Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.3% → 82.6%

Time: 19.6s

Alternatives: 25

Speedup: 2.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.3% 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: 82.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -3.5 \cdot 10^{-184}:\\ \;\;\;\;\frac{\left(\frac{t\_0}{\sqrt{-\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{d}{D\_m} \cdot \frac{2}{M}\right)}^{-2}, 1\right)\right) \cdot t\_0}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{-251}:\\ \;\;\;\;\left(1 - \frac{\left(\left(0.25 \cdot D\_m\right) \cdot M\right) \cdot h}{d} \cdot \frac{\left(\frac{0.5}{d} \cdot D\_m\right) \cdot M}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{-D\_m}{\ell}, \left(\left(\frac{M}{d} \cdot h\right) \cdot D\_m\right) \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot 0.25\right), 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (- d))))
   (if (<= l -3.5e-184)
     (/
      (*
       (*
        (/ t_0 (sqrt (- l)))
        (fma (* (/ h l) -0.5) (pow (* (/ d D_m) (/ 2.0 M)) -2.0) 1.0))
       t_0)
      (sqrt (- h)))
     (if (<= l 6.5e-251)
       (*
        (-
         1.0
         (* (/ (* (* (* 0.25 D_m) M) h) d) (/ (* (* (/ 0.5 d) D_m) M) l)))
        (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
       (/
        (*
         (*
          (fma
           (/ (- D_m) l)
           (* (* (* (/ M d) h) D_m) (* (* (/ 0.5 d) M) 0.25))
           1.0)
          (sqrt d))
         (sqrt d))
        (* (sqrt l) (sqrt h)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt(-d);
	double tmp;
	if (l <= -3.5e-184) {
		tmp = (((t_0 / sqrt(-l)) * fma(((h / l) * -0.5), pow(((d / D_m) * (2.0 / M)), -2.0), 1.0)) * t_0) / sqrt(-h);
	} else if (l <= 6.5e-251) {
		tmp = (1.0 - (((((0.25 * D_m) * M) * h) / d) * ((((0.5 / d) * D_m) * M) / l))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	} else {
		tmp = ((fma((-D_m / l), ((((M / d) * h) * D_m) * (((0.5 / d) * M) * 0.25)), 1.0) * sqrt(d)) * sqrt(d)) / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -3.5e-184)
		tmp = Float64(Float64(Float64(Float64(t_0 / sqrt(Float64(-l))) * fma(Float64(Float64(h / l) * -0.5), (Float64(Float64(d / D_m) * Float64(2.0 / M)) ^ -2.0), 1.0)) * t_0) / sqrt(Float64(-h)));
	elseif (l <= 6.5e-251)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(0.25 * D_m) * M) * h) / d) * Float64(Float64(Float64(Float64(0.5 / d) * D_m) * M) / l))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))));
	else
		tmp = Float64(Float64(Float64(fma(Float64(Float64(-D_m) / l), Float64(Float64(Float64(Float64(M / d) * h) * D_m) * Float64(Float64(Float64(0.5 / d) * M) * 0.25)), 1.0) * sqrt(d)) * sqrt(d)) / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -3.5e-184], N[(N[(N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(N[(d / D$95$m), $MachinePrecision] * N[(2.0 / M), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.5e-251], N[(N[(1.0 - N[(N[(N[(N[(N[(0.25 * D$95$m), $MachinePrecision] * M), $MachinePrecision] * h), $MachinePrecision] / d), $MachinePrecision] * N[(N[(N[(N[(0.5 / d), $MachinePrecision] * D$95$m), $MachinePrecision] * M), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[((-D$95$m) / l), $MachinePrecision] * N[(N[(N[(N[(M / d), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(N[(0.5 / d), $MachinePrecision] * M), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.49999999999999981e-184

   1. Initial program 60.5%

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

   4. Applied rewrites 73.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 82.8

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

   6. Applied rewrites 82.8%

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

   if -3.49999999999999981e-184 < l < 6.5000000000000002e-251

   1. Initial program 78.7%

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

   4. Applied rewrites 89.4%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-1 N/A

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

      5. remove-double-div N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 92.2

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 92.2

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

   6. Applied rewrites 92.2%

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

   if 6.5000000000000002e-251 < l

   1. Initial program 66.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 65.1%

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

   6. Applied rewrites 67.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 67.0

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

   8. Applied rewrites 67.0%

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

   10. Applied rewrites 85.4%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.5 \cdot 10^{-184}:\\ \;\;\;\;\frac{\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{d}{D} \cdot \frac{2}{M}\right)}^{-2}, 1\right)\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{-251}:\\ \;\;\;\;\left(1 - \frac{\left(\left(0.25 \cdot D\right) \cdot M\right) \cdot h}{d} \cdot \frac{\left(\frac{0.5}{d} \cdot D\right) \cdot M}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{-D}{\ell}, \left(\left(\frac{M}{d} \cdot h\right) \cdot D\right) \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot 0.25\right), 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_1 := \left(1 - \left({\left(\frac{D\_m \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-161}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(0.5 \cdot M\right) \cdot D\_m}{\left(-\ell\right) \cdot d}, \left(\frac{M}{d} \cdot h\right) \cdot \left(0.25 \cdot D\_m\right), 1\right) \cdot t\_2\right) \cdot t\_3\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+248}:\\ \;\;\;\;t\_2 \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (fabs (/ d (sqrt (* h l)))))
        (t_1
         (*
          (- 1.0 (* (* (pow (/ (* D_m M) (* d 2.0)) 2.0) (/ 1.0 2.0)) (/ h l)))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_2 (sqrt (/ d l)))
        (t_3 (sqrt (/ d h))))
   (if (<= t_1 -5e-161)
     (*
      (*
       (fma
        (/ (* (* 0.5 M) D_m) (* (- l) d))
        (* (* (/ M d) h) (* 0.25 D_m))
        1.0)
       t_2)
      t_3)
     (if (<= t_1 0.0) t_0 (if (<= t_1 1e+248) (* t_2 t_3) t_0)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = fabs((d / sqrt((h * l))));
	double t_1 = (1.0 - ((pow(((D_m * M) / (d * 2.0)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_2 = sqrt((d / l));
	double t_3 = sqrt((d / h));
	double tmp;
	if (t_1 <= -5e-161) {
		tmp = (fma((((0.5 * M) * D_m) / (-l * d)), (((M / d) * h) * (0.25 * D_m)), 1.0) * t_2) * t_3;
	} else if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 1e+248) {
		tmp = t_2 * t_3;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = abs(Float64(d / sqrt(Float64(h * l))))
	t_1 = Float64(Float64(1.0 - Float64(Float64((Float64(Float64(D_m * M) / Float64(d * 2.0)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	t_2 = sqrt(Float64(d / l))
	t_3 = sqrt(Float64(d / h))
	tmp = 0.0
	if (t_1 <= -5e-161)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(0.5 * M) * D_m) / Float64(Float64(-l) * d)), Float64(Float64(Float64(M / d) * h) * Float64(0.25 * D_m)), 1.0) * t_2) * t_3);
	elseif (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 1e+248)
		tmp = Float64(t_2 * t_3);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - N[(N[(N[Power[N[(N[(D$95$m * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -5e-161], N[(N[(N[(N[(N[(N[(0.5 * M), $MachinePrecision] * D$95$m), $MachinePrecision] / N[((-l) * d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M / d), $MachinePrecision] * h), $MachinePrecision] * N[(0.25 * D$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, 1e+248], N[(t$95$2 * t$95$3), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.9999999999999999e-161

   1. Initial program 81.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 78.3%

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

   6. Applied rewrites 82.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. frac-times N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. div-inv N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 77.5

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

   8. Applied rewrites 77.5%

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

   if -4.9999999999999999e-161 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 1.00000000000000005e248 < (*.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 23.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 28.7

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

   5. Applied rewrites 28.7%

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

   7. Applied rewrites 57.2%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.00000000000000005e248

   1. Initial program 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in d around inf

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 95.9

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

   7. Applied rewrites 95.9%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -5 \cdot 10^{-161}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(0.5 \cdot M\right) \cdot D}{\left(-\ell\right) \cdot d}, \left(\frac{M}{d} \cdot h\right) \cdot \left(0.25 \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+248}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left(1 - \left({\left(\frac{D\_m \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+102}:\\ \;\;\;\;\left(\left(\frac{\frac{D\_m}{d}}{d} \cdot \left(\left(-0.125 \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot D\_m\right)\right) \cdot t\_2\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(D\_m \cdot M\right) \cdot 0.5\right) \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\_m\right)}{d}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot t\_3\\ \mathbf{elif}\;t\_0 \leq 10^{+248}:\\ \;\;\;\;t\_2 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left|t\_3\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0
         (*
          (- 1.0 (* (* (pow (/ (* D_m M) (* d 2.0)) 2.0) (/ 1.0 2.0)) (/ h l)))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_1 (sqrt (/ d h)))
        (t_2 (sqrt (/ d l)))
        (t_3 (/ d (sqrt (* h l)))))
   (if (<= t_0 -1e+102)
     (* (* (* (/ (/ D_m d) d) (* (* -0.125 (/ (* (* M M) h) l)) D_m)) t_2) t_1)
     (if (<= t_0 0.0)
       (*
        (fma
         (/ (* (* (* D_m M) 0.5) (* (* (/ 0.5 d) M) D_m)) d)
         (* (/ h l) -0.5)
         1.0)
        t_3)
       (if (<= t_0 1e+248) (* t_2 t_1) (fabs t_3))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (1.0 - ((pow(((D_m * M) / (d * 2.0)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_1 = sqrt((d / h));
	double t_2 = sqrt((d / l));
	double t_3 = d / sqrt((h * l));
	double tmp;
	if (t_0 <= -1e+102) {
		tmp = ((((D_m / d) / d) * ((-0.125 * (((M * M) * h) / l)) * D_m)) * t_2) * t_1;
	} else if (t_0 <= 0.0) {
		tmp = fma(((((D_m * M) * 0.5) * (((0.5 / d) * M) * D_m)) / d), ((h / l) * -0.5), 1.0) * t_3;
	} else if (t_0 <= 1e+248) {
		tmp = t_2 * t_1;
	} else {
		tmp = fabs(t_3);
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(1.0 - Float64(Float64((Float64(Float64(D_m * M) / Float64(d * 2.0)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(d / l))
	t_3 = Float64(d / sqrt(Float64(h * l)))
	tmp = 0.0
	if (t_0 <= -1e+102)
		tmp = Float64(Float64(Float64(Float64(Float64(D_m / d) / d) * Float64(Float64(-0.125 * Float64(Float64(Float64(M * M) * h) / l)) * D_m)) * t_2) * t_1);
	elseif (t_0 <= 0.0)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(D_m * M) * 0.5) * Float64(Float64(Float64(0.5 / d) * M) * D_m)) / d), Float64(Float64(h / l) * -0.5), 1.0) * t_3);
	elseif (t_0 <= 1e+248)
		tmp = Float64(t_2 * t_1);
	else
		tmp = abs(t_3);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(1.0 - N[(N[(N[Power[N[(N[(D$95$m * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+102], N[(N[(N[(N[(N[(D$95$m / d), $MachinePrecision] / d), $MachinePrecision] * N[(N[(-0.125 * N[(N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(N[(N[(D$95$m * M), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(N[(0.5 / d), $MachinePrecision] * M), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[t$95$0, 1e+248], N[(t$95$2 * t$95$1), $MachinePrecision], N[Abs[t$95$3], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -9.99999999999999977e101

