Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.4% → 80.1%

Time: 15.1s

Alternatives: 24

Speedup: 3.0×

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: 66.4% 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: 80.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(t\_0 \cdot \mathsf{fma}\left(\left(\left(0.25 \cdot D\_m\right) \cdot \frac{M\_m}{d}\right) \cdot \frac{\left(M\_m \cdot \frac{0.5}{d}\right) \cdot D\_m}{-\ell}, h, 1\right)\right)\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{-137}:\\ \;\;\;\;\left(1 - \frac{\left(\left(D\_m \cdot 0.5\right) \cdot 0.5\right) \cdot \frac{M\_m}{d}}{{h}^{-1}} \cdot \frac{\left(D\_m \cdot \frac{0.5}{d}\right) \cdot M\_m}{\ell}\right) \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\left(\left(0.25 \cdot \frac{M\_m}{d}\right) \cdot D\_m\right) \cdot h, \frac{0.5}{-d} \cdot \left(\frac{D\_m}{\ell} \cdot M\_m\right), 1\right) \cdot t\_0\right) \cdot \sqrt{d}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))))
   (if (<= l -5e-310)
     (*
      (/ (sqrt (- d)) (sqrt (- h)))
      (*
       t_0
       (fma
        (* (* (* 0.25 D_m) (/ M_m d)) (/ (* (* M_m (/ 0.5 d)) D_m) (- l)))
        h
        1.0)))
     (if (<= l 4.2e-137)
       (*
        (-
         1.0
         (*
          (/ (* (* (* D_m 0.5) 0.5) (/ M_m d)) (pow h -1.0))
          (/ (* (* D_m (/ 0.5 d)) M_m) l)))
        (* (/ (sqrt d) (sqrt l)) (pow (/ d h) (/ 1.0 2.0))))
       (/
        (*
         (*
          (fma
           (* (* (* 0.25 (/ M_m d)) D_m) h)
           (* (/ 0.5 (- d)) (* (/ D_m l) M_m))
           1.0)
          t_0)
         (sqrt d))
        (sqrt h))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / l));
	double tmp;
	if (l <= -5e-310) {
		tmp = (sqrt(-d) / sqrt(-h)) * (t_0 * fma((((0.25 * D_m) * (M_m / d)) * (((M_m * (0.5 / d)) * D_m) / -l)), h, 1.0));
	} else if (l <= 4.2e-137) {
		tmp = (1.0 - (((((D_m * 0.5) * 0.5) * (M_m / d)) / pow(h, -1.0)) * (((D_m * (0.5 / d)) * M_m) / l))) * ((sqrt(d) / sqrt(l)) * pow((d / h), (1.0 / 2.0)));
	} else {
		tmp = ((fma((((0.25 * (M_m / d)) * D_m) * h), ((0.5 / -d) * ((D_m / l) * M_m)), 1.0) * t_0) * sqrt(d)) / sqrt(h);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(t_0 * fma(Float64(Float64(Float64(0.25 * D_m) * Float64(M_m / d)) * Float64(Float64(Float64(M_m * Float64(0.5 / d)) * D_m) / Float64(-l))), h, 1.0)));
	elseif (l <= 4.2e-137)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(D_m * 0.5) * 0.5) * Float64(M_m / d)) / (h ^ -1.0)) * Float64(Float64(Float64(D_m * Float64(0.5 / d)) * M_m) / l))) * Float64(Float64(sqrt(d) / sqrt(l)) * (Float64(d / h) ^ Float64(1.0 / 2.0))));
	else
		tmp = Float64(Float64(Float64(fma(Float64(Float64(Float64(0.25 * Float64(M_m / d)) * D_m) * h), Float64(Float64(0.5 / Float64(-d)) * Float64(Float64(D_m / l) * M_m)), 1.0) * t_0) * sqrt(d)) / sqrt(h));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -5e-310], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(N[(N[(N[(0.25 * D$95$m), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] / (-l)), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.2e-137], N[(N[(1.0 - N[(N[(N[(N[(N[(D$95$m * 0.5), $MachinePrecision] * 0.5), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] / N[Power[h, -1.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(0.25 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * h), $MachinePrecision] * N[(N[(0.5 / (-d)), $MachinePrecision] * N[(N[(D$95$m / l), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 69.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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 67.2%

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

   6. Applied rewrites 74.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 85.5

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

   8. Applied rewrites 85.5%

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

   if -4.999999999999985e-310 < l < 4.19999999999999983e-137

   1. Initial program 66.7%

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

   4. Applied rewrites 77.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 77.5

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 92.1

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

   6. Applied rewrites 92.1%

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

   if 4.19999999999999983e-137 < l

   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. 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 67.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 68.9%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

   10. Applied rewrites 80.3%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\left(\left(0.25 \cdot D\right) \cdot \frac{M}{d}\right) \cdot \frac{\left(M \cdot \frac{0.5}{d}\right) \cdot D}{-\ell}, h, 1\right)\right)\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{-137}:\\ \;\;\;\;\left(1 - \frac{\left(\left(D \cdot 0.5\right) \cdot 0.5\right) \cdot \frac{M}{d}}{{h}^{-1}} \cdot \frac{\left(D \cdot \frac{0.5}{d}\right) \cdot M}{\ell}\right) \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\left(\left(0.25 \cdot \frac{M}{d}\right) \cdot D\right) \cdot h, \frac{0.5}{-d} \cdot \left(\frac{D}{\ell} \cdot M\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 70.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+64}:\\ \;\;\;\;\left(\left(\left(\frac{M\_m}{\ell} \cdot M\_m\right) \cdot \left(-0.125 \cdot \left(\frac{\frac{D\_m \cdot D\_m}{d}}{d} \cdot h\right)\right)\right) \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{-70}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{\left(\left(D\_m \cdot M\_m\right) \cdot 0.5\right) \cdot \left(D\_m \cdot M\_m\right)}{\left(2 \cdot d\right) \cdot d}, \frac{h}{\ell} \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+235}:\\ \;\;\;\;t\_2 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (-
           1.0
           (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_1 (sqrt (/ d l)))
        (t_2 (sqrt (/ d h))))
   (if (<= t_0 -2e+64)
     (*
      (* (* (* (/ M_m l) M_m) (* -0.125 (* (/ (/ (* D_m D_m) d) d) h))) t_1)
      t_2)
     (if (<= t_0 -1e-70)
       (*
        (/ d (* (sqrt h) (sqrt l)))
        (fma
         (/ (* (* (* D_m M_m) 0.5) (* D_m M_m)) (* (* 2.0 d) d))
         (* (/ h l) -0.5)
         1.0))
       (if (<= t_0 5e+235) (* t_2 t_1) (fabs (/ d (sqrt (* h l)))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_1 = sqrt((d / l));
	double t_2 = sqrt((d / h));
	double tmp;
	if (t_0 <= -2e+64) {
		tmp = ((((M_m / l) * M_m) * (-0.125 * ((((D_m * D_m) / d) / d) * h))) * t_1) * t_2;
	} else if (t_0 <= -1e-70) {
		tmp = (d / (sqrt(h) * sqrt(l))) * fma(((((D_m * M_m) * 0.5) * (D_m * M_m)) / ((2.0 * d) * d)), ((h / l) * -0.5), 1.0);
	} else if (t_0 <= 5e+235) {
		tmp = t_2 * t_1;
	} else {
		tmp = fabs((d / sqrt((h * l))));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	t_1 = sqrt(Float64(d / l))
	t_2 = sqrt(Float64(d / h))
	tmp = 0.0
	if (t_0 <= -2e+64)
		tmp = Float64(Float64(Float64(Float64(Float64(M_m / l) * M_m) * Float64(-0.125 * Float64(Float64(Float64(Float64(D_m * D_m) / d) / d) * h))) * t_1) * t_2);
	elseif (t_0 <= -1e-70)
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * fma(Float64(Float64(Float64(Float64(D_m * M_m) * 0.5) * Float64(D_m * M_m)) / Float64(Float64(2.0 * d) * d)), Float64(Float64(h / l) * -0.5), 1.0));
	elseif (t_0 <= 5e+235)
		tmp = Float64(t_2 * t_1);
	else
		tmp = abs(Float64(d / sqrt(Float64(h * l))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -2e+64], N[(N[(N[(N[(N[(M$95$m / l), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(-0.125 * N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, -1e-70], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * 0.5), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+235], N[(t$95$2 * t$95$1), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $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)))) < -2.00000000000000004e64

