Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.0% → 83.4%

Time: 18.1s

Alternatives: 18

Speedup: 3.5×

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: 66.0% 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: 83.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\\ t_1 := \sqrt{-d}\\ t_2 := h \cdot \frac{M\_m}{d}\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\frac{t\_1}{\sqrt{-\ell}} \cdot \mathsf{fma}\left(\frac{t\_0}{-\ell}, \left(0.25 \cdot D\_m\right) \cdot t\_2, 1\right)\right) \cdot t\_1}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{-74}:\\ \;\;\;\;\left(1 - \frac{\left(\left(D\_m \cdot 0.5\right) \cdot 0.5\right) \cdot \frac{M\_m}{d}}{{h}^{-1}} \cdot \frac{\left(D\_m \cdot \frac{0.5}{d}\right) \cdot M\_m}{\ell}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{t\_0}{\ell}, \left(-0.25\right) \cdot \left(t\_2 \cdot D\_m\right), 1\right)\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* D_m (* M_m (/ 0.5 d))))
        (t_1 (sqrt (- d)))
        (t_2 (* h (/ M_m d))))
   (if (<= l -4e-310)
     (/
      (*
       (* (/ t_1 (sqrt (- l))) (fma (/ t_0 (- l)) (* (* 0.25 D_m) t_2) 1.0))
       t_1)
      (sqrt (- h)))
     (if (<= l 1.3e-74)
       (*
        (-
         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 l)) (/ (sqrt d) (sqrt h))))
       (*
        (sqrt (/ d h))
        (*
         (/ (sqrt d) (sqrt l))
         (fma (/ t_0 l) (* (- 0.25) (* t_2 D_m)) 1.0)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = D_m * (M_m * (0.5 / d));
	double t_1 = sqrt(-d);
	double t_2 = h * (M_m / d);
	double tmp;
	if (l <= -4e-310) {
		tmp = (((t_1 / sqrt(-l)) * fma((t_0 / -l), ((0.25 * D_m) * t_2), 1.0)) * t_1) / sqrt(-h);
	} else if (l <= 1.3e-74) {
		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 / l)) * (sqrt(d) / sqrt(h)));
	} else {
		tmp = sqrt((d / h)) * ((sqrt(d) / sqrt(l)) * fma((t_0 / l), (-0.25 * (t_2 * D_m)), 1.0));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(D_m * Float64(M_m * Float64(0.5 / d)))
	t_1 = sqrt(Float64(-d))
	t_2 = Float64(h * Float64(M_m / d))
	tmp = 0.0
	if (l <= -4e-310)
		tmp = Float64(Float64(Float64(Float64(t_1 / sqrt(Float64(-l))) * fma(Float64(t_0 / Float64(-l)), Float64(Float64(0.25 * D_m) * t_2), 1.0)) * t_1) / sqrt(Float64(-h)));
	elseif (l <= 1.3e-74)
		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(sqrt(Float64(d / l)) * Float64(sqrt(d) / sqrt(h))));
	else
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(sqrt(d) / sqrt(l)) * fma(Float64(t_0 / l), Float64(Float64(-0.25) * Float64(t_2 * D_m)), 1.0)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$2 = N[(h * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4e-310], N[(N[(N[(N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 / (-l)), $MachinePrecision] * N[(N[(0.25 * D$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.3e-74], 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[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 / l), $MachinePrecision] * N[((-0.25) * N[(t$95$2 * D$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.999999999999988e-310

   1. Initial program 67.8%

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

   4. Applied rewrites 72.7%

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

   6. Applied rewrites 73.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 82.6

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

   8. Applied rewrites 82.6%

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

   if -3.999999999999988e-310 < l < 1.3e-74

   1. Initial program 70.5%

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

   4. Applied rewrites 73.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{0.5}{d} \cdot D\right) \cdot M}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{{h}^{-1}}}\right) \]
   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 73.2

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

      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-sqrt.f64 73.2

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

   6. Applied rewrites 73.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 73.2

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. pow1/2 N/A

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

      8. pow1/2 N/A

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

      9. lower-/.f64 N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 95.8

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

   8. Applied rewrites 95.8%

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

   if 1.3e-74 < l

   1. Initial program 55.5%

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

   4. Applied rewrites 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 N/A

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

      11. unpow1/2 N/A

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

      12. lift-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-/.f64 N/A

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

      15. lower-*.f64 55.5

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

   6. Applied rewrites 55.5%

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

   8. Applied rewrites 58.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. lower-/.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 69.1