   1. Initial program 79.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 77.7%

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

   5. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 55.3

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

   7. Applied rewrites 55.3%

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

   9. Applied rewrites 63.1%

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

   10. Taylor expanded in d around 0

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

       1. associate-*r/ N/A

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

       2. associate-/l/ N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. associate-*l/ N/A

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

       6. associate-*r/ N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. associate-/l* N/A

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

       10. lower-*.f64 N/A

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

   12. Applied rewrites 66.6%

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

   if -9.99999999999999977e101 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 49.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 44.7%

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

   6. Applied rewrites 43.6%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow-pow N/A

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

      12. inv-pow N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. frac-times N/A

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

      17. clear-num N/A

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

      18. unpow2 N/A

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

      19. associate-/r* N/A

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

      20. associate-*l/ N/A

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

      21. lower-/.f64 N/A

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

   8. Applied rewrites 44.3%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.00000000000000005e248

   1. Initial program 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in d around inf

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 95.9

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

   7. Applied rewrites 95.9%

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

   if 1.00000000000000005e248 < (*.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 23.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 22.8

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

   5. Applied rewrites 22.8%

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

   7. Applied rewrites 55.0%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{+102}:\\ \;\;\;\;\left(\left(\frac{\frac{D}{d}}{d} \cdot \left(\left(-0.125 \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot D\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(D \cdot M\right) \cdot 0.5\right) \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right)}{d}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+248}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left(1 - \left({\left(\frac{D\_m \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\frac{\left(\left(\frac{\left(D\_m \cdot D\_m\right) \cdot h}{\ell} \cdot -0.125\right) \cdot M\right) \cdot M}{d \cdot d} \cdot t\_2\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(D\_m \cdot M\right) \cdot 0.5\right) \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\_m\right)}{d}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot t\_3\\ \mathbf{elif}\;t\_0 \leq 10^{+248}:\\ \;\;\;\;t\_2 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left|t\_3\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0
         (*
          (- 1.0 (* (* (pow (/ (* D_m M) (* d 2.0)) 2.0) (/ 1.0 2.0)) (/ h l)))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_1 (sqrt (/ d h)))
        (t_2 (sqrt (/ d l)))
        (t_3 (/ d (sqrt (* h l)))))
   (if (<= t_0 (- INFINITY))
     (* (* (/ (* (* (* (/ (* (* D_m D_m) h) l) -0.125) M) M) (* d d)) t_2) t_1)
     (if (<= t_0 0.0)
       (*
        (fma
         (/ (* (* (* D_m M) 0.5) (* (* (/ 0.5 d) M) D_m)) d)
         (* (/ h l) -0.5)
         1.0)
        t_3)
       (if (<= t_0 1e+248) (* t_2 t_1) (fabs t_3))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (1.0 - ((pow(((D_m * M) / (d * 2.0)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_1 = sqrt((d / h));
	double t_2 = sqrt((d / l));
	double t_3 = d / sqrt((h * l));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((((((((D_m * D_m) * h) / l) * -0.125) * M) * M) / (d * d)) * t_2) * t_1;
	} else if (t_0 <= 0.0) {
		tmp = fma(((((D_m * M) * 0.5) * (((0.5 / d) * M) * D_m)) / d), ((h / l) * -0.5), 1.0) * t_3;
	} else if (t_0 <= 1e+248) {
		tmp = t_2 * t_1;
	} else {
		tmp = fabs(t_3);
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(1.0 - Float64(Float64((Float64(Float64(D_m * M) / Float64(d * 2.0)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(d / l))
	t_3 = Float64(d / sqrt(Float64(h * l)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(D_m * D_m) * h) / l) * -0.125) * M) * M) / Float64(d * d)) * t_2) * t_1);
	elseif (t_0 <= 0.0)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(D_m * M) * 0.5) * Float64(Float64(Float64(0.5 / d) * M) * D_m)) / d), Float64(Float64(h / l) * -0.5), 1.0) * t_3);
	elseif (t_0 <= 1e+248)
		tmp = Float64(t_2 * t_1);
	else
		tmp = abs(t_3);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(1.0 - N[(N[(N[Power[N[(N[(D$95$m * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision] * -0.125), $MachinePrecision] * M), $MachinePrecision] * M), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(N[(N[(D$95$m * M), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(N[(0.5 / d), $MachinePrecision] * M), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[t$95$0, 1e+248], N[(t$95$2 * t$95$1), $MachinePrecision], N[Abs[t$95$3], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 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)))) < -inf.0

   1. Initial program 76.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 76.4%

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

   5. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 61.3

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

   7. Applied rewrites 61.3%

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

   8. Taylor expanded in d around 0

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

   10. Applied rewrites 67.9%

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

   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)))) < 0.0

   1. Initial program 66.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 59.2%

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

   6. Applied rewrites 46.1%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow-pow N/A

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

      12. inv-pow N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. frac-times N/A

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

      17. clear-num N/A

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

      18. unpow2 N/A

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

      19. associate-/r* N/A

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

      20. associate-*l/ N/A

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

      21. lower-/.f64 N/A

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

   8. Applied rewrites 46.5%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.00000000000000005e248

   1. Initial program 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in d around inf

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 95.9

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

   7. Applied rewrites 95.9%

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

   if 1.00000000000000005e248 < (*.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 23.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 22.8

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

   5. Applied rewrites 22.8%

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

   7. Applied rewrites 55.0%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -\infty:\\ \;\;\;\;\left(\frac{\left(\left(\frac{\left(D \cdot D\right) \cdot h}{\ell} \cdot -0.125\right) \cdot M\right) \cdot M}{d \cdot d} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(D \cdot M\right) \cdot 0.5\right) \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right)}{d}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+248}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left(1 - \left({\left(\frac{D\_m \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \frac{d}{\sqrt{h \cdot \ell}}\\ t_2 := \left|t\_1\right|\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-73}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(D\_m \cdot M\right) \cdot 0.5\right) \cdot \left(D\_m \cdot M\right)}{\left(d \cdot 2\right) \cdot d}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 10^{+248}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0
         (*
          (- 1.0 (* (* (pow (/ (* D_m M) (* d 2.0)) 2.0) (/ 1.0 2.0)) (/ h l)))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_1 (/ d (sqrt (* h l))))
        (t_2 (fabs t_1)))
   (if (<= t_0 -2e-73)
     (*
      (fma
       (/ (* (* (* D_m M) 0.5) (* D_m M)) (* (* d 2.0) d))
       (* (/ h l) -0.5)
       1.0)
      t_1)
     (if (<= t_0 0.0)
       t_2
       (if (<= t_0 1e+248) (* (sqrt (/ d l)) (sqrt (/ d h))) t_2)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (1.0 - ((pow(((D_m * M) / (d * 2.0)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_1 = d / sqrt((h * l));
	double t_2 = fabs(t_1);
	double tmp;
	if (t_0 <= -2e-73) {
		tmp = fma(((((D_m * M) * 0.5) * (D_m * M)) / ((d * 2.0) * d)), ((h / l) * -0.5), 1.0) * t_1;
	} else if (t_0 <= 0.0) {
		tmp = t_2;
	} else if (t_0 <= 1e+248) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(1.0 - Float64(Float64((Float64(Float64(D_m * M) / Float64(d * 2.0)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	t_1 = Float64(d / sqrt(Float64(h * l)))
	t_2 = abs(t_1)
	tmp = 0.0
	if (t_0 <= -2e-73)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(D_m * M) * 0.5) * Float64(D_m * M)) / Float64(Float64(d * 2.0) * d)), Float64(Float64(h / l) * -0.5), 1.0) * t_1);
	elseif (t_0 <= 0.0)
		tmp = t_2;
	elseif (t_0 <= 1e+248)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(1.0 - N[(N[(N[Power[N[(N[(D$95$m * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[t$95$1], $MachinePrecision]}, If[LessEqual[t$95$0, -2e-73], N[(N[(N[(N[(N[(N[(D$95$m * M), $MachinePrecision] * 0.5), $MachinePrecision] * N[(D$95$m * M), $MachinePrecision]), $MachinePrecision] / N[(N[(d * 2.0), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$2, If[LessEqual[t$95$0, 1e+248], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

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)))) < -1.99999999999999999e-73

   1. Initial program 80.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 77.9%

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

   6. Applied rewrites 38.9%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow-pow N/A

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

      12. inv-pow N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. frac-times N/A

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

      17. clear-num N/A

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

      18. unpow2 N/A

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

      19. associate-/r* N/A

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

      20. frac-times N/A

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

      21. lower-/.f64 N/A

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

   8. Applied rewrites 31.9%

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

   if -1.99999999999999999e-73 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 1.00000000000000005e248 < (*.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 25.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 28.1

