Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.6% → 80.5%

Time: 10.7s

Alternatives: 12

Speedup: 3.7×

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: 66.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{+36}:\\ \;\;\;\;\left(1 - \left(\left(\left(0.25 \cdot D\_m\right) \cdot \frac{M\_m}{d}\right) \cdot h\right) \cdot \frac{D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)}{\ell}\right) \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left(\left(\sqrt{\frac{-1}{\ell}} \cdot \sqrt{-d}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(D\_m, \sqrt{h} \cdot \left(\left(\frac{\frac{M\_m}{d} \cdot M\_m}{\ell} \cdot -0.125\right) \cdot D\_m\right), \frac{d}{\sqrt{h}}\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{h} \cdot -0.125, \frac{\frac{D\_m}{d}}{d} \cdot \frac{\left(M\_m \cdot M\_m\right) \cdot D\_m}{\ell}, \sqrt{\frac{1}{h}}\right) \cdot d}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -1e+36)
   (*
    (-
     1.0
     (* (* (* (* 0.25 D_m) (/ M_m d)) h) (/ (* D_m (* M_m (/ 0.5 d))) l)))
    (* (sqrt (/ 1.0 (* h l))) (- d)))
   (if (<= d -5e-310)
     (*
      (- 1.0 (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
      (* (* (sqrt (/ -1.0 l)) (sqrt (- d))) (pow (/ d h) (/ 1.0 2.0))))
     (if (<= d 3.8e-20)
       (/
        (fma
         D_m
         (* (sqrt h) (* (* (/ (* (/ M_m d) M_m) l) -0.125) D_m))
         (/ d (sqrt h)))
        (sqrt l))
       (/
        (*
         (fma
          (* (sqrt h) -0.125)
          (* (/ (/ D_m d) d) (/ (* (* M_m M_m) D_m) l))
          (sqrt (/ 1.0 h)))
         d)
        (sqrt l))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1e+36) {
		tmp = (1.0 - ((((0.25 * D_m) * (M_m / d)) * h) * ((D_m * (M_m * (0.5 / d))) / l))) * (sqrt((1.0 / (h * l))) * -d);
	} else if (d <= -5e-310) {
		tmp = (1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * ((sqrt((-1.0 / l)) * sqrt(-d)) * pow((d / h), (1.0 / 2.0)));
	} else if (d <= 3.8e-20) {
		tmp = fma(D_m, (sqrt(h) * (((((M_m / d) * M_m) / l) * -0.125) * D_m)), (d / sqrt(h))) / sqrt(l);
	} else {
		tmp = (fma((sqrt(h) * -0.125), (((D_m / d) / d) * (((M_m * M_m) * D_m) / l)), sqrt((1.0 / h))) * d) / sqrt(l);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -1e+36)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(0.25 * D_m) * Float64(M_m / d)) * h) * Float64(Float64(D_m * Float64(M_m * Float64(0.5 / d))) / l))) * Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(-d)));
	elseif (d <= -5e-310)
		tmp = 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(sqrt(Float64(-1.0 / l)) * sqrt(Float64(-d))) * (Float64(d / h) ^ Float64(1.0 / 2.0))));
	elseif (d <= 3.8e-20)
		tmp = Float64(fma(D_m, Float64(sqrt(h) * Float64(Float64(Float64(Float64(Float64(M_m / d) * M_m) / l) * -0.125) * D_m)), Float64(d / sqrt(h))) / sqrt(l));
	else
		tmp = Float64(Float64(fma(Float64(sqrt(h) * -0.125), Float64(Float64(Float64(D_m / d) / d) * Float64(Float64(Float64(M_m * M_m) * D_m) / l)), sqrt(Float64(1.0 / h))) * d) / sqrt(l));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -1e+36], N[(N[(1.0 - N[(N[(N[(N[(0.25 * D$95$m), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] * N[(N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], 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[(N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.8e-20], N[(N[(D$95$m * N[(N[Sqrt[h], $MachinePrecision] * N[(N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] / l), $MachinePrecision] * -0.125), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] + N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Sqrt[h], $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(N[(D$95$m / d), $MachinePrecision] / d), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.00000000000000004e36

   1. Initial program 70.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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 75.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-undiv N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. clear-num N/A

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

      11. un-div-inv N/A

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

      12. clear-num N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. sqrt-undiv N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-/.f64 76.4

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

      20. lift-/.f64 N/A

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

      21. metadata-eval 76.4

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

      22. lift-pow.f64 N/A

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

      23. unpow1/2 N/A

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

      24. lower-sqrt.f64 76.4

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

   6. Applied rewrites 76.4%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-1 N/A

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

      5. remove-double-div N/A

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

      6. lower-*.f64 76.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 76.4

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 76.4

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

   8. Applied rewrites 76.4%

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

   9. Taylor expanded in d around -inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 96.5

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

   11. Applied rewrites 96.5%

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

   if -1.00000000000000004e36 < d < -4.999999999999985e-310

   1. Initial program 74.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 83.5

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

   4. Applied rewrites 83.5%

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

   if -4.999999999999985e-310 < d < 3.7999999999999998e-20

   1. Initial program 61.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 64.8

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

   4. Applied rewrites 64.8%

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

   6. Applied rewrites 54.1%

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

   7. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   9. Applied rewrites 64.0%

