Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.1% → 81.3%

Time: 24.0s

Alternatives: 11

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.1% 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: 81.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;d \leq -1.65 \cdot 10^{+143}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{t\_0}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D\_m \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{t\_0}{\sqrt{-\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{-0.5}{\ell} \cdot {\left(0.5 \cdot \frac{M}{\frac{d}{D\_m}}\right)}^{2}, h, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (- d))))
   (if (<= d -1.65e+143)
     (*
      (sqrt (/ d l))
      (*
       (/ t_0 (sqrt (- h)))
       (+ 1.0 (* (/ h l) (* (pow (* D_m (/ (/ M 2.0) d)) 2.0) -0.5)))))
     (if (<= d -1e-310)
       (*
        (* (sqrt (/ d h)) (/ t_0 (sqrt (- l))))
        (- 1.0 (* 0.5 (/ (* h (pow (* M (* 0.5 (/ D_m d))) 2.0)) l))))
       (*
        d
        (/
         (fma (* (/ -0.5 l) (pow (* 0.5 (/ M (/ d D_m))) 2.0)) h 1.0)
         (* (sqrt l) (sqrt h))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt(-d);
	double tmp;
	if (d <= -1.65e+143) {
		tmp = sqrt((d / l)) * ((t_0 / sqrt(-h)) * (1.0 + ((h / l) * (pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))));
	} else if (d <= -1e-310) {
		tmp = (sqrt((d / h)) * (t_0 / sqrt(-l))) * (1.0 - (0.5 * ((h * pow((M * (0.5 * (D_m / d))), 2.0)) / l)));
	} else {
		tmp = d * (fma(((-0.5 / l) * pow((0.5 * (M / (d / D_m))), 2.0)), h, 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(-d))
	tmp = 0.0
	if (d <= -1.65e+143)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D_m * Float64(Float64(M / 2.0) / d)) ^ 2.0) * -0.5)))));
	elseif (d <= -1e-310)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(t_0 / sqrt(Float64(-l)))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(M * Float64(0.5 * Float64(D_m / d))) ^ 2.0)) / l))));
	else
		tmp = Float64(d * Float64(fma(Float64(Float64(-0.5 / l) * (Float64(0.5 * Float64(M / Float64(d / D_m))) ^ 2.0)), h, 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[d, -1.65e+143], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D$95$m * N[(N[(M / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-310], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(M * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(N[(-0.5 / l), $MachinePrecision] * N[Power[N[(0.5 * N[(M / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.65e143

