Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.4% → 82.0%

Time: 17.4s

Alternatives: 16

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left(\frac{0.5}{d} \cdot M\_m\right) \cdot D\_m\\ t_1 := \frac{t\_0}{\ell}\\ \mathbf{if}\;d \leq -1.1 \cdot 10^{-267}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\left(t\_1 \cdot h\right) \cdot t\_0, -0.5, 1\right)}{\frac{\sqrt{-h}}{\sqrt{-d}}}\\ \mathbf{elif}\;d \leq 5.2 \cdot 10^{-185}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{-0.5 \cdot \frac{D\_m \cdot M\_m}{d}}{\ell}, \left(D\_m \cdot h\right) \cdot \left(0.25 \cdot \frac{M\_m}{d}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{h}{\frac{d}{\left(0.25 \cdot D\_m\right) \cdot M\_m}} \cdot t\_1\right) \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \end{array} \end{array} \]

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 (* (* (/ 0.5 d) M_m) D_m)) (t_1 (/ t_0 l)))
   (if (<= d -1.1e-267)
     (/
      (* (sqrt (/ d l)) (fma (* (* t_1 h) t_0) -0.5 1.0))
      (/ (sqrt (- h)) (sqrt (- d))))
     (if (<= d 5.2e-185)
       (*
        (/ (fabs d) (sqrt (* l h)))
        (fma
         (/ (* -0.5 (/ (* D_m M_m) d)) l)
         (* (* D_m h) (* 0.25 (/ M_m d)))
         1.0))
       (*
        (- 1.0 (* (/ h (/ d (* (* 0.25 D_m) M_m))) t_1))
        (* (/ (sqrt d) (sqrt h)) (pow (/ d l) (/ 1.0 2.0))))))))

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 = ((0.5 / d) * M_m) * D_m;
	double t_1 = t_0 / l;
	double tmp;
	if (d <= -1.1e-267) {
		tmp = (sqrt((d / l)) * fma(((t_1 * h) * t_0), -0.5, 1.0)) / (sqrt(-h) / sqrt(-d));
	} else if (d <= 5.2e-185) {
		tmp = (fabs(d) / sqrt((l * h))) * fma(((-0.5 * ((D_m * M_m) / d)) / l), ((D_m * h) * (0.25 * (M_m / d))), 1.0);
	} else {
		tmp = (1.0 - ((h / (d / ((0.25 * D_m) * M_m))) * t_1)) * ((sqrt(d) / sqrt(h)) * pow((d / l), (1.0 / 2.0)));
	}
	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(Float64(Float64(0.5 / d) * M_m) * D_m)
	t_1 = Float64(t_0 / l)
	tmp = 0.0
	if (d <= -1.1e-267)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * fma(Float64(Float64(t_1 * h) * t_0), -0.5, 1.0)) / Float64(sqrt(Float64(-h)) / sqrt(Float64(-d))));
	elseif (d <= 5.2e-185)
		tmp = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * fma(Float64(Float64(-0.5 * Float64(Float64(D_m * M_m) / d)) / l), Float64(Float64(D_m * h) * Float64(0.25 * Float64(M_m / d))), 1.0));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(h / Float64(d / Float64(Float64(0.25 * D_m) * M_m))) * t_1)) * Float64(Float64(sqrt(d) / sqrt(h)) * (Float64(d / l) ^ Float64(1.0 / 2.0))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / l), $MachinePrecision]}, If[LessEqual[d, -1.1e-267], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(t$95$1 * h), $MachinePrecision] * t$95$0), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[(-h)], $MachinePrecision] / N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.2e-185], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.5 * N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(D$95$m * h), $MachinePrecision] * N[(0.25 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(h / N[(d / N[(N[(0.25 * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.09999999999999994e-267

   1. Initial program 75.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lift-/.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval 74.5

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

   4. Applied rewrites 74.5%

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

   6. Applied rewrites 75.3%

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

   8. Applied rewrites 75.4%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 87.5

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

   10. Applied rewrites 87.5%

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

   if -1.09999999999999994e-267 < d < 5.1999999999999997e-185

   1. Initial program 23.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 23.1

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

   4. Applied rewrites 23.1%

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

   6. Applied rewrites 24.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. pow1/2 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. lift-sqrt.f64 N/A

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

      14. sqrt-div N/A

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

      15. lift-/.f64 N/A

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

      16. clear-num N/A

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

      17. lift-/.f64 N/A

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

      18. pow1/2 N/A

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

      19. sqrt-unprod N/A

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

      20. lower-sqrt.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-sqrt.f64 14.4

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

   8. Applied rewrites 14.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 14.4

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

   10. Applied rewrites 57.6%

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

   if 5.1999999999999997e-185 < d

   1. Initial program 73.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 83.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-1 N/A

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

      6. remove-double-div N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. clear-num N/A

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

      11. un-div-inv N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 83.0

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. associate-*r* N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. lower-*.f64 N/A