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

   11. Applied rewrites 78.2%

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

   if 3.7999999999999998e-20 < d

   1. Initial program 76.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 77.7

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

   4. Applied rewrites 77.7%

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

   6. Applied rewrites 73.6%

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

   7. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 77.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.125 \cdot \sqrt{h}, \frac{\left(M \cdot M\right) \cdot D}{\ell} \cdot \frac{\frac{D}{d}}{d}, \sqrt{\frac{1}{h}}\right) \cdot d}}{\sqrt{\ell}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{+36}:\\ \;\;\;\;\left(1 - \left(\left(\left(0.25 \cdot D\right) \cdot \frac{M}{d}\right) \cdot h\right) \cdot \frac{D \cdot \left(M \cdot \frac{0.5}{d}\right)}{\ell}\right) \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left(\left(\sqrt{\frac{-1}{\ell}} \cdot \sqrt{-d}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(D, \sqrt{h} \cdot \left(\left(\frac{\frac{M}{d} \cdot M}{\ell} \cdot -0.125\right) \cdot D\right), \frac{d}{\sqrt{h}}\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{h} \cdot -0.125, \frac{\frac{D}{d}}{d} \cdot \frac{\left(M \cdot M\right) \cdot D}{\ell}, \sqrt{\frac{1}{h}}\right) \cdot d}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right)\\ t_1 := \frac{\left(\frac{\sqrt{h} \cdot M\_m}{d} \cdot \frac{M\_m}{\ell}\right) \cdot \left(\left(D\_m \cdot D\_m\right) \cdot -0.125\right)}{\sqrt{\ell}}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-66}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+238}:\\ \;\;\;\;1 \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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
         (*
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))
          (-
           1.0
           (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0))))))
        (t_1
         (/
          (* (* (/ (* (sqrt h) M_m) d) (/ M_m l)) (* (* D_m D_m) -0.125))
          (sqrt l))))
   (if (<= t_0 -4e-66)
     t_1
     (if (<= t_0 1e+238)
       (* 1.0 (* (sqrt (/ d h)) (sqrt (/ d l))))
       (if (<= t_0 INFINITY) (* (/ (sqrt (* d d)) (sqrt (* h l))) 1.0) t_1)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0))) * (1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0))));
	double t_1 = ((((sqrt(h) * M_m) / d) * (M_m / l)) * ((D_m * D_m) * -0.125)) / sqrt(l);
	double tmp;
	if (t_0 <= -4e-66) {
		tmp = t_1;
	} else if (t_0 <= 1e+238) {
		tmp = 1.0 * (sqrt((d / h)) * sqrt((d / l)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (sqrt((d * d)) / sqrt((h * l))) * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

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.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0))) * (1.0 - ((h / l) * (Math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0))));
	double t_1 = ((((Math.sqrt(h) * M_m) / d) * (M_m / l)) * ((D_m * D_m) * -0.125)) / Math.sqrt(l);
	double tmp;
	if (t_0 <= -4e-66) {
		tmp = t_1;
	} else if (t_0 <= 1e+238) {
		tmp = 1.0 * (Math.sqrt((d / h)) * Math.sqrt((d / l)));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (Math.sqrt((d * d)) / Math.sqrt((h * l))) * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0))) * (1.0 - ((h / l) * (math.pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0))))
	t_1 = ((((math.sqrt(h) * M_m) / d) * (M_m / l)) * ((D_m * D_m) * -0.125)) / math.sqrt(l)
	tmp = 0
	if t_0 <= -4e-66:
		tmp = t_1
	elif t_0 <= 1e+238:
		tmp = 1.0 * (math.sqrt((d / h)) * math.sqrt((d / l)))
	elif t_0 <= math.inf:
		tmp = (math.sqrt((d * d)) / math.sqrt((h * l))) * 1.0
	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((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))) * 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 = Float64(Float64(Float64(Float64(Float64(sqrt(h) * M_m) / d) * Float64(M_m / l)) * Float64(Float64(D_m * D_m) * -0.125)) / sqrt(l))
	tmp = 0.0
	if (t_0 <= -4e-66)
		tmp = t_1;
	elseif (t_0 <= 1e+238)
		tmp = Float64(1.0 * Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(d * d)) / sqrt(Float64(h * l))) * 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0))) * (1.0 - ((h / l) * ((((D_m * M_m) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0))));
	t_1 = ((((sqrt(h) * M_m) / d) * (M_m / l)) * ((D_m * D_m) * -0.125)) / sqrt(l);
	tmp = 0.0;
	if (t_0 <= -4e-66)
		tmp = t_1;
	elseif (t_0 <= 1e+238)
		tmp = 1.0 * (sqrt((d / h)) * sqrt((d / l)));
	elseif (t_0 <= Inf)
		tmp = (sqrt((d * d)) / sqrt((h * l))) * 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(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] * 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]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[Sqrt[h], $MachinePrecision] * M$95$m), $MachinePrecision] / d), $MachinePrecision] * N[(M$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-66], t$95$1, If[LessEqual[t$95$0, 1e+238], N[(1.0 * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[Sqrt[N[(d * d), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], t$95$1]]]]]

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)))) < -3.9999999999999999e-66 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 64.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 34.5

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

   4. Applied rewrites 34.5%

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

   6. Applied rewrites 29.5%

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

   7. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   9. Applied rewrites 34.2%

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

   10. Taylor expanded in h around -inf

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

   12. Applied rewrites 41.6%

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

   if -3.9999999999999999e-66 < (*.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)))) < 1e238

   1. Initial program 90.4%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 89.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-undiv N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 43.9

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

      13. lift-/.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. sqrt-undiv N/A

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

      17. lift-/.f64 N/A

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

      18. lower-sqrt.f64 89.3

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

      19. lift-/.f64 N/A

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

      20. metadata-eval 89.3

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

      21. lift-pow.f64 N/A

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

      22. unpow1/2 N/A

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

      23. lower-sqrt.f64 89.3

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

   7. Applied rewrites 89.3%

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

   if 1e238 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 49.8%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 49.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. unpow1/2 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-times N/A