   1. Initial program 65.5%

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

   3. Simplified 65.5%

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

      1. frac-2neg 65.5%

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

      2. sqrt-div 88.5%

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

   6. Applied egg-rr 88.5%

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

   if -1.65e143 < d < -9.999999999999969e-311

   1. Initial program 62.7%

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

   3. Simplified 62.7%

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

      1. *-commutative 62.7%

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

      2. unpow2 62.7%

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

      3. associate-*r* 63.8%

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

      4. *-commutative 63.8%

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

      5. associate-*l/ 63.8%

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

      6. associate-*r/ 62.7%

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

      7. *-commutative 62.7%

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

      8. associate-*l/ 62.7%

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

      9. associate-*r/ 62.7%

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

      10. associate-*r* 61.5%

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

      11. unpow2 61.5%

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

      12. associate-*l/ 64.8%

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

   6. Applied egg-rr 65.2%

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

      1. frac-2neg 65.2%

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

      2. sqrt-div 78.3%

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

   8. Applied egg-rr 78.3%

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

   if -9.999999999999969e-311 < d

   1. Initial program 63.3%

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

   3. Simplified 63.2%

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

      1. *-commutative 63.2%

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

      2. unpow2 63.2%

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

      3. associate-*r* 64.6%

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

      4. *-commutative 64.6%

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

      5. associate-*l/ 64.7%

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

      6. associate-*r/ 64.7%

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

      7. *-commutative 64.7%

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

      8. associate-*l/ 65.6%

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

      9. associate-*r/ 65.6%

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

      10. associate-*r* 63.3%

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

      11. unpow2 63.3%

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

      12. associate-*l/ 65.1%

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

   6. Applied egg-rr 65.0%

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

      1. add-sqr-sqrt 64.8%

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

      2. pow2 64.8%

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

      3. pow1/2 64.8%

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

      4. sqrt-pow1 64.9%

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

      5. metadata-eval 64.9%

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

   8. Applied egg-rr 64.9%

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

   10. Applied egg-rr 84.2%

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

       1. unpow1 84.2%

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

       2. *-commutative 84.2%

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

       3. associate-*l/ 85.1%

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

       4. associate-/l* 85.1%

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

   12. Simplified 86.8%

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

Alternative 2: 82.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\ell}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -5.5 \cdot 10^{+139}:\\ \;\;\;\;t\_1 \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D\_m \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;d \leq -2.05 \cdot 10^{-67}:\\ \;\;\;\;\left(1 - t\_0\right) \cdot \left(t\_1 \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(t\_0 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{-0.5}{\ell} \cdot {\left(0.5 \cdot \frac{M}{\frac{d}{D\_m}}\right)}^{2}, h, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ (* h (pow (* M (* 0.5 (/ D_m d))) 2.0)) l)))
        (t_1 (sqrt (/ d l))))
   (if (<= d -5.5e+139)
     (*
      t_1
      (*
       (/ (sqrt (- d)) (sqrt (- h)))
       (+ 1.0 (* (/ h l) (* (pow (* D_m (/ (/ M 2.0) d)) 2.0) -0.5)))))
     (if (<= d -2.05e-67)
       (* (- 1.0 t_0) (* t_1 (sqrt (/ d h))))
       (if (<= d -1e-310)
         (* (* d (sqrt (/ 1.0 (* l h)))) (+ t_0 -1.0))
         (*
          d
          (/
           (fma (* (/ -0.5 l) (pow (* 0.5 (/ M (/ d D_m))) 2.0)) h 1.0)
           (* (sqrt l) (sqrt h)))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 0.5 * ((h * pow((M * (0.5 * (D_m / d))), 2.0)) / l);
	double t_1 = sqrt((d / l));
	double tmp;
	if (d <= -5.5e+139) {
		tmp = t_1 * ((sqrt(-d) / sqrt(-h)) * (1.0 + ((h / l) * (pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))));
	} else if (d <= -2.05e-67) {
		tmp = (1.0 - t_0) * (t_1 * sqrt((d / h)));
	} else if (d <= -1e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (t_0 + -1.0);
	} else {
		tmp = d * (fma(((-0.5 / l) * pow((0.5 * (M / (d / D_m))), 2.0)), h, 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(0.5 * Float64(Float64(h * (Float64(M * Float64(0.5 * Float64(D_m / d))) ^ 2.0)) / l))
	t_1 = sqrt(Float64(d / l))
	tmp = 0.0
	if (d <= -5.5e+139)
		tmp = Float64(t_1 * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D_m * Float64(Float64(M / 2.0) / d)) ^ 2.0) * -0.5)))));
	elseif (d <= -2.05e-67)
		tmp = Float64(Float64(1.0 - t_0) * Float64(t_1 * sqrt(Float64(d / h))));
	elseif (d <= -1e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(t_0 + -1.0));
	else
		tmp = Float64(d * Float64(fma(Float64(Float64(-0.5 / l) * (Float64(0.5 * Float64(M / Float64(d / D_m))) ^ 2.0)), h, 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(0.5 * N[(N[(h * N[Power[N[(M * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -5.5e+139], N[(t$95$1 * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D$95$m * N[(N[(M / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2.05e-67], N[(N[(1.0 - t$95$0), $MachinePrecision] * N[(t$95$1 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(N[(-0.5 / l), $MachinePrecision] * N[Power[N[(0.5 * N[(M / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -5.4999999999999996e139