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

      23. metadata-eval 83.0

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

   6. Applied rewrites 83.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 83.0

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. pow1/2 N/A

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

      8. pow1/2 N/A

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

      9. lower-/.f64 N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 92.3

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

   8. Applied rewrites 92.3%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{-267}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\left(\frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot h\right) \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right), -0.5, 1\right)}{\frac{\sqrt{-h}}{\sqrt{-d}}}\\ \mathbf{elif}\;d \leq 5.2 \cdot 10^{-185}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{-0.5 \cdot \frac{D \cdot M}{d}}{\ell}, \left(D \cdot h\right) \cdot \left(0.25 \cdot \frac{M}{d}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{h}{\frac{d}{\left(0.25 \cdot D\right) \cdot M}} \cdot \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell}\right) \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \frac{D\_m \cdot M\_m}{d}\\ t_2 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{-0.5 \cdot t\_1}{\ell}, \left(D\_m \cdot h\right) \cdot \left(0.25 \cdot \frac{M\_m}{d}\right), 1\right)\\ t_3 := \sqrt{\frac{d}{h}}\\ t_4 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-134}:\\ \;\;\;\;\left(1 - \left(\left(\left(0.25 \cdot D\_m\right) \cdot h\right) \cdot \frac{M\_m}{d}\right) \cdot \frac{t\_1 \cdot 0.5}{\ell}\right) \cdot \left(t\_3 \cdot t\_4\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-264}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 10^{+215}:\\ \;\;\;\;\left(1 \cdot t\_4\right) \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (-
           1.0
           (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_1 (/ (* D_m M_m) d))
        (t_2
         (*
          (/ (fabs d) (sqrt (* l h)))
          (fma (/ (* -0.5 t_1) l) (* (* D_m h) (* 0.25 (/ M_m d))) 1.0)))
        (t_3 (sqrt (/ d h)))
        (t_4 (sqrt (/ d l))))
   (if (<= t_0 -4e-134)
     (*
      (- 1.0 (* (* (* (* 0.25 D_m) h) (/ M_m d)) (/ (* t_1 0.5) l)))
      (* t_3 t_4))
     (if (<= t_0 2e-264) t_2 (if (<= t_0 1e+215) (* (* 1.0 t_4) t_3) t_2)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_1 = (D_m * M_m) / d;
	double t_2 = (fabs(d) / sqrt((l * h))) * fma(((-0.5 * t_1) / l), ((D_m * h) * (0.25 * (M_m / d))), 1.0);
	double t_3 = sqrt((d / h));
	double t_4 = sqrt((d / l));
	double tmp;
	if (t_0 <= -4e-134) {
		tmp = (1.0 - ((((0.25 * D_m) * h) * (M_m / d)) * ((t_1 * 0.5) / l))) * (t_3 * t_4);
	} else if (t_0 <= 2e-264) {
		tmp = t_2;
	} else if (t_0 <= 1e+215) {
		tmp = (1.0 * t_4) * t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	t_1 = Float64(Float64(D_m * M_m) / d)
	t_2 = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * fma(Float64(Float64(-0.5 * t_1) / l), Float64(Float64(D_m * h) * Float64(0.25 * Float64(M_m / d))), 1.0))
	t_3 = sqrt(Float64(d / h))
	t_4 = sqrt(Float64(d / l))
	tmp = 0.0
	if (t_0 <= -4e-134)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(0.25 * D_m) * h) * Float64(M_m / d)) * Float64(Float64(t_1 * 0.5) / l))) * Float64(t_3 * t_4));
	elseif (t_0 <= 2e-264)
		tmp = t_2;
	elseif (t_0 <= 1e+215)
		tmp = Float64(Float64(1.0 * t_4) * t_3);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(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[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.5 * t$95$1), $MachinePrecision] / l), $MachinePrecision] * N[(N[(D$95$m * h), $MachinePrecision] * N[(0.25 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -4e-134], N[(N[(1.0 - N[(N[(N[(N[(0.25 * D$95$m), $MachinePrecision] * h), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * 0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 * t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-264], t$95$2, If[LessEqual[t$95$0, 1e+215], N[(N[(1.0 * t$95$4), $MachinePrecision] * t$95$3), $MachinePrecision], t$95$2]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.00000000000000016e-134

   1. Initial program 81.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 80.2

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

   4. Applied rewrites 80.2%

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

   6. Applied rewrites 79.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 79.7

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

      4. lift-/.f64 N/A

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

      5. metadata-eval 79.7

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

      6. lift-pow.f64 N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 79.7

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

      9. lift-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-div N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. lift-/.f64 N/A

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

      16. lower-sqrt.f64 80.8

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

   8. Applied rewrites 80.8%

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

   if -4.00000000000000016e-134 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2e-264 or 9.99999999999999907e214 < (*.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 28.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 29.0

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

   4. Applied rewrites 29.0%

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

   6. Applied rewrites 32.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. pow1/2 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. lift-sqrt.f64 N/A

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

      14. sqrt-div N/A

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

      15. lift-/.f64 N/A

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

      16. clear-num N/A

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

      17. lift-/.f64 N/A

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

      18. pow1/2 N/A

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

      19. sqrt-unprod N/A

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

      20. lower-sqrt.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-sqrt.f64 15.8

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

   8. Applied rewrites 15.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 15.8

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

   10. Applied rewrites 75.1%

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

   if 2e-264 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.99999999999999907e214

   1. Initial program 99.5%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 99.5%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -4 \cdot 10^{-134}:\\ \;\;\;\;\left(1 - \left(\left(\left(0.25 \cdot D\right) \cdot h\right) \cdot \frac{M}{d}\right) \cdot \frac{\frac{D \cdot M}{d} \cdot 0.5}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 2 \cdot 10^{-264}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{-0.5 \cdot \frac{D \cdot M}{d}}{\ell}, \left(D \cdot h\right) \cdot \left(0.25 \cdot \frac{M}{d}\right), 1\right)\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+215}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{-0.5 \cdot \frac{D \cdot M}{d}}{\ell}, \left(D \cdot h\right) \cdot \left(0.25 \cdot \frac{M}{d}\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{-0.5 \cdot \frac{D\_m \cdot M\_m}{d}}{\ell}, \left(D\_m \cdot h\right) \cdot \left(0.25 \cdot \frac{M\_m}{d}\right), 1\right)\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-264}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+215}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (-
           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
         (*
          (/ (fabs d) (sqrt (* l h)))
          (fma
           (/ (* -0.5 (/ (* D_m M_m) d)) l)
           (* (* D_m h) (* 0.25 (/ M_m d)))
           1.0))))
   (if (<= t_0 2e-264)
     t_1
     (if (<= t_0 1e+215) (* (* 1.0 (sqrt (/ d l))) (sqrt (/ d h))) t_1))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_1 = (fabs(d) / sqrt((l * h))) * fma(((-0.5 * ((D_m * M_m) / d)) / l), ((D_m * h) * (0.25 * (M_m / d))), 1.0);
	double tmp;
	if (t_0 <= 2e-264) {
		tmp = t_1;
	} else if (t_0 <= 1e+215) {
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	t_1 = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * fma(Float64(Float64(-0.5 * Float64(Float64(D_m * M_m) / d)) / l), Float64(Float64(D_m * h) * Float64(0.25 * Float64(M_m / d))), 1.0))
	tmp = 0.0
	if (t_0 <= 2e-264)
		tmp = t_1;
	elseif (t_0 <= 1e+215)
		tmp = Float64(Float64(1.0 * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.5 * N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(D$95$m * h), $MachinePrecision] * N[(0.25 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-264], t$95$1, If[LessEqual[t$95$0, 1e+215], N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2e-264 or 9.99999999999999907e214 < (*.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 54.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 54.3

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

   4. Applied rewrites 54.3%

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

   6. Applied rewrites 55.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. pow1/2 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. lift-sqrt.f64 N/A

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

      14. sqrt-div N/A

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

      15. lift-/.f64 N/A

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

      16. clear-num N/A

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

      17. lift-/.f64 N/A

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

      18. pow1/2 N/A

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

      19. sqrt-unprod N/A

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

      20. lower-sqrt.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-sqrt.f64 26.3