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

   5. Applied rewrites 28.1%

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

   7. Applied rewrites 56.0%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.00000000000000005e248

   1. Initial program 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in d around inf

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 95.9

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

   7. Applied rewrites 95.9%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -2 \cdot 10^{-73}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(D \cdot M\right) \cdot 0.5\right) \cdot \left(D \cdot M\right)}{\left(d \cdot 2\right) \cdot d}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+248}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left(1 - \left({\left(\frac{D\_m \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \frac{d}{\sqrt{h \cdot \ell}}\\ t_2 := \left|t\_1\right|\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-73}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{\left(D\_m \cdot D\_m\right) \cdot h}{\ell} \cdot -0.125\right) \cdot M, M, d \cdot d\right)}{d \cdot d} \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 10^{+248}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0
         (*
          (- 1.0 (* (* (pow (/ (* D_m M) (* d 2.0)) 2.0) (/ 1.0 2.0)) (/ h l)))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_1 (/ d (sqrt (* h l))))
        (t_2 (fabs t_1)))
   (if (<= t_0 -2e-73)
     (*
      (/ (fma (* (* (/ (* (* D_m D_m) h) l) -0.125) M) M (* d d)) (* d d))
      t_1)
     (if (<= t_0 0.0)
       t_2
       (if (<= t_0 1e+248) (* (sqrt (/ d l)) (sqrt (/ d h))) t_2)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (1.0 - ((pow(((D_m * M) / (d * 2.0)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_1 = d / sqrt((h * l));
	double t_2 = fabs(t_1);
	double tmp;
	if (t_0 <= -2e-73) {
		tmp = (fma((((((D_m * D_m) * h) / l) * -0.125) * M), M, (d * d)) / (d * d)) * t_1;
	} else if (t_0 <= 0.0) {
		tmp = t_2;
	} else if (t_0 <= 1e+248) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(1.0 - Float64(Float64((Float64(Float64(D_m * M) / Float64(d * 2.0)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	t_1 = Float64(d / sqrt(Float64(h * l)))
	t_2 = abs(t_1)
	tmp = 0.0
	if (t_0 <= -2e-73)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(Float64(D_m * D_m) * h) / l) * -0.125) * M), M, Float64(d * d)) / Float64(d * d)) * t_1);
	elseif (t_0 <= 0.0)
		tmp = t_2;
	elseif (t_0 <= 1e+248)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(1.0 - N[(N[(N[Power[N[(N[(D$95$m * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[t$95$1], $MachinePrecision]}, If[LessEqual[t$95$0, -2e-73], N[(N[(N[(N[(N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision] * -0.125), $MachinePrecision] * M), $MachinePrecision] * M + N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$2, If[LessEqual[t$95$0, 1e+248], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

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)))) < -1.99999999999999999e-73

   1. Initial program 80.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 77.9%

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

   6. Applied rewrites 38.9%

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

   7. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 26.6%

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

   if -1.99999999999999999e-73 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 1.00000000000000005e248 < (*.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 25.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 28.1

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

   5. Applied rewrites 28.1%

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

   7. Applied rewrites 56.0%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.00000000000000005e248

   1. Initial program 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in d around inf

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 95.9

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

   7. Applied rewrites 95.9%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -2 \cdot 10^{-73}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{\left(D \cdot D\right) \cdot h}{\ell} \cdot -0.125\right) \cdot M, M, d \cdot d\right)}{d \cdot d} \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+248}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left(1 - \left({\left(\frac{D\_m \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \frac{d}{\sqrt{h \cdot \ell}}\\ t_2 := \left|t\_1\right|\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+127}:\\ \;\;\;\;\left(\left(\frac{\frac{D\_m \cdot D\_m}{d}}{d} \cdot -0.125\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 10^{+248}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0
         (*
          (- 1.0 (* (* (pow (/ (* D_m M) (* d 2.0)) 2.0) (/ 1.0 2.0)) (/ h l)))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_1 (/ d (sqrt (* h l))))
        (t_2 (fabs t_1)))
   (if (<= t_0 -1e+127)
     (* (* (* (/ (/ (* D_m D_m) d) d) -0.125) (/ (* (* M M) h) l)) t_1)
     (if (<= t_0 0.0)
       t_2
       (if (<= t_0 1e+248) (* (sqrt (/ d l)) (sqrt (/ d h))) t_2)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (1.0 - ((pow(((D_m * M) / (d * 2.0)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_1 = d / sqrt((h * l));
	double t_2 = fabs(t_1);
	double tmp;
	if (t_0 <= -1e+127) {
		tmp = (((((D_m * D_m) / d) / d) * -0.125) * (((M * M) * h) / l)) * t_1;
	} else if (t_0 <= 0.0) {
		tmp = t_2;
	} else if (t_0 <= 1e+248) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (1.0d0 - (((((d_m * m) / (d * 2.0d0)) ** 2.0d0) * (1.0d0 / 2.0d0)) * (h / l))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))
    t_1 = d / sqrt((h * l))
    t_2 = abs(t_1)
    if (t_0 <= (-1d+127)) then
        tmp = (((((d_m * d_m) / d) / d) * (-0.125d0)) * (((m * m) * h) / l)) * t_1
    else if (t_0 <= 0.0d0) then
        tmp = t_2
    else if (t_0 <= 1d+248) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (1.0 - ((Math.pow(((D_m * M) / (d * 2.0)), 2.0) * (1.0 / 2.0)) * (h / l))) * (Math.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0)));
	double t_1 = d / Math.sqrt((h * l));
	double t_2 = Math.abs(t_1);
	double tmp;
	if (t_0 <= -1e+127) {
		tmp = (((((D_m * D_m) / d) / d) * -0.125) * (((M * M) * h) / l)) * t_1;
	} else if (t_0 <= 0.0) {
		tmp = t_2;
	} else if (t_0 <= 1e+248) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = (1.0 - ((math.pow(((D_m * M) / (d * 2.0)), 2.0) * (1.0 / 2.0)) * (h / l))) * (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0)))
	t_1 = d / math.sqrt((h * l))
	t_2 = math.fabs(t_1)
	tmp = 0
	if t_0 <= -1e+127:
		tmp = (((((D_m * D_m) / d) / d) * -0.125) * (((M * M) * h) / l)) * t_1
	elif t_0 <= 0.0:
		tmp = t_2
	elif t_0 <= 1e+248:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = t_2
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(1.0 - Float64(Float64((Float64(Float64(D_m * M) / Float64(d * 2.0)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	t_1 = Float64(d / sqrt(Float64(h * l)))
	t_2 = abs(t_1)
	tmp = 0.0
	if (t_0 <= -1e+127)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(D_m * D_m) / d) / d) * -0.125) * Float64(Float64(Float64(M * M) * h) / l)) * t_1);
	elseif (t_0 <= 0.0)
		tmp = t_2;
	elseif (t_0 <= 1e+248)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = (1.0 - (((((D_m * M) / (d * 2.0)) ^ 2.0) * (1.0 / 2.0)) * (h / l))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)));
	t_1 = d / sqrt((h * l));
	t_2 = abs(t_1);
	tmp = 0.0;
	if (t_0 <= -1e+127)
		tmp = (((((D_m * D_m) / d) / d) * -0.125) * (((M * M) * h) / l)) * t_1;
	elseif (t_0 <= 0.0)
		tmp = t_2;
	elseif (t_0 <= 1e+248)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(1.0 - N[(N[(N[Power[N[(N[(D$95$m * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[t$95$1], $MachinePrecision]}, If[LessEqual[t$95$0, -1e+127], N[(N[(N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$2, If[LessEqual[t$95$0, 1e+248], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -9.99999999999999955e126

   1. Initial program 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 77.3%

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

   6. Applied rewrites 40.5%

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

   7. Taylor expanded in d around 0

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

      1. times-frac N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 27.9

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

   9. Applied rewrites 27.9%

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

   if -9.99999999999999955e126 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 1.00000000000000005e248 < (*.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 31.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 26.1

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

   5. Applied rewrites 26.1%

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

   7. Applied rewrites 51.2%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.00000000000000005e248

   1. Initial program 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in d around inf

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 95.9

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

   7. Applied rewrites 95.9%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{+127}:\\ \;\;\;\;\left(\left(\frac{\frac{D \cdot D}{d}}{d} \cdot -0.125\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+248}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_3 := \left(1 - \left({\left(\frac{D\_m \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{-161}:\\ \;\;\;\;\left(-t\_0\right) \cdot t\_1\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 10^{+248}:\\ \;\;\;\;t\_0 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1 (sqrt (/ d h)))
        (t_2 (fabs (/ d (sqrt (* h l)))))
        (t_3
         (*
          (- 1.0 (* (* (pow (/ (* D_m M) (* d 2.0)) 2.0) (/ 1.0 2.0)) (/ h l)))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))))
   (if (<= t_3 -5e-161)
     (* (- t_0) t_1)
     (if (<= t_3 0.0) t_2 (if (<= t_3 1e+248) (* t_0 t_1) t_2)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt((d / l));
	double t_1 = sqrt((d / h));
	double t_2 = fabs((d / sqrt((h * l))));
	double t_3 = (1.0 - ((pow(((D_m * M) / (d * 2.0)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_3 <= -5e-161) {
		tmp = -t_0 * t_1;
	} else if (t_3 <= 0.0) {
		tmp = t_2;
	} else if (t_3 <= 1e+248) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = sqrt((d / h))
    t_2 = abs((d / sqrt((h * l))))
    t_3 = (1.0d0 - (((((d_m * m) / (d * 2.0d0)) ** 2.0d0) * (1.0d0 / 2.0d0)) * (h / l))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))
    if (t_3 <= (-5d-161)) then
        tmp = -t_0 * t_1
    else if (t_3 <= 0.0d0) then
        tmp = t_2
    else if (t_3 <= 1d+248) then
        tmp = t_0 * t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = Math.sqrt((d / h));
	double t_2 = Math.abs((d / Math.sqrt((h * l))));
	double t_3 = (1.0 - ((Math.pow(((D_m * M) / (d * 2.0)), 2.0) * (1.0 / 2.0)) * (h / l))) * (Math.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_3 <= -5e-161) {
		tmp = -t_0 * t_1;
	} else if (t_3 <= 0.0) {
		tmp = t_2;
	} else if (t_3 <= 1e+248) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.sqrt((d / l))
	t_1 = math.sqrt((d / h))
	t_2 = math.fabs((d / math.sqrt((h * l))))
	t_3 = (1.0 - ((math.pow(((D_m * M) / (d * 2.0)), 2.0) * (1.0 / 2.0)) * (h / l))) * (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0)))
	tmp = 0
	if t_3 <= -5e-161:
		tmp = -t_0 * t_1
	elif t_3 <= 0.0:
		tmp = t_2
	elif t_3 <= 1e+248:
		tmp = t_0 * t_1
	else:
		tmp = t_2
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(d / l))
	t_1 = sqrt(Float64(d / h))
	t_2 = abs(Float64(d / sqrt(Float64(h * l))))
	t_3 = Float64(Float64(1.0 - Float64(Float64((Float64(Float64(D_m * M) / Float64(d * 2.0)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	tmp = 0.0
	if (t_3 <= -5e-161)
		tmp = Float64(Float64(-t_0) * t_1);
	elseif (t_3 <= 0.0)
		tmp = t_2;
	elseif (t_3 <= 1e+248)
		tmp = Float64(t_0 * t_1);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = sqrt((d / l));
	t_1 = sqrt((d / h));
	t_2 = abs((d / sqrt((h * l))));
	t_3 = (1.0 - (((((D_m * M) / (d * 2.0)) ^ 2.0) * (1.0 / 2.0)) * (h / l))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)));
	tmp = 0.0;
	if (t_3 <= -5e-161)
		tmp = -t_0 * t_1;
	elseif (t_3 <= 0.0)
		tmp = t_2;
	elseif (t_3 <= 1e+248)
		tmp = t_0 * t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(1.0 - N[(N[(N[Power[N[(N[(D$95$m * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e-161], N[((-t$95$0) * t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 0.0], t$95$2, If[LessEqual[t$95$3, 1e+248], N[(t$95$0 * t$95$1), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.9999999999999999e-161