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

      13. sqrt-div N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-*.f64 63.9

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

   7. Applied rewrites 63.9%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \leq -4 \cdot 10^{-66}:\\ \;\;\;\;\frac{\left(\frac{\sqrt{h} \cdot M}{d} \cdot \frac{M}{\ell}\right) \cdot \left(\left(D \cdot D\right) \cdot -0.125\right)}{\sqrt{\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \leq 10^{+238}:\\ \;\;\;\;1 \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \leq \infty:\\ \;\;\;\;\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\sqrt{h} \cdot M}{d} \cdot \frac{M}{\ell}\right) \cdot \left(\left(D \cdot D\right) \cdot -0.125\right)}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{+36}:\\ \;\;\;\;\left(1 - \left(\left(\left(0.25 \cdot D\_m\right) \cdot \frac{M\_m}{d}\right) \cdot h\right) \cdot \frac{D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)}{\ell}\right) \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(D\_m, \sqrt{h} \cdot \left(\left(\frac{\frac{M\_m}{d} \cdot M\_m}{\ell} \cdot -0.125\right) \cdot D\_m\right), \frac{d}{\sqrt{h}}\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{h} \cdot -0.125, \frac{\frac{D\_m}{d}}{d} \cdot \frac{\left(M\_m \cdot M\_m\right) \cdot D\_m}{\ell}, \sqrt{\frac{1}{h}}\right) \cdot d}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -1e+36)
   (*
    (-
     1.0
     (* (* (* (* 0.25 D_m) (/ M_m d)) h) (/ (* D_m (* M_m (/ 0.5 d))) l)))
    (* (sqrt (/ 1.0 (* h l))) (- d)))
   (if (<= d -5e-310)
     (*
      (* (/ (sqrt (- d)) (sqrt (- l))) (pow (/ d h) (/ 1.0 2.0)))
      (- 1.0 (* (/ h l) (* (pow (/ (* D_m M_m) (* 2.0 d)) 2.0) (/ 1.0 2.0)))))
     (if (<= d 3.8e-20)
       (/
        (fma
         D_m
         (* (sqrt h) (* (* (/ (* (/ M_m d) M_m) l) -0.125) D_m))
         (/ d (sqrt h)))
        (sqrt l))
       (/
        (*
         (fma
          (* (sqrt h) -0.125)
          (* (/ (/ D_m d) d) (/ (* (* M_m M_m) D_m) l))
          (sqrt (/ 1.0 h)))
         d)
        (sqrt l))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1e+36) {
		tmp = (1.0 - ((((0.25 * D_m) * (M_m / d)) * h) * ((D_m * (M_m * (0.5 / d))) / l))) * (sqrt((1.0 / (h * l))) * -d);
	} else if (d <= -5e-310) {
		tmp = ((sqrt(-d) / sqrt(-l)) * pow((d / h), (1.0 / 2.0))) * (1.0 - ((h / l) * (pow(((D_m * M_m) / (2.0 * d)), 2.0) * (1.0 / 2.0))));
	} else if (d <= 3.8e-20) {
		tmp = fma(D_m, (sqrt(h) * (((((M_m / d) * M_m) / l) * -0.125) * D_m)), (d / sqrt(h))) / sqrt(l);
	} else {
		tmp = (fma((sqrt(h) * -0.125), (((D_m / d) / d) * (((M_m * M_m) * D_m) / l)), sqrt((1.0 / h))) * d) / sqrt(l);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -1e+36)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(0.25 * D_m) * Float64(M_m / d)) * h) * Float64(Float64(D_m * Float64(M_m * Float64(0.5 / d))) / l))) * Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(-d)));
	elseif (d <= -5e-310)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * (Float64(d / h) ^ Float64(1.0 / 2.0))) * 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)))));
	elseif (d <= 3.8e-20)
		tmp = Float64(fma(D_m, Float64(sqrt(h) * Float64(Float64(Float64(Float64(Float64(M_m / d) * M_m) / l) * -0.125) * D_m)), Float64(d / sqrt(h))) / sqrt(l));
	else
		tmp = Float64(Float64(fma(Float64(sqrt(h) * -0.125), Float64(Float64(Float64(D_m / d) / d) * Float64(Float64(Float64(M_m * M_m) * D_m) / l)), sqrt(Float64(1.0 / h))) * d) / sqrt(l));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -1e+36], N[(N[(1.0 - N[(N[(N[(N[(0.25 * D$95$m), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] * N[(N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision], If[LessEqual[d, 3.8e-20], N[(N[(D$95$m * N[(N[Sqrt[h], $MachinePrecision] * N[(N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] / l), $MachinePrecision] * -0.125), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] + N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Sqrt[h], $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(N[(D$95$m / d), $MachinePrecision] / d), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.00000000000000004e36

   1. Initial program 70.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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 75.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-undiv N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. clear-num N/A

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

      11. un-div-inv N/A

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

      12. clear-num N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. sqrt-undiv N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-/.f64 76.4

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

      20. lift-/.f64 N/A

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

      21. metadata-eval 76.4

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

      22. lift-pow.f64 N/A

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

      23. unpow1/2 N/A

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

      24. lower-sqrt.f64 76.4

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

   6. Applied rewrites 76.4%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-1 N/A

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

      5. remove-double-div N/A

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

      6. lower-*.f64 76.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 76.4

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 76.4

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

   8. Applied rewrites 76.4%

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

   9. Taylor expanded in d around -inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 96.5

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

   11. Applied rewrites 96.5%

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

   if -1.00000000000000004e36 < d < -4.999999999999985e-310

   1. Initial program 74.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 83.5

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

   4. Applied rewrites 83.5%

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

   if -4.999999999999985e-310 < d < 3.7999999999999998e-20

   1. Initial program 61.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 64.8

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

   4. Applied rewrites 64.8%

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

   6. Applied rewrites 54.1%

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

   7. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   9. Applied rewrites 64.0%