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

   8. Applied rewrites 26.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 26.3

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

   10. Applied rewrites 75.0%

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

   if 2e-264 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.99999999999999907e214

   1. Initial program 99.5%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 99.5%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 2 \cdot 10^{-264}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{-0.5 \cdot \frac{D \cdot M}{d}}{\ell}, \left(D \cdot h\right) \cdot \left(0.25 \cdot \frac{M}{d}\right), 1\right)\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+215}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{-0.5 \cdot \frac{D \cdot M}{d}}{\ell}, \left(D \cdot h\right) \cdot \left(0.25 \cdot \frac{M}{d}\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.5% accurate, 0.6× speedup

Math

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

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
   (if (<=
        (*
         (-
          1.0
          (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
         t_0)
        1e+215)
     (*
      (fma
       (* (* (* (/ 0.5 d) M_m) D_m) (/ (- h) l))
       (* (/ M_m d) (* (* 0.5 D_m) 0.5))
       1.0)
      t_0)
     (*
      (/ (fabs d) (sqrt (* l h)))
      (fma
       (/ (* -0.5 (/ (* D_m M_m) d)) l)
       (* (* D_m h) (* 0.25 (/ M_m d)))
       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 = pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0));
	double tmp;
	if (((1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * t_0) <= 1e+215) {
		tmp = fma(((((0.5 / d) * M_m) * D_m) * (-h / l)), ((M_m / d) * ((0.5 * D_m) * 0.5)), 1.0) * t_0;
	} else {
		tmp = (fabs(d) / sqrt((l * h))) * fma(((-0.5 * ((D_m * M_m) / d)) / l), ((D_m * h) * (0.25 * (M_m / d))), 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((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0)))
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * t_0) <= 1e+215)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(0.5 / d) * M_m) * D_m) * Float64(Float64(-h) / l)), Float64(Float64(M_m / d) * Float64(Float64(0.5 * D_m) * 0.5)), 1.0) * t_0);
	else
		tmp = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * fma(Float64(Float64(-0.5 * Float64(Float64(D_m * M_m) / d)) / l), Float64(Float64(D_m * h) * Float64(0.25 * Float64(M_m / d))), 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[(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]}, If[LessEqual[N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], 1e+215], N[(N[(N[(N[(N[(N[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * N[((-h) / l), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * N[(N[(0.5 * D$95$m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.5 * N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(D$95$m * h), $MachinePrecision] * N[(0.25 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.99999999999999907e214

   1. Initial program 84.3%

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

   4. Applied rewrites 87.3%

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

   if 9.99999999999999907e214 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 26.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 27.1

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

   4. Applied rewrites 27.1%

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

   6. Applied rewrites 31.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. pow1/2 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. lift-sqrt.f64 N/A

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

      14. sqrt-div N/A

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

      15. lift-/.f64 N/A

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

      16. clear-num N/A

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

      17. lift-/.f64 N/A

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

      18. pow1/2 N/A

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

      19. sqrt-unprod N/A

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

      20. lower-sqrt.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-sqrt.f64 13.7

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

   8. Applied rewrites 13.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 13.7

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

   10. Applied rewrites 74.3%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+215}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right) \cdot \frac{-h}{\ell}, \frac{M}{d} \cdot \left(\left(0.5 \cdot D\right) \cdot 0.5\right), 1\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{-0.5 \cdot \frac{D \cdot M}{d}}{\ell}, \left(D \cdot h\right) \cdot \left(0.25 \cdot \frac{M}{d}\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.5% accurate, 0.6× speedup

Math

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

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (* (/ 0.5 d) M_m) D_m))
        (t_1 (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
   (if (<=
        (*
         (-
          1.0
          (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
         t_1)
        1e+215)
     (* (- 1.0 (* (* (* 0.5 (/ h l)) t_0) t_0)) t_1)
     (*
      (/ (fabs d) (sqrt (* l h)))
      (fma
       (/ (* -0.5 (/ (* D_m M_m) d)) l)
       (* (* D_m h) (* 0.25 (/ M_m d)))
       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 = ((0.5 / d) * M_m) * D_m;
	double t_1 = pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0));
	double tmp;
	if (((1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * t_1) <= 1e+215) {
		tmp = (1.0 - (((0.5 * (h / l)) * t_0) * t_0)) * t_1;
	} else {
		tmp = (fabs(d) / sqrt((l * h))) * fma(((-0.5 * ((D_m * M_m) / d)) / l), ((D_m * h) * (0.25 * (M_m / d))), 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(Float64(Float64(0.5 / d) * M_m) * D_m)
	t_1 = Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0)))
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * t_1) <= 1e+215)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(0.5 * Float64(h / l)) * t_0) * t_0)) * t_1);
	else
		tmp = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * fma(Float64(Float64(-0.5 * Float64(Float64(D_m * M_m) / d)) / l), Float64(Float64(D_m * h) * Float64(0.25 * Float64(M_m / d))), 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[(N[(N[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], 1e+215], N[(N[(1.0 - N[(N[(N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.5 * N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(D$95$m * h), $MachinePrecision] * N[(0.25 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.99999999999999907e214

   1. Initial program 84.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 87.3%

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

   if 9.99999999999999907e214 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 26.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 27.1

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

   4. Applied rewrites 27.1%

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

   6. Applied rewrites 31.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. pow1/2 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. lift-sqrt.f64 N/A

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

      14. sqrt-div N/A

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

      15. lift-/.f64 N/A

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

      16. clear-num N/A

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

      17. lift-/.f64 N/A

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

      18. pow1/2 N/A

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

      19. sqrt-unprod N/A

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

      20. lower-sqrt.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-sqrt.f64 13.7

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

   8. Applied rewrites 13.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 13.7

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

   10. Applied rewrites 74.3%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+215}:\\ \;\;\;\;\left(1 - \left(\left(0.5 \cdot \frac{h}{\ell}\right) \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right)\right) \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{-0.5 \cdot \frac{D \cdot M}{d}}{\ell}, \left(D \cdot h\right) \cdot \left(0.25 \cdot \frac{M}{d}\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.6% accurate, 0.7× speedup

