0.125\right) \cdot \frac{D\_m \cdot D\_m}{d}, \frac{\frac{h}{\ell}}{\sqrt{\frac{\ell}{h}}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(t\_1 \cdot \left(\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot M\_m\right) \cdot t\_0\right), t\_0 \cdot M\_m, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* D_m (/ 0.5 d))) (t_1 (/ d (sqrt (* l h)))))
   (if (<= d -1.32e+154)
     (* (- d) (sqrt (pow (* l h) -1.0)))
     (if (<= d -1.5e-186)
       (/
        (*
         (*
          (/
           (fma (* -0.125 (* D_m D_m)) (* h (/ (* M_m M_m) l)) (* d d))
           (* d d))
          (sqrt (/ d l)))
         (sqrt (- d)))
        (sqrt (- h)))
       (if (<= d 1.1e-206)
         (/
          (fma
           (* (* (* M_m M_m) -0.125) (/ (* D_m D_m) d))
           (/ (/ h l) (sqrt (/ l h)))
           (* (sqrt (/ h l)) d))
          h)
         (if (<= d 1.6e+115)
           (fma (* t_1 (* (* (* (/ h l) -0.5) M_m) t_0)) (* t_0 M_m) t_1)
           (/ d (* (sqrt l) (sqrt h)))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = D_m * (0.5 / d);
	double t_1 = d / sqrt((l * h));
	double tmp;
	if (d <= -1.32e+154) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else if (d <= -1.5e-186) {
		tmp = (((fma((-0.125 * (D_m * D_m)), (h * ((M_m * M_m) / l)), (d * d)) / (d * d)) * sqrt((d / l))) * sqrt(-d)) / sqrt(-h);
	} else if (d <= 1.1e-206) {
		tmp = fma((((M_m * M_m) * -0.125) * ((D_m * D_m) / d)), ((h / l) / sqrt((l / h))), (sqrt((h / l)) * d)) / h;
	} else if (d <= 1.6e+115) {
		tmp = fma((t_1 * ((((h / l) * -0.5) * M_m) * t_0)), (t_0 * M_m), t_1);
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(D_m * Float64(0.5 / d))
	t_1 = Float64(d / sqrt(Float64(l * h)))
	tmp = 0.0
	if (d <= -1.32e+154)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	elseif (d <= -1.5e-186)
		tmp = Float64(Float64(Float64(Float64(fma(Float64(-0.125 * Float64(D_m * D_m)), Float64(h * Float64(Float64(M_m * M_m) / l)), Float64(d * d)) / Float64(d * d)) * sqrt(Float64(d / l))) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (d <= 1.1e-206)
		tmp = Float64(fma(Float64(Float64(Float64(M_m * M_m) * -0.125) * Float64(Float64(D_m * D_m) / d)), Float64(Float64(h / l) / sqrt(Float64(l / h))), Float64(sqrt(Float64(h / l)) * d)) / h);
	elseif (d <= 1.6e+115)
		tmp = fma(Float64(t_1 * Float64(Float64(Float64(Float64(h / l) * -0.5) * M_m) * t_0)), Float64(t_0 * M_m), t_1);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.32e+154], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.5e-186], N[(N[(N[(N[(N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(M$95$m * M$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.1e-206], N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] / N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 1.6e+115], N[(N[(t$95$1 * N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * M$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * M$95$m), $MachinePrecision] + t$95$1), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.31999999999999998e154

   1. Initial program 76.3%

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

   3. Taylor expanded in 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 62.6

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

   5. Applied rewrites 62.6%

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

   if -1.31999999999999998e154 < d < -1.5000000000000001e-186

   1. Initial program 73.0%

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

   4. Applied rewrites 83.1%

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

   5. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 70.0

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

   7. Applied rewrites 70.0%

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

   if -1.5000000000000001e-186 < d < 1.0999999999999999e-206

   1. Initial program 39.8%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 52.5%

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

   7. Applied rewrites 59.4%

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

   if 1.0999999999999999e-206 < d < 1.6e115

   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-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 71.9%