   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 78.1%

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

   6. Applied rewrites 83.7%

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

   7. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   9. Applied rewrites 66.8%

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

   if -2.00000000000000004e64 < (*.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)))) < -9.99999999999999996e-71

   1. Initial program 98.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 98.3%

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

   6. Applied rewrites 32.4%

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

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

      9. metadata-eval N/A

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

      10. pow-pow N/A

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

      11. inv-pow N/A

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

      12. frac-times N/A

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

      13. clear-num N/A

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

      14. unpow2 N/A

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

      15. associate-/r* N/A

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

      16. frac-times N/A

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

      17. 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 31.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 45.5

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

   10. Applied rewrites 45.5%

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

   if -9.99999999999999996e-71 < (*.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)))) < 5.00000000000000027e235

   1. Initial program 92.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 37.4

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

   5. Applied rewrites 37.4%

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

   7. Applied rewrites 91.7%

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

   if 5.00000000000000027e235 < (*.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 26.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 34.6

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

   5. Applied rewrites 34.6%

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

   7. Applied rewrites 59.3%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -2 \cdot 10^{+64}:\\ \;\;\;\;\left(\left(\left(\frac{M}{\ell} \cdot M\right) \cdot \left(-0.125 \cdot \left(\frac{\frac{D \cdot D}{d}}{d} \cdot h\right)\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{-70}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{\left(\left(D \cdot M\right) \cdot 0.5\right) \cdot \left(D \cdot M\right)}{\left(2 \cdot d\right) \cdot d}, \frac{h}{\ell} \cdot -0.5, 1\right)\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+235}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_1 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-169}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+135}:\\ \;\;\;\;\left|\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (fabs (/ d (sqrt (* h l)))))
        (t_1
         (*
          (-
           1.0
           (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))))
   (if (<= t_1 -1e-31)
     (* (sqrt (/ 1.0 (* h l))) (- d))
     (if (<= t_1 1e-169)
       t_0
       (if (<= t_1 1e+135) (fabs (sqrt (* (/ d h) (/ d l)))) t_0)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = fabs((d / sqrt((h * l))));
	double t_1 = (1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_1 <= -1e-31) {
		tmp = sqrt((1.0 / (h * l))) * -d;
	} else if (t_1 <= 1e-169) {
		tmp = t_0;
	} else if (t_1 <= 1e+135) {
		tmp = fabs(sqrt(((d / h) * (d / l))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((d / sqrt((h * l))))
    t_1 = (1.0d0 - ((h / l) * ((((d_m * m_m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))
    if (t_1 <= (-1d-31)) then
        tmp = sqrt((1.0d0 / (h * l))) * -d
    else if (t_1 <= 1d-169) then
        tmp = t_0
    else if (t_1 <= 1d+135) then
        tmp = abs(sqrt(((d / h) * (d / l))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.abs((d / Math.sqrt((h * l))));
	double t_1 = (1.0 - ((h / l) * (Math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (Math.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_1 <= -1e-31) {
		tmp = Math.sqrt((1.0 / (h * l))) * -d;
	} else if (t_1 <= 1e-169) {
		tmp = t_0;
	} else if (t_1 <= 1e+135) {
		tmp = Math.abs(Math.sqrt(((d / h) * (d / l))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.fabs((d / math.sqrt((h * l))))
	t_1 = (1.0 - ((h / l) * (math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0)))
	tmp = 0
	if t_1 <= -1e-31:
		tmp = math.sqrt((1.0 / (h * l))) * -d
	elif t_1 <= 1e-169:
		tmp = t_0
	elif t_1 <= 1e+135:
		tmp = math.fabs(math.sqrt(((d / h) * (d / l))))
	else:
		tmp = t_0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = abs(Float64(d / sqrt(Float64(h * l))))
	t_1 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	tmp = 0.0
	if (t_1 <= -1e-31)
		tmp = Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(-d));
	elseif (t_1 <= 1e-169)
		tmp = t_0;
	elseif (t_1 <= 1e+135)
		tmp = abs(sqrt(Float64(Float64(d / h) * Float64(d / l))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = abs((d / sqrt((h * l))));
	t_1 = (1.0 - ((h / l) * ((((D_m * M_m) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)));
	tmp = 0.0;
	if (t_1 <= -1e-31)
		tmp = sqrt((1.0 / (h * l))) * -d;
	elseif (t_1 <= 1e-169)
		tmp = t_0;
	elseif (t_1 <= 1e+135)
		tmp = abs(sqrt(((d / h) * (d / l))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-31], N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], If[LessEqual[t$95$1, 1e-169], t$95$0, If[LessEqual[t$95$1, 1e+135], N[Abs[N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $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)))) < -1e-31

   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. 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 12.3

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

   5. Applied rewrites 12.3%

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

   if -1e-31 < (*.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)))) < 1.00000000000000002e-169 or 9.99999999999999962e134 < (*.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 39.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 40.2

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

   5. Applied rewrites 40.2%

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

   7. Applied rewrites 63.4%

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

   if 1.00000000000000002e-169 < (*.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)))) < 9.99999999999999962e134

   1. Initial program 99.6%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 26.6

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

   5. Applied rewrites 26.6%

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

   7. Applied rewrites 66.2%

      \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \]
   8. Step-by-step derivation

   9. Applied rewrites 96.1%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{-169}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+135}:\\ \;\;\;\;\left|\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_1 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-169}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+135}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{h}}{\ell} \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (fabs (/ d (sqrt (* h l)))))
        (t_1
         (*
          (-
           1.0
           (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))))
   (if (<= t_1 -1e-31)
     (* (sqrt (/ 1.0 (* h l))) (- d))
     (if (<= t_1 1e-169)
       t_0
       (if (<= t_1 1e+135) (sqrt (* (/ (/ d h) l) d)) t_0)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = fabs((d / sqrt((h * l))));
	double t_1 = (1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_1 <= -1e-31) {
		tmp = sqrt((1.0 / (h * l))) * -d;
	} else if (t_1 <= 1e-169) {
		tmp = t_0;
	} else if (t_1 <= 1e+135) {
		tmp = sqrt((((d / h) / l) * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((d / sqrt((h * l))))
    t_1 = (1.0d0 - ((h / l) * ((((d_m * m_m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))
    if (t_1 <= (-1d-31)) then
        tmp = sqrt((1.0d0 / (h * l))) * -d
    else if (t_1 <= 1d-169) then
        tmp = t_0
    else if (t_1 <= 1d+135) then
        tmp = sqrt((((d / h) / l) * d))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.abs((d / Math.sqrt((h * l))));
	double t_1 = (1.0 - ((h / l) * (Math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (Math.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_1 <= -1e-31) {
		tmp = Math.sqrt((1.0 / (h * l))) * -d;
	} else if (t_1 <= 1e-169) {
		tmp = t_0;
	} else if (t_1 <= 1e+135) {
		tmp = Math.sqrt((((d / h) / l) * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.fabs((d / math.sqrt((h * l))))
	t_1 = (1.0 - ((h / l) * (math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0)))
	tmp = 0
	if t_1 <= -1e-31:
		tmp = math.sqrt((1.0 / (h * l))) * -d
	elif t_1 <= 1e-169:
		tmp = t_0
	elif t_1 <= 1e+135:
		tmp = math.sqrt((((d / h) / l) * d))
	else:
		tmp = t_0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = abs(Float64(d / sqrt(Float64(h * l))))
	t_1 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	tmp = 0.0
	if (t_1 <= -1e-31)
		tmp = Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(-d));
	elseif (t_1 <= 1e-169)
		tmp = t_0;
	elseif (t_1 <= 1e+135)
		tmp = sqrt(Float64(Float64(Float64(d / h) / l) * d));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = abs((d / sqrt((h * l))));
	t_1 = (1.0 - ((h / l) * ((((D_m * M_m) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)));
	tmp = 0.0;
	if (t_1 <= -1e-31)
		tmp = sqrt((1.0 / (h * l))) * -d;
	elseif (t_1 <= 1e-169)
		tmp = t_0;
	elseif (t_1 <= 1e+135)
		tmp = sqrt((((d / h) / l) * d));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-31], N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], If[LessEqual[t$95$1, 1e-169], t$95$0, If[LessEqual[t$95$1, 1e+135], N[Sqrt[N[(N[(N[(d / h), $MachinePrecision] / l), $MachinePrecision] * d), $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)))) < -1e-31