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

   11. Applied rewrites 78.2%

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

   if 3.7999999999999998e-20 < d

   1. Initial program 76.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 77.7

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

   4. Applied rewrites 77.7%

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

   6. Applied rewrites 73.6%

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

   7. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 77.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.125 \cdot \sqrt{h}, \frac{\left(M \cdot M\right) \cdot D}{\ell} \cdot \frac{\frac{D}{d}}{d}, \sqrt{\frac{1}{h}}\right) \cdot d}}{\sqrt{\ell}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{+36}:\\ \;\;\;\;\left(1 - \left(\left(\left(0.25 \cdot D\right) \cdot \frac{M}{d}\right) \cdot h\right) \cdot \frac{D \cdot \left(M \cdot \frac{0.5}{d}\right)}{\ell}\right) \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(D, \sqrt{h} \cdot \left(\left(\frac{\frac{M}{d} \cdot M}{\ell} \cdot -0.125\right) \cdot D\right), \frac{d}{\sqrt{h}}\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{h} \cdot -0.125, \frac{\frac{D}{d}}{d} \cdot \frac{\left(M \cdot M\right) \cdot D}{\ell}, \sqrt{\frac{1}{h}}\right) \cdot d}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-D\_m\right) \cdot \left(M\_m \cdot \frac{0.5}{d}\right)}{\ell} \cdot \left(\left(0.25 \cdot D\_m\right) \cdot \frac{M\_m}{d}\right), h, 1\right) \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(D\_m, \sqrt{h} \cdot \left(\left(\frac{\frac{M\_m}{d} \cdot M\_m}{\ell} \cdot -0.125\right) \cdot D\_m\right), \frac{d}{\sqrt{h}}\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{h} \cdot -0.125, \frac{\frac{D\_m}{d}}{d} \cdot \frac{\left(M\_m \cdot M\_m\right) \cdot D\_m}{\ell}, \sqrt{\frac{1}{h}}\right) \cdot d}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -5e-310)
   (*
    (fma
     (* (/ (* (- D_m) (* M_m (/ 0.5 d))) l) (* (* 0.25 D_m) (/ M_m d)))
     h
     1.0)
    (* (sqrt (/ 1.0 (* h l))) (- d)))
   (if (<= d 3.8e-20)
     (/
      (fma
       D_m
       (* (sqrt h) (* (* (/ (* (/ M_m d) M_m) l) -0.125) D_m))
       (/ d (sqrt h)))
      (sqrt l))
     (/
      (*
       (fma
        (* (sqrt h) -0.125)
        (* (/ (/ D_m d) d) (/ (* (* M_m M_m) D_m) l))
        (sqrt (/ 1.0 h)))
       d)
      (sqrt l)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -5e-310) {
		tmp = fma((((-D_m * (M_m * (0.5 / d))) / l) * ((0.25 * D_m) * (M_m / d))), h, 1.0) * (sqrt((1.0 / (h * l))) * -d);
	} else if (d <= 3.8e-20) {
		tmp = fma(D_m, (sqrt(h) * (((((M_m / d) * M_m) / l) * -0.125) * D_m)), (d / sqrt(h))) / sqrt(l);
	} else {
		tmp = (fma((sqrt(h) * -0.125), (((D_m / d) / d) * (((M_m * M_m) * D_m) / l)), sqrt((1.0 / h))) * d) / sqrt(l);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -5e-310)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(-D_m) * Float64(M_m * Float64(0.5 / d))) / l) * Float64(Float64(0.25 * D_m) * Float64(M_m / d))), h, 1.0) * Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(-d)));
	elseif (d <= 3.8e-20)
		tmp = Float64(fma(D_m, Float64(sqrt(h) * Float64(Float64(Float64(Float64(Float64(M_m / d) * M_m) / l) * -0.125) * D_m)), Float64(d / sqrt(h))) / sqrt(l));
	else
		tmp = Float64(Float64(fma(Float64(sqrt(h) * -0.125), Float64(Float64(Float64(D_m / d) / d) * Float64(Float64(Float64(M_m * M_m) * D_m) / l)), sqrt(Float64(1.0 / h))) * d) / sqrt(l));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -5e-310], N[(N[(N[(N[(N[((-D$95$m) * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(0.25 * D$95$m), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.8e-20], N[(N[(D$95$m * N[(N[Sqrt[h], $MachinePrecision] * N[(N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] / l), $MachinePrecision] * -0.125), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] + N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Sqrt[h], $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(N[(D$95$m / d), $MachinePrecision] / d), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.999999999999985e-310

   1. Initial program 72.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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 75.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-undiv N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. clear-num N/A

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

      11. un-div-inv N/A

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

      12. clear-num N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. sqrt-undiv N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-/.f64 74.1

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

      20. lift-/.f64 N/A

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

      21. metadata-eval 74.1

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

      22. lift-pow.f64 N/A

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

      23. unpow1/2 N/A

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

      24. lower-sqrt.f64 74.1

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

   6. Applied rewrites 74.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow-1 N/A

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

      10. remove-double-div N/A

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

      11. associate-*r* N/A

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

   8. Applied rewrites 74.1%

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

   9. Taylor expanded in d around -inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 84.0

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

   11. Applied rewrites 84.0%

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

   if -4.999999999999985e-310 < d < 3.7999999999999998e-20

   1. Initial program 61.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 64.8

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

   4. Applied rewrites 64.8%

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

   6. Applied rewrites 54.1%

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

   7. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   9. Applied rewrites 64.0%