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

   6. Applied rewrites 75.3%

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

   8. Applied rewrites 69.5%

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

   if 1.6e115 < d

   1. Initial program 69.4%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 79.2

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

   5. Applied rewrites 79.2%

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

   7. Applied rewrites 79.1%

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

   9. Applied rewrites 89.1%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;d \leq -1.5 \cdot 10^{-186}:\\ \;\;\;\;\frac{\left(\frac{\mathsf{fma}\left(-0.125 \cdot \left(D \cdot D\right), h \cdot \frac{M \cdot M}{\ell}, d \cdot d\right)}{d \cdot d} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{-206}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \frac{D \cdot D}{d}, \frac{\frac{h}{\ell}}{\sqrt{\frac{\ell}{h}}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot M\right) \cdot \left(D \cdot \frac{0.5}{d}\right)\right), \left(D \cdot \frac{0.5}{d}\right) \cdot M, \frac{d}{\sqrt{\ell \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{\ell \cdot h}}\\ t_1 := D\_m \cdot \frac{0.5}{d}\\ \mathbf{if}\;d \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;d \leq -1.5 \cdot 10^{-186}:\\ \;\;\;\;\frac{\left(\frac{\mathsf{fma}\left(-0.125 \cdot \left(D\_m \cdot D\_m\right), h \cdot \frac{M\_m \cdot M\_m}{\ell}, d \cdot d\right)}{d \cdot d} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{-206}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(M\_m \cdot M\_m\right) \cdot -0.125\right) \cdot \frac{D\_m \cdot D\_m}{d}, \frac{\frac{h}{\ell}}{\sqrt{\frac{\ell}{h}}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \left(t\_1 \cdot M\_m\right), \left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot M\_m\right) \cdot t\_1, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ d (sqrt (* l h)))) (t_1 (* D_m (/ 0.5 d))))
   (if (<= d -1.32e+154)
     (* (- d) (sqrt (pow (* l h) -1.0)))
     (if (<= d -1.5e-186)
       (/
        (*
         (*
          (/
           (fma (* -0.125 (* D_m D_m)) (* h (/ (* M_m M_m) l)) (* d d))
           (* d d))
          (sqrt (/ d l)))
         (sqrt (- d)))
        (sqrt (- h)))
       (if (<= d 1.1e-206)
         (/
          (fma
           (* (* (* M_m M_m) -0.125) (/ (* D_m D_m) d))
           (/ (/ h l) (sqrt (/ l h)))
           (* (sqrt (/ h l)) d))
          h)
         (if (<= d 1.6e+115)
           (fma (* t_0 (* t_1 M_m)) (* (* (* (/ h l) -0.5) M_m) t_1) t_0)
           (/ d (* (sqrt l) (sqrt h)))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = d / sqrt((l * h));
	double t_1 = D_m * (0.5 / d);
	double tmp;
	if (d <= -1.32e+154) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else if (d <= -1.5e-186) {
		tmp = (((fma((-0.125 * (D_m * D_m)), (h * ((M_m * M_m) / l)), (d * d)) / (d * d)) * sqrt((d / l))) * sqrt(-d)) / sqrt(-h);
	} else if (d <= 1.1e-206) {
		tmp = fma((((M_m * M_m) * -0.125) * ((D_m * D_m) / d)), ((h / l) / sqrt((l / h))), (sqrt((h / l)) * d)) / h;
	} else if (d <= 1.6e+115) {
		tmp = fma((t_0 * (t_1 * M_m)), ((((h / l) * -0.5) * M_m) * t_1), t_0);
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(d / sqrt(Float64(l * h)))
	t_1 = Float64(D_m * Float64(0.5 / d))
	tmp = 0.0
	if (d <= -1.32e+154)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	elseif (d <= -1.5e-186)
		tmp = Float64(Float64(Float64(Float64(fma(Float64(-0.125 * Float64(D_m * D_m)), Float64(h * Float64(Float64(M_m * M_m) / l)), Float64(d * d)) / Float64(d * d)) * sqrt(Float64(d / l))) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (d <= 1.1e-206)
		tmp = Float64(fma(Float64(Float64(Float64(M_m * M_m) * -0.125) * Float64(Float64(D_m * D_m) / d)), Float64(Float64(h / l) / sqrt(Float64(l / h))), Float64(sqrt(Float64(h / l)) * d)) / h);
	elseif (d <= 1.6e+115)
		tmp = fma(Float64(t_0 * Float64(t_1 * M_m)), Float64(Float64(Float64(Float64(h / l) * -0.5) * M_m) * t_1), t_0);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.32e+154], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.5e-186], N[(N[(N[(N[(N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(M$95$m * M$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.1e-206], N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] / N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 1.6e+115], N[(N[(t$95$0 * N[(t$95$1 * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * M$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] + t$95$0), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.31999999999999998e154