   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. 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 12.3

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

   5. Applied rewrites 12.3%

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

   if -1e-31 < (*.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)))) < 1.00000000000000002e-169 or 9.99999999999999962e134 < (*.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 39.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 40.2

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

   5. Applied rewrites 40.2%

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

   7. Applied rewrites 63.4%

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

   if 1.00000000000000002e-169 < (*.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)))) < 9.99999999999999962e134

   1. Initial program 99.6%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 26.6

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

   5. Applied rewrites 26.6%

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

   7. Applied rewrites 28.3%

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

   9. Applied rewrites 92.1%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{-169}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+135}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{h}}{\ell} \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-273}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(0.25 \cdot \frac{M\_m}{d}\right) \cdot D\_m, \frac{\left(D\_m \cdot M\_m\right) \cdot 0.5}{d \cdot \ell} \cdot \left(-h\right), 1\right) \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+235}:\\ \;\;\;\;t\_2 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (-
           1.0
           (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_1 (sqrt (/ d l)))
        (t_2 (sqrt (/ d h))))
   (if (<= t_0 5e-273)
     (*
      (*
       (fma
        (* (* 0.25 (/ M_m d)) D_m)
        (* (/ (* (* D_m M_m) 0.5) (* d l)) (- h))
        1.0)
       t_1)
      t_2)
     (if (<= t_0 5e+235) (* t_2 t_1) (fabs (/ d (sqrt (* h l))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_1 = sqrt((d / l));
	double t_2 = sqrt((d / h));
	double tmp;
	if (t_0 <= 5e-273) {
		tmp = (fma(((0.25 * (M_m / d)) * D_m), ((((D_m * M_m) * 0.5) / (d * l)) * -h), 1.0) * t_1) * t_2;
	} else if (t_0 <= 5e+235) {
		tmp = t_2 * t_1;
	} else {
		tmp = fabs((d / sqrt((h * l))));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	t_1 = sqrt(Float64(d / l))
	t_2 = sqrt(Float64(d / h))
	tmp = 0.0
	if (t_0 <= 5e-273)
		tmp = Float64(Float64(fma(Float64(Float64(0.25 * Float64(M_m / d)) * D_m), Float64(Float64(Float64(Float64(D_m * M_m) * 0.5) / Float64(d * l)) * Float64(-h)), 1.0) * t_1) * t_2);
	elseif (t_0 <= 5e+235)
		tmp = Float64(t_2 * t_1);
	else
		tmp = abs(Float64(d / sqrt(Float64(h * l))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 5e-273], N[(N[(N[(N[(N[(0.25 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * (-h)), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 5e+235], N[(t$95$2 * t$95$1), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.99999999999999965e-273

   1. Initial program 78.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 77.1%

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

   6. Applied rewrites 81.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 80.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. frac-times N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 71.7

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

   10. Applied rewrites 71.7%

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

   if 4.99999999999999965e-273 < (*.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)))) < 5.00000000000000027e235

   1. Initial program 99.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 30.7

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

   5. Applied rewrites 30.7%

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

   7. Applied rewrites 98.4%

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

   if 5.00000000000000027e235 < (*.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 26.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 34.6

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

   5. Applied rewrites 34.6%

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

   7. Applied rewrites 59.3%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{-273}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(0.25 \cdot \frac{M}{d}\right) \cdot D, \frac{\left(D \cdot M\right) \cdot 0.5}{d \cdot \ell} \cdot \left(-h\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+235}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\left(\left(\left(\frac{\frac{M\_m}{d} \cdot M\_m}{d} \cdot h\right) \cdot \frac{-0.125 \cdot \left(D\_m \cdot D\_m\right)}{\ell}\right) \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+235}:\\ \;\;\;\;t\_2 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (-
           1.0
           (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_1 (sqrt (/ d l)))
        (t_2 (sqrt (/ d h))))
   (if (<= t_0 -1e-31)
     (*
      (* (* (* (/ (* (/ M_m d) M_m) d) h) (/ (* -0.125 (* D_m D_m)) l)) t_1)
      t_2)
     (if (<= t_0 5e+235) (* t_2 t_1) (fabs (/ d (sqrt (* h l))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_1 = sqrt((d / l));
	double t_2 = sqrt((d / h));
	double tmp;
	if (t_0 <= -1e-31) {
		tmp = ((((((M_m / d) * M_m) / d) * h) * ((-0.125 * (D_m * D_m)) / l)) * t_1) * t_2;
	} else if (t_0 <= 5e+235) {
		tmp = t_2 * t_1;
	} else {
		tmp = fabs((d / sqrt((h * l))));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (1.0d0 - ((h / l) * ((((d_m * m_m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))
    t_1 = sqrt((d / l))
    t_2 = sqrt((d / h))
    if (t_0 <= (-1d-31)) then
        tmp = ((((((m_m / d) * m_m) / d) * h) * (((-0.125d0) * (d_m * d_m)) / l)) * t_1) * t_2
    else if (t_0 <= 5d+235) then
        tmp = t_2 * t_1
    else
        tmp = abs((d / sqrt((h * l))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (1.0 - ((h / l) * (Math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (Math.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0)));
	double t_1 = Math.sqrt((d / l));
	double t_2 = Math.sqrt((d / h));
	double tmp;
	if (t_0 <= -1e-31) {
		tmp = ((((((M_m / d) * M_m) / d) * h) * ((-0.125 * (D_m * D_m)) / l)) * t_1) * t_2;
	} else if (t_0 <= 5e+235) {
		tmp = t_2 * t_1;
	} else {
		tmp = Math.abs((d / Math.sqrt((h * l))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (1.0 - ((h / l) * (math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0)))
	t_1 = math.sqrt((d / l))
	t_2 = math.sqrt((d / h))
	tmp = 0
	if t_0 <= -1e-31:
		tmp = ((((((M_m / d) * M_m) / d) * h) * ((-0.125 * (D_m * D_m)) / l)) * t_1) * t_2
	elif t_0 <= 5e+235:
		tmp = t_2 * t_1
	else:
		tmp = math.fabs((d / math.sqrt((h * l))))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	t_1 = sqrt(Float64(d / l))
	t_2 = sqrt(Float64(d / h))
	tmp = 0.0
	if (t_0 <= -1e-31)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(M_m / d) * M_m) / d) * h) * Float64(Float64(-0.125 * Float64(D_m * D_m)) / l)) * t_1) * t_2);
	elseif (t_0 <= 5e+235)
		tmp = Float64(t_2 * t_1);
	else
		tmp = abs(Float64(d / sqrt(Float64(h * l))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (1.0 - ((h / l) * ((((D_m * M_m) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)));
	t_1 = sqrt((d / l));
	t_2 = sqrt((d / h));
	tmp = 0.0;
	if (t_0 <= -1e-31)
		tmp = ((((((M_m / d) * M_m) / d) * h) * ((-0.125 * (D_m * D_m)) / l)) * t_1) * t_2;
	elseif (t_0 <= 5e+235)
		tmp = t_2 * t_1;
	else
		tmp = abs((d / sqrt((h * l))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -1e-31], N[(N[(N[(N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] / d), $MachinePrecision] * h), $MachinePrecision] * N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 5e+235], N[(t$95$2 * t$95$1), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1e-31

   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 80.6%

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

   6. Applied rewrites 84.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 83.5%

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

   9. Taylor expanded in d around 0

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

       1. associate-*r/ N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. associate-*l/ N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. associate-/r* N/A