   1. Initial program 81.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 78.3%

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

   5. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 15.8

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

   7. Applied rewrites 15.8%

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

   if -4.9999999999999999e-161 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 1.00000000000000005e248 < (*.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 23.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 28.7

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

   5. Applied rewrites 28.7%

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

   7. Applied rewrites 57.2%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.00000000000000005e248

   1. Initial program 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in d around inf

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 95.9

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

   7. Applied rewrites 95.9%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -5 \cdot 10^{-161}:\\ \;\;\;\;\left(-\sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+248}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_1 := \left(1 - \left({\left(\frac{D\_m \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+248}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (fabs (/ d (sqrt (* h l)))))
        (t_1
         (*
          (- 1.0 (* (* (pow (/ (* D_m M) (* d 2.0)) 2.0) (/ 1.0 2.0)) (/ h l)))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))))
   (if (<= t_1 -2e-73)
     (* (sqrt (/ 1.0 (* h l))) (- d))
     (if (<= t_1 0.0)
       t_0
       (if (<= t_1 1e+248) (* (sqrt (/ d l)) (sqrt (/ d h))) t_0)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = fabs((d / sqrt((h * l))));
	double t_1 = (1.0 - ((pow(((D_m * M) / (d * 2.0)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_1 <= -2e-73) {
		tmp = sqrt((1.0 / (h * l))) * -d;
	} else if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 1e+248) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((d / sqrt((h * l))))
    t_1 = (1.0d0 - (((((d_m * m) / (d * 2.0d0)) ** 2.0d0) * (1.0d0 / 2.0d0)) * (h / l))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))
    if (t_1 <= (-2d-73)) then
        tmp = sqrt((1.0d0 / (h * l))) * -d
    else if (t_1 <= 0.0d0) then
        tmp = t_0
    else if (t_1 <= 1d+248) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.abs((d / Math.sqrt((h * l))));
	double t_1 = (1.0 - ((Math.pow(((D_m * M) / (d * 2.0)), 2.0) * (1.0 / 2.0)) * (h / l))) * (Math.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_1 <= -2e-73) {
		tmp = Math.sqrt((1.0 / (h * l))) * -d;
	} else if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 1e+248) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.fabs((d / math.sqrt((h * l))))
	t_1 = (1.0 - ((math.pow(((D_m * M) / (d * 2.0)), 2.0) * (1.0 / 2.0)) * (h / l))) * (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0)))
	tmp = 0
	if t_1 <= -2e-73:
		tmp = math.sqrt((1.0 / (h * l))) * -d
	elif t_1 <= 0.0:
		tmp = t_0
	elif t_1 <= 1e+248:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = t_0
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = abs(Float64(d / sqrt(Float64(h * l))))
	t_1 = Float64(Float64(1.0 - Float64(Float64((Float64(Float64(D_m * M) / Float64(d * 2.0)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	tmp = 0.0
	if (t_1 <= -2e-73)
		tmp = Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(-d));
	elseif (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 1e+248)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = abs((d / sqrt((h * l))));
	t_1 = (1.0 - (((((D_m * M) / (d * 2.0)) ^ 2.0) * (1.0 / 2.0)) * (h / l))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)));
	tmp = 0.0;
	if (t_1 <= -2e-73)
		tmp = sqrt((1.0 / (h * l))) * -d;
	elseif (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 1e+248)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - N[(N[(N[Power[N[(N[(D$95$m * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-73], N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, 1e+248], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

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)))) < -1.99999999999999999e-73

   1. Initial program 80.8%

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

   3. Taylor expanded in 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 9.5

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

   5. Applied rewrites 9.5%

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

   if -1.99999999999999999e-73 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 1.00000000000000005e248 < (*.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 25.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 28.1

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

   5. Applied rewrites 28.1%

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

   7. Applied rewrites 56.0%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.00000000000000005e248

   1. Initial program 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in d around inf

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 95.9

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

   7. Applied rewrites 95.9%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -2 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+248}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 49.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_1 := \left(1 - \left({\left(\frac{D\_m \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-150}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+96}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (fabs (/ d (sqrt (* h l)))))
        (t_1
         (*
          (- 1.0 (* (* (pow (/ (* D_m M) (* d 2.0)) 2.0) (/ 1.0 2.0)) (/ h l)))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))))
   (if (<= t_1 -2e-73)
     (* (sqrt (/ 1.0 (* h l))) (- d))
     (if (<= t_1 2e-150)
       t_0
       (if (<= t_1 1e+96) (sqrt (* (/ d l) (/ d h))) t_0)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = fabs((d / sqrt((h * l))));
	double t_1 = (1.0 - ((pow(((D_m * M) / (d * 2.0)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_1 <= -2e-73) {
		tmp = sqrt((1.0 / (h * l))) * -d;
	} else if (t_1 <= 2e-150) {
		tmp = t_0;
	} else if (t_1 <= 1e+96) {
		tmp = sqrt(((d / l) * (d / h)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((d / sqrt((h * l))))
    t_1 = (1.0d0 - (((((d_m * m) / (d * 2.0d0)) ** 2.0d0) * (1.0d0 / 2.0d0)) * (h / l))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))
    if (t_1 <= (-2d-73)) then
        tmp = sqrt((1.0d0 / (h * l))) * -d
    else if (t_1 <= 2d-150) then
        tmp = t_0
    else if (t_1 <= 1d+96) then
        tmp = sqrt(((d / l) * (d / h)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.abs((d / Math.sqrt((h * l))));
	double t_1 = (1.0 - ((Math.pow(((D_m * M) / (d * 2.0)), 2.0) * (1.0 / 2.0)) * (h / l))) * (Math.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_1 <= -2e-73) {
		tmp = Math.sqrt((1.0 / (h * l))) * -d;
	} else if (t_1 <= 2e-150) {
		tmp = t_0;
	} else if (t_1 <= 1e+96) {
		tmp = Math.sqrt(((d / l) * (d / h)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.fabs((d / math.sqrt((h * l))))
	t_1 = (1.0 - ((math.pow(((D_m * M) / (d * 2.0)), 2.0) * (1.0 / 2.0)) * (h / l))) * (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0)))
	tmp = 0
	if t_1 <= -2e-73:
		tmp = math.sqrt((1.0 / (h * l))) * -d
	elif t_1 <= 2e-150:
		tmp = t_0
	elif t_1 <= 1e+96:
		tmp = math.sqrt(((d / l) * (d / h)))
	else:
		tmp = t_0
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = abs(Float64(d / sqrt(Float64(h * l))))
	t_1 = Float64(Float64(1.0 - Float64(Float64((Float64(Float64(D_m * M) / Float64(d * 2.0)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	tmp = 0.0
	if (t_1 <= -2e-73)
		tmp = Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(-d));
	elseif (t_1 <= 2e-150)
		tmp = t_0;
	elseif (t_1 <= 1e+96)
		tmp = sqrt(Float64(Float64(d / l) * Float64(d / h)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = abs((d / sqrt((h * l))));
	t_1 = (1.0 - (((((D_m * M) / (d * 2.0)) ^ 2.0) * (1.0 / 2.0)) * (h / l))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)));
	tmp = 0.0;
	if (t_1 <= -2e-73)
		tmp = sqrt((1.0 / (h * l))) * -d;
	elseif (t_1 <= 2e-150)
		tmp = t_0;
	elseif (t_1 <= 1e+96)
		tmp = sqrt(((d / l) * (d / h)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - N[(N[(N[Power[N[(N[(D$95$m * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-73], N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], If[LessEqual[t$95$1, 2e-150], t$95$0, If[LessEqual[t$95$1, 1e+96], N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]

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)))) < -1.99999999999999999e-73

   1. Initial program 80.8%

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

   3. Taylor expanded in 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 9.5

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

   5. Applied rewrites 9.5%

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

   if -1.99999999999999999e-73 < (*.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)))) < 2.00000000000000001e-150 or 1.00000000000000005e96 < (*.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 39.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 28.7

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

   5. Applied rewrites 28.7%

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

   7. Applied rewrites 61.0%

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

   if 2.00000000000000001e-150 < (*.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)))) < 1.00000000000000005e96

   1. Initial program 99.5%

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

   3. 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 38.8

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

   5. Applied rewrites 38.8%

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

   7. Applied rewrites 38.7%

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

   9. Applied rewrites 95.4%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -2 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 2 \cdot 10^{-150}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+96}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left(\frac{0.5}{d} \cdot M\right) \cdot D\_m\\ t_1 := \left(1 - \left({\left(\frac{D\_m \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_2 := \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_2 \cdot \mathsf{fma}\left(t\_0 \cdot \left(\frac{h}{\ell} \cdot -0.5\right), t\_0, 1\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+248}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|t\_2\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* (* (/ 0.5 d) M) D_m))
        (t_1
         (*
          (- 1.0 (* (* (pow (/ (* D_m M) (* d 2.0)) 2.0) (/ 1.0 2.0)) (/ h l)))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_2 (/ d (sqrt (* h l)))))
   (if (<= t_1 0.0)
     (* t_2 (fma (* t_0 (* (/ h l) -0.5)) t_0 1.0))
     (if (<= t_1 1e+248) (* (sqrt (/ d l)) (sqrt (/ d h))) (fabs t_2)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = ((0.5 / d) * M) * D_m;
	double t_1 = (1.0 - ((pow(((D_m * M) / (d * 2.0)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_2 = d / sqrt((h * l));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_2 * fma((t_0 * ((h / l) * -0.5)), t_0, 1.0);
	} else if (t_1 <= 1e+248) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = fabs(t_2);
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(Float64(0.5 / d) * M) * D_m)
	t_1 = Float64(Float64(1.0 - Float64(Float64((Float64(Float64(D_m * M) / Float64(d * 2.0)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	t_2 = Float64(d / sqrt(Float64(h * l)))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(t_2 * fma(Float64(t_0 * Float64(Float64(h / l) * -0.5)), t_0, 1.0));
	elseif (t_1 <= 1e+248)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = abs(t_2);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(N[(0.5 / d), $MachinePrecision] * M), $MachinePrecision] * D$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - N[(N[(N[Power[N[(N[(D$95$m * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(t$95$2 * N[(N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+248], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[t$95$2], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 70.6%