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

   10. Applied rewrites 69.1%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \mathsf{fma}\left(\frac{D \cdot \left(M \cdot \frac{0.5}{d}\right)}{-\ell}, \left(0.25 \cdot D\right) \cdot \left(h \cdot \frac{M}{d}\right), 1\right)\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{-74}:\\ \;\;\;\;\left(1 - \frac{\left(\left(D \cdot 0.5\right) \cdot 0.5\right) \cdot \frac{M}{d}}{{h}^{-1}} \cdot \frac{\left(D \cdot \frac{0.5}{d}\right) \cdot M}{\ell}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{D \cdot \left(M \cdot \frac{0.5}{d}\right)}{\ell}, \left(-0.25\right) \cdot \left(\left(h \cdot \frac{M}{d}\right) \cdot D\right), 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 60.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+15}:\\ \;\;\;\;\left(\frac{\left(\left(\frac{M\_m}{d} \cdot M\_m\right) \cdot h\right) \cdot \frac{-0.125 \cdot \left(D\_m \cdot D\_m\right)}{\ell}}{d} \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+296}:\\ \;\;\;\;t\_1 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (-
           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+15)
     (*
      (* (/ (* (* (* (/ M_m d) M_m) h) (/ (* -0.125 (* D_m D_m)) l)) d) t_1)
      t_2)
     (if (<= t_0 5e+296) (* t_1 t_2) (/ d (* (sqrt l) (sqrt h)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (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+15) {
		tmp = ((((((M_m / d) * M_m) * h) * ((-0.125 * (D_m * D_m)) / l)) / d) * t_1) * t_2;
	} else if (t_0 <= 5e+296) {
		tmp = t_1 * t_2;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: 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 <= (-2d+15)) then
        tmp = ((((((m_m / d) * m_m) * h) * (((-0.125d0) * (d_m * d_m)) / l)) / d) * t_1) * t_2
    else if (t_0 <= 5d+296) then
        tmp = t_1 * t_2
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double 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 <= -2e+15) {
		tmp = ((((((M_m / d) * M_m) * h) * ((-0.125 * (D_m * D_m)) / l)) / d) * t_1) * t_2;
	} else if (t_0 <= 5e+296) {
		tmp = t_1 * t_2;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	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 <= -2e+15:
		tmp = ((((((M_m / d) * M_m) * h) * ((-0.125 * (D_m * D_m)) / l)) / d) * t_1) * t_2
	elif t_0 <= 5e+296:
		tmp = t_1 * t_2
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	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+15)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(M_m / d) * M_m) * h) * Float64(Float64(-0.125 * Float64(D_m * D_m)) / l)) / d) * t_1) * t_2);
	elseif (t_0 <= 5e+296)
		tmp = Float64(t_1 * t_2);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	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 <= -2e+15)
		tmp = ((((((M_m / d) * M_m) * h) * ((-0.125 * (D_m * D_m)) / l)) / d) * t_1) * t_2;
	elseif (t_0 <= 5e+296)
		tmp = t_1 * t_2;
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := 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+15], N[(N[(N[(N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] * h), $MachinePrecision] * N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 5e+296], N[(t$95$1 * t$95$2), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $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)))) < -2e15

   1. Initial program 87.4%

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

   4. Applied rewrites 36.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 N/A

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

      11. unpow1/2 N/A

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

      12. lift-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-/.f64 N/A

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

      15. lower-*.f64 85.0

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

   6. Applied rewrites 85.1%

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

   7. Taylor expanded in h around inf

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

      1. 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 61.9

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

   9. Applied rewrites 61.9%

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

   11. Applied rewrites 68.1%

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

   if -2e15 < (*.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)))) < 5.0000000000000001e296

   1. Initial program 90.5%

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

   4. Applied rewrites 53.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 N/A

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

      11. unpow1/2 N/A

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

      12. lift-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-/.f64 N/A

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

      15. lower-*.f64 89.4

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

   6. Applied rewrites 89.4%

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

   7. Taylor expanded in h around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 86.9

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

   9. Applied rewrites 86.9%

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

   if 5.0000000000000001e296 < (*.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 12.0%

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

   3. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 24.7

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

   5. Applied rewrites 24.7%

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

   7. Applied rewrites 24.7%

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

   9. Applied rewrites 24.9%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -2 \cdot 10^{+15}:\\ \;\;\;\;\left(\frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot h\right) \cdot \frac{-0.125 \cdot \left(D \cdot D\right)}{\ell}}{d} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+296}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+15}:\\ \;\;\;\;\left(\frac{\left(\left(\frac{M\_m}{d} \cdot M\_m\right) \cdot h\right) \cdot \left(-0.125 \cdot \left(D\_m \cdot D\_m\right)\right)}{d \cdot \ell} \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+296}:\\ \;\;\;\;t\_1 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (-
           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+15)
     (*
      (* (/ (* (* (* (/ M_m d) M_m) h) (* -0.125 (* D_m D_m))) (* d l)) t_1)
      t_2)
     (if (<= t_0 5e+296) (* t_1 t_2) (/ d (* (sqrt l) (sqrt h)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (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+15) {
		tmp = ((((((M_m / d) * M_m) * h) * (-0.125 * (D_m * D_m))) / (d * l)) * t_1) * t_2;
	} else if (t_0 <= 5e+296) {
		tmp = t_1 * t_2;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: 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 <= (-2d+15)) then
        tmp = ((((((m_m / d) * m_m) * h) * ((-0.125d0) * (d_m * d_m))) / (d * l)) * t_1) * t_2
    else if (t_0 <= 5d+296) then
        tmp = t_1 * t_2
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double 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 <= -2e+15) {
		tmp = ((((((M_m / d) * M_m) * h) * (-0.125 * (D_m * D_m))) / (d * l)) * t_1) * t_2;
	} else if (t_0 <= 5e+296) {
		tmp = t_1 * t_2;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	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 <= -2e+15:
		tmp = ((((((M_m / d) * M_m) * h) * (-0.125 * (D_m * D_m))) / (d * l)) * t_1) * t_2
	elif t_0 <= 5e+296:
		tmp = t_1 * t_2
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	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+15)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(M_m / d) * M_m) * h) * Float64(-0.125 * Float64(D_m * D_m))) / Float64(d * l)) * t_1) * t_2);
	elseif (t_0 <= 5e+296)
		tmp = Float64(t_1 * t_2);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	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 <= -2e+15)
		tmp = ((((((M_m / d) * M_m) * h) * (-0.125 * (D_m * D_m))) / (d * l)) * t_1) * t_2;
	elseif (t_0 <= 5e+296)
		tmp = t_1 * t_2;
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := 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+15], N[(N[(N[(N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] * h), $MachinePrecision] * N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 5e+296], N[(t$95$1 * t$95$2), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $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)))) < -2e15