   1. Initial program 76.3%

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

   3. Taylor expanded in 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 62.6

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

   5. Applied rewrites 62.6%

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

   if -1.31999999999999998e154 < d < -1.5000000000000001e-186

   1. Initial program 73.0%

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

   4. Applied rewrites 83.1%

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

   5. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 70.0

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

   7. Applied rewrites 70.0%

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

   if -1.5000000000000001e-186 < d < 1.0999999999999999e-206

   1. Initial program 39.8%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 52.5%

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

   7. Applied rewrites 59.4%

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

   if 1.0999999999999999e-206 < d < 1.6e115

   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-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 71.9%

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

   6. Applied rewrites 75.3%

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

   8. Applied rewrites 69.5%

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

   if 1.6e115 < d

   1. Initial program 69.4%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 79.2

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

   5. Applied rewrites 79.2%

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

   7. Applied rewrites 79.1%

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

   9. Applied rewrites 89.1%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;d \leq -1.5 \cdot 10^{-186}:\\ \;\;\;\;\frac{\left(\frac{\mathsf{fma}\left(-0.125 \cdot \left(D \cdot D\right), h \cdot \frac{M \cdot M}{\ell}, d \cdot d\right)}{d \cdot d} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{-206}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \frac{D \cdot D}{d}, \frac{\frac{h}{\ell}}{\sqrt{\frac{\ell}{h}}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(\left(D \cdot \frac{0.5}{d}\right) \cdot M\right), \left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot M\right) \cdot \left(D \cdot \frac{0.5}{d}\right), \frac{d}{\sqrt{\ell \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 72.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M\_m}{d}\\ t_1 := \frac{d}{\sqrt{\ell \cdot h}}\\ t_2 := D\_m \cdot \frac{0.5}{d}\\ \mathbf{if}\;d \leq -5.5 \cdot 10^{-291}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{0.5 \cdot t\_0}{2}, \frac{\left(t\_0 \cdot h\right) \cdot -0.5}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{-206}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(M\_m \cdot M\_m\right) \cdot -0.125\right) \cdot \frac{D\_m \cdot D\_m}{d}, \frac{\frac{h}{\ell}}{\sqrt{\frac{\ell}{h}}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(t\_1 \cdot \left(\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot M\_m\right) \cdot t\_2\right), t\_2 \cdot M\_m, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* D_m (/ M_m d)))
        (t_1 (/ d (sqrt (* l h))))
        (t_2 (* D_m (/ 0.5 d))))
   (if (<= d -5.5e-291)
     (/
      (*
       (*
        (fma (/ (* 0.5 t_0) 2.0) (/ (* (* t_0 h) -0.5) l) 1.0)
        (sqrt (/ d l)))
       (sqrt (- d)))
      (sqrt (- h)))
     (if (<= d 1.1e-206)
       (/
        (fma
         (* (* (* M_m M_m) -0.125) (/ (* D_m D_m) d))
         (/ (/ h l) (sqrt (/ l h)))
         (* (sqrt (/ h l)) d))
        h)
       (if (<= d 1.6e+115)
         (fma (* t_1 (* (* (* (/ h l) -0.5) M_m) t_2)) (* t_2 M_m) t_1)
         (/ d (* (sqrt l) (sqrt h))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = D_m * (M_m / d);
	double t_1 = d / sqrt((l * h));
	double t_2 = D_m * (0.5 / d);
	double tmp;
	if (d <= -5.5e-291) {
		tmp = ((fma(((0.5 * t_0) / 2.0), (((t_0 * h) * -0.5) / l), 1.0) * sqrt((d / l))) * sqrt(-d)) / sqrt(-h);
	} else if (d <= 1.1e-206) {
		tmp = fma((((M_m * M_m) * -0.125) * ((D_m * D_m) / d)), ((h / l) / sqrt((l / h))), (sqrt((h / l)) * d)) / h;
	} else if (d <= 1.6e+115) {
		tmp = fma((t_1 * ((((h / l) * -0.5) * M_m) * t_2)), (t_2 * M_m), t_1);
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(D_m * Float64(M_m / d))
	t_1 = Float64(d / sqrt(Float64(l * h)))
	t_2 = Float64(D_m * Float64(0.5 / d))
	tmp = 0.0
	if (d <= -5.5e-291)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(0.5 * t_0) / 2.0), Float64(Float64(Float64(t_0 * h) * -0.5) / l), 1.0) * sqrt(Float64(d / l))) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (d <= 1.1e-206)
		tmp = Float64(fma(Float64(Float64(Float64(M_m * M_m) * -0.125) * Float64(Float64(D_m * D_m) / d)), Float64(Float64(h / l) / sqrt(Float64(l / h))), Float64(sqrt(Float64(h / l)) * d)) / h);
	elseif (d <= 1.6e+115)
		tmp = fma(Float64(t_1 * Float64(Float64(Float64(Float64(h / l) * -0.5) * M_m) * t_2)), Float64(t_2 * M_m), t_1);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5.5e-291], N[(N[(N[(N[(N[(N[(0.5 * t$95$0), $MachinePrecision] / 2.0), $MachinePrecision] * N[(N[(N[(t$95$0 * h), $MachinePrecision] * -0.5), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.1e-206], N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] / N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 1.6e+115], N[(N[(t$95$1 * N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * M$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * M$95$m), $MachinePrecision] + t$95$1), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -5.5000000000000002e-291