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

       14. lower-/.f64 N/A

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

       15. unpow2 N/A

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

       16. associate-/l* N/A

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

       17. lower-*.f64 N/A

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

       18. lower-/.f64 56.4

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

   11. Applied rewrites 56.4%

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

   if -1e-31 < (*.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)))) < 5.00000000000000027e235

   1. Initial program 92.6%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 36.9

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

   5. Applied rewrites 36.9%

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

   7. Applied rewrites 90.6%

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

   if 5.00000000000000027e235 < (*.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 26.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 34.6

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

   5. Applied rewrites 34.6%

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

   7. Applied rewrites 59.3%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\left(\left(\left(\frac{\frac{M}{d} \cdot M}{d} \cdot h\right) \cdot \frac{-0.125 \cdot \left(D \cdot D\right)}{\ell}\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+235}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\left(\left(\left(\frac{\frac{M\_m \cdot M\_m}{d}}{d} \cdot h\right) \cdot \frac{-0.125 \cdot \left(D\_m \cdot D\_m\right)}{\ell}\right) \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+235}:\\ \;\;\;\;t\_2 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (-
           1.0
           (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_1 (sqrt (/ d l)))
        (t_2 (sqrt (/ d h))))
   (if (<= t_0 -1e-31)
     (*
      (* (* (* (/ (/ (* M_m M_m) d) d) h) (/ (* -0.125 (* D_m D_m)) l)) t_1)
      t_2)
     (if (<= t_0 5e+235) (* t_2 t_1) (fabs (/ d (sqrt (* h l))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_1 = sqrt((d / l));
	double t_2 = sqrt((d / h));
	double tmp;
	if (t_0 <= -1e-31) {
		tmp = ((((((M_m * M_m) / d) / d) * h) * ((-0.125 * (D_m * D_m)) / l)) * t_1) * t_2;
	} else if (t_0 <= 5e+235) {
		tmp = t_2 * t_1;
	} else {
		tmp = fabs((d / sqrt((h * l))));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (1.0d0 - ((h / l) * ((((d_m * m_m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))
    t_1 = sqrt((d / l))
    t_2 = sqrt((d / h))
    if (t_0 <= (-1d-31)) then
        tmp = ((((((m_m * m_m) / d) / d) * h) * (((-0.125d0) * (d_m * d_m)) / l)) * t_1) * t_2
    else if (t_0 <= 5d+235) then
        tmp = t_2 * t_1
    else
        tmp = abs((d / sqrt((h * l))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (1.0 - ((h / l) * (Math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (Math.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0)));
	double t_1 = Math.sqrt((d / l));
	double t_2 = Math.sqrt((d / h));
	double tmp;
	if (t_0 <= -1e-31) {
		tmp = ((((((M_m * M_m) / d) / d) * h) * ((-0.125 * (D_m * D_m)) / l)) * t_1) * t_2;
	} else if (t_0 <= 5e+235) {
		tmp = t_2 * t_1;
	} else {
		tmp = Math.abs((d / Math.sqrt((h * l))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (1.0 - ((h / l) * (math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0)))
	t_1 = math.sqrt((d / l))
	t_2 = math.sqrt((d / h))
	tmp = 0
	if t_0 <= -1e-31:
		tmp = ((((((M_m * M_m) / d) / d) * h) * ((-0.125 * (D_m * D_m)) / l)) * t_1) * t_2
	elif t_0 <= 5e+235:
		tmp = t_2 * t_1
	else:
		tmp = math.fabs((d / math.sqrt((h * l))))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	t_1 = sqrt(Float64(d / l))
	t_2 = sqrt(Float64(d / h))
	tmp = 0.0
	if (t_0 <= -1e-31)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(M_m * M_m) / d) / d) * h) * Float64(Float64(-0.125 * Float64(D_m * D_m)) / l)) * t_1) * t_2);
	elseif (t_0 <= 5e+235)
		tmp = Float64(t_2 * t_1);
	else
		tmp = abs(Float64(d / sqrt(Float64(h * l))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (1.0 - ((h / l) * ((((D_m * M_m) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)));
	t_1 = sqrt((d / l));
	t_2 = sqrt((d / h));
	tmp = 0.0;
	if (t_0 <= -1e-31)
		tmp = ((((((M_m * M_m) / d) / d) * h) * ((-0.125 * (D_m * D_m)) / l)) * t_1) * t_2;
	elseif (t_0 <= 5e+235)
		tmp = t_2 * t_1;
	else
		tmp = abs((d / sqrt((h * l))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -1e-31], N[(N[(N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] * h), $MachinePrecision] * N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 5e+235], N[(t$95$2 * t$95$1), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1e-31

   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 80.6%

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

   5. Taylor expanded in d around 0

      \[\leadsto \left(\color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]
   6. 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-*r* N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 52.4

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

   7. Applied rewrites 52.4%

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

   if -1e-31 < (*.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)))) < 5.00000000000000027e235

   1. Initial program 92.6%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 36.9

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

   5. Applied rewrites 36.9%

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

   7. Applied rewrites 90.6%

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

   if 5.00000000000000027e235 < (*.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 26.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 34.6

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

   5. Applied rewrites 34.6%

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

   7. Applied rewrites 59.3%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\left(\left(\left(\frac{\frac{M \cdot M}{d}}{d} \cdot h\right) \cdot \frac{-0.125 \cdot \left(D \cdot D\right)}{\ell}\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+235}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;t\_0 \leq -20:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{\left(\left(D\_m \cdot M\_m\right) \cdot 0.5\right) \cdot \left(D\_m \cdot M\_m\right)}{\left(2 \cdot d\right) \cdot d}, \frac{h}{\ell} \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+235}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (-
           1.0
           (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))))
   (if (<= t_0 -20.0)
     (*
      (sqrt (* (/ d h) (/ d l)))
      (fma
       (/ (* (* (* D_m M_m) 0.5) (* D_m M_m)) (* (* 2.0 d) d))
       (* (/ h l) -0.5)
       1.0))
     (if (<= t_0 5e+235)
       (* (sqrt (/ d h)) (sqrt (/ d l)))
       (fabs (/ d (sqrt (* h l))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_0 <= -20.0) {
		tmp = sqrt(((d / h) * (d / l))) * fma(((((D_m * M_m) * 0.5) * (D_m * M_m)) / ((2.0 * d) * d)), ((h / l) * -0.5), 1.0);
	} else if (t_0 <= 5e+235) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = fabs((d / sqrt((h * l))));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	tmp = 0.0
	if (t_0 <= -20.0)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * fma(Float64(Float64(Float64(Float64(D_m * M_m) * 0.5) * Float64(D_m * M_m)) / Float64(Float64(2.0 * d) * d)), Float64(Float64(h / l) * -0.5), 1.0));
	elseif (t_0 <= 5e+235)
		tmp = Float64(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]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -20.0], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * 0.5), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+235], N[(N[Sqrt[N[(d / h), $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 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)))) < -20

   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 79.9%

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

   6. Applied rewrites 36.3%

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

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

      9. metadata-eval N/A

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

      10. pow-pow N/A

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

      11. inv-pow N/A

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

      12. frac-times N/A

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

      13. clear-num N/A

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

      14. unpow2 N/A

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

      15. associate-/r* N/A

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

      16. frac-times N/A

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

      17. 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 27.6%

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

      1. lift-/.f64 N/A

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

      2. rem-square-sqrt N/A

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

      3. sqrt-unprod N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-div N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. frac-times N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-*.f64 53.5

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

   10. Applied rewrites 53.5%

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

   if -20 < (*.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)))) < 5.00000000000000027e235

   1. Initial program 92.8%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 35.3

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

   5. Applied rewrites 35.3%

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

   7. Applied rewrites 86.3%

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

   if 5.00000000000000027e235 < (*.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 26.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 34.6