   1. Initial program 65.5%

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

   3. Simplified 65.5%

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

      1. frac-2neg 65.5%

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

      2. sqrt-div 88.5%

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

   6. Applied egg-rr 88.5%

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

   if -5.4999999999999996e139 < d < -2.0499999999999999e-67

   1. Initial program 71.3%

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

   3. Simplified 71.3%

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

      1. *-commutative 71.3%

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

      2. unpow2 71.3%

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

      3. associate-*r* 73.8%

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

      4. *-commutative 73.8%

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

      5. associate-*l/ 73.8%

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

      6. associate-*r/ 71.4%

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

      7. *-commutative 71.4%

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

      8. associate-*l/ 71.4%

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

      9. associate-*r/ 71.3%

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

      10. associate-*r* 68.8%

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

      11. unpow2 68.8%

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

      12. associate-*l/ 78.6%

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

   6. Applied egg-rr 79.5%

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

   if -2.0499999999999999e-67 < d < -9.999999999999969e-311

   1. Initial program 56.3%

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

   3. Simplified 56.3%

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

      1. *-commutative 56.3%

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

      2. unpow2 56.3%

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

      3. associate-*r* 56.4%

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

      4. *-commutative 56.4%

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

      5. associate-*l/ 56.4%

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

      6. associate-*r/ 56.2%

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

      7. *-commutative 56.2%

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

      8. associate-*l/ 56.3%

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

      9. associate-*r/ 56.3%

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

      10. associate-*r* 56.2%

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

      11. unpow2 56.2%

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

      12. associate-*l/ 54.5%

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

   6. Applied egg-rr 54.7%

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

      1. add-sqr-sqrt 54.5%

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

      2. pow2 54.5%

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

      3. pow1/2 54.5%

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

      4. sqrt-pow1 54.6%

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

      5. metadata-eval 54.6%

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

   8. Applied egg-rr 54.6%

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

   9. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. *-commutative 0.0%

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

       3. *-commutative 0.0%

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

       4. unpow2 0.0%

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

       5. rem-square-sqrt 69.5%

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

       6. neg-mul-1 69.5%

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

   11. Simplified 69.5%

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

   if -9.999999999999969e-311 < d

   1. Initial program 63.3%

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

   3. Simplified 63.2%

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

      1. *-commutative 63.2%

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

      2. unpow2 63.2%

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

      3. associate-*r* 64.6%

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

      4. *-commutative 64.6%

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

      5. associate-*l/ 64.7%

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

      6. associate-*r/ 64.7%

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

      7. *-commutative 64.7%

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

      8. associate-*l/ 65.6%

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

      9. associate-*r/ 65.6%

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

      10. associate-*r* 63.3%

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

      11. unpow2 63.3%

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

      12. associate-*l/ 65.1%

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

   6. Applied egg-rr 65.0%

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

      1. add-sqr-sqrt 64.8%

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

      2. pow2 64.8%

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

      3. pow1/2 64.8%

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

      4. sqrt-pow1 64.9%

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

      5. metadata-eval 64.9%

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

   8. Applied egg-rr 64.9%

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

   10. Applied egg-rr 84.2%

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

       1. unpow1 84.2%

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

       2. *-commutative 84.2%

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

       3. associate-*l/ 85.1%

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

       4. associate-/l* 85.1%

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

   12. Simplified 86.8%

       \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(\frac{-0.5}{\ell} \cdot {\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right)}^{2}, h, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.5 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;d \leq -2.05 \cdot 10^{-67}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{-0.5}{\ell} \cdot {\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right)}^{2}, h, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\ell}\\ \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+138}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-229}:\\ \;\;\;\;\left(1 - t\_0\right) \cdot \frac{1}{\frac{\sqrt{\frac{h}{d}}}{\sqrt{\frac{d}{\ell}}}}\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(t\_0 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{-0.5}{\ell} \cdot {\left(0.5 \cdot \frac{M}{\frac{d}{D\_m}}\right)}^{2}, h, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ (* h (pow (* M (* 0.5 (/ D_m d))) 2.0)) l))))
   (if (<= l -2.5e+138)
     (* (- d) (sqrt (/ (/ 1.0 h) l)))
     (if (<= l -4e-229)
       (* (- 1.0 t_0) (/ 1.0 (/ (sqrt (/ h d)) (sqrt (/ d l)))))
       (if (<= l -4e-310)
         (* (* d (pow (* l h) -0.5)) (+ t_0 -1.0))
         (*
          d
          (/
           (fma (* (/ -0.5 l) (pow (* 0.5 (/ M (/ d D_m))) 2.0)) h 1.0)
           (* (sqrt l) (sqrt h)))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 0.5 * ((h * pow((M * (0.5 * (D_m / d))), 2.0)) / l);
	double tmp;
	if (l <= -2.5e+138) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else if (l <= -4e-229) {
		tmp = (1.0 - t_0) * (1.0 / (sqrt((h / d)) / sqrt((d / l))));
	} else if (l <= -4e-310) {
		tmp = (d * pow((l * h), -0.5)) * (t_0 + -1.0);
	} else {
		tmp = d * (fma(((-0.5 / l) * pow((0.5 * (M / (d / D_m))), 2.0)), h, 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(0.5 * Float64(Float64(h * (Float64(M * Float64(0.5 * Float64(D_m / d))) ^ 2.0)) / l))
	tmp = 0.0
	if (l <= -2.5e+138)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	elseif (l <= -4e-229)
		tmp = Float64(Float64(1.0 - t_0) * Float64(1.0 / Float64(sqrt(Float64(h / d)) / sqrt(Float64(d / l)))));
	elseif (l <= -4e-310)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(t_0 + -1.0));
	else
		tmp = Float64(d * Float64(fma(Float64(Float64(-0.5 / l) * (Float64(0.5 * Float64(M / Float64(d / D_m))) ^ 2.0)), h, 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(0.5 * N[(N[(h * N[Power[N[(M * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.5e+138], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -4e-229], N[(N[(1.0 - t$95$0), $MachinePrecision] * N[(1.0 / N[(N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -4e-310], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(N[(-0.5 / l), $MachinePrecision] * N[Power[N[(0.5 * N[(M / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.50000000000000008e138