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

   11. Applied rewrites 78.2%

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

   if 3.7999999999999998e-20 < d

   1. Initial program 76.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 77.7

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

   4. Applied rewrites 77.7%

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

   6. Applied rewrites 73.6%

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

   7. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 77.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.125 \cdot \sqrt{h}, \frac{\left(M \cdot M\right) \cdot D}{\ell} \cdot \frac{\frac{D}{d}}{d}, \sqrt{\frac{1}{h}}\right) \cdot d}}{\sqrt{\ell}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-D\right) \cdot \left(M \cdot \frac{0.5}{d}\right)}{\ell} \cdot \left(\left(0.25 \cdot D\right) \cdot \frac{M}{d}\right), h, 1\right) \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(D, \sqrt{h} \cdot \left(\left(\frac{\frac{M}{d} \cdot M}{\ell} \cdot -0.125\right) \cdot D\right), \frac{d}{\sqrt{h}}\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{h} \cdot -0.125, \frac{\frac{D}{d}}{d} \cdot \frac{\left(M \cdot M\right) \cdot D}{\ell}, \sqrt{\frac{1}{h}}\right) \cdot d}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;\ell \leq -1.7 \cdot 10^{-220}:\\ \;\;\;\;t\_0 \cdot \left(-d\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_0 \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(D\_m, \sqrt{h} \cdot \left(\left(\frac{\frac{M\_m}{d} \cdot M\_m}{\ell} \cdot -0.125\right) \cdot D\_m\right), \frac{d}{\sqrt{h}}\right)}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* h l)))))
   (if (<= l -1.7e-220)
     (* t_0 (- d))
     (if (<= l -5e-310)
       (* t_0 d)
       (/
        (fma
         D_m
         (* (sqrt h) (* (* (/ (* (/ M_m d) M_m) l) -0.125) D_m))
         (/ d (sqrt h)))
        (sqrt l))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((1.0 / (h * l)));
	double tmp;
	if (l <= -1.7e-220) {
		tmp = t_0 * -d;
	} else if (l <= -5e-310) {
		tmp = t_0 * d;
	} else {
		tmp = fma(D_m, (sqrt(h) * (((((M_m / d) * M_m) / l) * -0.125) * D_m)), (d / sqrt(h))) / sqrt(l);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(1.0 / Float64(h * l)))
	tmp = 0.0
	if (l <= -1.7e-220)
		tmp = Float64(t_0 * Float64(-d));
	elseif (l <= -5e-310)
		tmp = Float64(t_0 * d);
	else
		tmp = Float64(fma(D_m, Float64(sqrt(h) * Float64(Float64(Float64(Float64(Float64(M_m / d) * M_m) / l) * -0.125) * D_m)), Float64(d / sqrt(h))) / sqrt(l));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.7e-220], N[(t$95$0 * (-d)), $MachinePrecision], If[LessEqual[l, -5e-310], N[(t$95$0 * d), $MachinePrecision], N[(N[(D$95$m * N[(N[Sqrt[h], $MachinePrecision] * N[(N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] / l), $MachinePrecision] * -0.125), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] + N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.69999999999999997e-220

   1. Initial program 70.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 78.0

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

   4. Applied rewrites 78.0%

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

   5. Taylor expanded in d around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 61.4

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

   7. Applied rewrites 61.4%

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

   if -1.69999999999999997e-220 < l < -4.999999999999985e-310

   1. Initial program 80.2%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 16.6%

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

   6. Taylor expanded in d around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 33.9

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

   8. Applied rewrites 33.9%

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

   if -4.999999999999985e-310 < l

   1. Initial program 69.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 71.5

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

   4. Applied rewrites 71.5%

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

   6. Applied rewrites 64.2%

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

   7. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   9. Applied rewrites 63.1%

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

   11. Applied rewrites 77.1%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.7 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(D, \sqrt{h} \cdot \left(\left(\frac{\frac{M}{d} \cdot M}{\ell} \cdot -0.125\right) \cdot D\right), \frac{d}{\sqrt{h}}\right)}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-D\_m\right) \cdot \left(M\_m \cdot \frac{0.5}{d}\right)}{\ell} \cdot \left(\left(0.25 \cdot D\_m\right) \cdot \frac{M\_m}{d}\right), h, 1\right) \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(D\_m, \sqrt{h} \cdot \left(\left(\frac{\frac{M\_m}{d} \cdot M\_m}{\ell} \cdot -0.125\right) \cdot D\_m\right), \frac{d}{\sqrt{h}}\right)}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -5e-310)
   (*
    (fma
     (* (/ (* (- D_m) (* M_m (/ 0.5 d))) l) (* (* 0.25 D_m) (/ M_m d)))
     h
     1.0)
    (* (sqrt (/ 1.0 (* h l))) (- d)))
   (/
    (fma
     D_m
     (* (sqrt h) (* (* (/ (* (/ M_m d) M_m) l) -0.125) D_m))
     (/ d (sqrt h)))
    (sqrt l))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -5e-310) {
		tmp = fma((((-D_m * (M_m * (0.5 / d))) / l) * ((0.25 * D_m) * (M_m / d))), h, 1.0) * (sqrt((1.0 / (h * l))) * -d);
	} else {
		tmp = fma(D_m, (sqrt(h) * (((((M_m / d) * M_m) / l) * -0.125) * D_m)), (d / sqrt(h))) / sqrt(l);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(-D_m) * Float64(M_m * Float64(0.5 / d))) / l) * Float64(Float64(0.25 * D_m) * Float64(M_m / d))), h, 1.0) * Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(-d)));
	else
		tmp = Float64(fma(D_m, Float64(sqrt(h) * Float64(Float64(Float64(Float64(Float64(M_m / d) * M_m) / l) * -0.125) * D_m)), Float64(d / sqrt(h))) / sqrt(l));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -5e-310], N[(N[(N[(N[(N[((-D$95$m) * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(0.25 * D$95$m), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision]), $MachinePrecision], N[(N[(D$95$m * N[(N[Sqrt[h], $MachinePrecision] * N[(N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] / l), $MachinePrecision] * -0.125), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] + N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 72.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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 75.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-undiv N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. clear-num N/A