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

   5. Applied rewrites 34.6%

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

   7. Applied rewrites 59.3%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -20:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{\left(\left(D \cdot M\right) \cdot 0.5\right) \cdot \left(D \cdot M\right)}{\left(2 \cdot d\right) \cdot d}, \frac{h}{\ell} \cdot -0.5, 1\right)\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+235}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\left(\left(0.25 \cdot \frac{M\_m}{d}\right) \cdot D\_m\right) \cdot h, \frac{0.5}{-d} \cdot \left(\frac{D\_m}{\ell} \cdot M\_m\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+235}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (-
           1.0
           (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_1 (/ d (sqrt (* h l)))))
   (if (<= t_0 0.0)
     (*
      t_1
      (fma
       (* (* (* 0.25 (/ M_m d)) D_m) h)
       (* (/ 0.5 (- d)) (* (/ D_m l) M_m))
       1.0))
     (if (<= t_0 5e+235) (* (sqrt (/ d h)) (sqrt (/ d l))) (fabs t_1)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * 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((((0.25 * (M_m / d)) * D_m) * h), ((0.5 / -d) * ((D_m / l) * M_m)), 1.0);
	} else if (t_0 <= 5e+235) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = fabs(t_1);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(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(0.25 * Float64(M_m / d)) * D_m) * h), Float64(Float64(0.5 / Float64(-d)) * Float64(Float64(D_m / l) * M_m)), 1.0));
	elseif (t_0 <= 5e+235)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = abs(t_1);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[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[(0.25 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * h), $MachinePrecision] * N[(N[(0.5 / (-d)), $MachinePrecision] * N[(N[(D$95$m / l), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+235], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $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 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 77.4%

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

   6. Applied rewrites 81.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 80.9%

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

   10. Applied rewrites 40.1%

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

   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)))) < 5.00000000000000027e235

   1. Initial program 98.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 31.4

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

   5. Applied rewrites 31.4%

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

   7. Applied rewrites 97.4%

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

   if 5.00000000000000027e235 < (*.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 26.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 34.6

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

   5. Applied rewrites 34.6%

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

   7. Applied rewrites 59.3%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\left(\left(0.25 \cdot \frac{M}{d}\right) \cdot D\right) \cdot h, \frac{0.5}{-d} \cdot \left(\frac{D}{\ell} \cdot M\right), 1\right)\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+235}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\left(h \cdot D\_m\right) \cdot \left(0.25 \cdot \frac{M\_m}{d}\right), \left(M\_m \cdot \frac{0.5}{d}\right) \cdot \frac{-D\_m}{\ell}, 1\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+235}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (-
           1.0
           (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_1 (/ d (sqrt (* h l)))))
   (if (<= t_0 0.0)
     (*
      (fma
       (* (* h D_m) (* 0.25 (/ M_m d)))
       (* (* M_m (/ 0.5 d)) (/ (- D_m) l))
       1.0)
      t_1)
     (if (<= t_0 5e+235) (* (sqrt (/ d h)) (sqrt (/ d l))) (fabs t_1)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_1 = d / sqrt((h * l));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = fma(((h * D_m) * (0.25 * (M_m / d))), ((M_m * (0.5 / d)) * (-D_m / l)), 1.0) * t_1;
	} else if (t_0 <= 5e+235) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = fabs(t_1);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(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(fma(Float64(Float64(h * D_m) * Float64(0.25 * Float64(M_m / d))), Float64(Float64(M_m * Float64(0.5 / d)) * Float64(Float64(-D_m) / l)), 1.0) * t_1);
	elseif (t_0 <= 5e+235)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = abs(t_1);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[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[(h * D$95$m), $MachinePrecision] * N[(0.25 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * N[((-D$95$m) / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 5e+235], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $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 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 77.4%

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

   6. Applied rewrites 81.8%

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

   8. Applied rewrites 38.0%

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

   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)))) < 5.00000000000000027e235

   1. Initial program 98.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 31.4

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

   5. Applied rewrites 31.4%

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

   7. Applied rewrites 97.4%

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

   if 5.00000000000000027e235 < (*.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 26.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 34.6

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

   5. Applied rewrites 34.6%

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

   7. Applied rewrites 59.3%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\left(h \cdot D\right) \cdot \left(0.25 \cdot \frac{M}{d}\right), \left(M \cdot \frac{0.5}{d}\right) \cdot \frac{-D}{\ell}, 1\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+235}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-70}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{D\_m}{d} \cdot M\_m\right) \cdot \left(\left(0.25 \cdot \frac{M\_m}{d}\right) \cdot D\_m\right), \frac{h}{\ell} \cdot -0.5, 1\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+235}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (-
           1.0
           (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_1 (/ d (sqrt (* h l)))))
   (if (<= t_0 -1e-70)
     (*
      (fma
       (* (* (/ D_m d) M_m) (* (* 0.25 (/ M_m d)) D_m))
       (* (/ h l) -0.5)
       1.0)
      t_1)
     (if (<= t_0 5e+235) (* (sqrt (/ d h)) (sqrt (/ d l))) (fabs t_1)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_1 = d / sqrt((h * l));
	double tmp;
	if (t_0 <= -1e-70) {
		tmp = fma((((D_m / d) * M_m) * ((0.25 * (M_m / d)) * D_m)), ((h / l) * -0.5), 1.0) * t_1;
	} else if (t_0 <= 5e+235) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = fabs(t_1);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	t_1 = Float64(d / sqrt(Float64(h * l)))
	tmp = 0.0
	if (t_0 <= -1e-70)
		tmp = Float64(fma(Float64(Float64(Float64(D_m / d) * M_m) * Float64(Float64(0.25 * Float64(M_m / d)) * D_m)), Float64(Float64(h / l) * -0.5), 1.0) * t_1);
	elseif (t_0 <= 5e+235)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = abs(t_1);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[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, -1e-70], N[(N[(N[(N[(N[(D$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(N[(0.25 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 5e+235], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $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)))) < -9.99999999999999996e-71

   1. Initial program 81.7%

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

      1. lift-*.f64 N/A

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

      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 80.8%

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

   6. Applied rewrites 36.6%

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

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

      5. pow-pow N/A

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

      6. unpow-1 N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      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. lift-/.f64 N/A

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

      12. associate-/r/ N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 N/A

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

   8. Applied rewrites 35.6%

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

   if -9.99999999999999996e-71 < (*.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)))) < 5.00000000000000027e235

   1. Initial program 92.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 37.4

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

   5. Applied rewrites 37.4%

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

   7. Applied rewrites 91.7%

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

   if 5.00000000000000027e235 < (*.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 26.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 34.6

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

   5. Applied rewrites 34.6%

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

   7. Applied rewrites 59.3%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{-70}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{D}{d} \cdot M\right) \cdot \left(\left(0.25 \cdot \frac{M}{d}\right) \cdot D\right), \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+235}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\left(\frac{\left(\frac{\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot -0.125}{d} \cdot D\_m\right) \cdot \frac{D\_m}{d}}{\ell} + 1\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+235}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (-
           1.0
           (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_1 (/ d (sqrt (* h l)))))
   (if (<= t_0 -5e-10)
     (*
      (+ (/ (* (* (/ (* (* (* M_m M_m) h) -0.125) d) D_m) (/ D_m d)) l) 1.0)
      t_1)
     (if (<= t_0 5e+235) (* (sqrt (/ d h)) (sqrt (/ d l))) (fabs t_1)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_1 = d / sqrt((h * l));
	double tmp;
	if (t_0 <= -5e-10) {
		tmp = ((((((((M_m * M_m) * h) * -0.125) / d) * D_m) * (D_m / d)) / l) + 1.0) * t_1;
	} else if (t_0 <= 5e+235) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = fabs(t_1);
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 - ((h / l) * ((((d_m * m_m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))
    t_1 = d / sqrt((h * l))
    if (t_0 <= (-5d-10)) then
        tmp = ((((((((m_m * m_m) * h) * (-0.125d0)) / d) * d_m) * (d_m / d)) / l) + 1.0d0) * t_1
    else if (t_0 <= 5d+235) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else
        tmp = abs(t_1)
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (1.0 - ((h / l) * (Math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (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 tmp;
	if (t_0 <= -5e-10) {
		tmp = ((((((((M_m * M_m) * h) * -0.125) / d) * D_m) * (D_m / d)) / l) + 1.0) * t_1;
	} else if (t_0 <= 5e+235) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else {
		tmp = Math.abs(t_1);
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (1.0 - ((h / l) * (math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0)))
	t_1 = d / math.sqrt((h * l))
	tmp = 0
	if t_0 <= -5e-10:
		tmp = ((((((((M_m * M_m) * h) * -0.125) / d) * D_m) * (D_m / d)) / l) + 1.0) * t_1
	elif t_0 <= 5e+235:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	else:
		tmp = math.fabs(t_1)
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	t_1 = Float64(d / sqrt(Float64(h * l)))
	tmp = 0.0
	if (t_0 <= -5e-10)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(M_m * M_m) * h) * -0.125) / d) * D_m) * Float64(D_m / d)) / l) + 1.0) * t_1);
	elseif (t_0 <= 5e+235)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = abs(t_1);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (1.0 - ((h / l) * ((((D_m * M_m) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)));
	t_1 = d / sqrt((h * l));
	tmp = 0.0;
	if (t_0 <= -5e-10)
		tmp = ((((((((M_m * M_m) * h) * -0.125) / d) * D_m) * (D_m / d)) / l) + 1.0) * t_1;
	elseif (t_0 <= 5e+235)
		tmp = sqrt((d / h)) * sqrt((d / l));
	else
		tmp = abs(t_1);
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[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, -5e-10], N[(N[(N[(N[(N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * -0.125), $MachinePrecision] / d), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 5e+235], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $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)))) < -5.00000000000000031e-10