   1. Initial program 45.3%

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

   3. Simplified 45.3%

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

      1. add-sqr-sqrt 45.3%

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

      2. pow2 45.3%

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

      3. sqrt-prod 45.3%

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

      4. sqrt-pow1 49.2%

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

      5. metadata-eval 49.2%

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

      6. pow1 49.2%

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

      7. div-inv 49.2%

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

      8. metadata-eval 49.2%

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

      9. associate-*l* 49.2%

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

   6. Applied egg-rr 49.2%

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

   7. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 63.6%

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

      6. neg-mul-1 63.6%

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

   9. Simplified 63.6%

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

   if -2.50000000000000008e138 < l < -4.00000000000000028e-229

   1. Initial program 80.2%

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

   3. Simplified 80.2%

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

      1. *-commutative 80.2%

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

      2. unpow2 80.2%

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

      3. associate-*r* 80.2%

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

      4. *-commutative 80.2%

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

      5. associate-*l/ 80.2%

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

      6. associate-*r/ 78.8%

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

      7. *-commutative 78.8%

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

      8. associate-*l/ 78.8%

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

      9. associate-*r/ 78.7%

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

      10. associate-*r* 78.7%

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

      11. unpow2 78.7%

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

      12. associate-*l/ 83.1%

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

   6. Applied egg-rr 83.6%

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

      1. add-sqr-sqrt 83.2%

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

      2. pow2 83.2%

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

      3. pow1/2 83.2%

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

      4. sqrt-pow1 83.3%

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

      5. metadata-eval 83.3%

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

   8. Applied egg-rr 83.3%

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

      1. pow-pow 83.6%

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

      2. metadata-eval 83.6%

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

      3. pow1/2 83.6%

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

      4. *-commutative 83.6%

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

      5. clear-num 83.6%

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

      6. sqrt-div 85.3%

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

      7. metadata-eval 85.3%

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

      8. div-inv 85.2%

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

      9. clear-num 85.3%

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

   10. Applied egg-rr 85.3%

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

   if -4.00000000000000028e-229 < l < -3.999999999999988e-310

   1. Initial program 52.0%

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

   3. Simplified 52.0%

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

      1. *-commutative 52.0%

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

      2. unpow2 52.0%

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

      3. associate-*r* 52.0%

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

      4. *-commutative 52.0%

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

      5. associate-*l/ 52.0%

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

      6. associate-*r/ 52.0%

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

      7. *-commutative 52.0%

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

      8. associate-*l/ 57.6%

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

      9. associate-*r/ 57.6%

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

      10. associate-*r* 52.0%

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

      11. unpow2 52.0%

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

      12. associate-*l/ 52.6%

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

   6. Applied egg-rr 52.6%

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

      1. add-sqr-sqrt 52.6%

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

      2. pow2 52.6%

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

      3. pow1/2 52.6%

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

      4. sqrt-pow1 52.6%

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

      5. metadata-eval 52.6%

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

   8. Applied egg-rr 52.6%

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

      1. pow-pow 52.6%

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

      2. metadata-eval 52.6%

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

      3. pow1/2 52.6%

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

      4. *-commutative 52.6%

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

      5. clear-num 52.6%

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

      6. sqrt-div 52.5%

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

      7. metadata-eval 52.5%

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

      8. div-inv 52.6%

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

      9. clear-num 52.6%

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

   10. Applied egg-rr 52.6%

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

   11. Taylor expanded in d around -inf 78.9%

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

       1. associate-*r* 78.9%

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

       2. rem-exp-log 77.5%

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

       3. exp-neg 77.5%

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

       4. unpow1/2 77.5%

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

       5. exp-prod 77.5%

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

       6. distribute-lft-neg-out 77.5%

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