   1. Initial program 87.4%

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

   4. Applied rewrites 36.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 N/A

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

      11. unpow1/2 N/A

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

      12. lift-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-/.f64 N/A

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

      15. lower-*.f64 85.0

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

   6. Applied rewrites 85.1%

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

   7. Taylor expanded in h around inf

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

      1. 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 61.9

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

   9. Applied rewrites 61.9%

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

   11. Applied rewrites 65.6%

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

   if -2e15 < (*.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)))) < 5.0000000000000001e296

   1. Initial program 90.5%

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

   4. Applied rewrites 53.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 N/A

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

      11. unpow1/2 N/A

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

      12. lift-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-/.f64 N/A

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

      15. lower-*.f64 89.4

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

   6. Applied rewrites 89.4%

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

   7. Taylor expanded in h around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 86.9

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

   9. Applied rewrites 86.9%

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

   if 5.0000000000000001e296 < (*.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 12.0%

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

   3. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 24.7

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

   5. Applied rewrites 24.7%

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

   7. Applied rewrites 24.7%

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

   9. Applied rewrites 24.9%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -2 \cdot 10^{+15}:\\ \;\;\;\;\left(\frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot h\right) \cdot \left(-0.125 \cdot \left(D \cdot D\right)\right)}{d \cdot \ell} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+296}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 48.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+25}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}}}{\left|\ell\right|} \cdot \left(\frac{M\_m \cdot M\_m}{d} \cdot \left(0.125 \cdot \left(D\_m \cdot D\_m\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+296}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (-
           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 -5e+25)
     (*
      (/ (sqrt (/ h l)) (fabs l))
      (* (/ (* M_m M_m) d) (* 0.125 (* D_m D_m))))
     (if (<= t_0 5e+296)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (/ d (* (sqrt l) (sqrt h)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double 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 <= -5e+25) {
		tmp = (sqrt((h / l)) / fabs(l)) * (((M_m * M_m) / d) * (0.125 * (D_m * D_m)));
	} else if (t_0 <= 5e+296) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: 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 <= (-5d+25)) then
        tmp = (sqrt((h / l)) / abs(l)) * (((m_m * m_m) / d) * (0.125d0 * (d_m * d_m)))
    else if (t_0 <= 5d+296) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double 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 <= -5e+25) {
		tmp = (Math.sqrt((h / l)) / Math.abs(l)) * (((M_m * M_m) / d) * (0.125 * (D_m * D_m)));
	} else if (t_0 <= 5e+296) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	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 <= -5e+25:
		tmp = (math.sqrt((h / l)) / math.fabs(l)) * (((M_m * M_m) / d) * (0.125 * (D_m * D_m)))
	elif t_0 <= 5e+296:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	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 <= -5e+25)
		tmp = Float64(Float64(sqrt(Float64(h / l)) / abs(l)) * Float64(Float64(Float64(M_m * M_m) / d) * Float64(0.125 * Float64(D_m * D_m))));
	elseif (t_0 <= 5e+296)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	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 <= -5e+25)
		tmp = (sqrt((h / l)) / abs(l)) * (((M_m * M_m) / d) * (0.125 * (D_m * D_m)));
	elseif (t_0 <= 5e+296)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := 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, -5e+25], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision] * N[(0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+296], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $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)))) < -5.00000000000000024e25

   1. Initial program 87.1%

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

   3. Taylor expanded in h around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 26.7%

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

   7. Applied rewrites 33.0%

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

   if -5.00000000000000024e25 < (*.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)))) < 5.0000000000000001e296

   1. Initial program 90.7%

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

   4. Applied rewrites 52.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 N/A

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

      11. unpow1/2 N/A

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

      12. lift-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-/.f64 N/A

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

      15. lower-*.f64 89.6

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

   6. Applied rewrites 89.6%

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

   7. Taylor expanded in h around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 85.1

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

   9. Applied rewrites 85.1%

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

   if 5.0000000000000001e296 < (*.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 12.0%

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

   3. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 24.7

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

   5. Applied rewrites 24.7%

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

   7. Applied rewrites 24.7%

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

   9. Applied rewrites 24.9%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -5 \cdot 10^{+25}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}}}{\left|\ell\right|} \cdot \left(\frac{M \cdot M}{d} \cdot \left(0.125 \cdot \left(D \cdot D\right)\right)\right)\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{+296}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.8% accurate, 2.7× speedup

Math

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

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 (* D_m (* M_m (/ 0.5 d))))
        (t_1 (sqrt (- d)))
        (t_2 (* h (/ M_m d))))
   (if (<= l -4e-310)
     (/
      (*
       (* (/ t_1 (sqrt (- l))) (fma (/ t_0 (- l)) (* (* 0.25 D_m) t_2) 1.0))
       t_1)
      (sqrt (- h)))
     (*
      (* (fma (/ t_0 l) (* (- 0.25) (* t_2 D_m)) 1.0) (sqrt (/ d l)))
      (/ (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 = D_m * (M_m * (0.5 / d));
	double t_1 = sqrt(-d);
	double t_2 = h * (M_m / d);
	double tmp;
	if (l <= -4e-310) {
		tmp = (((t_1 / sqrt(-l)) * fma((t_0 / -l), ((0.25 * D_m) * t_2), 1.0)) * t_1) / sqrt(-h);
	} else {
		tmp = (fma((t_0 / l), (-0.25 * (t_2 * D_m)), 1.0) * sqrt((d / l))) * (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 = Float64(D_m * Float64(M_m * Float64(0.5 / d)))
	t_1 = sqrt(Float64(-d))
	t_2 = Float64(h * Float64(M_m / d))
	tmp = 0.0
	if (l <= -4e-310)
		tmp = Float64(Float64(Float64(Float64(t_1 / sqrt(Float64(-l))) * fma(Float64(t_0 / Float64(-l)), Float64(Float64(0.25 * D_m) * t_2), 1.0)) * t_1) / sqrt(Float64(-h)));
	else
		tmp = Float64(Float64(fma(Float64(t_0 / l), Float64(Float64(-0.25) * Float64(t_2 * D_m)), 1.0) * sqrt(Float64(d / l))) * Float64(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[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$2 = N[(h * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4e-310], N[(N[(N[(N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 / (-l)), $MachinePrecision] * N[(N[(0.25 * D$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$0 / l), $MachinePrecision] * N[((-0.25) * N[(t$95$2 * D$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if l < -3.999999999999988e-310