   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

   4. Applied rewrites 77.2%

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

   6. Applied rewrites 83.3%

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

   if -5.5000000000000002e-291 < d < 1.0999999999999999e-206

   1. Initial program 46.6%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 62.4%

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

   7. Applied rewrites 69.1%

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

   if 1.0999999999999999e-206 < d < 1.6e115

   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-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 71.9%

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

   6. Applied rewrites 75.3%

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

   8. Applied rewrites 69.5%

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

   if 1.6e115 < d

   1. Initial program 69.4%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 79.2

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

   5. Applied rewrites 79.2%

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

   7. Applied rewrites 79.1%

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

   9. Applied rewrites 89.1%

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

Alternative 15: 72.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{\ell \cdot h}}\\ t_1 := D\_m \cdot \frac{0.5}{d}\\ t_2 := D\_m \cdot \frac{M\_m}{d}\\ \mathbf{if}\;d \leq -5.5 \cdot 10^{-291}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\left(t\_2 \cdot h\right) \cdot -0.5, \frac{0.5}{\ell} \cdot \left(0.5 \cdot t\_2\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{-206}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(M\_m \cdot M\_m\right) \cdot -0.125\right) \cdot \frac{D\_m \cdot D\_m}{d}, \frac{\frac{h}{\ell}}{\sqrt{\frac{\ell}{h}}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \left(\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot M\_m\right) \cdot t\_1\right), t\_1 \cdot M\_m, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ d (sqrt (* l h))))
        (t_1 (* D_m (/ 0.5 d)))
        (t_2 (* D_m (/ M_m d))))
   (if (<= d -5.5e-291)
     (/
      (*
       (*
        (fma (* (* t_2 h) -0.5) (* (/ 0.5 l) (* 0.5 t_2)) 1.0)
        (sqrt (/ d l)))
       (sqrt (- d)))
      (sqrt (- h)))
     (if (<= d 1.1e-206)
       (/
        (fma
         (* (* (* M_m M_m) -0.125) (/ (* D_m D_m) d))
         (/ (/ h l) (sqrt (/ l h)))
         (* (sqrt (/ h l)) d))
        h)
       (if (<= d 1.6e+115)
         (fma (* t_0 (* (* (* (/ h l) -0.5) M_m) t_1)) (* t_1 M_m) t_0)
         (/ d (* (sqrt l) (sqrt h))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = d / sqrt((l * h));
	double t_1 = D_m * (0.5 / d);
	double t_2 = D_m * (M_m / d);
	double tmp;
	if (d <= -5.5e-291) {
		tmp = ((fma(((t_2 * h) * -0.5), ((0.5 / l) * (0.5 * t_2)), 1.0) * sqrt((d / l))) * sqrt(-d)) / sqrt(-h);
	} else if (d <= 1.1e-206) {
		tmp = fma((((M_m * M_m) * -0.125) * ((D_m * D_m) / d)), ((h / l) / sqrt((l / h))), (sqrt((h / l)) * d)) / h;
	} else if (d <= 1.6e+115) {
		tmp = fma((t_0 * ((((h / l) * -0.5) * M_m) * t_1)), (t_1 * M_m), t_0);
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(d / sqrt(Float64(l * h)))
	t_1 = Float64(D_m * Float64(0.5 / d))
	t_2 = Float64(D_m * Float64(M_m / d))
	tmp = 0.0
	if (d <= -5.5e-291)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(t_2 * h) * -0.5), Float64(Float64(0.5 / l) * Float64(0.5 * t_2)), 1.0) * sqrt(Float64(d / l))) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (d <= 1.1e-206)
		tmp = Float64(fma(Float64(Float64(Float64(M_m * M_m) * -0.125) * Float64(Float64(D_m * D_m) / d)), Float64(Float64(h / l) / sqrt(Float64(l / h))), Float64(sqrt(Float64(h / l)) * d)) / h);
	elseif (d <= 1.6e+115)
		tmp = fma(Float64(t_0 * Float64(Float64(Float64(Float64(h / l) * -0.5) * M_m) * t_1)), Float64(t_1 * M_m), t_0);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5.5e-291], N[(N[(N[(N[(N[(N[(t$95$2 * h), $MachinePrecision] * -0.5), $MachinePrecision] * N[(N[(0.5 / l), $MachinePrecision] * N[(0.5 * t$95$2), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.1e-206], N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] / N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 1.6e+115], N[(N[(t$95$0 * N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * M$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * M$95$m), $MachinePrecision] + t$95$0), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -5.5000000000000002e-291