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

      11. un-div-inv N/A

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

      12. clear-num N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. sqrt-undiv N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-/.f64 74.1

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

      20. lift-/.f64 N/A

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

      21. metadata-eval 74.1

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

      22. lift-pow.f64 N/A

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

      23. unpow1/2 N/A

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

      24. lower-sqrt.f64 74.1

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

   6. Applied rewrites 74.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow-1 N/A

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

      10. remove-double-div N/A

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

      11. associate-*r* N/A

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

   8. Applied rewrites 74.1%

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

   9. Taylor expanded in d around -inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 84.0

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

   11. Applied rewrites 84.0%

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

   if -4.999999999999985e-310 < l

   1. Initial program 69.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 71.5

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

   4. Applied rewrites 71.5%

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

   6. Applied rewrites 64.2%

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

   7. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   9. Applied rewrites 63.1%

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

   11. Applied rewrites 77.1%

       \[\leadsto \frac{\mathsf{fma}\left(D, \color{blue}{\left(\left(-0.125 \cdot \frac{\frac{M}{d} \cdot M}{\ell}\right) \cdot D\right) \cdot \sqrt{h}}, \frac{d}{\sqrt{h}}\right)}{\sqrt{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-D\right) \cdot \left(M \cdot \frac{0.5}{d}\right)}{\ell} \cdot \left(\left(0.25 \cdot D\right) \cdot \frac{M}{d}\right), h, 1\right) \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(D, \sqrt{h} \cdot \left(\left(\frac{\frac{M}{d} \cdot M}{\ell} \cdot -0.125\right) \cdot D\right), \frac{d}{\sqrt{h}}\right)}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \left(\left(\left(0.25 \cdot D\_m\right) \cdot \frac{M\_m}{d}\right) \cdot h\right) \cdot \frac{D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)}{\ell}\right) \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(D\_m, \sqrt{h} \cdot \left(\left(\frac{\frac{M\_m}{d} \cdot M\_m}{\ell} \cdot -0.125\right) \cdot D\_m\right), \frac{d}{\sqrt{h}}\right)}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -5e-310)
   (*
    (-
     1.0
     (* (* (* (* 0.25 D_m) (/ M_m d)) h) (/ (* D_m (* M_m (/ 0.5 d))) l)))
    (* (sqrt (/ 1.0 (* h l))) (- d)))
   (/
    (fma
     D_m
     (* (sqrt h) (* (* (/ (* (/ M_m d) M_m) l) -0.125) D_m))
     (/ d (sqrt h)))
    (sqrt l))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -5e-310) {
		tmp = (1.0 - ((((0.25 * D_m) * (M_m / d)) * h) * ((D_m * (M_m * (0.5 / d))) / l))) * (sqrt((1.0 / (h * l))) * -d);
	} else {
		tmp = fma(D_m, (sqrt(h) * (((((M_m / d) * M_m) / l) * -0.125) * D_m)), (d / sqrt(h))) / sqrt(l);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(0.25 * D_m) * Float64(M_m / d)) * h) * Float64(Float64(D_m * Float64(M_m * Float64(0.5 / d))) / l))) * Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(-d)));
	else
		tmp = Float64(fma(D_m, Float64(sqrt(h) * Float64(Float64(Float64(Float64(Float64(M_m / d) * M_m) / l) * -0.125) * D_m)), Float64(d / sqrt(h))) / sqrt(l));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -5e-310], N[(N[(1.0 - N[(N[(N[(N[(0.25 * D$95$m), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] * N[(N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision]), $MachinePrecision], N[(N[(D$95$m * N[(N[Sqrt[h], $MachinePrecision] * N[(N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] / l), $MachinePrecision] * -0.125), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] + N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 72.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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 75.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-undiv N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. clear-num N/A

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

      11. un-div-inv N/A

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

      12. clear-num N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. sqrt-undiv N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-/.f64 74.1

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

      20. lift-/.f64 N/A

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

      21. metadata-eval 74.1

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

      22. lift-pow.f64 N/A

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

      23. unpow1/2 N/A

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

      24. lower-sqrt.f64 74.1

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

   6. Applied rewrites 74.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-1 N/A

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

      5. remove-double-div N/A

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

      6. lower-*.f64 74.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 74.1

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 74.1

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

   8. Applied rewrites 74.1%

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

   9. Taylor expanded in d around -inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 84.8

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

   11. Applied rewrites 84.8%

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

   if -4.999999999999985e-310 < l

   1. Initial program 69.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 71.5

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

   4. Applied rewrites 71.5%

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

   6. Applied rewrites 64.2%

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

   7. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   9. Applied rewrites 63.1%