   1. Initial program 81.0%

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

   4. Applied rewrites 55.6%

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

   5. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 39.0

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

   7. Applied rewrites 39.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-*.f64 N/A

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

   9. Applied rewrites 33.0%

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

   if -5.00000000000000031e-10 < (*.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)))) < 5.00000000000000027e235

   1. Initial program 92.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 35.7

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

   5. Applied rewrites 35.7%

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

   7. Applied rewrites 87.4%

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

   if 5.00000000000000027e235 < (*.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 26.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 34.6

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

   5. Applied rewrites 34.6%

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

   7. Applied rewrites 59.3%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\left(\frac{\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot -0.125}{d} \cdot D\right) \cdot \frac{D}{d}}{\ell} + 1\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+235}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-70}:\\ \;\;\;\;\left(-t\_0\right) \cdot t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+235}:\\ \;\;\;\;t\_1 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1 (sqrt (/ d h)))
        (t_2
         (*
          (-
           1.0
           (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))))
   (if (<= t_2 -1e-70)
     (* (- t_0) t_1)
     (if (<= t_2 5e+235) (* t_1 t_0) (fabs (/ d (sqrt (* h l))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / l));
	double t_1 = sqrt((d / h));
	double t_2 = (1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_2 <= -1e-70) {
		tmp = -t_0 * t_1;
	} else if (t_2 <= 5e+235) {
		tmp = t_1 * t_0;
	} else {
		tmp = fabs((d / sqrt((h * l))));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = sqrt((d / h))
    t_2 = (1.0d0 - ((h / l) * ((((d_m * m_m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))
    if (t_2 <= (-1d-70)) then
        tmp = -t_0 * t_1
    else if (t_2 <= 5d+235) then
        tmp = t_1 * t_0
    else
        tmp = abs((d / sqrt((h * l))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = Math.sqrt((d / h));
	double t_2 = (1.0 - ((h / l) * (Math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (Math.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_2 <= -1e-70) {
		tmp = -t_0 * t_1;
	} else if (t_2 <= 5e+235) {
		tmp = t_1 * t_0;
	} else {
		tmp = Math.abs((d / Math.sqrt((h * l))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((d / l))
	t_1 = math.sqrt((d / h))
	t_2 = (1.0 - ((h / l) * (math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0)))
	tmp = 0
	if t_2 <= -1e-70:
		tmp = -t_0 * t_1
	elif t_2 <= 5e+235:
		tmp = t_1 * t_0
	else:
		tmp = math.fabs((d / math.sqrt((h * l))))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / l))
	t_1 = sqrt(Float64(d / h))
	t_2 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	tmp = 0.0
	if (t_2 <= -1e-70)
		tmp = Float64(Float64(-t_0) * t_1);
	elseif (t_2 <= 5e+235)
		tmp = Float64(t_1 * t_0);
	else
		tmp = abs(Float64(d / sqrt(Float64(h * l))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((d / l));
	t_1 = sqrt((d / h));
	t_2 = (1.0 - ((h / l) * ((((D_m * M_m) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)));
	tmp = 0.0;
	if (t_2 <= -1e-70)
		tmp = -t_0 * t_1;
	elseif (t_2 <= 5e+235)
		tmp = t_1 * t_0;
	else
		tmp = abs((d / sqrt((h * l))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := 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[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-70], N[((-t$95$0) * t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 5e+235], N[(t$95$1 * t$95$0), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -9.99999999999999996e-71

   1. Initial program 81.7%

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

      1. lift-*.f64 N/A

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

      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 80.8%

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

   6. Applied rewrites 84.7%

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

   7. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \cdot \sqrt{\frac{d}{h}} \]
   8. 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 13.7

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

   9. Applied rewrites 13.7%

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

   if -9.99999999999999996e-71 < (*.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)))) < 5.00000000000000027e235

   1. Initial program 92.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 37.4

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

   5. Applied rewrites 37.4%

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

   7. Applied rewrites 91.7%

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

   if 5.00000000000000027e235 < (*.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 26.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 34.6

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

   5. Applied rewrites 34.6%

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

   7. Applied rewrites 59.3%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{-70}:\\ \;\;\;\;\left(-\sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+235}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+235}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (-
           1.0
           (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))))
   (if (<= t_0 0.0)
     (* (sqrt (/ 1.0 (* h l))) (- d))
     (if (<= t_0 5e+235)
       (* (sqrt (/ d h)) (sqrt (/ d l)))
       (fabs (/ d (sqrt (* h l))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = sqrt((1.0 / (h * l))) * -d;
	} else if (t_0 <= 5e+235) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = fabs((d / sqrt((h * l))));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - ((h / l) * ((((d_m * m_m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))
    if (t_0 <= 0.0d0) then
        tmp = sqrt((1.0d0 / (h * l))) * -d
    else if (t_0 <= 5d+235) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else
        tmp = abs((d / sqrt((h * l))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (1.0 - ((h / l) * (Math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (Math.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.sqrt((1.0 / (h * l))) * -d;
	} else if (t_0 <= 5e+235) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else {
		tmp = Math.abs((d / Math.sqrt((h * l))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (1.0 - ((h / l) * (math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.sqrt((1.0 / (h * l))) * -d
	elif t_0 <= 5e+235:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	else:
		tmp = math.fabs((d / math.sqrt((h * l))))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(-d));
	elseif (t_0 <= 5e+235)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = abs(Float64(d / sqrt(Float64(h * l))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (1.0 - ((h / l) * ((((D_m * M_m) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = sqrt((1.0 / (h * l))) * -d;
	elseif (t_0 <= 5e+235)
		tmp = sqrt((d / h)) * sqrt((d / l));
	else
		tmp = abs((d / sqrt((h * l))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], If[LessEqual[t$95$0, 5e+235], N[(N[Sqrt[N[(d / h), $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 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 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. 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 17.6

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

   5. Applied rewrites 17.6%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.00000000000000027e235

   1. Initial program 98.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 31.4

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

   5. Applied rewrites 31.4%

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

   7. Applied rewrites 97.4%

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

   if 5.00000000000000027e235 < (*.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 26.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 34.6