Math

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

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (* (/ 0.5 d) M_m) D_m)))
   (if (<=
        (*
         (-
          1.0
          (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
         (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
        1e+215)
     (/
      (* (sqrt (/ d l)) (fma (* (* (/ t_0 l) h) t_0) -0.5 1.0))
      (/ 1.0 (sqrt (/ d h))))
     (*
      (/ (fabs d) (sqrt (* l h)))
      (fma
       (/ (* -0.5 (/ (* D_m M_m) d)) l)
       (* (* D_m h) (* 0.25 (/ M_m d)))
       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 = ((0.5 / d) * M_m) * D_m;
	double tmp;
	if (((1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)))) <= 1e+215) {
		tmp = (sqrt((d / l)) * fma((((t_0 / l) * h) * t_0), -0.5, 1.0)) / (1.0 / sqrt((d / h)));
	} else {
		tmp = (fabs(d) / sqrt((l * h))) * fma(((-0.5 * ((D_m * M_m) / d)) / l), ((D_m * h) * (0.25 * (M_m / d))), 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(Float64(Float64(0.5 / d) * M_m) * D_m)
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0)))) <= 1e+215)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * fma(Float64(Float64(Float64(t_0 / l) * h) * t_0), -0.5, 1.0)) / Float64(1.0 / sqrt(Float64(d / h))));
	else
		tmp = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * fma(Float64(Float64(-0.5 * Float64(Float64(D_m * M_m) / d)) / l), Float64(Float64(D_m * h) * Float64(0.25 * Float64(M_m / d))), 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[(N[(N[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+215], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[(t$95$0 / l), $MachinePrecision] * h), $MachinePrecision] * t$95$0), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.5 * N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(D$95$m * h), $MachinePrecision] * N[(0.25 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.99999999999999907e214

   1. Initial program 84.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lift-/.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval 83.7

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

   4. Applied rewrites 83.7%

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

   6. Applied rewrites 83.7%

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

   8. Applied rewrites 86.9%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. sqrt-div N/A

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

      8. clear-num N/A

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

      9. lift-/.f64 N/A

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

      10. lower-sqrt.f64 87.4

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

   10. Applied rewrites 87.4%

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

   if 9.99999999999999907e214 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 26.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 27.1

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

   4. Applied rewrites 27.1%

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

   6. Applied rewrites 31.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. pow1/2 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. lift-sqrt.f64 N/A

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

      14. sqrt-div N/A

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

      15. lift-/.f64 N/A

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

      16. clear-num N/A

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

      17. lift-/.f64 N/A

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

      18. pow1/2 N/A

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

      19. sqrt-unprod N/A

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

      20. lower-sqrt.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-sqrt.f64 13.7

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

   8. Applied rewrites 13.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 13.7

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

   10. Applied rewrites 74.3%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+215}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\left(\frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot h\right) \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right), -0.5, 1\right)}{\frac{1}{\sqrt{\frac{d}{h}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{-0.5 \cdot \frac{D \cdot M}{d}}{\ell}, \left(D \cdot h\right) \cdot \left(0.25 \cdot \frac{M}{d}\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.1% accurate, 0.8× speedup

Math

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

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (* (/ 0.5 d) M_m) D_m)))
   (if (<=
        (*
         (-
          1.0
          (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
         (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
        1e+215)
     (/
      (* (sqrt (/ d l)) (fma (* (* (/ t_0 l) h) t_0) -0.5 1.0))
      (sqrt (/ h d)))
     (*
      (/ (fabs d) (sqrt (* l h)))
      (fma
       (/ (* -0.5 (/ (* D_m M_m) d)) l)
       (* (* D_m h) (* 0.25 (/ M_m d)))
       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 = ((0.5 / d) * M_m) * D_m;
	double tmp;
	if (((1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)))) <= 1e+215) {
		tmp = (sqrt((d / l)) * fma((((t_0 / l) * h) * t_0), -0.5, 1.0)) / sqrt((h / d));
	} else {
		tmp = (fabs(d) / sqrt((l * h))) * fma(((-0.5 * ((D_m * M_m) / d)) / l), ((D_m * h) * (0.25 * (M_m / d))), 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(Float64(Float64(0.5 / d) * M_m) * D_m)
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0)))) <= 1e+215)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * fma(Float64(Float64(Float64(t_0 / l) * h) * t_0), -0.5, 1.0)) / sqrt(Float64(h / d)));
	else
		tmp = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * fma(Float64(Float64(-0.5 * Float64(Float64(D_m * M_m) / d)) / l), Float64(Float64(D_m * h) * Float64(0.25 * Float64(M_m / d))), 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[(N[(N[(0.5 / d), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+215], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[(t$95$0 / l), $MachinePrecision] * h), $MachinePrecision] * t$95$0), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.5 * N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(D$95$m * h), $MachinePrecision] * N[(0.25 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.99999999999999907e214

   1. Initial program 84.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lift-/.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval 83.7

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

   4. Applied rewrites 83.7%

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

   6. Applied rewrites 83.7%

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

   8. Applied rewrites 86.9%

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

   if 9.99999999999999907e214 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 26.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 27.1

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

   4. Applied rewrites 27.1%

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

   6. Applied rewrites 31.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. pow1/2 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. lift-sqrt.f64 N/A

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

      14. sqrt-div N/A

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

      15. lift-/.f64 N/A

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

      16. clear-num N/A

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

      17. lift-/.f64 N/A

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

      18. pow1/2 N/A

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

      19. sqrt-unprod N/A

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

      20. lower-sqrt.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-sqrt.f64 13.7

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

   8. Applied rewrites 13.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 13.7

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

   10. Applied rewrites 74.3%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+215}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\left(\frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell} \cdot h\right) \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right), -0.5, 1\right)}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{-0.5 \cdot \frac{D \cdot M}{d}}{\ell}, \left(D \cdot h\right) \cdot \left(0.25 \cdot \frac{M}{d}\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.8% accurate, 0.8× speedup

Math

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

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ M_m d) D_m)))
   (if (<=
        (*
         (-
          1.0
          (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
         (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
        1e+215)
     (/
      (* (fma (* (* t_0 (* 0.25 (/ h l))) t_0) -0.5 1.0) (sqrt (/ d l)))
      (sqrt (/ h d)))
     (*
      (/ (fabs d) (sqrt (* l h)))
      (fma
       (/ (* -0.5 (/ (* D_m M_m) d)) l)
       (* (* D_m h) (* 0.25 (/ M_m d)))
       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 = (M_m / d) * D_m;
	double tmp;
	if (((1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)))) <= 1e+215) {
		tmp = (fma(((t_0 * (0.25 * (h / l))) * t_0), -0.5, 1.0) * sqrt((d / l))) / sqrt((h / d));
	} else {
		tmp = (fabs(d) / sqrt((l * h))) * fma(((-0.5 * ((D_m * M_m) / d)) / l), ((D_m * h) * (0.25 * (M_m / d))), 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(Float64(M_m / d) * D_m)
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0)))) <= 1e+215)
		tmp = Float64(Float64(fma(Float64(Float64(t_0 * Float64(0.25 * Float64(h / l))) * t_0), -0.5, 1.0) * sqrt(Float64(d / l))) / sqrt(Float64(h / d)));
	else
		tmp = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * fma(Float64(Float64(-0.5 * Float64(Float64(D_m * M_m) / d)) / l), Float64(Float64(D_m * h) * Float64(0.25 * Float64(M_m / d))), 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[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+215], N[(N[(N[(N[(N[(t$95$0 * N[(0.25 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.5 * N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(D$95$m * h), $MachinePrecision] * N[(0.25 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.99999999999999907e214