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

   11. Applied rewrites 77.1%

       \[\leadsto \frac{\mathsf{fma}\left(D, \color{blue}{\left(\left(-0.125 \cdot \frac{\frac{M}{d} \cdot M}{\ell}\right) \cdot D\right) \cdot \sqrt{h}}, \frac{d}{\sqrt{h}}\right)}{\sqrt{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \left(\left(\left(0.25 \cdot D\right) \cdot \frac{M}{d}\right) \cdot h\right) \cdot \frac{D \cdot \left(M \cdot \frac{0.5}{d}\right)}{\ell}\right) \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(D, \sqrt{h} \cdot \left(\left(\frac{\frac{M}{d} \cdot M}{\ell} \cdot -0.125\right) \cdot D\right), \frac{d}{\sqrt{h}}\right)}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;\ell \leq -1.7 \cdot 10^{-220}:\\ \;\;\;\;t\_0 \cdot \left(-d\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_0 \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\left(0.25 \cdot D\_m\right) \cdot \frac{M\_m}{d}\right) \cdot h\right) \cdot \frac{D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* h l)))))
   (if (<= l -1.7e-220)
     (* t_0 (- d))
     (if (<= l -5e-310)
       (* t_0 d)
       (*
        (/ d (sqrt (* h l)))
        (-
         1.0
         (*
          (* (* (* 0.25 D_m) (/ M_m d)) h)
          (/ (* D_m (* M_m (/ 0.5 d))) l))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((1.0 / (h * l)));
	double tmp;
	if (l <= -1.7e-220) {
		tmp = t_0 * -d;
	} else if (l <= -5e-310) {
		tmp = t_0 * d;
	} else {
		tmp = (d / sqrt((h * l))) * (1.0 - ((((0.25 * D_m) * (M_m / d)) * h) * ((D_m * (M_m * (0.5 / d))) / l)));
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / (h * l)))
    if (l <= (-1.7d-220)) then
        tmp = t_0 * -d
    else if (l <= (-5d-310)) then
        tmp = t_0 * d
    else
        tmp = (d / sqrt((h * l))) * (1.0d0 - ((((0.25d0 * d_m) * (m_m / d)) * h) * ((d_m * (m_m * (0.5d0 / d))) / l)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((1.0 / (h * l)));
	double tmp;
	if (l <= -1.7e-220) {
		tmp = t_0 * -d;
	} else if (l <= -5e-310) {
		tmp = t_0 * d;
	} else {
		tmp = (d / Math.sqrt((h * l))) * (1.0 - ((((0.25 * D_m) * (M_m / d)) * h) * ((D_m * (M_m * (0.5 / d))) / l)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((1.0 / (h * l)))
	tmp = 0
	if l <= -1.7e-220:
		tmp = t_0 * -d
	elif l <= -5e-310:
		tmp = t_0 * d
	else:
		tmp = (d / math.sqrt((h * l))) * (1.0 - ((((0.25 * D_m) * (M_m / d)) * h) * ((D_m * (M_m * (0.5 / d))) / l)))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(1.0 / Float64(h * l)))
	tmp = 0.0
	if (l <= -1.7e-220)
		tmp = Float64(t_0 * Float64(-d));
	elseif (l <= -5e-310)
		tmp = Float64(t_0 * d);
	else
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * Float64(1.0 - Float64(Float64(Float64(Float64(0.25 * D_m) * Float64(M_m / d)) * h) * Float64(Float64(D_m * Float64(M_m * Float64(0.5 / d))) / l))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((1.0 / (h * l)));
	tmp = 0.0;
	if (l <= -1.7e-220)
		tmp = t_0 * -d;
	elseif (l <= -5e-310)
		tmp = t_0 * d;
	else
		tmp = (d / sqrt((h * l))) * (1.0 - ((((0.25 * D_m) * (M_m / d)) * h) * ((D_m * (M_m * (0.5 / d))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.7e-220], N[(t$95$0 * (-d)), $MachinePrecision], If[LessEqual[l, -5e-310], N[(t$95$0 * d), $MachinePrecision], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(0.25 * D$95$m), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] * N[(N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.69999999999999997e-220

   1. Initial program 70.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 78.0

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

   4. Applied rewrites 78.0%

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

   5. Taylor expanded in d around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 61.4

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

   7. Applied rewrites 61.4%

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

   if -1.69999999999999997e-220 < l < -4.999999999999985e-310

   1. Initial program 80.2%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 16.6%

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

   6. Taylor expanded in d around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 33.9

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

   8. Applied rewrites 33.9%

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

   if -4.999999999999985e-310 < l

   1. Initial program 69.0%

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

      1. lift-*.f64 N/A

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

      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 71.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-undiv N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. clear-num N/A

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

      11. un-div-inv N/A

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

      12. clear-num N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. sqrt-undiv N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-/.f64 71.6

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

      20. lift-/.f64 N/A

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

      21. metadata-eval 71.6

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

      22. lift-pow.f64 N/A

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

      23. unpow1/2 N/A

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

      24. lower-sqrt.f64 71.6

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

   6. Applied rewrites 71.6%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-1 N/A

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

      5. remove-double-div N/A

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

      6. lower-*.f64 71.6

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 71.6

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 71.6

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

   8. Applied rewrites 71.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-undiv N/A

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

      7. un-div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. sqrt-div N/A

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

      15. sqrt-prod N/A

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

      16. rem-square-sqrt N/A

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

      17. lift-sqrt.f64 N/A

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

      18. lift-/.f64 77.9

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 77.9

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

   10. Applied rewrites 77.9%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.7 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\left(0.25 \cdot D\right) \cdot \frac{M}{d}\right) \cdot h\right) \cdot \frac{D \cdot \left(M \cdot \frac{0.5}{d}\right)}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.5% accurate, 8.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;\ell \leq -1.7 \cdot 10^{-220}:\\ \;\;\;\;t\_0 \cdot \left(-d\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_0 \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 (/ 1.0 (* h l)))))
   (if (<= l -1.7e-220)
     (* t_0 (- d))
     (if (<= l -5e-310) (* t_0 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((1.0 / (h * l)));
	double tmp;
	if (l <= -1.7e-220) {
		tmp = t_0 * -d;
	} else if (l <= -5e-310) {
		tmp = t_0 * 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((1.0d0 / (h * l)))
    if (l <= (-1.7d-220)) then
        tmp = t_0 * -d
    else if (l <= (-5d-310)) then
        tmp = t_0 * 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((1.0 / (h * l)));
	double tmp;
	if (l <= -1.7e-220) {
		tmp = t_0 * -d;
	} else if (l <= -5e-310) {
		tmp = t_0 * 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((1.0 / (h * l)))
	tmp = 0
	if l <= -1.7e-220:
		tmp = t_0 * -d
	elif l <= -5e-310:
		tmp = t_0 * 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(1.0 / Float64(h * l)))
	tmp = 0.0
	if (l <= -1.7e-220)
		tmp = Float64(t_0 * Float64(-d));
	elseif (l <= -5e-310)
		tmp = Float64(t_0 * 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((1.0 / (h * l)));
	tmp = 0.0;
	if (l <= -1.7e-220)
		tmp = t_0 * -d;
	elseif (l <= -5e-310)
		tmp = t_0 * 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[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.7e-220], N[(t$95$0 * (-d)), $MachinePrecision], If[LessEqual[l, -5e-310], N[(t$95$0 * d), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.69999999999999997e-220