   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

   4. Applied rewrites 77.2%

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

   6. Applied rewrites 83.4%

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

   if -5.5000000000000002e-291 < d < 1.0999999999999999e-206

   1. Initial program 46.6%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 62.4%

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

   7. Applied rewrites 69.1%

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

   if 1.0999999999999999e-206 < d < 1.6e115

   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-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 71.9%

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

   6. Applied rewrites 75.3%

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

   8. Applied rewrites 69.5%

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

   if 1.6e115 < d

   1. Initial program 69.4%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 79.2

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

   5. Applied rewrites 79.2%

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

   7. Applied rewrites 79.1%

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

   9. Applied rewrites 89.1%

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

Alternative 16: 70.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{\ell \cdot h}}\\ t_1 := D\_m \cdot \frac{0.5}{d}\\ \mathbf{if}\;d \leq -5.5 \cdot 10^{-291}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\left(\left(-0.25 \cdot \frac{h}{\ell}\right) \cdot \left(D\_m \cdot \frac{M\_m}{d}\right)\right) \cdot \frac{M\_m}{d}, 0.5 \cdot D\_m, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{-206}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(M\_m \cdot M\_m\right) \cdot -0.125\right) \cdot \frac{D\_m \cdot D\_m}{d}, \frac{\frac{h}{\ell}}{\sqrt{\frac{\ell}{h}}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \left(\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot M\_m\right) \cdot t\_1\right), t\_1 \cdot M\_m, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ d (sqrt (* l h)))) (t_1 (* D_m (/ 0.5 d))))
   (if (<= d -5.5e-291)
     (/
      (*
       (*
        (fma
         (* (* (* -0.25 (/ h l)) (* D_m (/ M_m d))) (/ M_m d))
         (* 0.5 D_m)
         1.0)
        (sqrt (/ d l)))
       (sqrt (- d)))
      (sqrt (- h)))
     (if (<= d 1.1e-206)
       (/
        (fma
         (* (* (* M_m M_m) -0.125) (/ (* D_m D_m) d))
         (/ (/ h l) (sqrt (/ l h)))
         (* (sqrt (/ h l)) d))
        h)
       (if (<= d 1.6e+115)
         (fma (* t_0 (* (* (* (/ h l) -0.5) M_m) t_1)) (* t_1 M_m) t_0)
         (/ d (* (sqrt l) (sqrt h))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = d / sqrt((l * h));
	double t_1 = D_m * (0.5 / d);
	double tmp;
	if (d <= -5.5e-291) {
		tmp = ((fma((((-0.25 * (h / l)) * (D_m * (M_m / d))) * (M_m / d)), (0.5 * D_m), 1.0) * sqrt((d / l))) * sqrt(-d)) / sqrt(-h);
	} else if (d <= 1.1e-206) {
		tmp = fma((((M_m * M_m) * -0.125) * ((D_m * D_m) / d)), ((h / l) / sqrt((l / h))), (sqrt((h / l)) * d)) / h;
	} else if (d <= 1.6e+115) {
		tmp = fma((t_0 * ((((h / l) * -0.5) * M_m) * t_1)), (t_1 * M_m), t_0);
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(d / sqrt(Float64(l * h)))
	t_1 = Float64(D_m * Float64(0.5 / d))
	tmp = 0.0
	if (d <= -5.5e-291)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(Float64(-0.25 * Float64(h / l)) * Float64(D_m * Float64(M_m / d))) * Float64(M_m / d)), Float64(0.5 * D_m), 1.0) * sqrt(Float64(d / l))) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (d <= 1.1e-206)
		tmp = Float64(fma(Float64(Float64(Float64(M_m * M_m) * -0.125) * Float64(Float64(D_m * D_m) / d)), Float64(Float64(h / l) / sqrt(Float64(l / h))), Float64(sqrt(Float64(h / l)) * d)) / h);
	elseif (d <= 1.6e+115)
		tmp = fma(Float64(t_0 * Float64(Float64(Float64(Float64(h / l) * -0.5) * M_m) * t_1)), Float64(t_1 * M_m), t_0);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5.5e-291], N[(N[(N[(N[(N[(N[(N[(-0.25 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(0.5 * D$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.1e-206], N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] / N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 1.6e+115], N[(N[(t$95$0 * N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * M$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * M$95$m), $MachinePrecision] + t$95$0), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -5.5000000000000002e-291