   1. Initial program 67.8%

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

   4. Applied rewrites 72.7%

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

   6. Applied rewrites 73.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 82.6

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

   8. Applied rewrites 82.6%

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

   if -3.999999999999988e-310 < l

   1. Initial program 60.9%

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

   4. Applied rewrites 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 N/A

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

      11. unpow1/2 N/A

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

      12. lift-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-/.f64 N/A

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

      15. lower-*.f64 60.2

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

   6. Applied rewrites 60.2%

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

   8. Applied rewrites 62.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. lower-/.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 74.8

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

   10. Applied rewrites 74.8%

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

4. Final simplification 78.7%

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

Alternative 6: 78.4% accurate, 3.0× speedup

Math

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

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
         (fma
          (/ (* D_m (* M_m (/ 0.5 d))) l)
          (* (- 0.25) (* (* h (/ M_m d)) D_m))
          1.0)))
   (if (<= l -4e-310)
     (* (* t_0 (/ (sqrt (- d)) (sqrt (- l)))) (sqrt (/ d h)))
     (* (* t_0 (sqrt (/ d l))) (/ (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 = fma(((D_m * (M_m * (0.5 / d))) / l), (-0.25 * ((h * (M_m / d)) * D_m)), 1.0);
	double tmp;
	if (l <= -4e-310) {
		tmp = (t_0 * (sqrt(-d) / sqrt(-l))) * sqrt((d / h));
	} else {
		tmp = (t_0 * sqrt((d / l))) * (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 = fma(Float64(Float64(D_m * Float64(M_m * Float64(0.5 / d))) / l), Float64(Float64(-0.25) * Float64(Float64(h * Float64(M_m / d)) * D_m)), 1.0)
	tmp = 0.0
	if (l <= -4e-310)
		tmp = Float64(Float64(t_0 * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l)))) * sqrt(Float64(d / h)));
	else
		tmp = Float64(Float64(t_0 * sqrt(Float64(d / l))) * Float64(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[(N[(N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[((-0.25) * N[(N[(h * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[l, -4e-310], N[(N[(t$95$0 * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -3.999999999999988e-310

   1. Initial program 67.8%

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

   4. Applied rewrites 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 N/A

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

      11. unpow1/2 N/A

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

      12. lift-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-/.f64 N/A

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

      15. lower-*.f64 67.1

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

   6. Applied rewrites 67.1%

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

   8. Applied rewrites 67.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. sqrt-div N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 75.9

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

   10. Applied rewrites 75.9%

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

   if -3.999999999999988e-310 < l

   1. Initial program 60.9%

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

   4. Applied rewrites 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 N/A

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

      11. unpow1/2 N/A

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

      12. lift-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-/.f64 N/A

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

      15. lower-*.f64 60.2

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

   6. Applied rewrites 60.2%

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

   8. Applied rewrites 62.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. lower-/.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 74.8

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

   10. Applied rewrites 74.8%

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

4. Final simplification 75.3%

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

Alternative 7: 78.8% accurate, 3.0× speedup

Math

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

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 (* (* h (/ M_m d)) D_m)))
   (if (<= h -5e-310)
     (*
      (fma (/ (* (* D_m (/ 0.5 d)) M_m) l) (* -0.25 t_1) 1.0)
      (* (/ (sqrt (- d)) (sqrt (- h))) t_0))
     (*
      (* (fma (/ (* D_m (* M_m (/ 0.5 d))) l) (* (- 0.25) t_1) 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 = (h * (M_m / d)) * D_m;
	double tmp;
	if (h <= -5e-310) {
		tmp = fma((((D_m * (0.5 / d)) * M_m) / l), (-0.25 * t_1), 1.0) * ((sqrt(-d) / sqrt(-h)) * t_0);
	} else {
		tmp = (fma(((D_m * (M_m * (0.5 / d))) / l), (-0.25 * t_1), 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(h * Float64(M_m / d)) * D_m)
	tmp = 0.0
	if (h <= -5e-310)
		tmp = Float64(fma(Float64(Float64(Float64(D_m * Float64(0.5 / d)) * M_m) / l), Float64(-0.25 * t_1), 1.0) * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_0));
	else
		tmp = Float64(Float64(fma(Float64(Float64(D_m * Float64(M_m * Float64(0.5 / d))) / l), Float64(Float64(-0.25) * t_1), 1.0) * t_0) * Float64(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[(h * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision]}, If[LessEqual[h, -5e-310], N[(N[(N[(N[(N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(-0.25 * t$95$1), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[((-0.25) * t$95$1), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 67.8%