   1. Initial program 84.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lift-/.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval 83.7

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

   4. Applied rewrites 83.7%

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

   6. Applied rewrites 83.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 86.2

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 86.2

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 86.2

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

   8. Applied rewrites 86.2%

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

   if 9.99999999999999907e214 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 26.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 27.1

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

   4. Applied rewrites 27.1%

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

   6. Applied rewrites 31.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. pow1/2 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. lift-sqrt.f64 N/A

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

      14. sqrt-div N/A

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

      15. lift-/.f64 N/A

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

      16. clear-num N/A

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

      17. lift-/.f64 N/A

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

      18. pow1/2 N/A

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

      19. sqrt-unprod N/A

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

      20. lower-sqrt.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-sqrt.f64 13.7

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

   8. Applied rewrites 13.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 13.7

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

   10. Applied rewrites 74.3%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+215}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(0.25 \cdot \frac{h}{\ell}\right)\right) \cdot \left(\frac{M}{d} \cdot D\right), -0.5, 1\right) \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{-0.5 \cdot \frac{D \cdot M}{d}}{\ell}, \left(D \cdot h\right) \cdot \left(0.25 \cdot \frac{M}{d}\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{t\_0}{\sqrt{-\ell}} \cdot \frac{t\_0}{\sqrt{-h}}\right) \cdot 1\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(D\_m \cdot D\_m\right) \cdot -0.125}{\ell}, \frac{\frac{M\_m}{d} \cdot M\_m}{d} \cdot h, 1\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (sqrt (- d))))
   (if (<= d -5e-310)
     (* (* (/ t_0 (sqrt (- l))) (/ t_0 (sqrt (- h)))) 1.0)
     (if (<= d 2.2e+138)
       (*
        (fma (/ (* (* D_m D_m) -0.125) l) (* (/ (* (/ M_m d) M_m) d) h) 1.0)
        (* (sqrt (/ 1.0 (* l h))) 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 t_0 = sqrt(-d);
	double tmp;
	if (d <= -5e-310) {
		tmp = ((t_0 / sqrt(-l)) * (t_0 / sqrt(-h))) * 1.0;
	} else if (d <= 2.2e+138) {
		tmp = fma((((D_m * D_m) * -0.125) / l), ((((M_m / d) * M_m) / d) * h), 1.0) * (sqrt((1.0 / (l * h))) * d);
	} 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 = sqrt(Float64(-d))
	tmp = 0.0
	if (d <= -5e-310)
		tmp = Float64(Float64(Float64(t_0 / sqrt(Float64(-l))) * Float64(t_0 / sqrt(Float64(-h)))) * 1.0);
	elseif (d <= 2.2e+138)
		tmp = Float64(fma(Float64(Float64(Float64(D_m * D_m) * -0.125) / l), Float64(Float64(Float64(Float64(M_m / d) * M_m) / d) * h), 1.0) * Float64(sqrt(Float64(1.0 / Float64(l * h))) * d));
	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[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[d, -5e-310], N[(N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[d, 2.2e+138], N[(N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] / d), $MachinePrecision] * h), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.999999999999985e-310

   1. Initial program 71.4%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 42.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 42.0

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 49.7

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

   7. Applied rewrites 49.7%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 49.7

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-neg.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 54.1

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

   9. Applied rewrites 54.1%

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

   if -4.999999999999985e-310 < d < 2.2000000000000001e138

   1. Initial program 55.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 55.7

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

   4. Applied rewrites 55.7%

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

   5. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 64.6

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

   7. Applied rewrites 64.6%

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

   8. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

   10. Applied rewrites 56.1%

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

   if 2.2000000000000001e138 < d

   1. Initial program 74.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lift-/.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval 75.3

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

   4. Applied rewrites 75.3%

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

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. 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 59.8

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

   7. Applied rewrites 59.8%

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

   9. Applied rewrites 59.8%

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

   11. Applied rewrites 75.3%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right) \cdot 1\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(D \cdot D\right) \cdot -0.125}{\ell}, \frac{\frac{M}{d} \cdot M}{d} \cdot h, 1\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{t\_0}{\sqrt{-\ell}} \cdot \frac{t\_0}{\sqrt{-h}}\right) \cdot 1\\ \mathbf{elif}\;d \leq 8.2:\\ \;\;\;\;\left(\left(\frac{\frac{M\_m}{d} \cdot M\_m}{d} \cdot h\right) \cdot \frac{\left(D\_m \cdot D\_m\right) \cdot -0.125}{\ell}\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (sqrt (- d))))
   (if (<= d -5e-310)
     (* (* (/ t_0 (sqrt (- l))) (/ t_0 (sqrt (- h)))) 1.0)
     (if (<= d 8.2)
       (*
        (* (* (/ (* (/ M_m d) M_m) d) h) (/ (* (* D_m D_m) -0.125) l))
        (* (sqrt (/ 1.0 (* l h))) 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 t_0 = sqrt(-d);
	double tmp;
	if (d <= -5e-310) {
		tmp = ((t_0 / sqrt(-l)) * (t_0 / sqrt(-h))) * 1.0;
	} else if (d <= 8.2) {
		tmp = (((((M_m / d) * M_m) / d) * h) * (((D_m * D_m) * -0.125) / l)) * (sqrt((1.0 / (l * h))) * 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt(-d)
    if (d <= (-5d-310)) then
        tmp = ((t_0 / sqrt(-l)) * (t_0 / sqrt(-h))) * 1.0d0
    else if (d <= 8.2d0) then
        tmp = (((((m_m / d) * m_m) / d) * h) * (((d_m * d_m) * (-0.125d0)) / l)) * (sqrt((1.0d0 / (l * h))) * 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 t_0 = Math.sqrt(-d);
	double tmp;
	if (d <= -5e-310) {
		tmp = ((t_0 / Math.sqrt(-l)) * (t_0 / Math.sqrt(-h))) * 1.0;
	} else if (d <= 8.2) {
		tmp = (((((M_m / d) * M_m) / d) * h) * (((D_m * D_m) * -0.125) / l)) * (Math.sqrt((1.0 / (l * h))) * 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):
	t_0 = math.sqrt(-d)
	tmp = 0
	if d <= -5e-310:
		tmp = ((t_0 / math.sqrt(-l)) * (t_0 / math.sqrt(-h))) * 1.0
	elif d <= 8.2:
		tmp = (((((M_m / d) * M_m) / d) * h) * (((D_m * D_m) * -0.125) / l)) * (math.sqrt((1.0 / (l * h))) * 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)
	t_0 = sqrt(Float64(-d))
	tmp = 0.0
	if (d <= -5e-310)
		tmp = Float64(Float64(Float64(t_0 / sqrt(Float64(-l))) * Float64(t_0 / sqrt(Float64(-h)))) * 1.0);
	elseif (d <= 8.2)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M_m / d) * M_m) / d) * h) * Float64(Float64(Float64(D_m * D_m) * -0.125) / l)) * Float64(sqrt(Float64(1.0 / Float64(l * h))) * 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)
	t_0 = sqrt(-d);
	tmp = 0.0;
	if (d <= -5e-310)
		tmp = ((t_0 / sqrt(-l)) * (t_0 / sqrt(-h))) * 1.0;
	elseif (d <= 8.2)
		tmp = (((((M_m / d) * M_m) / d) * h) * (((D_m * D_m) * -0.125) / l)) * (sqrt((1.0 / (l * h))) * 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_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[d, -5e-310], N[(N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[d, 8.2], N[(N[(N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] / d), $MachinePrecision] * h), $MachinePrecision] * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.999999999999985e-310