   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

   4. Applied rewrites 77.2%

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

   6. Applied rewrites 78.7%

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

   if -5.5000000000000002e-291 < d < 1.0999999999999999e-206

   1. Initial program 46.6%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 62.4%

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

   7. Applied rewrites 69.1%

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

   if 1.0999999999999999e-206 < d < 1.6e115

   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-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 71.9%

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

   6. Applied rewrites 75.3%

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

   8. Applied rewrites 69.5%

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

   if 1.6e115 < d

   1. Initial program 69.4%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 79.2

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

   5. Applied rewrites 79.2%

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

   7. Applied rewrites 79.1%

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

   9. Applied rewrites 89.1%

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

Alternative 17: 60.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;d \leq -1.5 \cdot 10^{-186}:\\ \;\;\;\;\frac{\left(\frac{\mathsf{fma}\left(-0.125 \cdot \left(D\_m \cdot D\_m\right), h \cdot \frac{M\_m \cdot M\_m}{\ell}, d \cdot d\right)}{d \cdot d} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 4.7 \cdot 10^{+124}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(M\_m \cdot M\_m\right) \cdot -0.125\right) \cdot \frac{D\_m \cdot D\_m}{d}, \frac{\frac{h}{\ell}}{\sqrt{\frac{\ell}{h}}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -1.32e+154)
   (* (- d) (sqrt (pow (* l h) -1.0)))
   (if (<= d -1.5e-186)
     (/
      (*
       (*
        (/
         (fma (* -0.125 (* D_m D_m)) (* h (/ (* M_m M_m) l)) (* d d))
         (* d d))
        (sqrt (/ d l)))
       (sqrt (- d)))
      (sqrt (- h)))
     (if (<= d 4.7e+124)
       (/
        (fma
         (* (* (* M_m M_m) -0.125) (/ (* D_m D_m) d))
         (/ (/ h l) (sqrt (/ l h)))
         (* (sqrt (/ h l)) d))
        h)
       (/ d (* (sqrt l) (sqrt h)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1.32e+154) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else if (d <= -1.5e-186) {
		tmp = (((fma((-0.125 * (D_m * D_m)), (h * ((M_m * M_m) / l)), (d * d)) / (d * d)) * sqrt((d / l))) * sqrt(-d)) / sqrt(-h);
	} else if (d <= 4.7e+124) {
		tmp = fma((((M_m * M_m) * -0.125) * ((D_m * D_m) / d)), ((h / l) / sqrt((l / h))), (sqrt((h / l)) * d)) / h;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -1.32e+154)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	elseif (d <= -1.5e-186)
		tmp = Float64(Float64(Float64(Float64(fma(Float64(-0.125 * Float64(D_m * D_m)), Float64(h * Float64(Float64(M_m * M_m) / l)), Float64(d * d)) / Float64(d * d)) * sqrt(Float64(d / l))) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (d <= 4.7e+124)
		tmp = Float64(fma(Float64(Float64(Float64(M_m * M_m) * -0.125) * Float64(Float64(D_m * D_m) / d)), Float64(Float64(h / l) / sqrt(Float64(l / h))), Float64(sqrt(Float64(h / l)) * d)) / h);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -1.32e+154], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.5e-186], N[(N[(N[(N[(N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(M$95$m * M$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.7e+124], N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] / N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.31999999999999998e154