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

   4. Applied rewrites 68.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{0.5}{d} \cdot D\right) \cdot M}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{{h}^{-1}}}\right) \]
   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 68.8

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

      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-sqrt.f64 68.8

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

   6. Applied rewrites 68.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 68.8

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sqrt-div N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lower-/.f64 75.7

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

   8. Applied rewrites 75.7%

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

   10. Applied rewrites 73.4%

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

   if -4.999999999999985e-310 < h

   1. Initial program 60.9%

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

   4. Applied rewrites 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 N/A

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

      11. unpow1/2 N/A

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

      12. lift-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-/.f64 N/A

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

      15. lower-*.f64 60.2

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

   6. Applied rewrites 60.2%

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

   8. Applied rewrites 62.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. lower-/.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 74.8

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

   10. Applied rewrites 74.8%

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

4. Final simplification 74.1%

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

Alternative 8: 77.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := h \cdot \frac{M\_m}{d}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -2.1 \cdot 10^{-303}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{\left(M\_m \cdot 0.5\right) \cdot D\_m}{\left(-d\right) \cdot \ell}, \left(0.25 \cdot D\_m\right) \cdot t\_0, 1\right) \cdot t\_1\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)}{\ell}, \left(-0.25\right) \cdot \left(t\_0 \cdot D\_m\right), 1\right) \cdot t\_1\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\\ \end{array} \end{array} \]

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 (* h (/ M_m d))) (t_1 (sqrt (/ d l))))
   (if (<= d -2.1e-303)
     (/
      (*
       (*
        (fma (/ (* (* M_m 0.5) D_m) (* (- d) l)) (* (* 0.25 D_m) t_0) 1.0)
        t_1)
       (sqrt (- d)))
      (sqrt (- h)))
     (*
      (*
       (fma (/ (* D_m (* M_m (/ 0.5 d))) l) (* (- 0.25) (* t_0 D_m)) 1.0)
       t_1)
      (/ (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 = h * (M_m / d);
	double t_1 = sqrt((d / l));
	double tmp;
	if (d <= -2.1e-303) {
		tmp = ((fma((((M_m * 0.5) * D_m) / (-d * l)), ((0.25 * D_m) * t_0), 1.0) * t_1) * sqrt(-d)) / sqrt(-h);
	} else {
		tmp = (fma(((D_m * (M_m * (0.5 / d))) / l), (-0.25 * (t_0 * D_m)), 1.0) * t_1) * (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 = Float64(h * Float64(M_m / d))
	t_1 = sqrt(Float64(d / l))
	tmp = 0.0
	if (d <= -2.1e-303)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(Float64(M_m * 0.5) * D_m) / Float64(Float64(-d) * l)), Float64(Float64(0.25 * D_m) * t_0), 1.0) * t_1) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	else
		tmp = Float64(Float64(fma(Float64(Float64(D_m * Float64(M_m * Float64(0.5 / d))) / l), Float64(Float64(-0.25) * Float64(t_0 * D_m)), 1.0) * t_1) * Float64(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[(h * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -2.1e-303], N[(N[(N[(N[(N[(N[(N[(M$95$m * 0.5), $MachinePrecision] * D$95$m), $MachinePrecision] / N[((-d) * l), $MachinePrecision]), $MachinePrecision] * N[(N[(0.25 * D$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[((-0.25) * N[(t$95$0 * D$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if d < -2.1e-303

   1. Initial program 68.3%

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

   4. Applied rewrites 73.2%

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

   6. Applied rewrites 74.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. frac-times N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. div-inv N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 70.9

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

   8. Applied rewrites 70.9%

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

   if -2.1e-303 < d

   1. Initial program 60.5%

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

   4. Applied rewrites 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 N/A

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

      11. unpow1/2 N/A

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

      12. lift-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-/.f64 N/A

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

      15. lower-*.f64 59.8

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

   6. Applied rewrites 59.8%

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

   8. Applied rewrites 61.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. lower-/.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 74.2

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

   10. Applied rewrites 74.2%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.1 \cdot 10^{-303}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{\left(M \cdot 0.5\right) \cdot D}{\left(-d\right) \cdot \ell}, \left(0.25 \cdot D\right) \cdot \left(h \cdot \frac{M}{d}\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{D \cdot \left(M \cdot \frac{0.5}{d}\right)}{\ell}, \left(-0.25\right) \cdot \left(\left(h \cdot \frac{M}{d}\right) \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := h \cdot \frac{M\_m}{d}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -1.45 \cdot 10^{-125}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{\left(M\_m \cdot 0.5\right) \cdot D\_m}{\left(-d\right) \cdot \ell}, \left(0.25 \cdot D\_m\right) \cdot t\_0, 1\right) \cdot t\_1\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{+217}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)}{\ell}, \left(-0.25 \cdot D\_m\right) \cdot t\_0, 1\right) \cdot t\_1\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* h (/ M_m d))) (t_1 (sqrt (/ d l))))
   (if (<= l -1.45e-125)
     (/
      (*
       (*
        (fma (/ (* (* M_m 0.5) D_m) (* (- d) l)) (* (* 0.25 D_m) t_0) 1.0)
        t_1)
       (sqrt (- d)))
      (sqrt (- h)))
     (if (<= l 4.2e+217)
       (*
        (* (fma (/ (* D_m (* M_m (/ 0.5 d))) l) (* (* -0.25 D_m) t_0) 1.0) t_1)
        (sqrt (/ d h)))
       (/ d (* (sqrt l) (sqrt h)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = h * (M_m / d);
	double t_1 = sqrt((d / l));
	double tmp;
	if (l <= -1.45e-125) {
		tmp = ((fma((((M_m * 0.5) * D_m) / (-d * l)), ((0.25 * D_m) * t_0), 1.0) * t_1) * sqrt(-d)) / sqrt(-h);
	} else if (l <= 4.2e+217) {
		tmp = (fma(((D_m * (M_m * (0.5 / d))) / l), ((-0.25 * D_m) * t_0), 1.0) * t_1) * sqrt((d / h));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(h * Float64(M_m / d))
	t_1 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -1.45e-125)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(Float64(M_m * 0.5) * D_m) / Float64(Float64(-d) * l)), Float64(Float64(0.25 * D_m) * t_0), 1.0) * t_1) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (l <= 4.2e+217)
		tmp = Float64(Float64(fma(Float64(Float64(D_m * Float64(M_m * Float64(0.5 / d))) / l), Float64(Float64(-0.25 * D_m) * t_0), 1.0) * t_1) * sqrt(Float64(d / h)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(h * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.45e-125], N[(N[(N[(N[(N[(N[(N[(M$95$m * 0.5), $MachinePrecision] * D$95$m), $MachinePrecision] / N[((-d) * l), $MachinePrecision]), $MachinePrecision] * N[(N[(0.25 * D$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.2e+217], N[(N[(N[(N[(N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(-0.25 * D$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.4500000000000001e-125