   1. Initial program 71.4%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 42.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 42.0

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 49.7

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

   7. Applied rewrites 49.7%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 49.7

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-neg.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 54.1

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

   9. Applied rewrites 54.1%

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

   if -4.999999999999985e-310 < d < 8.1999999999999993

   1. Initial program 43.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 43.1

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

   4. Applied rewrites 43.1%

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

   5. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 59.4

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

   7. Applied rewrites 59.4%

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

   8. Taylor expanded in h around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 37.7

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

   10. Applied rewrites 37.7%

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

   if 8.1999999999999993 < d

   1. Initial program 81.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lift-/.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval 82.2

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

   4. Applied rewrites 82.2%

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

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. 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 53.1

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

   7. Applied rewrites 53.1%

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

   9. Applied rewrites 53.1%

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

   11. Applied rewrites 66.4%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right) \cdot 1\\ \mathbf{elif}\;d \leq 8.2:\\ \;\;\;\;\left(\left(\frac{\frac{M}{d} \cdot M}{d} \cdot h\right) \cdot \frac{\left(D \cdot D\right) \cdot -0.125}{\ell}\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.4% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(-d)
    if (d <= (-5d-310)) then
        tmp = ((t_0 / sqrt(-l)) * (t_0 / sqrt(-h))) * 1.0d0
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt(-d);
	double tmp;
	if (d <= -5e-310) {
		tmp = ((t_0 / Math.sqrt(-l)) * (t_0 / Math.sqrt(-h))) * 1.0;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt(-d)
	tmp = 0
	if d <= -5e-310:
		tmp = ((t_0 / math.sqrt(-l)) * (t_0 / math.sqrt(-h))) * 1.0
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

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

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt(-d);
	tmp = 0.0;
	if (d <= -5e-310)
		tmp = ((t_0 / sqrt(-l)) * (t_0 / sqrt(-h))) * 1.0;
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if d < -4.999999999999985e-310

   1. Initial program 71.4%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 42.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 42.0

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 49.7

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

   7. Applied rewrites 49.7%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 49.7

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-neg.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 54.1

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

   9. Applied rewrites 54.1%

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

   if -4.999999999999985e-310 < d

   1. Initial program 61.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lift-/.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval 61.8

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

   4. Applied rewrites 61.8%

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

   5. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 37.2

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   7. Applied rewrites 37.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   8. Step-by-step derivation

   9. Applied rewrites 37.2%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 45.2%

       \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -3.3 \cdot 10^{-284}:\\ \;\;\;\;\frac{\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;h \leq 9 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 (<= h -3.3e-284)
   (/ (* (* 1.0 (sqrt (/ d l))) (sqrt (- d))) (sqrt (- h)))
   (if (<= h 9e-276)
     (* (sqrt (/ 1.0 (* l h))) (- 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 (h <= -3.3e-284) {
		tmp = ((1.0 * sqrt((d / l))) * sqrt(-d)) / sqrt(-h);
	} else if (h <= 9e-276) {
		tmp = sqrt((1.0 / (l * h))) * -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 (h <= (-3.3d-284)) then
        tmp = ((1.0d0 * sqrt((d / l))) * sqrt(-d)) / sqrt(-h)
    else if (h <= 9d-276) then
        tmp = sqrt((1.0d0 / (l * h))) * -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 (h <= -3.3e-284) {
		tmp = ((1.0 * Math.sqrt((d / l))) * Math.sqrt(-d)) / Math.sqrt(-h);
	} else if (h <= 9e-276) {
		tmp = Math.sqrt((1.0 / (l * h))) * -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 h <= -3.3e-284:
		tmp = ((1.0 * math.sqrt((d / l))) * math.sqrt(-d)) / math.sqrt(-h)
	elif h <= 9e-276:
		tmp = math.sqrt((1.0 / (l * h))) * -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 (h <= -3.3e-284)
		tmp = Float64(Float64(Float64(1.0 * sqrt(Float64(d / l))) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (h <= 9e-276)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * 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 (h <= -3.3e-284)
		tmp = ((1.0 * sqrt((d / l))) * sqrt(-d)) / sqrt(-h);
	elseif (h <= 9e-276)
		tmp = sqrt((1.0 / (l * h))) * -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[h, -3.3e-284], N[(N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 9e-276], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -3.30000000000000008e-284

   1. Initial program 71.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 42.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      2. metadata-eval 42.2

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      6. frac-2neg N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      7. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      10. lower-neg.f64 N/A

          \[\leadsto \left(\frac{\sqrt{\color{blue}{-d}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\sqrt{-d}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      12. lower-neg.f64 49.6

          \[\leadsto \left(\frac{\sqrt{-d}}{\sqrt{\color{blue}{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