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

   6. Applied rewrites 40.5%

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

   8. Applied rewrites 43.2%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.00000000000000005e248

   1. Initial program 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in d around inf

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 95.9

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

   7. Applied rewrites 95.9%

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

   if 1.00000000000000005e248 < (*.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 23.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 22.8

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

   5. Applied rewrites 22.8%

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

   7. Applied rewrites 55.0%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right) \cdot \left(\frac{h}{\ell} \cdot -0.5\right), \left(\frac{0.5}{d} \cdot M\right) \cdot D, 1\right)\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+248}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left(1 - \left({\left(\frac{D\_m \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\left(\left(\left(\frac{M}{d} \cdot h\right) \cdot D\_m\right) \cdot 0.25\right) \cdot \left(\frac{0.5}{d} \cdot M\right), \frac{-D\_m}{\ell}, 1\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+248}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0
         (*
          (- 1.0 (* (* (pow (/ (* D_m M) (* d 2.0)) 2.0) (/ 1.0 2.0)) (/ h l)))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_1 (/ d (sqrt (* h l)))))
   (if (<= t_0 0.0)
     (*
      t_1
      (fma
       (* (* (* (* (/ M d) h) D_m) 0.25) (* (/ 0.5 d) M))
       (/ (- D_m) l)
       1.0))
     (if (<= t_0 1e+248) (* (sqrt (/ d l)) (sqrt (/ d h))) (fabs t_1)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (1.0 - ((pow(((D_m * M) / (d * 2.0)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_1 = d / sqrt((h * l));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1 * fma((((((M / d) * h) * D_m) * 0.25) * ((0.5 / d) * M)), (-D_m / l), 1.0);
	} else if (t_0 <= 1e+248) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = fabs(t_1);
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(1.0 - Float64(Float64((Float64(Float64(D_m * M) / Float64(d * 2.0)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	t_1 = Float64(d / sqrt(Float64(h * l)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(t_1 * fma(Float64(Float64(Float64(Float64(Float64(M / d) * h) * D_m) * 0.25) * Float64(Float64(0.5 / d) * M)), Float64(Float64(-D_m) / l), 1.0));
	elseif (t_0 <= 1e+248)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = abs(t_1);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(1.0 - N[(N[(N[Power[N[(N[(D$95$m * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(t$95$1 * N[(N[(N[(N[(N[(N[(M / d), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] * 0.25), $MachinePrecision] * N[(N[(0.5 / d), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision] * N[((-D$95$m) / l), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+248], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[t$95$1], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 70.6%

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

   6. Applied rewrites 74.5%

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

   8. Applied rewrites 42.9%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.00000000000000005e248

   1. Initial program 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in d around inf

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 95.9

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

   7. Applied rewrites 95.9%

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

   if 1.00000000000000005e248 < (*.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 23.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 22.8

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

   5. Applied rewrites 22.8%

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

   7. Applied rewrites 55.0%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\left(\left(\left(\frac{M}{d} \cdot h\right) \cdot D\right) \cdot 0.25\right) \cdot \left(\frac{0.5}{d} \cdot M\right), \frac{-D}{\ell}, 1\right)\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+248}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{D\_m}{d} \cdot M\\ t_1 := \left(1 - \left({\left(\frac{D\_m \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_2 := \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\left(t\_0 \cdot 0.25\right) \cdot t\_0, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+248}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|t\_2\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* (/ D_m d) M))
        (t_1
         (*
          (- 1.0 (* (* (pow (/ (* D_m M) (* d 2.0)) 2.0) (/ 1.0 2.0)) (/ h l)))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_2 (/ d (sqrt (* h l)))))
   (if (<= t_1 0.0)
     (* (fma (* (* t_0 0.25) t_0) (* (/ h l) -0.5) 1.0) t_2)
     (if (<= t_1 1e+248) (* (sqrt (/ d l)) (sqrt (/ d h))) (fabs t_2)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (D_m / d) * M;
	double t_1 = (1.0 - ((pow(((D_m * M) / (d * 2.0)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_2 = d / sqrt((h * l));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = fma(((t_0 * 0.25) * t_0), ((h / l) * -0.5), 1.0) * t_2;
	} else if (t_1 <= 1e+248) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = fabs(t_2);
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(D_m / d) * M)
	t_1 = Float64(Float64(1.0 - Float64(Float64((Float64(Float64(D_m * M) / Float64(d * 2.0)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	t_2 = Float64(d / sqrt(Float64(h * l)))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(fma(Float64(Float64(t_0 * 0.25) * t_0), Float64(Float64(h / l) * -0.5), 1.0) * t_2);
	elseif (t_1 <= 1e+248)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = abs(t_2);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(D$95$m / d), $MachinePrecision] * M), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - N[(N[(N[Power[N[(N[(D$95$m * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[(N[(N[(t$95$0 * 0.25), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$1, 1e+248], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[t$95$2], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 70.6%

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

   6. Applied rewrites 40.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. inv-pow N/A

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

      7. pow-pow N/A

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

      8. metadata-eval N/A

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

      9. pow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

   8. Applied rewrites 40.4%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.00000000000000005e248

   1. Initial program 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in d around inf

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 95.9

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

   7. Applied rewrites 95.9%

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

   if 1.00000000000000005e248 < (*.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 23.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 22.8

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

   5. Applied rewrites 22.8%

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

   7. Applied rewrites 55.0%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\frac{D}{d} \cdot M\right) \cdot 0.25\right) \cdot \left(\frac{D}{d} \cdot M\right), \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+248}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left(1 - \left({\left(\frac{D\_m \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{D\_m \cdot D\_m}{d} \cdot -0.125, \frac{\left(M \cdot M\right) \cdot h}{d}, \ell\right)}{\ell} \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+248}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0
         (*
          (- 1.0 (* (* (pow (/ (* D_m M) (* d 2.0)) 2.0) (/ 1.0 2.0)) (/ h l)))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_1 (/ d (sqrt (* h l)))))
   (if (<= t_0 0.0)
     (* (/ (fma (* (/ (* D_m D_m) d) -0.125) (/ (* (* M M) h) d) l) l) t_1)
     (if (<= t_0 1e+248) (* (sqrt (/ d l)) (sqrt (/ d h))) (fabs t_1)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (1.0 - ((pow(((D_m * M) / (d * 2.0)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_1 = d / sqrt((h * l));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (fma((((D_m * D_m) / d) * -0.125), (((M * M) * h) / d), l) / l) * t_1;
	} else if (t_0 <= 1e+248) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = fabs(t_1);
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(1.0 - Float64(Float64((Float64(Float64(D_m * M) / Float64(d * 2.0)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	t_1 = Float64(d / sqrt(Float64(h * l)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(D_m * D_m) / d) * -0.125), Float64(Float64(Float64(M * M) * h) / d), l) / l) * t_1);
	elseif (t_0 <= 1e+248)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = abs(t_1);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(1.0 - N[(N[(N[Power[N[(N[(D$95$m * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision] / d), $MachinePrecision] + l), $MachinePrecision] / l), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 1e+248], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[t$95$1], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 70.6%

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

   6. Applied rewrites 40.5%

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

   7. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 33.2

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

   9. Applied rewrites 33.2%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.00000000000000005e248

   1. Initial program 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in d around inf

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 95.9

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

   7. Applied rewrites 95.9%

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

   if 1.00000000000000005e248 < (*.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 23.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 22.8

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

   5. Applied rewrites 22.8%

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

   7. Applied rewrites 55.0%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{D \cdot D}{d} \cdot -0.125, \frac{\left(M \cdot M\right) \cdot h}{d}, \ell\right)}{\ell} \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+248}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 75.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D\_m \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+248}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\frac{\left(0.5 \cdot M\right) \cdot D\_m}{d}}{-\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(0.25 \cdot D\_m\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<=
      (*
       (- 1.0 (* (* (pow (/ (* D_m M) (* d 2.0)) 2.0) (/ 1.0 2.0)) (/ h l)))
       (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
      1e+248)
   (*
    (*
     (fma (/ (/ (* (* 0.5 M) D_m) d) (- l)) (* (* (/ M d) h) (* 0.25 D_m)) 1.0)
     (sqrt (/ d l)))
    (sqrt (/ d h)))
   (fabs (/ d (sqrt (* h l))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (((1.0 - ((pow(((D_m * M) / (d * 2.0)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)))) <= 1e+248) {
		tmp = (fma(((((0.5 * M) * D_m) / d) / -l), (((M / d) * h) * (0.25 * D_m)), 1.0) * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = fabs((d / sqrt((h * l))));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64((Float64(Float64(D_m * M) / Float64(d * 2.0)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0)))) <= 1e+248)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(0.5 * M) * D_m) / d) / Float64(-l)), Float64(Float64(Float64(M / d) * h) * Float64(0.25 * D_m)), 1.0) * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	else
		tmp = abs(Float64(d / sqrt(Float64(h * l))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[N[(N[(1.0 - N[(N[(N[Power[N[(N[(D$95$m * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+248], N[(N[(N[(N[(N[(N[(N[(0.5 * M), $MachinePrecision] * D$95$m), $MachinePrecision] / d), $MachinePrecision] / (-l)), $MachinePrecision] * N[(N[(N[(M / d), $MachinePrecision] * h), $MachinePrecision] * N[(0.25 * D$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $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)))) < 1.00000000000000005e248

   1. Initial program 82.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 80.9%

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

   6. Applied rewrites 82.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. div-inv N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 82.1

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

   8. Applied rewrites 82.1%

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

   if 1.00000000000000005e248 < (*.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 23.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 22.8

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

   5. Applied rewrites 22.8%

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

   7. Applied rewrites 55.0%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+248}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\frac{\left(0.5 \cdot M\right) \cdot D}{d}}{-\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(0.25 \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 75.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D\_m \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+248}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D\_m}{-\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(0.25 \cdot D\_m\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<=
      (*
       (- 1.0 (* (* (pow (/ (* D_m M) (* d 2.0)) 2.0) (/ 1.0 2.0)) (/ h l)))
       (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
      1e+248)
   (*
    (*
     (fma (/ (* (* (/ 0.5 d) M) D_m) (- l)) (* (* (/ M d) h) (* 0.25 D_m)) 1.0)
     (sqrt (/ d l)))
    (sqrt (/ d h)))
   (fabs (/ d (sqrt (* h l))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (((1.0 - ((pow(((D_m * M) / (d * 2.0)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)))) <= 1e+248) {
		tmp = (fma(((((0.5 / d) * M) * D_m) / -l), (((M / d) * h) * (0.25 * D_m)), 1.0) * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = fabs((d / sqrt((h * l))));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64((Float64(Float64(D_m * M) / Float64(d * 2.0)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0)))) <= 1e+248)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(0.5 / d) * M) * D_m) / Float64(-l)), Float64(Float64(Float64(M / d) * h) * Float64(0.25 * D_m)), 1.0) * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	else
		tmp = abs(Float64(d / sqrt(Float64(h * l))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[N[(N[(1.0 - N[(N[(N[Power[N[(N[(D$95$m * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+248], N[(N[(N[(N[(N[(N[(N[(0.5 / d), $MachinePrecision] * M), $MachinePrecision] * D$95$m), $MachinePrecision] / (-l)), $MachinePrecision] * N[(N[(N[(M / d), $MachinePrecision] * h), $MachinePrecision] * N[(0.25 * D$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $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)))) < 1.00000000000000005e248