   1. Initial program 64.1%

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

   4. Applied rewrites 70.4%

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

   6. Applied rewrites 70.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. frac-times N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. div-inv N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 69.3

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

   8. Applied rewrites 69.3%

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

   if -1.4500000000000001e-125 < l < 4.2000000000000002e217

   1. Initial program 69.6%

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

   4. Applied rewrites 16.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 N/A

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

      11. unpow1/2 N/A

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

      12. lift-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-/.f64 N/A

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

      15. lower-*.f64 68.9

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

   6. Applied rewrites 68.9%

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

   8. Applied rewrites 71.4%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 71.4

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

   10. Applied rewrites 71.4%

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

   if 4.2000000000000002e217 < l

   1. Initial program 19.3%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 61.2

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

   5. Applied rewrites 61.2%

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

   7. Applied rewrites 61.1%

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

   9. Applied rewrites 63.1%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.45 \cdot 10^{-125}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{\left(M \cdot 0.5\right) \cdot D}{\left(-d\right) \cdot \ell}, \left(0.25 \cdot D\right) \cdot \left(h \cdot \frac{M}{d}\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{+217}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{D \cdot \left(M \cdot \frac{0.5}{d}\right)}{\ell}, \left(-0.25 \cdot D\right) \cdot \left(h \cdot \frac{M}{d}\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.2 \cdot 10^{+217}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)}{\ell}, \left(-0.25 \cdot D\_m\right) \cdot \left(h \cdot \frac{M\_m}{d}\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l 4.2e+217)
   (*
    (*
     (fma
      (/ (* D_m (* M_m (/ 0.5 d))) l)
      (* (* -0.25 D_m) (* h (/ M_m d)))
      1.0)
     (sqrt (/ d l)))
    (sqrt (/ d h)))
   (/ d (* (sqrt l) (sqrt h)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 4.2e+217) {
		tmp = (fma(((D_m * (M_m * (0.5 / d))) / l), ((-0.25 * D_m) * (h * (M_m / d))), 1.0) * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= 4.2e+217)
		tmp = Float64(Float64(fma(Float64(Float64(D_m * Float64(M_m * Float64(0.5 / d))) / l), Float64(Float64(-0.25 * D_m) * Float64(h * Float64(M_m / d))), 1.0) * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, 4.2e+217], N[(N[(N[(N[(N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(-0.25 * D$95$m), $MachinePrecision] * N[(h * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.2000000000000002e217

   1. Initial program 67.3%

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

   4. Applied rewrites 38.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 N/A

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

      11. unpow1/2 N/A

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

      12. lift-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-/.f64 N/A

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

      15. lower-*.f64 66.6

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

   6. Applied rewrites 66.6%

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

   8. Applied rewrites 67.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 67.7

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

   10. Applied rewrites 67.7%

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

   if 4.2000000000000002e217 < l

   1. Initial program 19.3%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 61.2

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

   5. Applied rewrites 61.2%

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

   7. Applied rewrites 61.1%

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

   9. Applied rewrites 63.1%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.2 \cdot 10^{+217}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{D \cdot \left(M \cdot \frac{0.5}{d}\right)}{\ell}, \left(-0.25 \cdot D\right) \cdot \left(h \cdot \frac{M}{d}\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.6 \cdot 10^{+214}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(D\_m \cdot \frac{0.5}{d}\right) \cdot M\_m, \frac{-0.25 \cdot \left(\left(h \cdot \frac{M\_m}{d}\right) \cdot D\_m\right)}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l 2.6e+214)
   (*
    (*
     (fma
      (* (* D_m (/ 0.5 d)) M_m)
      (/ (* -0.25 (* (* h (/ M_m d)) D_m)) l)
      1.0)
     (sqrt (/ d l)))
    (sqrt (/ d h)))
   (/ d (* (sqrt l) (sqrt h)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 2.6e+214) {
		tmp = (fma(((D_m * (0.5 / d)) * M_m), ((-0.25 * ((h * (M_m / d)) * D_m)) / l), 1.0) * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= 2.6e+214)
		tmp = Float64(Float64(fma(Float64(Float64(D_m * Float64(0.5 / d)) * M_m), Float64(Float64(-0.25 * Float64(Float64(h * Float64(M_m / d)) * D_m)) / l), 1.0) * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, 2.6e+214], N[(N[(N[(N[(N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(N[(-0.25 * N[(N[(h * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.59999999999999993e214