   7. Applied rewrites 49.6%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot 1 \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot 1\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot 1\right) \]

   9. Applied rewrites 49.6%

      \[\leadsto \color{blue}{\frac{\sqrt{-d} \cdot \left(1 \cdot \sqrt{\frac{d}{\ell}}\right)}{\sqrt{-h}}} \]

   if -3.30000000000000008e-284 < h < 8.99999999999999925e-276

   1. Initial program 55.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. clear-num N/A

         \[\leadsto \left({\color{blue}{\left(\frac{1}{\frac{h}{d}}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. inv-pow N/A

         \[\leadsto \left({\color{blue}{\left({\left(\frac{h}{d}\right)}^{-1}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. pow-pow N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{h}{d}\right)}^{\left(-1 \cdot \frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(-1 \cdot \color{blue}{\frac{1}{2}}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(-1 \cdot \color{blue}{\frac{1}{2}}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{h}{d}\right)}^{\color{blue}{\frac{-1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left({\left(\frac{h}{d}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. metadata-eval N/A

          \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-pow.f64 N/A

          \[\leadsto \left(\color{blue}{{\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \left({\color{blue}{\left(\frac{h}{d}\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. metadata-eval 58.2

          \[\leadsto \left({\left(\frac{h}{d}\right)}^{\color{blue}{-0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 58.2%

      \[\leadsto \left(\color{blue}{{\left(\frac{h}{d}\right)}^{-0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 62.9

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Applied rewrites 62.9%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if 8.99999999999999925e-276 < h

   1. Initial program 62.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. clear-num N/A

         \[\leadsto \left({\color{blue}{\left(\frac{1}{\frac{h}{d}}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. inv-pow N/A

         \[\leadsto \left({\color{blue}{\left({\left(\frac{h}{d}\right)}^{-1}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. pow-pow N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{h}{d}\right)}^{\left(-1 \cdot \frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(-1 \cdot \color{blue}{\frac{1}{2}}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(-1 \cdot \color{blue}{\frac{1}{2}}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{h}{d}\right)}^{\color{blue}{\frac{-1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left({\left(\frac{h}{d}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. metadata-eval N/A

          \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-pow.f64 N/A

          \[\leadsto \left(\color{blue}{{\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \left({\color{blue}{\left(\frac{h}{d}\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. metadata-eval 62.7

          \[\leadsto \left({\left(\frac{h}{d}\right)}^{\color{blue}{-0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 62.7%

      \[\leadsto \left(\color{blue}{{\left(\frac{h}{d}\right)}^{-0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 38.8

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   7. Applied rewrites 38.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   8. Step-by-step derivation

   9. Applied rewrites 38.8%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 47.1%

       \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -3.3 \cdot 10^{-284}:\\ \;\;\;\;\frac{\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;h \leq 9 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 45.0% accurate, 9.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq 9 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 (<= h 9e-276)
   (* (sqrt (/ 1.0 (* l h))) (- 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 (h <= 9e-276) {
		tmp = sqrt((1.0 / (l * h))) * -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 (h <= 9d-276) then
        tmp = sqrt((1.0d0 / (l * h))) * -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 (h <= 9e-276) {
		tmp = Math.sqrt((1.0 / (l * h))) * -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 h <= 9e-276:
		tmp = math.sqrt((1.0 / (l * h))) * -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 (h <= 9e-276)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * 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 (h <= 9e-276)
		tmp = sqrt((1.0 / (l * h))) * -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[h, 9e-276], N[(N[Sqrt[N[(1.0 / N[(l * h), $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 h < 8.99999999999999925e-276

   1. Initial program 70.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. clear-num N/A

         \[\leadsto \left({\color{blue}{\left(\frac{1}{\frac{h}{d}}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. inv-pow N/A

         \[\leadsto \left({\color{blue}{\left({\left(\frac{h}{d}\right)}^{-1}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. pow-pow N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{h}{d}\right)}^{\left(-1 \cdot \frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(-1 \cdot \color{blue}{\frac{1}{2}}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(-1 \cdot \color{blue}{\frac{1}{2}}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{h}{d}\right)}^{\color{blue}{\frac{-1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left({\left(\frac{h}{d}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. metadata-eval N/A

          \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-pow.f64 N/A

          \[\leadsto \left(\color{blue}{{\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \left({\color{blue}{\left(\frac{h}{d}\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. metadata-eval 69.9

          \[\leadsto \left({\left(\frac{h}{d}\right)}^{\color{blue}{-0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 69.9%

      \[\leadsto \left(\color{blue}{{\left(\frac{h}{d}\right)}^{-0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. 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 47.8

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Applied rewrites 47.8%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if 8.99999999999999925e-276 < h

   1. Initial program 62.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. clear-num N/A

         \[\leadsto \left({\color{blue}{\left(\frac{1}{\frac{h}{d}}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. inv-pow N/A

         \[\leadsto \left({\color{blue}{\left({\left(\frac{h}{d}\right)}^{-1}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. pow-pow N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{h}{d}\right)}^{\left(-1 \cdot \frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(-1 \cdot \color{blue}{\frac{1}{2}}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(-1 \cdot \color{blue}{\frac{1}{2}}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{h}{d}\right)}^{\color{blue}{\frac{-1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left({\left(\frac{h}{d}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. metadata-eval N/A

          \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-pow.f64 N/A

          \[\leadsto \left(\color{blue}{{\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \left({\color{blue}{\left(\frac{h}{d}\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. metadata-eval 62.7

          \[\leadsto \left({\left(\frac{h}{d}\right)}^{\color{blue}{-0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 62.7%

      \[\leadsto \left(\color{blue}{{\left(\frac{h}{d}\right)}^{-0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 38.8

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   7. Applied rewrites 38.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   8. Step-by-step derivation

   9. Applied rewrites 38.8%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 47.1%

       \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 9 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 41.6% accurate, 10.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{if}\;h \leq 9.5 \cdot 10^{-276}:\\ \;\;\;\;t\_0 \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot d\\ \end{array} \end{array} \]

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 (* l h)))))
   (if (<= h 9.5e-276) (* 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 / (l * h)));
	double tmp;
	if (h <= 9.5e-276) {
		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 / (l * h)))
    if (h <= 9.5d-276) 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 / (l * h)));
	double tmp;
	if (h <= 9.5e-276) {
		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 / (l * h)))
	tmp = 0
	if h <= 9.5e-276:
		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(l * h)))
	tmp = 0.0
	if (h <= 9.5e-276)
		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 / (l * h)));
	tmp = 0.0;
	if (h <= 9.5e-276)
		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[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, 9.5e-276], N[(t$95$0 * (-d)), $MachinePrecision], N[(t$95$0 * d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < 9.49999999999999929e-276