       7. distribute-rgt-neg-in 77.5%

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

       8. metadata-eval 77.5%

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

       9. exp-to-pow 79.0%

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

       10. neg-mul-1 79.0%

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

       11. distribute-lft-neg-in 79.0%

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

       12. distribute-rgt-neg-in 79.0%

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

   13. Simplified 79.0%

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

   if -3.999999999999988e-310 < l

   1. Initial program 63.3%

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

   3. Simplified 63.2%

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

      1. *-commutative 63.2%

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

      2. unpow2 63.2%

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

      3. associate-*r* 64.6%

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

      4. *-commutative 64.6%

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

      5. associate-*l/ 64.7%

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

      6. associate-*r/ 64.7%

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

      7. *-commutative 64.7%

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

      8. associate-*l/ 65.6%

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

      9. associate-*r/ 65.6%

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

      10. associate-*r* 63.3%

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

      11. unpow2 63.3%

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

      12. associate-*l/ 65.1%

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

   6. Applied egg-rr 65.0%

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

      1. add-sqr-sqrt 64.8%

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

      2. pow2 64.8%

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

      3. pow1/2 64.8%

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

      4. sqrt-pow1 64.9%

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

      5. metadata-eval 64.9%

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

   8. Applied egg-rr 64.9%

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

   10. Applied egg-rr 84.2%

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

       1. unpow1 84.2%

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

       2. *-commutative 84.2%

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

       3. associate-*l/ 85.1%

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

       4. associate-/l* 85.1%

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

   12. Simplified 86.8%

       \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left(\frac{-0.5}{\ell} \cdot {\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right)}^{2}, h, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+138}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-229}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right) \cdot \frac{1}{\frac{\sqrt{\frac{h}{d}}}{\sqrt{\frac{d}{\ell}}}}\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{-0.5}{\ell} \cdot {\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right)}^{2}, h, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\ell}\\ t_1 := 1 - t\_0\\ \mathbf{if}\;\ell \leq -2.55 \cdot 10^{+138}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq -7.2 \cdot 10^{-230}:\\ \;\;\;\;t\_1 \cdot \frac{1}{\frac{\sqrt{\frac{h}{d}}}{\sqrt{\frac{d}{\ell}}}}\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(t\_0 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ (* h (pow (* M (* 0.5 (/ D_m d))) 2.0)) l)))
        (t_1 (- 1.0 t_0)))
   (if (<= l -2.55e+138)
     (* (- d) (sqrt (/ (/ 1.0 h) l)))
     (if (<= l -7.2e-230)
       (* t_1 (/ 1.0 (/ (sqrt (/ h d)) (sqrt (/ d l)))))
       (if (<= l -4e-310)
         (* (* d (pow (* l h) -0.5)) (+ t_0 -1.0))
         (* t_1 (/ d (* (sqrt l) (sqrt h)))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 0.5 * ((h * pow((M * (0.5 * (D_m / d))), 2.0)) / l);
	double t_1 = 1.0 - t_0;
	double tmp;
	if (l <= -2.55e+138) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else if (l <= -7.2e-230) {
		tmp = t_1 * (1.0 / (sqrt((h / d)) / sqrt((d / l))));
	} else if (l <= -4e-310) {
		tmp = (d * pow((l * h), -0.5)) * (t_0 + -1.0);
	} else {
		tmp = t_1 * (d / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * ((h * ((m * (0.5d0 * (d_m / d))) ** 2.0d0)) / l)
    t_1 = 1.0d0 - t_0
    if (l <= (-2.55d+138)) then
        tmp = -d * sqrt(((1.0d0 / h) / l))
    else if (l <= (-7.2d-230)) then
        tmp = t_1 * (1.0d0 / (sqrt((h / d)) / sqrt((d / l))))
    else if (l <= (-4d-310)) then
        tmp = (d * ((l * h) ** (-0.5d0))) * (t_0 + (-1.0d0))
    else
        tmp = t_1 * (d / (sqrt(l) * sqrt(h)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 0.5 * ((h * Math.pow((M * (0.5 * (D_m / d))), 2.0)) / l);
	double t_1 = 1.0 - t_0;
	double tmp;
	if (l <= -2.55e+138) {
		tmp = -d * Math.sqrt(((1.0 / h) / l));
	} else if (l <= -7.2e-230) {
		tmp = t_1 * (1.0 / (Math.sqrt((h / d)) / Math.sqrt((d / l))));
	} else if (l <= -4e-310) {
		tmp = (d * Math.pow((l * h), -0.5)) * (t_0 + -1.0);
	} else {
		tmp = t_1 * (d / (Math.sqrt(l) * Math.sqrt(h)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = 0.5 * ((h * math.pow((M * (0.5 * (D_m / d))), 2.0)) / l)
	t_1 = 1.0 - t_0
	tmp = 0
	if l <= -2.55e+138:
		tmp = -d * math.sqrt(((1.0 / h) / l))
	elif l <= -7.2e-230:
		tmp = t_1 * (1.0 / (math.sqrt((h / d)) / math.sqrt((d / l))))
	elif l <= -4e-310:
		tmp = (d * math.pow((l * h), -0.5)) * (t_0 + -1.0)
	else:
		tmp = t_1 * (d / (math.sqrt(l) * math.sqrt(h)))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(0.5 * Float64(Float64(h * (Float64(M * Float64(0.5 * Float64(D_m / d))) ^ 2.0)) / l))
	t_1 = Float64(1.0 - t_0)
	tmp = 0.0
	if (l <= -2.55e+138)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	elseif (l <= -7.2e-230)
		tmp = Float64(t_1 * Float64(1.0 / Float64(sqrt(Float64(h / d)) / sqrt(Float64(d / l)))));
	elseif (l <= -4e-310)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(t_0 + -1.0));
	else
		tmp = Float64(t_1 * Float64(d / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = 0.5 * ((h * ((M * (0.5 * (D_m / d))) ^ 2.0)) / l);
	t_1 = 1.0 - t_0;
	tmp = 0.0;
	if (l <= -2.55e+138)
		tmp = -d * sqrt(((1.0 / h) / l));
	elseif (l <= -7.2e-230)
		tmp = t_1 * (1.0 / (sqrt((h / d)) / sqrt((d / l))));
	elseif (l <= -4e-310)
		tmp = (d * ((l * h) ^ -0.5)) * (t_0 + -1.0);
	else
		tmp = t_1 * (d / (sqrt(l) * sqrt(h)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(0.5 * N[(N[(h * N[Power[N[(M * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - t$95$0), $MachinePrecision]}, If[LessEqual[l, -2.55e+138], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -7.2e-230], N[(t$95$1 * N[(1.0 / N[(N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -4e-310], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.5499999999999999e138