   1. Initial program 82.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 80.9%

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

   6. Applied rewrites 82.1%

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

   if 1.00000000000000005e248 < (*.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 23.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 22.8

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

   5. Applied rewrites 22.8%

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

   7. Applied rewrites 55.0%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+248}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{-\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(0.25 \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 76.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D\_m \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+248}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\left(\frac{-M}{d} \cdot \left(0.25 \cdot D\_m\right)\right) \cdot \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D\_m}{\ell}, h, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<=
      (*
       (- 1.0 (* (* (pow (/ (* D_m M) (* d 2.0)) 2.0) (/ 1.0 2.0)) (/ h l)))
       (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
      1e+248)
   (*
    (sqrt (/ d h))
    (*
     (sqrt (/ d l))
     (fma
      (* (* (/ (- M) d) (* 0.25 D_m)) (/ (* (* (/ 0.5 d) M) D_m) l))
      h
      1.0)))
   (fabs (/ d (sqrt (* h l))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (((1.0 - ((pow(((D_m * M) / (d * 2.0)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)))) <= 1e+248) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * fma((((-M / d) * (0.25 * D_m)) * ((((0.5 / d) * M) * D_m) / l)), h, 1.0));
	} else {
		tmp = fabs((d / sqrt((h * l))));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64((Float64(Float64(D_m * M) / Float64(d * 2.0)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0)))) <= 1e+248)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * fma(Float64(Float64(Float64(Float64(-M) / d) * Float64(0.25 * D_m)) * Float64(Float64(Float64(Float64(0.5 / d) * M) * D_m) / l)), h, 1.0)));
	else
		tmp = abs(Float64(d / sqrt(Float64(h * l))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[N[(N[(1.0 - N[(N[(N[Power[N[(N[(D$95$m * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+248], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[((-M) / d), $MachinePrecision] * N[(0.25 * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.5 / d), $MachinePrecision] * M), $MachinePrecision] * D$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $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)))) < 1.00000000000000005e248

   1. Initial program 82.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 80.9%

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

   6. Applied rewrites 83.0%

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

   if 1.00000000000000005e248 < (*.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 23.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 22.8

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

   5. Applied rewrites 22.8%

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

   7. Applied rewrites 55.0%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+248}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\left(\frac{-M}{d} \cdot \left(0.25 \cdot D\right)\right) \cdot \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell}, h, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 73.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D\_m \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+96}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\left(\left(\frac{M}{d} \cdot h\right) \cdot D\_m\right) \cdot 0.25\right) \cdot \left(\frac{0.5}{d} \cdot M\right), \frac{-D\_m}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<=
      (*
       (- 1.0 (* (* (pow (/ (* D_m M) (* d 2.0)) 2.0) (/ 1.0 2.0)) (/ h l)))
       (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
      1e+96)
   (*
    (*
     (fma (* (* (* (* (/ M d) h) D_m) 0.25) (* (/ 0.5 d) M)) (/ (- D_m) l) 1.0)
     (sqrt (/ d h)))
    (sqrt (/ d l)))
   (fabs (/ d (sqrt (* h l))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (((1.0 - ((pow(((D_m * M) / (d * 2.0)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)))) <= 1e+96) {
		tmp = (fma((((((M / d) * h) * D_m) * 0.25) * ((0.5 / d) * M)), (-D_m / l), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else {
		tmp = fabs((d / sqrt((h * l))));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64((Float64(Float64(D_m * M) / Float64(d * 2.0)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0)))) <= 1e+96)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(Float64(M / d) * h) * D_m) * 0.25) * Float64(Float64(0.5 / d) * M)), Float64(Float64(-D_m) / l), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	else
		tmp = abs(Float64(d / sqrt(Float64(h * l))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[N[(N[(1.0 - N[(N[(N[Power[N[(N[(D$95$m * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+96], N[(N[(N[(N[(N[(N[(N[(N[(M / d), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] * 0.25), $MachinePrecision] * N[(N[(0.5 / d), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision] * N[((-D$95$m) / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $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)))) < 1.00000000000000005e96

   1. Initial program 81.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 79.7%

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

   6. Applied rewrites 80.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

   8. Applied rewrites 80.4%

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

   if 1.00000000000000005e96 < (*.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 33.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 22.3

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

   5. Applied rewrites 22.3%

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

   7. Applied rewrites 59.7%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+96}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\left(\left(\frac{M}{d} \cdot h\right) \cdot D\right) \cdot 0.25\right) \cdot \left(\frac{0.5}{d} \cdot M\right), \frac{-D}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 45.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D\_m \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -2 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<=
      (*
       (- 1.0 (* (* (pow (/ (* D_m M) (* d 2.0)) 2.0) (/ 1.0 2.0)) (/ h l)))
       (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
      -2e-73)
   (* (sqrt (/ 1.0 (* h l))) (- d))
   (fabs (/ d (sqrt (* h l))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (((1.0 - ((pow(((D_m * M) / (d * 2.0)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)))) <= -2e-73) {
		tmp = sqrt((1.0 / (h * l))) * -d;
	} else {
		tmp = fabs((d / sqrt((h * l))));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: tmp
    if (((1.0d0 - (((((d_m * m) / (d * 2.0d0)) ** 2.0d0) * (1.0d0 / 2.0d0)) * (h / l))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))) <= (-2d-73)) then
        tmp = sqrt((1.0d0 / (h * l))) * -d
    else
        tmp = abs((d / sqrt((h * l))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (((1.0 - ((Math.pow(((D_m * M) / (d * 2.0)), 2.0) * (1.0 / 2.0)) * (h / l))) * (Math.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0)))) <= -2e-73) {
		tmp = Math.sqrt((1.0 / (h * l))) * -d;
	} else {
		tmp = Math.abs((d / Math.sqrt((h * l))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if ((1.0 - ((math.pow(((D_m * M) / (d * 2.0)), 2.0) * (1.0 / 2.0)) * (h / l))) * (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0)))) <= -2e-73:
		tmp = math.sqrt((1.0 / (h * l))) * -d
	else:
		tmp = math.fabs((d / math.sqrt((h * l))))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64((Float64(Float64(D_m * M) / Float64(d * 2.0)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0)))) <= -2e-73)
		tmp = Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(-d));
	else
		tmp = abs(Float64(d / sqrt(Float64(h * l))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (((1.0 - (((((D_m * M) / (d * 2.0)) ^ 2.0) * (1.0 / 2.0)) * (h / l))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)))) <= -2e-73)
		tmp = sqrt((1.0 / (h * l))) * -d;
	else
		tmp = abs((d / sqrt((h * l))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[N[(N[(1.0 - N[(N[(N[Power[N[(N[(D$95$m * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-73], N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $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)))) < -1.99999999999999999e-73

   1. Initial program 80.8%

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

   3. Taylor expanded in 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 9.5

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

   5. Applied rewrites 9.5%

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

   if -1.99999999999999999e-73 < (*.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 56.0%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 31.4

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

   5. Applied rewrites 31.4%

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

   7. Applied rewrites 61.7%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -2 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 45.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;\left(1 - \left({\left(\frac{D\_m \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{-41}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|t\_0\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (/ d (sqrt (* h l)))))
   (if (<=
        (*
         (- 1.0 (* (* (pow (/ (* D_m M) (* d 2.0)) 2.0) (/ 1.0 2.0)) (/ h l)))
         (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
        -1e-41)
     t_0
     (fabs t_0))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = d / sqrt((h * l));
	double tmp;
	if (((1.0 - ((pow(((D_m * M) / (d * 2.0)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)))) <= -1e-41) {
		tmp = t_0;
	} else {
		tmp = fabs(t_0);
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d / sqrt((h * l))
    if (((1.0d0 - (((((d_m * m) / (d * 2.0d0)) ** 2.0d0) * (1.0d0 / 2.0d0)) * (h / l))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))) <= (-1d-41)) then
        tmp = t_0
    else
        tmp = abs(t_0)
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = d / Math.sqrt((h * l));
	double tmp;
	if (((1.0 - ((Math.pow(((D_m * M) / (d * 2.0)), 2.0) * (1.0 / 2.0)) * (h / l))) * (Math.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0)))) <= -1e-41) {
		tmp = t_0;
	} else {
		tmp = Math.abs(t_0);
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = d / math.sqrt((h * l))
	tmp = 0
	if ((1.0 - ((math.pow(((D_m * M) / (d * 2.0)), 2.0) * (1.0 / 2.0)) * (h / l))) * (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0)))) <= -1e-41:
		tmp = t_0
	else:
		tmp = math.fabs(t_0)
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(d / sqrt(Float64(h * l)))
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64((Float64(Float64(D_m * M) / Float64(d * 2.0)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0)))) <= -1e-41)
		tmp = t_0;
	else
		tmp = abs(t_0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = d / sqrt((h * l));
	tmp = 0.0;
	if (((1.0 - (((((D_m * M) / (d * 2.0)) ^ 2.0) * (1.0 / 2.0)) * (h / l))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)))) <= -1e-41)
		tmp = t_0;
	else
		tmp = abs(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 - N[(N[(N[Power[N[(N[(D$95$m * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-41], t$95$0, N[Abs[t$95$0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.00000000000000001e-41

   1. Initial program 80.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 5.4

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

   5. Applied rewrites 5.4%

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

   7. Applied rewrites 5.4%

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

   if -1.00000000000000001e-41 < (*.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 56.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 31.3