   1. Initial program 70.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. clear-num N/A

         \[\leadsto \left({\color{blue}{\left(\frac{1}{\frac{h}{d}}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. inv-pow N/A

         \[\leadsto \left({\color{blue}{\left({\left(\frac{h}{d}\right)}^{-1}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. pow-pow N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{h}{d}\right)}^{\left(-1 \cdot \frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(-1 \cdot \color{blue}{\frac{1}{2}}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(-1 \cdot \color{blue}{\frac{1}{2}}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{h}{d}\right)}^{\color{blue}{\frac{-1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left({\left(\frac{h}{d}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. metadata-eval N/A

          \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-pow.f64 N/A

          \[\leadsto \left(\color{blue}{{\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \left({\color{blue}{\left(\frac{h}{d}\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. metadata-eval 69.9

          \[\leadsto \left({\left(\frac{h}{d}\right)}^{\color{blue}{-0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 69.9%

      \[\leadsto \left(\color{blue}{{\left(\frac{h}{d}\right)}^{-0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. 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 47.8

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Applied rewrites 47.8%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if 9.49999999999999929e-276 < h

   1. Initial program 62.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. clear-num N/A

         \[\leadsto \left({\color{blue}{\left(\frac{1}{\frac{h}{d}}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. inv-pow N/A

         \[\leadsto \left({\color{blue}{\left({\left(\frac{h}{d}\right)}^{-1}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. pow-pow N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{h}{d}\right)}^{\left(-1 \cdot \frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(-1 \cdot \color{blue}{\frac{1}{2}}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(-1 \cdot \color{blue}{\frac{1}{2}}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{h}{d}\right)}^{\color{blue}{\frac{-1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left({\left(\frac{h}{d}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. metadata-eval N/A

          \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-pow.f64 N/A

          \[\leadsto \left(\color{blue}{{\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \left({\color{blue}{\left(\frac{h}{d}\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. metadata-eval 62.7

          \[\leadsto \left({\left(\frac{h}{d}\right)}^{\color{blue}{-0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 62.7%

      \[\leadsto \left(\color{blue}{{\left(\frac{h}{d}\right)}^{-0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 38.8

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   7. Applied rewrites 38.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 9.5 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 26.7% accurate, 12.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \sqrt{\frac{1}{\ell \cdot h}} \cdot d \end{array} \]

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 (* l h))) d))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return sqrt((1.0 / (l * h))) * 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 / (l * h))) * 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 / (l * h))) * 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 / (l * h))) * 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(l * h))) * 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 / (l * h))) * 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[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]

Derivation

1. Initial program 66.8%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. clear-num N/A

      \[\leadsto \left({\color{blue}{\left(\frac{1}{\frac{h}{d}}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. inv-pow N/A

      \[\leadsto \left({\color{blue}{\left({\left(\frac{h}{d}\right)}^{-1}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. pow-pow N/A

      \[\leadsto \left(\color{blue}{{\left(\frac{h}{d}\right)}^{\left(-1 \cdot \frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. lift-/.f64 N/A

      \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(-1 \cdot \color{blue}{\frac{1}{2}}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   7. metadata-eval N/A

      \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(-1 \cdot \color{blue}{\frac{1}{2}}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   8. metadata-eval N/A

      \[\leadsto \left({\left(\frac{h}{d}\right)}^{\color{blue}{\frac{-1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   9. metadata-eval N/A

      \[\leadsto \left({\left(\frac{h}{d}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   10. metadata-eval N/A

       \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   11. lift-/.f64 N/A

       \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   12. lower-pow.f64 N/A

       \[\leadsto \left(\color{blue}{{\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   13. lower-/.f64 N/A

       \[\leadsto \left({\color{blue}{\left(\frac{h}{d}\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   14. lift-/.f64 N/A

       \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   15. metadata-eval N/A

       \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   16. metadata-eval 66.7

       \[\leadsto \left({\left(\frac{h}{d}\right)}^{\color{blue}{-0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

4. Applied rewrites 66.7%

   \[\leadsto \left(\color{blue}{{\left(\frac{h}{d}\right)}^{-0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

5. Taylor expanded in h around 0

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   6. lower-*.f64 22.1

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

7. Applied rewrites 22.1%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
8. Add Preprocessing

Alternative 16: 26.6% accurate, 15.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (/ d (sqrt (* l h))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return d / sqrt((l * h));
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = d / sqrt((l * h))
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return d / Math.sqrt((l * h));
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return d / math.sqrt((l * h))

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = d / sqrt((l * h));
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.8%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. clear-num N/A

      \[\leadsto \left({\color{blue}{\left(\frac{1}{\frac{h}{d}}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. inv-pow N/A

      \[\leadsto \left({\color{blue}{\left({\left(\frac{h}{d}\right)}^{-1}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. pow-pow N/A

      \[\leadsto \left(\color{blue}{{\left(\frac{h}{d}\right)}^{\left(-1 \cdot \frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. lift-/.f64 N/A

      \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(-1 \cdot \color{blue}{\frac{1}{2}}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   7. metadata-eval N/A

      \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(-1 \cdot \color{blue}{\frac{1}{2}}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   8. metadata-eval N/A

      \[\leadsto \left({\left(\frac{h}{d}\right)}^{\color{blue}{\frac{-1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   9. metadata-eval N/A

      \[\leadsto \left({\left(\frac{h}{d}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   10. metadata-eval N/A

       \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   11. lift-/.f64 N/A

       \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   12. lower-pow.f64 N/A

       \[\leadsto \left(\color{blue}{{\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   13. lower-/.f64 N/A

       \[\leadsto \left({\color{blue}{\left(\frac{h}{d}\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   14. lift-/.f64 N/A

       \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   15. metadata-eval N/A

       \[\leadsto \left({\left(\frac{h}{d}\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   16. metadata-eval 66.7

       \[\leadsto \left({\left(\frac{h}{d}\right)}^{\color{blue}{-0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

4. Applied rewrites 66.7%

   \[\leadsto \left(\color{blue}{{\left(\frac{h}{d}\right)}^{-0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

5. Taylor expanded in h around 0

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   6. lower-*.f64 22.1

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

7. Applied rewrites 22.1%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
8. Step-by-step derivation

9. Applied rewrites 21.7%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024242 
(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)))))