   1. Initial program 45.3%

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

   3. Simplified 45.3%

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

      1. add-sqr-sqrt 45.3%

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

      2. pow2 45.3%

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

      3. sqrt-prod 45.3%

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

      4. sqrt-pow1 49.2%

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

      5. metadata-eval 49.2%

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

      6. pow1 49.2%

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

      7. div-inv 49.2%

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

      8. metadata-eval 49.2%

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

      9. associate-*l* 49.2%

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

   6. Applied egg-rr 49.2%

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

   7. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 63.6%

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

      6. neg-mul-1 63.6%

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

   9. Simplified 63.6%

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

   if -2.5499999999999999e138 < l < -7.1999999999999997e-230

   1. Initial program 80.2%

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

   3. Simplified 80.2%

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

      1. *-commutative 80.2%

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

      2. unpow2 80.2%

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

      3. associate-*r* 80.2%

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

      4. *-commutative 80.2%

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

      5. associate-*l/ 80.2%

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

      6. associate-*r/ 78.8%

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

      7. *-commutative 78.8%

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

      8. associate-*l/ 78.8%

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

      9. associate-*r/ 78.7%

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

      10. associate-*r* 78.7%

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

      11. unpow2 78.7%

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

      12. associate-*l/ 83.1%

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

   6. Applied egg-rr 83.6%

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

      1. add-sqr-sqrt 83.2%

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

      2. pow2 83.2%

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

      3. pow1/2 83.2%

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

      4. sqrt-pow1 83.3%

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

      5. metadata-eval 83.3%

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

   8. Applied egg-rr 83.3%

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

      1. pow-pow 83.6%

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

      2. metadata-eval 83.6%

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

      3. pow1/2 83.6%

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

      4. *-commutative 83.6%

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

      5. clear-num 83.6%

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

      6. sqrt-div 85.3%

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

      7. metadata-eval 85.3%

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

      8. div-inv 85.2%

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

      9. clear-num 85.3%

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

   10. Applied egg-rr 85.3%

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

   if -7.1999999999999997e-230 < l < -3.999999999999988e-310

   1. Initial program 52.0%

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

   3. Simplified 52.0%

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

      1. *-commutative 52.0%

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

      2. unpow2 52.0%

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

      3. associate-*r* 52.0%

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

      4. *-commutative 52.0%

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

      5. associate-*l/ 52.0%

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

      6. associate-*r/ 52.0%

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

      7. *-commutative 52.0%

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

      8. associate-*l/ 57.6%

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

      9. associate-*r/ 57.6%

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

      10. associate-*r* 52.0%

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

      11. unpow2 52.0%

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

      12. associate-*l/ 52.6%

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

   6. Applied egg-rr 52.6%

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

      1. add-sqr-sqrt 52.6%

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

      2. pow2 52.6%

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

      3. pow1/2 52.6%

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

      4. sqrt-pow1 52.6%

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

      5. metadata-eval 52.6%

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

   8. Applied egg-rr 52.6%

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

      1. pow-pow 52.6%

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

      2. metadata-eval 52.6%

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

      3. pow1/2 52.6%

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

      4. *-commutative 52.6%

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

      5. clear-num 52.6%

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

      6. sqrt-div 52.5%

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

      7. metadata-eval 52.5%

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

      8. div-inv 52.6%

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

      9. clear-num 52.6%

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

   10. Applied egg-rr 52.6%

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

   11. Taylor expanded in d around -inf 78.9%

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

       1. associate-*r* 78.9%

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

       2. rem-exp-log 77.5%

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

       3. exp-neg 77.5%

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

       4. unpow1/2 77.5%

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

       5. exp-prod 77.5%

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

       6. distribute-lft-neg-out 77.5%

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

       7. distribute-rgt-neg-in 77.5%

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

       8. metadata-eval 77.5%

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

       9. exp-to-pow 79.0%

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

       10. neg-mul-1 79.0%

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

       11. distribute-lft-neg-in 79.0%

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

       12. distribute-rgt-neg-in 79.0%

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

   13. Simplified 79.0%

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

   if -3.999999999999988e-310 < l

   1. Initial program 63.3%

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

   3. Simplified 63.2%

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

      1. *-commutative 63.2%

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

      2. unpow2 63.2%

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

      3. associate-*r* 64.6%

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

      4. *-commutative 64.6%

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

      5. associate-*l/ 64.7%

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

      6. associate-*r/ 64.7%

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

      7. *-commutative 64.7%

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

      8. associate-*l/ 65.6%

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

      9. associate-*r/ 65.6%

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

      10. associate-*r* 63.3%

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

      11. unpow2 63.3%

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

      12. associate-*l/ 65.1%

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

   6. Applied egg-rr 65.0%

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

      1. add-sqr-sqrt 64.8%

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

      2. pow2 64.8%

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

      3. pow1/2 64.8%

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

      4. sqrt-pow1 64.9%

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

      5. metadata-eval 64.9%

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

   8. Applied egg-rr 64.9%

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

      1. sqrt-div 75.5%

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

      2. pow-pow 75.6%

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

      3. metadata-eval 75.6%

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

      4. pow1/2 75.6%

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

      5. sqrt-div 84.1%

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

      6. frac-times 84.1%

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

      7. add-sqr-sqrt 84.2%

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

   10. Applied egg-rr 84.2%

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

       1. *-commutative 84.2%

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

   12. Simplified 84.2%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.55 \cdot 10^{+138}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq -7.2 \cdot 10^{-230}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right) \cdot \frac{1}{\frac{\sqrt{\frac{h}{d}}}{\sqrt{\frac{d}{\ell}}}}\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * ((h * ((m * (0.5d0 * (d_m / d))) ** 2.0d0)) / l)
    if (h <= (-5d-310)) then
        tmp = (d * ((l * h) ** (-0.5d0))) * (t_0 + (-1.0d0))
    else
        tmp = (1.0d0 - t_0) * (d / (sqrt(l) * sqrt(h)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 0.5 * ((h * Math.pow((M * (0.5 * (D_m / d))), 2.0)) / l);
	double tmp;
	if (h <= -5e-310) {
		tmp = (d * Math.pow((l * h), -0.5)) * (t_0 + -1.0);
	} else {
		tmp = (1.0 - t_0) * (d / (Math.sqrt(l) * Math.sqrt(h)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = 0.5 * ((h * math.pow((M * (0.5 * (D_m / d))), 2.0)) / l)
	tmp = 0
	if h <= -5e-310:
		tmp = (d * math.pow((l * h), -0.5)) * (t_0 + -1.0)
	else:
		tmp = (1.0 - t_0) * (d / (math.sqrt(l) * math.sqrt(h)))
	return tmp

Julia

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

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = 0.5 * ((h * ((M * (0.5 * (D_m / d))) ^ 2.0)) / l);
	tmp = 0.0;
	if (h <= -5e-310)
		tmp = (d * ((l * h) ^ -0.5)) * (t_0 + -1.0);
	else
		tmp = (1.0 - t_0) * (d / (sqrt(l) * sqrt(h)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 63.6%