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

   5. Applied rewrites 31.3%

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

   7. Applied rewrites 61.4%

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

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{-41}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 82.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{0.5}{d} \cdot M\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -6.8 \cdot 10^{-187}:\\ \;\;\;\;\frac{\left(\frac{t\_1}{\sqrt{-\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{d}{D\_m} \cdot \frac{2}{M}\right)}^{-2}, 1\right)\right) \cdot t\_1}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-271}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\left(\frac{-M}{d} \cdot \left(0.25 \cdot D\_m\right)\right) \cdot \frac{t\_0 \cdot D\_m}{\ell}, h, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{-D\_m}{\ell}, \left(\left(\frac{M}{d} \cdot h\right) \cdot D\_m\right) \cdot \left(t\_0 \cdot 0.25\right), 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* (/ 0.5 d) M)) (t_1 (sqrt (- d))))
   (if (<= l -6.8e-187)
     (/
      (*
       (*
        (/ t_1 (sqrt (- l)))
        (fma (* (/ h l) -0.5) (pow (* (/ d D_m) (/ 2.0 M)) -2.0) 1.0))
       t_1)
      (sqrt (- h)))
     (if (<= l 5e-271)
       (*
        (sqrt (/ d h))
        (*
         (sqrt (/ d l))
         (fma (* (* (/ (- M) d) (* 0.25 D_m)) (/ (* t_0 D_m) l)) h 1.0)))
       (/
        (*
         (*
          (fma (/ (- D_m) l) (* (* (* (/ M d) h) D_m) (* t_0 0.25)) 1.0)
          (sqrt d))
         (sqrt d))
        (* (sqrt l) (sqrt h)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (0.5 / d) * M;
	double t_1 = sqrt(-d);
	double tmp;
	if (l <= -6.8e-187) {
		tmp = (((t_1 / sqrt(-l)) * fma(((h / l) * -0.5), pow(((d / D_m) * (2.0 / M)), -2.0), 1.0)) * t_1) / sqrt(-h);
	} else if (l <= 5e-271) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * fma((((-M / d) * (0.25 * D_m)) * ((t_0 * D_m) / l)), h, 1.0));
	} else {
		tmp = ((fma((-D_m / l), ((((M / d) * h) * D_m) * (t_0 * 0.25)), 1.0) * sqrt(d)) * sqrt(d)) / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(0.5 / d) * M)
	t_1 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -6.8e-187)
		tmp = Float64(Float64(Float64(Float64(t_1 / sqrt(Float64(-l))) * fma(Float64(Float64(h / l) * -0.5), (Float64(Float64(d / D_m) * Float64(2.0 / M)) ^ -2.0), 1.0)) * t_1) / sqrt(Float64(-h)));
	elseif (l <= 5e-271)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * fma(Float64(Float64(Float64(Float64(-M) / d) * Float64(0.25 * D_m)) * Float64(Float64(t_0 * D_m) / l)), h, 1.0)));
	else
		tmp = Float64(Float64(Float64(fma(Float64(Float64(-D_m) / l), Float64(Float64(Float64(Float64(M / d) * h) * D_m) * Float64(t_0 * 0.25)), 1.0) * sqrt(d)) * sqrt(d)) / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(0.5 / d), $MachinePrecision] * M), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -6.8e-187], N[(N[(N[(N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(N[(d / D$95$m), $MachinePrecision] * N[(2.0 / M), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5e-271], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[((-M) / d), $MachinePrecision] * N[(0.25 * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * D$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[((-D$95$m) / l), $MachinePrecision] * N[(N[(N[(N[(M / d), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(t$95$0 * 0.25), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.8000000000000003e-187

   1. Initial program 60.5%

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

   4. Applied rewrites 73.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 82.8

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

   6. Applied rewrites 82.8%

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

   if -6.8000000000000003e-187 < l < 5.0000000000000002e-271

   1. Initial program 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 76.3%

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

   6. Applied rewrites 94.3%

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

   if 5.0000000000000002e-271 < l

   1. Initial program 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 65.4%

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

   6. Applied rewrites 67.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 66.3

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

   8. Applied rewrites 66.3%

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

   10. Applied rewrites 85.0%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.8 \cdot 10^{-187}:\\ \;\;\;\;\frac{\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{d}{D} \cdot \frac{2}{M}\right)}^{-2}, 1\right)\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-271}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\left(\frac{-M}{d} \cdot \left(0.25 \cdot D\right)\right) \cdot \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell}, h, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{-D}{\ell}, \left(\left(\frac{M}{d} \cdot h\right) \cdot D\right) \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot 0.25\right), 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 82.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M}{d} \cdot h\\ t_1 := \frac{0.5}{d} \cdot M\\ t_2 := t\_1 \cdot D\_m\\ t_3 := \sqrt{-d}\\ \mathbf{if}\;h \leq -7.6 \cdot 10^{-72}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{t\_2}{-\ell}, t\_0 \cdot \left(0.25 \cdot D\_m\right), 1\right) \cdot \frac{t\_3}{\sqrt{-\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(t\_2 \cdot \left(\frac{h}{\ell} \cdot -0.5\right), t\_2, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot t\_3}{\sqrt{-h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{-D\_m}{\ell}, \left(t\_0 \cdot D\_m\right) \cdot \left(t\_1 \cdot 0.25\right), 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* (/ M d) h))
        (t_1 (* (/ 0.5 d) M))
        (t_2 (* t_1 D_m))
        (t_3 (sqrt (- d))))
   (if (<= h -7.6e-72)
     (*
      (* (fma (/ t_2 (- l)) (* t_0 (* 0.25 D_m)) 1.0) (/ t_3 (sqrt (- l))))
      (sqrt (/ d h)))
     (if (<= h -5e-310)
       (/
        (* (* (fma (* t_2 (* (/ h l) -0.5)) t_2 1.0) (sqrt (/ d l))) t_3)
        (sqrt (- h)))
       (/
        (*
         (* (fma (/ (- D_m) l) (* (* t_0 D_m) (* t_1 0.25)) 1.0) (sqrt d))
         (sqrt d))
        (* (sqrt l) (sqrt h)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (M / d) * h;
	double t_1 = (0.5 / d) * M;
	double t_2 = t_1 * D_m;
	double t_3 = sqrt(-d);
	double tmp;
	if (h <= -7.6e-72) {
		tmp = (fma((t_2 / -l), (t_0 * (0.25 * D_m)), 1.0) * (t_3 / sqrt(-l))) * sqrt((d / h));
	} else if (h <= -5e-310) {
		tmp = ((fma((t_2 * ((h / l) * -0.5)), t_2, 1.0) * sqrt((d / l))) * t_3) / sqrt(-h);
	} else {
		tmp = ((fma((-D_m / l), ((t_0 * D_m) * (t_1 * 0.25)), 1.0) * sqrt(d)) * sqrt(d)) / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(M / d) * h)
	t_1 = Float64(Float64(0.5 / d) * M)
	t_2 = Float64(t_1 * D_m)
	t_3 = sqrt(Float64(-d))
	tmp = 0.0
	if (h <= -7.6e-72)
		tmp = Float64(Float64(fma(Float64(t_2 / Float64(-l)), Float64(t_0 * Float64(0.25 * D_m)), 1.0) * Float64(t_3 / sqrt(Float64(-l)))) * sqrt(Float64(d / h)));
	elseif (h <= -5e-310)
		tmp = Float64(Float64(Float64(fma(Float64(t_2 * Float64(Float64(h / l) * -0.5)), t_2, 1.0) * sqrt(Float64(d / l))) * t_3) / sqrt(Float64(-h)));
	else
		tmp = Float64(Float64(Float64(fma(Float64(Float64(-D_m) / l), Float64(Float64(t_0 * D_m) * Float64(t_1 * 0.25)), 1.0) * sqrt(d)) * sqrt(d)) / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(M / d), $MachinePrecision] * h), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 / d), $MachinePrecision] * M), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * D$95$m), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[h, -7.6e-72], N[(N[(N[(N[(t$95$2 / (-l)), $MachinePrecision] * N[(t$95$0 * N[(0.25 * D$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(t$95$3 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-310], N[(N[(N[(N[(N[(t$95$2 * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[((-D$95$m) / l), $MachinePrecision] * N[(N[(t$95$0 * D$95$m), $MachinePrecision] * N[(t$95$1 * 0.25), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -7.60000000000000004e-72