   1. Initial program 70.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 78.0

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

   4. Applied rewrites 78.0%

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

   5. Taylor expanded in d around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 61.4

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

   7. Applied rewrites 61.4%

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

   if -1.69999999999999997e-220 < l < -4.999999999999985e-310

   1. Initial program 80.2%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 16.6%

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

   6. Taylor expanded in d around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 33.9

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

   8. Applied rewrites 33.9%

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

   if -4.999999999999985e-310 < l

   1. Initial program 69.0%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 35.4%

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

   6. Taylor expanded in d around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 40.5

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

   8. Applied rewrites 40.5%

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

   10. Applied rewrites 40.5%

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

   12. Applied rewrites 46.3%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.7 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.7% accurate, 10.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;\ell \leq -1.7 \cdot 10^{-220}:\\ \;\;\;\;t\_0 \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot d\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* h l)))))
   (if (<= l -1.7e-220) (* t_0 (- d)) (* t_0 d))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((1.0 / (h * l)));
	double tmp;
	if (l <= -1.7e-220) {
		tmp = t_0 * -d;
	} else {
		tmp = t_0 * d;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / (h * l)))
    if (l <= (-1.7d-220)) then
        tmp = t_0 * -d
    else
        tmp = t_0 * d
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((1.0 / (h * l)));
	double tmp;
	if (l <= -1.7e-220) {
		tmp = t_0 * -d;
	} else {
		tmp = t_0 * d;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((1.0 / (h * l)))
	tmp = 0
	if l <= -1.7e-220:
		tmp = t_0 * -d
	else:
		tmp = t_0 * d
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(1.0 / Float64(h * l)))
	tmp = 0.0
	if (l <= -1.7e-220)
		tmp = Float64(t_0 * Float64(-d));
	else
		tmp = Float64(t_0 * d);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((1.0 / (h * l)));
	tmp = 0.0;
	if (l <= -1.7e-220)
		tmp = t_0 * -d;
	else
		tmp = t_0 * d;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.7e-220], N[(t$95$0 * (-d)), $MachinePrecision], N[(t$95$0 * d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -1.69999999999999997e-220

   1. Initial program 70.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 78.0

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

   4. Applied rewrites 78.0%

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

   5. Taylor expanded in d around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 61.4

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

   7. Applied rewrites 61.4%

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

   if -1.69999999999999997e-220 < l

   1. Initial program 70.9%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 32.3%

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

   6. Taylor expanded in d around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      5. rem-square-sqrt N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{-1} \cdot d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. remove-double-neg N/A

         \[\leadsto \color{blue}{d} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      13. lower-*.f64 39.4

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 39.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.7 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 26.7% accurate, 12.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \sqrt{\frac{1}{h \cdot \ell}} \cdot d \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (* (sqrt (/ 1.0 (* h l))) d))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return sqrt((1.0 / (h * l))) * d;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = sqrt((1.0d0 / (h * l))) * d
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return Math.sqrt((1.0 / (h * l))) * d;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return math.sqrt((1.0 / (h * l))) * d

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(sqrt(Float64(1.0 / Float64(h * l))) * d)
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = sqrt((1.0 / (h * l))) * d;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]

Derivation

1. Initial program 70.7%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing

3. Taylor expanded in d around inf

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 39.6%

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

6. Taylor expanded in d around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
7. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   2. mul-1-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   3. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot d}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   4. unpow2 N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   5. rem-square-sqrt N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{-1} \cdot d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   6. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   7. remove-double-neg N/A

      \[\leadsto \color{blue}{d} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   8. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   9. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

   11. lower-/.f64 N/A

       \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   12. *-commutative N/A

       \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   13. lower-*.f64 26.2

       \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

8. Applied rewrites 26.2%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

9. Final simplification 26.2%

   \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]
10. Add Preprocessing

Alternative 12: 26.5% accurate, 15.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \frac{d}{\sqrt{h \cdot \ell}} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (/ d (sqrt (* h l))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return d / sqrt((h * l));
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = d / sqrt((h * l))
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return d / Math.sqrt((h * l));
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return d / math.sqrt((h * l))

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(d / sqrt(Float64(h * l)))
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = d / sqrt((h * l));
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.7%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing

3. Taylor expanded in d around inf

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 39.6%

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

6. Taylor expanded in d around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
7. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   2. mul-1-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   3. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot d}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   4. unpow2 N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   5. rem-square-sqrt N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{-1} \cdot d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   6. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   7. remove-double-neg N/A

      \[\leadsto \color{blue}{d} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   8. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   9. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

   11. lower-/.f64 N/A

       \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   12. *-commutative N/A

       \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   13. lower-*.f64 26.2

       \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

8. Applied rewrites 26.2%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
9. Step-by-step derivation

10. Applied rewrites 25.5%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

11. Final simplification 25.5%

    \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024304 
(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)))))