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

   3. Simplified 63.6%

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

      1. *-commutative 63.6%

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

      2. unpow2 63.6%

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

      3. associate-*r* 65.1%

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

      4. *-commutative 65.1%

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

      5. associate-*l/ 64.4%

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

      6. associate-*r/ 64.3%

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

      7. *-commutative 64.3%

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

      8. associate-*l/ 64.3%

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

      9. associate-*r/ 65.0%

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

      10. associate-*r* 62.8%

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

      11. unpow2 62.8%

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

      12. associate-*l/ 64.4%

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

   6. Applied egg-rr 64.7%

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

      1. add-sqr-sqrt 64.5%

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

      2. pow2 64.5%

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

      3. pow1/2 64.5%

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

      4. sqrt-pow1 64.6%

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

      5. metadata-eval 64.6%

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

   8. Applied egg-rr 64.6%

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

      1. pow-pow 64.7%

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

      2. metadata-eval 64.7%

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

      3. pow1/2 64.7%

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

      4. *-commutative 64.7%

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

      5. clear-num 64.7%

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

      6. sqrt-div 65.6%

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

      7. metadata-eval 65.6%

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

      8. div-inv 65.6%

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

      9. clear-num 65.7%

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

   10. Applied egg-rr 65.7%

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

   11. Taylor expanded in d around -inf 71.3%

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

       1. associate-*r* 71.3%

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

       2. rem-exp-log 68.9%

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

       3. exp-neg 68.9%

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

       4. unpow1/2 68.9%

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

       5. exp-prod 68.9%

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

       6. distribute-lft-neg-out 68.9%

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

       7. distribute-rgt-neg-in 68.9%

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

       8. metadata-eval 68.9%

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

       9. exp-to-pow 71.3%

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

       10. neg-mul-1 71.3%

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

       11. distribute-lft-neg-in 71.3%

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

       12. distribute-rgt-neg-in 71.3%

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

   13. Simplified 71.3%

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

   if -4.999999999999985e-310 < h

   1. Initial program 63.3%

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

   3. Simplified 63.2%

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

      1. *-commutative 63.2%

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

      2. unpow2 63.2%

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

      3. associate-*r* 64.6%

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

      4. *-commutative 64.6%

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

      5. associate-*l/ 64.7%

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

      6. associate-*r/ 64.7%

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

      7. *-commutative 64.7%

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

      8. associate-*l/ 65.6%

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

      9. associate-*r/ 65.6%

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

      10. associate-*r* 63.3%

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

      11. unpow2 63.3%

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

      12. associate-*l/ 65.1%

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

   6. Applied egg-rr 65.0%

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

      1. add-sqr-sqrt 64.8%

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

      2. pow2 64.8%

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

      3. pow1/2 64.8%

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

      4. sqrt-pow1 64.9%

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

      5. metadata-eval 64.9%

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

   8. Applied egg-rr 64.9%

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

      1. sqrt-div 75.5%

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

      2. pow-pow 75.6%

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

      3. metadata-eval 75.6%

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

      4. pow1/2 75.6%

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

      5. sqrt-div 84.1%

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

      6. frac-times 84.1%

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

      7. add-sqr-sqrt 84.2%

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

   10. Applied egg-rr 84.2%

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

       1. *-commutative 84.2%

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

   12. Simplified 84.2%

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

4. Final simplification 77.3%

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

Alternative 6: 72.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ t_1 := 1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\ell}\\ \mathbf{if}\;h \leq -4 \cdot 10^{+133}:\\ \;\;\;\;t\_1 \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* d (pow (* l h) -0.5)))
        (t_1 (- 1.0 (* 0.5 (/ (* h (pow (* M (* 0.5 (/ D_m d))) 2.0)) l)))))
   (if (<= h -4e+133)
     (* t_1 (sqrt (* (/ d l) (/ d h))))
     (if (<= h -5e-310)
       (* t_0 (+ (* 0.5 (* (/ h l) (pow (* (/ M 2.0) (/ D_m d)) 2.0))) -1.0))
       (* t_1 t_0)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = d * pow((l * h), -0.5);
	double t_1 = 1.0 - (0.5 * ((h * pow((M * (0.5 * (D_m / d))), 2.0)) / l));
	double tmp;
	if (h <= -4e+133) {
		tmp = t_1 * sqrt(((d / l) * (d / h)));
	} else if (h <= -5e-310) {
		tmp = t_0 * ((0.5 * ((h / l) * pow(((M / 2.0) * (D_m / d)), 2.0))) + -1.0);
	} else {
		tmp = t_1 * t_0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = d * ((l * h) ** (-0.5d0))
    t_1 = 1.0d0 - (0.5d0 * ((h * ((m * (0.5d0 * (d_m / d))) ** 2.0d0)) / l))
    if (h <= (-4d+133)) then
        tmp = t_1 * sqrt(((d / l) * (d / h)))
    else if (h <= (-5d-310)) then
        tmp = t_0 * ((0.5d0 * ((h / l) * (((m / 2.0d0) * (d_m / d)) ** 2.0d0))) + (-1.0d0))
    else
        tmp = t_1 * t_0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = d * Math.pow((l * h), -0.5);
	double t_1 = 1.0 - (0.5 * ((h * Math.pow((M * (0.5 * (D_m / d))), 2.0)) / l));
	double tmp;
	if (h <= -4e+133) {
		tmp = t_1 * Math.sqrt(((d / l) * (d / h)));
	} else if (h <= -5e-310) {
		tmp = t_0 * ((0.5 * ((h / l) * Math.pow(((M / 2.0) * (D_m / d)), 2.0))) + -1.0);
	} else {
		tmp = t_1 * t_0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = d * math.pow((l * h), -0.5)
	t_1 = 1.0 - (0.5 * ((h * math.pow((M * (0.5 * (D_m / d))), 2.0)) / l))
	tmp = 0
	if h <= -4e+133:
		tmp = t_1 * math.sqrt(((d / l) * (d / h)))
	elif h <= -5e-310:
		tmp = t_0 * ((0.5 * ((h / l) * math.pow(((M / 2.0) * (D_m / d)), 2.0))) + -1.0)
	else:
		tmp = t_1 * t_0
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(d * (Float64(l * h) ^ -0.5))
	t_1 = Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(M * Float64(0.5 * Float64(D_m / d))) ^ 2.0)) / l)))
	tmp = 0.0
	if (h <= -4e+133)
		tmp = Float64(t_1 * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	elseif (h <= -5e-310)
		tmp = Float64(t_0 * Float64(Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M / 2.0) * Float64(D_m / d)) ^ 2.0))) + -1.0));
	else
		tmp = Float64(t_1 * t_0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = d * ((l * h) ^ -0.5);
	t_1 = 1.0 - (0.5 * ((h * ((M * (0.5 * (D_m / d))) ^ 2.0)) / l));
	tmp = 0.0;
	if (h <= -4e+133)
		tmp = t_1 * sqrt(((d / l) * (d / h)));
	elseif (h <= -5e-310)
		tmp = t_0 * ((0.5 * ((h / l) * (((M / 2.0) * (D_m / d)) ^ 2.0))) + -1.0);
	else
		tmp = t_1 * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(M * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -4e+133], N[(t$95$1 * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-310], N[(t$95$0 * N[(N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -4.0000000000000001e133