   1. Initial program 64.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 64.9%

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

   6. Applied rewrites 67.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 76.2

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

   8. Applied rewrites 76.2%

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

   if -7.60000000000000004e-72 < h < -4.999999999999985e-310

   1. Initial program 67.1%

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

   4. Applied rewrites 83.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

   6. Applied rewrites 88.5%

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

   if -4.999999999999985e-310 < h

   1. Initial program 65.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 64.4%

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

   6. Applied rewrites 68.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 67.6

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

   8. Applied rewrites 67.6%

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

   10. Applied rewrites 84.6%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -7.6 \cdot 10^{-72}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{-\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(0.25 \cdot D\right), 1\right) \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right) \cdot \left(\frac{h}{\ell} \cdot -0.5\right), \left(\frac{0.5}{d} \cdot M\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{-D}{\ell}, \left(\left(\frac{M}{d} \cdot h\right) \cdot D\right) \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot 0.25\right), 1\right) \cdot \sqrt{d}\right) \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 79.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M}{d} \cdot h\\ t_1 := \frac{0.5}{d} \cdot M\\ t_2 := t\_1 \cdot D\_m\\ t_3 := \mathsf{fma}\left(\frac{t\_2}{-\ell}, t\_0 \cdot \left(0.25 \cdot D\_m\right), 1\right)\\ t_4 := \sqrt{-d}\\ t_5 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -7.6 \cdot 10^{-72}:\\ \;\;\;\;\left(t\_3 \cdot \frac{t\_4}{\sqrt{-\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(t\_2 \cdot \left(\frac{h}{\ell} \cdot -0.5\right), t\_2, 1\right) \cdot t\_5\right) \cdot t\_4}{\sqrt{-h}}\\ \mathbf{elif}\;h \leq 1.3 \cdot 10^{+82}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\left(\left(t\_0 \cdot D\_m\right) \cdot 0.25\right) \cdot t\_1, \frac{-D\_m}{\ell}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t\_3 \cdot t\_5\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* (/ M d) h))
        (t_1 (* (/ 0.5 d) M))
        (t_2 (* t_1 D_m))
        (t_3 (fma (/ t_2 (- l)) (* t_0 (* 0.25 D_m)) 1.0))
        (t_4 (sqrt (- d)))
        (t_5 (sqrt (/ d l))))
   (if (<= h -7.6e-72)
     (* (* t_3 (/ t_4 (sqrt (- l)))) (sqrt (/ d h)))
     (if (<= h -5e-310)
       (/ (* (* (fma (* t_2 (* (/ h l) -0.5)) t_2 1.0) t_5) t_4) (sqrt (- h)))
       (if (<= h 1.3e+82)
         (*
          (/ d (sqrt (* h l)))
          (fma (* (* (* t_0 D_m) 0.25) t_1) (/ (- D_m) l) 1.0))
         (* (/ (sqrt d) (sqrt h)) (* t_3 t_5)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (M / d) * h;
	double t_1 = (0.5 / d) * M;
	double t_2 = t_1 * D_m;
	double t_3 = fma((t_2 / -l), (t_0 * (0.25 * D_m)), 1.0);
	double t_4 = sqrt(-d);
	double t_5 = sqrt((d / l));
	double tmp;
	if (h <= -7.6e-72) {
		tmp = (t_3 * (t_4 / sqrt(-l))) * sqrt((d / h));
	} else if (h <= -5e-310) {
		tmp = ((fma((t_2 * ((h / l) * -0.5)), t_2, 1.0) * t_5) * t_4) / sqrt(-h);
	} else if (h <= 1.3e+82) {
		tmp = (d / sqrt((h * l))) * fma((((t_0 * D_m) * 0.25) * t_1), (-D_m / l), 1.0);
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (t_3 * t_5);
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(M / d) * h)
	t_1 = Float64(Float64(0.5 / d) * M)
	t_2 = Float64(t_1 * D_m)
	t_3 = fma(Float64(t_2 / Float64(-l)), Float64(t_0 * Float64(0.25 * D_m)), 1.0)
	t_4 = sqrt(Float64(-d))
	t_5 = sqrt(Float64(d / l))
	tmp = 0.0
	if (h <= -7.6e-72)
		tmp = Float64(Float64(t_3 * Float64(t_4 / sqrt(Float64(-l)))) * sqrt(Float64(d / h)));
	elseif (h <= -5e-310)
		tmp = Float64(Float64(Float64(fma(Float64(t_2 * Float64(Float64(h / l) * -0.5)), t_2, 1.0) * t_5) * t_4) / sqrt(Float64(-h)));
	elseif (h <= 1.3e+82)
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * fma(Float64(Float64(Float64(t_0 * D_m) * 0.25) * t_1), Float64(Float64(-D_m) / l), 1.0));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(t_3 * t_5));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(M / d), $MachinePrecision] * h), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 / d), $MachinePrecision] * M), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * D$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 / (-l)), $MachinePrecision] * N[(t$95$0 * N[(0.25 * D$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -7.6e-72], N[(N[(t$95$3 * N[(t$95$4 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-310], N[(N[(N[(N[(N[(t$95$2 * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision] * t$95$5), $MachinePrecision] * t$95$4), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 1.3e+82], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(t$95$0 * D$95$m), $MachinePrecision] * 0.25), $MachinePrecision] * t$95$1), $MachinePrecision] * N[((-D$95$m) / l), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$3 * t$95$5), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -7.60000000000000004e-72

   1. Initial program 64.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 64.9%

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

   6. Applied rewrites 67.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 76.2

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

   8. Applied rewrites 76.2%

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

   if -7.60000000000000004e-72 < h < -4.999999999999985e-310

   1. Initial program 67.1%

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

   4. Applied rewrites 83.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

   6. Applied rewrites 88.5%

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

   if -4.999999999999985e-310 < h < 1.2999999999999999e82

   1. Initial program 61.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 61.8%

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

   6. Applied rewrites 71.5%

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

   8. Applied rewrites 84.0%

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

   if 1.2999999999999999e82 < h

   1. Initial program 71.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 68.9%

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

   6. Applied rewrites 62.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. pow1/2 N/A

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

      6. lower-/.f64 N/A

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

      7. pow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 80.8

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

   8. Applied rewrites 80.8%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -7.6 \cdot 10^{-72}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{-\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(0.25 \cdot D\right), 1\right) \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right) \cdot \left(\frac{h}{\ell} \cdot -0.5\right), \left(\frac{0.5}{d} \cdot M\right) \cdot D, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;h \leq 1.3 \cdot 10^{+82}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\left(\left(\left(\frac{M}{d} \cdot h\right) \cdot D\right) \cdot 0.25\right) \cdot \left(\frac{0.5}{d} \cdot M\right), \frac{-D}{\ell}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\mathsf{fma}\left(\frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{-\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(0.25 \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 77.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M}{d} \cdot h\\ t_1 := \frac{0.5}{d} \cdot M\\ t_2 := \mathsf{fma}\left(\frac{t\_1 \cdot D\_m}{-\ell}, t\_0 \cdot \left(0.25 \cdot D\_m\right), 1\right)\\ \mathbf{if}\;h \leq -8 \cdot 10^{-242}:\\ \;\;\;\;\left(t\_2 \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{elif}\;h \leq 1.3 \cdot 10^{+82}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\left(\left(t\_0 \cdot D\_m\right) \cdot 0.25\right) \cdot t\_1, \frac{-D\_m}{\ell}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t\_2 \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* (/ M d) h))
        (t_1 (* (/ 0.5 d) M))
        (t_2 (fma (/ (* t_1 D_m) (- l)) (* t_0 (* 0.25 D_m)) 1.0)))
   (if (<= h -8e-242)
     (* (* t_2 (/ (sqrt (- d)) (sqrt (- l)))) (sqrt (/ d h)))
     (if (<= h -5e-310)
       (* (sqrt (/ 1.0 (* h l))) (- d))
       (if (<= h 1.3e+82)
         (*
          (/ d (sqrt (* h l)))
          (fma (* (* (* t_0 D_m) 0.25) t_1) (/ (- D_m) l) 1.0))
         (* (/ (sqrt d) (sqrt h)) (* t_2 (sqrt (/ d l)))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (M / d) * h;
	double t_1 = (0.5 / d) * M;
	double t_2 = fma(((t_1 * D_m) / -l), (t_0 * (0.25 * D_m)), 1.0);
	double tmp;
	if (h <= -8e-242) {
		tmp = (t_2 * (sqrt(-d) / sqrt(-l))) * sqrt((d / h));
	} else if (h <= -5e-310) {
		tmp = sqrt((1.0 / (h * l))) * -d;
	} else if (h <= 1.3e+82) {
		tmp = (d / sqrt((h * l))) * fma((((t_0 * D_m) * 0.25) * t_1), (-D_m / l), 1.0);
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (t_2 * sqrt((d / l)));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(M / d) * h)
	t_1 = Float64(Float64(0.5 / d) * M)
	t_2 = fma(Float64(Float64(t_1 * D_m) / Float64(-l)), Float64(t_0 * Float64(0.25 * D_m)), 1.0)
	tmp = 0.0
	if (h <= -8e-242)
		tmp = Float64(Float64(t_2 * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l)))) * sqrt(Float64(d / h)));
	elseif (h <= -5e-310)
		tmp = Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(-d));
	elseif (h <= 1.3e+82)
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * fma(Float64(Float64(Float64(t_0 * D_m) * 0.25) * t_1), Float64(Float64(-D_m) / l), 1.0));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(t_2 * sqrt(Float64(d / l))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(M / d), $MachinePrecision] * h), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 / d), $MachinePrecision] * M), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$1 * D$95$m), $MachinePrecision] / (-l)), $MachinePrecision] * N[(t$95$0 * N[(0.25 * D$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[h, -8e-242], N[(N[(t$95$2 * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-310], N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], If[LessEqual[h, 1.3e+82], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(t$95$0 * D$95$m), $MachinePrecision] * 0.25), $MachinePrecision] * t$95$1), $MachinePrecision] * N[((-D$95$m) / l), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -8e-242

   1. Initial program 70.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 69.1%

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

   6. Applied rewrites 70.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 78.1

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

   8. Applied rewrites 78.1%

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

   if -8e-242 < h < -4.999999999999985e-310

   1. Initial program 27.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 75.1

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

   5. Applied rewrites 75.1%

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

   if -4.999999999999985e-310 < h < 1.2999999999999999e82

   1. Initial program 61.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 61.8%

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

   6. Applied rewrites 71.5%

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

   8. Applied rewrites 84.0%

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

   if 1.2999999999999999e82 < h

   1. Initial program 71.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 68.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{2}{M} \cdot \frac{d}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

   6. Applied rewrites 62.7%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{-\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(0.25 \cdot D\right), 1\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{-\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(\frac{1}{4} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{-\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(\frac{1}{4} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\color{blue}{\frac{d}{h}}} \]

      3. sqrt-div N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{-\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(\frac{1}{4} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \]

      4. pow1/2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{-\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(\frac{1}{4} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{\color{blue}{{d}^{\frac{1}{2}}}}{\sqrt{h}} \]

      5. pow1/2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{-\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(\frac{1}{4} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{{d}^{\frac{1}{2}}}{\color{blue}{{h}^{\frac{1}{2}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{-\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(\frac{1}{4} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\frac{{d}^{\frac{1}{2}}}{{h}^{\frac{1}{2}}}} \]

      7. pow1/2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{-\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(\frac{1}{4} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{\color{blue}{\sqrt{d}}}{{h}^{\frac{1}{2}}} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{-\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(\frac{1}{4} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{\color{blue}{\sqrt{d}}}{{h}^{\frac{1}{2}}} \]

      9. pow1/2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\left(\frac{\frac{1}{2}}{d} \cdot M\right) \cdot D}{-\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(\frac{1}{4} \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \]

      10. lower-sqrt.f64 80.8

          \[\leadsto \left(\mathsf{fma}\left(\frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{-\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(0.25 \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \]

   8. Applied rewrites 80.8%

      \[\leadsto \left(\mathsf{fma}\left(\frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{-\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(0.25 \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -8 \cdot 10^{-242}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{-\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(0.25 \cdot D\right), 1\right) \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{elif}\;h \leq 1.3 \cdot 10^{+82}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\left(\left(\left(\frac{M}{d} \cdot h\right) \cdot D\right) \cdot 0.25\right) \cdot \left(\frac{0.5}{d} \cdot M\right), \frac{-D}{\ell}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\mathsf{fma}\left(\frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{-\ell}, \left(\frac{M}{d} \cdot h\right) \cdot \left(0.25 \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 26.2% accurate, 15.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \frac{d}{\sqrt{h \cdot \ell}} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m) :precision binary64 (/ d (sqrt (* h l))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	return d / sqrt((h * l));
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    code = d / sqrt((h * l))
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	return d / Math.sqrt((h * l));
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	return d / math.sqrt((h * l))

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	return Float64(d / sqrt(Float64(h * l)))
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp = code(d, h, l, M, D_m)
	tmp = d / sqrt((h * l));
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.4%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
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 21.6

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

5. Applied rewrites 21.6%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
6. Step-by-step derivation

7. Applied rewrites 21.6%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

8. Final simplification 21.6%

   \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024276 
(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)))))