   1. Initial program 76.3%

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

   3. Taylor expanded in 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 62.6

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

   5. Applied rewrites 62.6%

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

   if -1.31999999999999998e154 < d < -1.5000000000000001e-186

   1. Initial program 73.0%

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

   4. Applied rewrites 83.1%

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

   5. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 70.0

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

   7. Applied rewrites 70.0%

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

   if -1.5000000000000001e-186 < d < 4.69999999999999991e124

   1. Initial program 56.7%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 51.0%

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

   7. Applied rewrites 58.7%

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

   if 4.69999999999999991e124 < d

   1. Initial program 68.6%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 78.6

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

   5. Applied rewrites 78.6%

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

   7. Applied rewrites 78.5%

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

   9. Applied rewrites 88.9%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;d \leq -1.5 \cdot 10^{-186}:\\ \;\;\;\;\frac{\left(\frac{\mathsf{fma}\left(-0.125 \cdot \left(D \cdot D\right), h \cdot \frac{M \cdot M}{\ell}, d \cdot d\right)}{d \cdot d} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 4.7 \cdot 10^{+124}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \frac{D \cdot D}{d}, \frac{\frac{h}{\ell}}{\sqrt{\frac{\ell}{h}}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 56.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2.7 \cdot 10^{-78}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{h}{\ell \cdot d}, \left(-0.125 \cdot \left(D\_m \cdot D\_m\right)\right) \cdot \frac{M\_m \cdot M\_m}{d}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq 4.7 \cdot 10^{+124}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(M\_m \cdot M\_m\right) \cdot -0.125\right) \cdot \frac{D\_m \cdot D\_m}{d}, \frac{\frac{h}{\ell}}{\sqrt{\frac{\ell}{h}}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -2.7e-78)
   (/
    (*
     (*
      (fma (/ h (* l d)) (* (* -0.125 (* D_m D_m)) (/ (* M_m M_m) d)) 1.0)
      (sqrt (/ d h)))
     (sqrt (- d)))
    (sqrt (- l)))
   (if (<= d 4.7e+124)
     (/
      (fma
       (* (* (* M_m M_m) -0.125) (/ (* D_m D_m) d))
       (/ (/ h l) (sqrt (/ l h)))
       (* (sqrt (/ h l)) d))
      h)
     (/ d (* (sqrt l) (sqrt h))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -2.7e-78) {
		tmp = ((fma((h / (l * d)), ((-0.125 * (D_m * D_m)) * ((M_m * M_m) / d)), 1.0) * sqrt((d / h))) * sqrt(-d)) / sqrt(-l);
	} else if (d <= 4.7e+124) {
		tmp = fma((((M_m * M_m) * -0.125) * ((D_m * D_m) / d)), ((h / l) / sqrt((l / h))), (sqrt((h / l)) * d)) / h;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -2.7e-78)
		tmp = Float64(Float64(Float64(fma(Float64(h / Float64(l * d)), Float64(Float64(-0.125 * Float64(D_m * D_m)) * Float64(Float64(M_m * M_m) / d)), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(-d))) / sqrt(Float64(-l)));
	elseif (d <= 4.7e+124)
		tmp = Float64(fma(Float64(Float64(Float64(M_m * M_m) * -0.125) * Float64(Float64(D_m * D_m) / d)), Float64(Float64(h / l) / sqrt(Float64(l / h))), Float64(sqrt(Float64(h / l)) * d)) / h);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -2.7e-78], N[(N[(N[(N[(N[(h / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.7e+124], N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] / N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.69999999999999994e-78