   1. Initial program 67.9%

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

   4. Applied rewrites 38.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 N/A

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

      11. unpow1/2 N/A

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

      12. lift-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-/.f64 N/A

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

      15. lower-*.f64 67.1

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

   6. Applied rewrites 67.1%

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

   8. Applied rewrites 68.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   10. Applied rewrites 67.4%

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

   if 2.59999999999999993e214 < l

   1. Initial program 17.4%

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

   3. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 54.7

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

   5. Applied rewrites 54.7%

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

   7. Applied rewrites 54.6%

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

   9. Applied rewrites 56.4%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.6 \cdot 10^{+214}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(D \cdot \frac{0.5}{d}\right) \cdot M, \frac{-0.25 \cdot \left(\left(h \cdot \frac{M}{d}\right) \cdot D\right)}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.2 \cdot 10^{+217}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(M\_m \cdot 0.5\right) \cdot D\_m}{d \cdot \ell}, \left(-0.25\right) \cdot \left(\left(h \cdot \frac{M\_m}{d}\right) \cdot D\_m\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l 4.2e+217)
   (*
    (*
     (fma
      (/ (* (* M_m 0.5) D_m) (* d l))
      (* (- 0.25) (* (* h (/ M_m d)) D_m))
      1.0)
     (sqrt (/ d l)))
    (sqrt (/ d h)))
   (/ d (* (sqrt l) (sqrt h)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 4.2e+217) {
		tmp = (fma((((M_m * 0.5) * D_m) / (d * l)), (-0.25 * ((h * (M_m / d)) * D_m)), 1.0) * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= 4.2e+217)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(M_m * 0.5) * D_m) / Float64(d * l)), Float64(Float64(-0.25) * Float64(Float64(h * Float64(M_m / d)) * D_m)), 1.0) * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, 4.2e+217], N[(N[(N[(N[(N[(N[(M$95$m * 0.5), $MachinePrecision] * D$95$m), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[((-0.25) * N[(N[(h * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.2000000000000002e217

   1. Initial program 67.3%

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

   4. Applied rewrites 38.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 N/A

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

      11. unpow1/2 N/A

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

      12. lift-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-/.f64 N/A

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

      15. lower-*.f64 66.6

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

   6. Applied rewrites 66.6%

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

   8. Applied rewrites 67.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. div-inv N/A

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

      12. lower-*.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 63.9

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

   10. Applied rewrites 63.9%

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

   if 4.2000000000000002e217 < l

   1. Initial program 19.3%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 61.2

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

   5. Applied rewrites 61.2%

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

   7. Applied rewrites 61.1%

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

   9. Applied rewrites 63.1%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.2 \cdot 10^{+217}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(M \cdot 0.5\right) \cdot D}{d \cdot \ell}, \left(-0.25\right) \cdot \left(\left(h \cdot \frac{M}{d}\right) \cdot D\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.8% accurate, 6.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-291}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{-262}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -1.55e-291)
   (/ (* (sqrt (/ d l)) (sqrt (- d))) (sqrt (- h)))
   (if (<= l 1.8e-262)
     (/ (* (sqrt (/ h l)) (- d)) h)
     (/ d (* (sqrt l) (sqrt h))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1.55e-291) {
		tmp = (sqrt((d / l)) * sqrt(-d)) / sqrt(-h);
	} else if (l <= 1.8e-262) {
		tmp = (sqrt((h / l)) * -d) / h;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-1.55d-291)) then
        tmp = (sqrt((d / l)) * sqrt(-d)) / sqrt(-h)
    else if (l <= 1.8d-262) then
        tmp = (sqrt((h / l)) * -d) / h
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1.55e-291) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt(-d)) / Math.sqrt(-h);
	} else if (l <= 1.8e-262) {
		tmp = (Math.sqrt((h / l)) * -d) / h;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -1.55e-291:
		tmp = (math.sqrt((d / l)) * math.sqrt(-d)) / math.sqrt(-h)
	elif l <= 1.8e-262:
		tmp = (math.sqrt((h / l)) * -d) / h
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -1.55e-291)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (l <= 1.8e-262)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * Float64(-d)) / h);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -1.55e-291)
		tmp = (sqrt((d / l)) * sqrt(-d)) / sqrt(-h);
	elseif (l <= 1.8e-262)
		tmp = (sqrt((h / l)) * -d) / h;
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -1.55e-291], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.8e-262], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision] / h), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.55000000000000006e-291

   1. Initial program 67.3%

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in h around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 47.3