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

   5. Applied rewrites 34.6%

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

   7. Applied rewrites 59.3%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+235}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 75.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+235}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\left(\left(0.25 \cdot D\_m\right) \cdot \frac{M\_m}{d}\right) \cdot \frac{\left(M\_m \cdot \frac{0.5}{d}\right) \cdot D\_m}{-\ell}, h, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<=
      (*
       (- 1.0 (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
       (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
      5e+235)
   (*
    (sqrt (/ d h))
    (*
     (sqrt (/ d l))
     (fma
      (* (* (* 0.25 D_m) (/ M_m d)) (/ (* (* M_m (/ 0.5 d)) D_m) (- l)))
      h
      1.0)))
   (fabs (/ d (sqrt (* h l))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (((1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)))) <= 5e+235) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * fma((((0.25 * D_m) * (M_m / d)) * (((M_m * (0.5 / d)) * D_m) / -l)), h, 1.0));
	} else {
		tmp = fabs((d / sqrt((h * l))));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0)))) <= 5e+235)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * fma(Float64(Float64(Float64(0.25 * D_m) * Float64(M_m / d)) * Float64(Float64(Float64(M_m * Float64(0.5 / d)) * D_m) / Float64(-l))), h, 1.0)));
	else
		tmp = abs(Float64(d / sqrt(Float64(h * l))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+235], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[(0.25 * D$95$m), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $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)))) < 5.00000000000000027e235

   1. Initial program 86.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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 84.8%

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

   6. Applied rewrites 88.1%

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

   if 5.00000000000000027e235 < (*.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 26.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 34.6

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

   5. Applied rewrites 34.6%

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

   7. Applied rewrites 59.3%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+235}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\left(\left(0.25 \cdot D\right) \cdot \frac{M}{d}\right) \cdot \frac{\left(M \cdot \frac{0.5}{d}\right) \cdot D}{-\ell}, h, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 75.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+184}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(0.25 \cdot \frac{M\_m}{d}\right) \cdot D\_m, \left(\frac{D\_m}{\ell} \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right) \cdot \left(-h\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<=
      (*
       (- 1.0 (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
       (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
      5e+184)
   (*
    (*
     (fma
      (* (* 0.25 (/ M_m d)) D_m)
      (* (* (/ D_m l) (* M_m (/ 0.5 d))) (- h))
      1.0)
     (sqrt (/ d l)))
    (sqrt (/ d h)))
   (fabs (/ d (sqrt (* h l))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (((1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)))) <= 5e+184) {
		tmp = (fma(((0.25 * (M_m / d)) * D_m), (((D_m / l) * (M_m * (0.5 / d))) * -h), 1.0) * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = fabs((d / sqrt((h * l))));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0)))) <= 5e+184)
		tmp = Float64(Float64(fma(Float64(Float64(0.25 * Float64(M_m / d)) * D_m), Float64(Float64(Float64(D_m / l) * Float64(M_m * Float64(0.5 / d))) * Float64(-h)), 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]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+184], N[(N[(N[(N[(N[(0.25 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(N[(D$95$m / l), $MachinePrecision] * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-h)), $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)))) < 4.9999999999999999e184

   1. Initial program 86.1%

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

      1. lift-*.f64 N/A

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

      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 84.4%

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

   6. Applied rewrites 87.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 87.2%

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

   if 4.9999999999999999e184 < (*.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 30.6%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 35.0

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

   5. Applied rewrites 35.0%

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

   7. Applied rewrites 60.6%

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

4. Final simplification 78.4%

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

Alternative 17: 75.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+184}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(0.25 \cdot \frac{M\_m}{d}\right) \cdot D\_m, \left(\frac{D\_m}{\ell} \cdot \frac{-0.5 \cdot M\_m}{d}\right) \cdot h, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<=
      (*
       (- 1.0 (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
       (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
      5e+184)
   (*
    (*
     (fma
      (* (* 0.25 (/ M_m d)) D_m)
      (* (* (/ D_m l) (/ (* -0.5 M_m) d)) h)
      1.0)
     (sqrt (/ d l)))
    (sqrt (/ d h)))
   (fabs (/ d (sqrt (* h l))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (((1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)))) <= 5e+184) {
		tmp = (fma(((0.25 * (M_m / d)) * D_m), (((D_m / l) * ((-0.5 * M_m) / d)) * h), 1.0) * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = fabs((d / sqrt((h * l))));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0)))) <= 5e+184)
		tmp = Float64(Float64(fma(Float64(Float64(0.25 * Float64(M_m / d)) * D_m), Float64(Float64(Float64(D_m / l) * Float64(Float64(-0.5 * M_m) / d)) * h), 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]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+184], N[(N[(N[(N[(N[(0.25 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(N[(D$95$m / l), $MachinePrecision] * N[(N[(-0.5 * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * h), $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)))) < 4.9999999999999999e184

   1. Initial program 86.1%

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

      1. lift-*.f64 N/A

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

      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 84.4%

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

   6. Applied rewrites 87.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 87.2%

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

   9. Taylor expanded in d around 0

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 87.2

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

   11. Applied rewrites 87.2%

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

   if 4.9999999999999999e184 < (*.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 30.6%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 35.0

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

   5. Applied rewrites 35.0%

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

   7. Applied rewrites 60.6%

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

4. Final simplification 78.3%

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

Alternative 18: 74.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+235}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\frac{D\_m}{d} \cdot \frac{-0.5 \cdot M\_m}{\ell}\right) \cdot \left(\left(0.25 \cdot D\_m\right) \cdot \frac{M\_m}{d}\right), h, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<=
      (*
       (- 1.0 (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
       (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
      5e+235)
   (*
    (*
     (fma
      (* (* (/ D_m d) (/ (* -0.5 M_m) l)) (* (* 0.25 D_m) (/ M_m d)))
      h
      1.0)
     (sqrt (/ d l)))
    (sqrt (/ d h)))
   (fabs (/ d (sqrt (* h l))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (((1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)))) <= 5e+235) {
		tmp = (fma((((D_m / d) * ((-0.5 * M_m) / l)) * ((0.25 * D_m) * (M_m / d))), h, 1.0) * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = fabs((d / sqrt((h * l))));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0)))) <= 5e+235)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(D_m / d) * Float64(Float64(-0.5 * M_m) / l)) * Float64(Float64(0.25 * D_m) * Float64(M_m / d))), h, 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]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+235], N[(N[(N[(N[(N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(-0.5 * M$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(0.25 * D$95$m), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h + 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)))) < 5.00000000000000027e235

   1. Initial program 86.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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 84.8%

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

   6. Applied rewrites 88.1%

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

   7. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 85.2

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

   9. Applied rewrites 85.2%

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

   if 5.00000000000000027e235 < (*.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 26.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 34.6

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

   5. Applied rewrites 34.6%

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

   7. Applied rewrites 59.3%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+235}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\frac{D}{d} \cdot \frac{-0.5 \cdot M}{\ell}\right) \cdot \left(\left(0.25 \cdot D\right) \cdot \frac{M}{d}\right), h, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 45.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<=
      (*
       (- 1.0 (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
       (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
      -1e-31)
   (* (sqrt (/ 1.0 (* h l))) (- d))
   (fabs (/ d (sqrt (* h l))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (((1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)))) <= -1e-31) {
		tmp = sqrt((1.0 / (h * l))) * -d;
	} else {
		tmp = fabs((d / sqrt((h * l))));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (((1.0d0 - ((h / l) * ((((d_m * m_m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))) <= (-1d-31)) 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);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (((1.0 - ((h / l) * (Math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (Math.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0)))) <= -1e-31) {
		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)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if ((1.0 - ((h / l) * (math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0)))) <= -1e-31:
		tmp = math.sqrt((1.0 / (h * l))) * -d
	else:
		tmp = math.fabs((d / math.sqrt((h * l))))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0)))) <= -1e-31)
		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);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (((1.0 - ((h / l) * ((((D_m * M_m) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)))) <= -1e-31)
		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]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-31], 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)))) < -1e-31

   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. 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 12.3

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

   5. Applied rewrites 12.3%

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

   if -1e-31 < (*.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 59.3%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 35.8

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

   5. Applied rewrites 35.8%

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

   7. Applied rewrites 64.3%

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

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 45.7% accurate, 0.9× speedup