   1. Initial program 78.4%

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

   4. Applied rewrites 79.4%

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

   5. Taylor expanded in d around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. div-sub N/A

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

      5. *-inverses N/A

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

      6. associate-/l* N/A

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

      7. associate-/l/ N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. associate-*r* N/A

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

   7. Applied rewrites 62.1%

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

   if -2.69999999999999994e-78 < d < 4.69999999999999991e124

   1. Initial program 56.5%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 50.2%

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

   7. Applied rewrites 57.5%

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

   if 4.69999999999999991e124 < d

   1. Initial program 68.6%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 78.6

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

   5. Applied rewrites 78.6%

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

   7. Applied rewrites 78.5%

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

   9. Applied rewrites 88.9%

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

Alternative 19: 47.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \cdot 10^{-139}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{-298}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -2.5e-139)
   (* (- d) (sqrt (pow (* l h) -1.0)))
   (if (<= l 5.8e-298)
     (/ (* (- d) (sqrt (/ h l))) h)
     (/ d (* (sqrt l) (sqrt h))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -2.5e-139) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else if (l <= 5.8e-298) {
		tmp = (-d * sqrt((h / l))) / h;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-2.5d-139)) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else if (l <= 5.8d-298) then
        tmp = (-d * sqrt((h / l))) / h
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -2.5e-139) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else if (l <= 5.8e-298) {
		tmp = (-d * Math.sqrt((h / l))) / h;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -2.5e-139:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	elif l <= 5.8e-298:
		tmp = (-d * math.sqrt((h / l))) / h
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -2.5e-139)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	elseif (l <= 5.8e-298)
		tmp = Float64(Float64(Float64(-d) * sqrt(Float64(h / l))) / h);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -2.5e-139)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	elseif (l <= 5.8e-298)
		tmp = (-d * sqrt((h / l))) / h;
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -2.5e-139], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.8e-298], N[(N[((-d) * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.50000000000000017e-139

   1. Initial program 60.2%

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

   3. Taylor expanded in 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 45.2

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

   5. Applied rewrites 45.2%

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

   if -2.50000000000000017e-139 < l < 5.8000000000000003e-298

   1. Initial program 71.9%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 31.4%

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

   6. Taylor expanded in l around -inf

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

   8. Applied rewrites 44.9%

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

   if 5.8000000000000003e-298 < l

   1. Initial program 67.3%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 48.6

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

   5. Applied rewrites 48.6%

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

   7. Applied rewrites 49.3%

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

   9. Applied rewrites 54.9%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \cdot 10^{-139}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{-298}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 46.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 7.5 \cdot 10^{-208}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d 7.5e-208)
   (* (- d) (sqrt (pow (* l h) -1.0)))
   (/ d (* (sqrt l) (sqrt h)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 7.5e-208) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= 7.5d-208) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 7.5e-208) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= 7.5e-208:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= 7.5e-208)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= 7.5e-208)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, 7.5e-208], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 7.4999999999999999e-208

   1. Initial program 61.0%

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

   3. Taylor expanded in 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 36.3

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

   5. Applied rewrites 36.3%

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

   if 7.4999999999999999e-208 < d

   1. Initial program 72.1%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 56.2

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

   5. Applied rewrites 56.2%

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

   7. Applied rewrites 56.2%

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

   9. Applied rewrites 63.0%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 7.5 \cdot 10^{-208}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 43.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-201}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

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-201) (* (- d) (sqrt (pow (* l h) -1.0))) (/ d (sqrt (* l h)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1e-201) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} 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-201)) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    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-201) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} 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-201:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	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-201)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	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-201)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	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-201], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -9.99999999999999946e-202

   1. Initial program 61.6%

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

   3. Taylor expanded in 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 41.8

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

   5. Applied rewrites 41.8%

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

   if -9.99999999999999946e-202 < l

   1. Initial program 68.1%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 46.7

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

   5. Applied rewrites 46.7%

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

   7. Applied rewrites 47.3%

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

4. Final simplification 44.8%

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

Alternative 22: 26.7% 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 65.2%

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

3. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
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 27.9

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

5. Applied rewrites 27.9%

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

7. Applied rewrites 28.2%

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

Reproduce

herbie shell --seed 2024296 
(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)))))