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

   7. Applied rewrites 47.3%

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

   if -1.55000000000000006e-291 < l < 1.7999999999999999e-262

   1. Initial program 85.7%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 42.9%

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

   6. Taylor expanded in l around -inf

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

   8. Applied rewrites 58.5%

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

   if 1.7999999999999999e-262 < l

   1. Initial program 58.6%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 42.2

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

   5. Applied rewrites 42.2%

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

   7. Applied rewrites 42.2%

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

   9. Applied rewrites 47.4%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-291}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{-262}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 45.8% accurate, 8.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-291}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{-262}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -1.55e-291)
   (* (sqrt (/ 1.0 (* h l))) (- d))
   (if (<= l 1.8e-262)
     (/ (* (sqrt (/ h l)) (- d)) h)
     (/ d (* (sqrt l) (sqrt h))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1.55e-291) {
		tmp = sqrt((1.0 / (h * l))) * -d;
	} else if (l <= 1.8e-262) {
		tmp = (sqrt((h / l)) * -d) / h;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-1.55d-291)) then
        tmp = sqrt((1.0d0 / (h * l))) * -d
    else if (l <= 1.8d-262) then
        tmp = (sqrt((h / l)) * -d) / h
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1.55e-291) {
		tmp = Math.sqrt((1.0 / (h * l))) * -d;
	} else if (l <= 1.8e-262) {
		tmp = (Math.sqrt((h / l)) * -d) / h;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -1.55e-291:
		tmp = math.sqrt((1.0 / (h * l))) * -d
	elif l <= 1.8e-262:
		tmp = (math.sqrt((h / l)) * -d) / h
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -1.55e-291)
		tmp = Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(-d));
	elseif (l <= 1.8e-262)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * Float64(-d)) / h);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -1.55e-291)
		tmp = sqrt((1.0 / (h * l))) * -d;
	elseif (l <= 1.8e-262)
		tmp = (sqrt((h / l)) * -d) / h;
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -1.55e-291], N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], If[LessEqual[l, 1.8e-262], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision] / h), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.55000000000000006e-291

   1. Initial program 67.3%

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

   3. Taylor expanded in 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 46.9

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

   5. Applied rewrites 46.9%

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

   if -1.55000000000000006e-291 < l < 1.7999999999999999e-262

   1. Initial program 85.7%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 42.9%

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

   6. Taylor expanded in l around -inf

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

   8. Applied rewrites 58.5%

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

   if 1.7999999999999999e-262 < l

   1. Initial program 58.6%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 42.2

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

   5. Applied rewrites 42.2%

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

   7. Applied rewrites 42.2%

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

   9. Applied rewrites 47.4%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-291}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{-262}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 45.4% accurate, 9.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.8 \cdot 10^{-248}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l 4.8e-248)
   (* (sqrt (/ 1.0 (* h l))) (- d))
   (/ d (* (sqrt l) (sqrt h)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 4.8e-248) {
		tmp = sqrt((1.0 / (h * l))) * -d;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= 4.8d-248) then
        tmp = sqrt((1.0d0 / (h * l))) * -d
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 4.8e-248) {
		tmp = Math.sqrt((1.0 / (h * l))) * -d;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= 4.8e-248:
		tmp = math.sqrt((1.0 / (h * l))) * -d
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= 4.8e-248)
		tmp = Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(-d));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= 4.8e-248)
		tmp = sqrt((1.0 / (h * l))) * -d;
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, 4.8e-248], N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.80000000000000006e-248

   1. Initial program 69.1%

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

   3. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 44.3

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

   5. Applied rewrites 44.3%

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

   if 4.80000000000000006e-248 < l

   1. Initial program 58.4%

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

   3. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 42.5

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

   5. Applied rewrites 42.5%

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

   7. Applied rewrites 42.4%

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

   9. Applied rewrites 47.8%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.8 \cdot 10^{-248}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 42.0% accurate, 10.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;\ell \leq -7 \cdot 10^{-278}:\\ \;\;\;\;t\_0 \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot d\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* h l)))))
   (if (<= l -7e-278) (* t_0 (- d)) (* t_0 d))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((1.0 / (h * l)));
	double tmp;
	if (l <= -7e-278) {
		tmp = t_0 * -d;
	} else {
		tmp = t_0 * d;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / (h * l)))
    if (l <= (-7d-278)) then
        tmp = t_0 * -d
    else
        tmp = t_0 * d
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((1.0 / (h * l)));
	double tmp;
	if (l <= -7e-278) {
		tmp = t_0 * -d;
	} else {
		tmp = t_0 * d;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((1.0 / (h * l)))
	tmp = 0
	if l <= -7e-278:
		tmp = t_0 * -d
	else:
		tmp = t_0 * d
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(1.0 / Float64(h * l)))
	tmp = 0.0
	if (l <= -7e-278)
		tmp = Float64(t_0 * Float64(-d));
	else
		tmp = Float64(t_0 * d);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((1.0 / (h * l)));
	tmp = 0.0;
	if (l <= -7e-278)
		tmp = t_0 * -d;
	else
		tmp = t_0 * d;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -7e-278], N[(t$95$0 * (-d)), $MachinePrecision], N[(t$95$0 * d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -6.99999999999999941e-278

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 48.3

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

   5. Applied rewrites 48.3%

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

   if -6.99999999999999941e-278 < l

   1. Initial program 62.7%

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

   3. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 40.3

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

   5. Applied rewrites 40.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 25.2% accurate, 12.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \sqrt{\frac{1}{h \cdot \ell}} \cdot d \end{array} \]

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 (* (sqrt (/ 1.0 (* h l))) d))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return sqrt((1.0 / (h * l))) * d;
}

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 = sqrt((1.0d0 / (h * l))) * d
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 Math.sqrt((1.0 / (h * l))) * d;
}

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 math.sqrt((1.0 / (h * l))) * d

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(sqrt(Float64(1.0 / Float64(h * l))) * d)
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 = sqrt((1.0 / (h * l))) * d;
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[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]

Derivation

1. Initial program 64.3%

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

3. Taylor expanded in h around 0

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
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 24.9

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

5. Applied rewrites 24.9%

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

6. Final simplification 24.9%

   \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]
7. Add Preprocessing

Alternative 18: 25.2% accurate, 15.3× speedup

Math

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

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 64.3%

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

3. Taylor expanded in h around 0

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
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 24.9

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

5. Applied rewrites 24.9%

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

7. Applied rewrites 24.9%

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

8. Final simplification 24.9%

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

Reproduce

herbie shell --seed 2024284 
(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)))))