Math

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

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<=
      (*
       (- 1.0 (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
       (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
      -5e-10)
   (* (sqrt (/ 1.0 (* h l))) d)
   (fabs (/ d (sqrt (* h l))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (((1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)))) <= -5e-10) {
		tmp = sqrt((1.0 / (h * l))) * d;
	} else {
		tmp = fabs((d / sqrt((h * l))));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (((1.0d0 - ((h / l) * ((((d_m * m_m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))) <= (-5d-10)) 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);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (((1.0 - ((h / l) * (Math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (Math.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0)))) <= -5e-10) {
		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)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if ((1.0 - ((h / l) * (math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0)))) <= -5e-10:
		tmp = math.sqrt((1.0 / (h * l))) * d
	else:
		tmp = math.fabs((d / math.sqrt((h * l))))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0)))) <= -5e-10)
		tmp = Float64(sqrt(Float64(1.0 / Float64(h * l))) * d);
	else
		tmp = abs(Float64(d / sqrt(Float64(h * l))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (((1.0 - ((h / l) * ((((D_m * M_m) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)))) <= -5e-10)
		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]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-10], 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)))) < -5.00000000000000031e-10

   1. Initial program 81.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 9.9

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

   5. Applied rewrites 9.9%

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

   if -5.00000000000000031e-10 < (*.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 60.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 35.2

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

   5. Applied rewrites 35.2%

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

   7. Applied rewrites 63.2%

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

4. Final simplification 43.6%

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

Alternative 21: 45.5% accurate, 0.9× speedup

Math

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

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ d (sqrt (* h l)))))
   (if (<=
        (*
         (-
          1.0
          (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
         (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
        -5e-10)
     t_0
     (fabs t_0))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = d / sqrt((h * l));
	double tmp;
	if (((1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)))) <= -5e-10) {
		tmp = t_0;
	} else {
		tmp = fabs(t_0);
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d / sqrt((h * l))
    if (((1.0d0 - ((h / l) * ((((d_m * m_m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))) <= (-5d-10)) then
        tmp = t_0
    else
        tmp = abs(t_0)
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = d / Math.sqrt((h * l));
	double tmp;
	if (((1.0 - ((h / l) * (Math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (Math.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0)))) <= -5e-10) {
		tmp = t_0;
	} else {
		tmp = Math.abs(t_0);
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = d / math.sqrt((h * l))
	tmp = 0
	if ((1.0 - ((h / l) * (math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0)))) <= -5e-10:
		tmp = t_0
	else:
		tmp = math.fabs(t_0)
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(d / sqrt(Float64(h * l)))
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0)))) <= -5e-10)
		tmp = t_0;
	else
		tmp = abs(t_0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = d / sqrt((h * l));
	tmp = 0.0;
	if (((1.0 - ((h / l) * ((((D_m * M_m) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)))) <= -5e-10)
		tmp = t_0;
	else
		tmp = abs(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-10], 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)))) < -5.00000000000000031e-10

   1. Initial program 81.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 9.9

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

   5. Applied rewrites 9.9%

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

   7. Applied rewrites 7.9%

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

   if -5.00000000000000031e-10 < (*.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 60.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 35.2

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

   5. Applied rewrites 35.2%

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

   7. Applied rewrites 63.2%

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

4. Final simplification 42.9%

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

Alternative 22: 80.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \left(M\_m \cdot \frac{0.5}{d}\right) \cdot D\_m\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(t\_0 \cdot \mathsf{fma}\left(\left(\left(0.25 \cdot D\_m\right) \cdot \frac{M\_m}{d}\right) \cdot \frac{t\_1}{-\ell}, h, 1\right)\right)\\ \mathbf{elif}\;\ell \leq 7.6 \cdot 10^{-137}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{{t\_1}^{2} \cdot h}{\ell}, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\left(\left(0.25 \cdot \frac{M\_m}{d}\right) \cdot D\_m\right) \cdot h, \frac{0.5}{-d} \cdot \left(\frac{D\_m}{\ell} \cdot M\_m\right), 1\right) \cdot t\_0\right) \cdot \sqrt{d}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))) (t_1 (* (* M_m (/ 0.5 d)) D_m)))
   (if (<= l -5e-310)
     (*
      (/ (sqrt (- d)) (sqrt (- h)))
      (* t_0 (fma (* (* (* 0.25 D_m) (/ M_m d)) (/ t_1 (- l))) h 1.0)))
     (if (<= l 7.6e-137)
       (* (/ d (sqrt (* h l))) (fma (/ (* (pow t_1 2.0) h) l) -0.5 1.0))
       (/
        (*
         (*
          (fma
           (* (* (* 0.25 (/ M_m d)) D_m) h)
           (* (/ 0.5 (- d)) (* (/ D_m l) M_m))
           1.0)
          t_0)
         (sqrt d))
        (sqrt h))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / l));
	double t_1 = (M_m * (0.5 / d)) * D_m;
	double tmp;
	if (l <= -5e-310) {
		tmp = (sqrt(-d) / sqrt(-h)) * (t_0 * fma((((0.25 * D_m) * (M_m / d)) * (t_1 / -l)), h, 1.0));
	} else if (l <= 7.6e-137) {
		tmp = (d / sqrt((h * l))) * fma(((pow(t_1, 2.0) * h) / l), -0.5, 1.0);
	} else {
		tmp = ((fma((((0.25 * (M_m / d)) * D_m) * h), ((0.5 / -d) * ((D_m / l) * M_m)), 1.0) * t_0) * sqrt(d)) / sqrt(h);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(Float64(M_m * Float64(0.5 / d)) * D_m)
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(t_0 * fma(Float64(Float64(Float64(0.25 * D_m) * Float64(M_m / d)) * Float64(t_1 / Float64(-l))), h, 1.0)));
	elseif (l <= 7.6e-137)
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * fma(Float64(Float64((t_1 ^ 2.0) * h) / l), -0.5, 1.0));
	else
		tmp = Float64(Float64(Float64(fma(Float64(Float64(Float64(0.25 * Float64(M_m / d)) * D_m) * h), Float64(Float64(0.5 / Float64(-d)) * Float64(Float64(D_m / l) * M_m)), 1.0) * t_0) * sqrt(d)) / sqrt(h));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision]}, If[LessEqual[l, -5e-310], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(N[(N[(N[(0.25 * D$95$m), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / (-l)), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.6e-137], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(0.25 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * h), $MachinePrecision] * N[(N[(0.5 / (-d)), $MachinePrecision] * N[(N[(D$95$m / l), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 69.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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 67.2%

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

   6. Applied rewrites 74.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 85.5

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

   8. Applied rewrites 85.5%

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

   if -4.999999999999985e-310 < l < 7.59999999999999997e-137

   1. Initial program 66.7%

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

      1. lift-*.f64 N/A

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

      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 66.7%

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

   6. Applied rewrites 69.2%

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

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

   8. Applied rewrites 87.2%

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

   if 7.59999999999999997e-137 < l

   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. 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 67.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 68.9%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

   10. Applied rewrites 80.3%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\left(\left(0.25 \cdot D\right) \cdot \frac{M}{d}\right) \cdot \frac{\left(M \cdot \frac{0.5}{d}\right) \cdot D}{-\ell}, h, 1\right)\right)\\ \mathbf{elif}\;\ell \leq 7.6 \cdot 10^{-137}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{{\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right)}^{2} \cdot h}{\ell}, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\left(\left(0.25 \cdot \frac{M}{d}\right) \cdot D\right) \cdot h, \frac{0.5}{-d} \cdot \left(\frac{D}{\ell} \cdot M\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 79.3% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if d < -3.999999999999988e-310

   1. Initial program 69.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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 67.2%

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

   6. Applied rewrites 74.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 85.5

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

   8. Applied rewrites 85.5%

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

   if -3.999999999999988e-310 < d

   1. Initial program 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 65.8%

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

   6. Applied rewrites 70.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 69.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

   10. Applied rewrites 78.6%

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

4. Final simplification 82.2%

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

Alternative 24: 26.1% accurate, 15.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \frac{d}{\sqrt{h \cdot \ell}} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (/ d (sqrt (* h l))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return d / sqrt((h * l));
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = d / sqrt((h * l))
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return d / Math.sqrt((h * l));
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return d / math.sqrt((h * l))

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(d / sqrt(Float64(h * l)))
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = d / sqrt((h * l));
end

Wolfram

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

Derivation

1. Initial program 67.7%

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

3. 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 25.9

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

5. Applied rewrites 25.9%

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

7. Applied rewrites 25.5%

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

8. Final simplification 25.5%

   \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024294 
(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)))))