   1. Initial program 56.8%

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

   3. Simplified 56.8%

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

      1. *-commutative 56.8%

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

      2. unpow2 56.8%

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

      3. associate-*r* 56.9%

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

      4. *-commutative 56.9%

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

      5. associate-*l/ 56.9%

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

      6. associate-*r/ 54.3%

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

      7. *-commutative 54.3%

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

      8. associate-*l/ 54.3%

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

      9. associate-*r/ 54.3%

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

      10. associate-*r* 54.3%

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

      11. unpow2 54.3%

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

      12. associate-*l/ 59.6%

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

   6. Applied egg-rr 60.4%

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

      1. add-sqr-sqrt 60.3%

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

      2. pow2 60.3%

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

      3. pow1/2 60.3%

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

      4. sqrt-pow1 60.3%

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

      5. metadata-eval 60.3%

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

   8. Applied egg-rr 60.3%

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

      1. pow1/2 60.3%

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

      2. pow-pow 60.4%

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

      3. metadata-eval 60.4%

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

      4. pow-prod-down 58.0%

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

   10. Applied egg-rr 58.0%

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

       1. unpow1/2 58.0%

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

       2. *-commutative 58.0%

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

   12. Simplified 58.0%

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

   if -4.0000000000000001e133 < h < -4.999999999999985e-310

   1. Initial program 66.3%

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

   3. Simplified 66.3%

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

      1. clear-num 66.3%

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

      2. sqrt-div 67.6%

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

      3. metadata-eval 67.6%

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

   6. Applied egg-rr 67.6%

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

   7. Taylor expanded in d around -inf 75.4%

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

      1. mul-1-neg 75.4%

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

      2. *-commutative 75.4%

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

      3. inv-pow 75.4%

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

      4. sqrt-pow1 75.5%

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

      5. metadata-eval 75.5%

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

   9. Applied egg-rr 75.5%

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

   if -4.999999999999985e-310 < h

   1. Initial program 63.3%

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

   3. Simplified 63.2%

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

      1. *-commutative 63.2%

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

      2. unpow2 63.2%

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

      3. associate-*r* 64.6%

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

      4. *-commutative 64.6%

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

      5. associate-*l/ 64.7%

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

      6. associate-*r/ 64.7%

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

      7. *-commutative 64.7%

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

      8. associate-*l/ 65.6%

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

      9. associate-*r/ 65.6%

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

      10. associate-*r* 63.3%

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

      11. unpow2 63.3%

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

      12. associate-*l/ 65.1%

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

   6. Applied egg-rr 65.0%

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

      1. add-sqr-sqrt 64.8%

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

      2. pow2 64.8%

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

      3. pow1/2 64.8%

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

      4. sqrt-pow1 64.9%

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

      5. metadata-eval 64.9%

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

   8. Applied egg-rr 64.9%

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

      1. pow-pow 65.0%

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

      2. metadata-eval 65.0%

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

      3. pow1/2 65.0%

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

      4. *-commutative 65.0%

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

      5. clear-num 64.9%

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

      6. sqrt-div 64.9%

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

      7. metadata-eval 64.9%

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

      8. div-inv 64.9%

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

      9. clear-num 64.9%

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

   10. Applied egg-rr 64.9%

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

   11. Taylor expanded in h around 0 78.5%

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

       1. rem-exp-log 76.2%

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

       2. exp-neg 76.2%

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

       3. unpow1/2 76.2%

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

       4. exp-prod 76.2%

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

       5. distribute-lft-neg-out 76.2%

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

       6. distribute-rgt-neg-in 76.2%

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

       7. metadata-eval 76.2%

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

       8. exp-to-pow 78.5%

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

   13. Simplified 78.5%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -4 \cdot 10^{+133}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{-141}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq -2.8 \cdot 10^{-289}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\ell}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= l -1.8e-141)
   (* (- d) (sqrt (/ (/ 1.0 h) l)))
   (if (<= l -2.8e-289)
     (* (sqrt (/ d l)) (sqrt (/ d h)))
     (*
      (- 1.0 (* 0.5 (/ (* h (pow (* M (* 0.5 (/ D_m d))) 2.0)) l)))
      (* d (pow (* l h) -0.5))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= -1.8e-141) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else if (l <= -2.8e-289) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = (1.0 - (0.5 * ((h * pow((M * (0.5 * (D_m / d))), 2.0)) / l))) * (d * pow((l * h), -0.5));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-1.8d-141)) then
        tmp = -d * sqrt(((1.0d0 / h) / l))
    else if (l <= (-2.8d-289)) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = (1.0d0 - (0.5d0 * ((h * ((m * (0.5d0 * (d_m / d))) ** 2.0d0)) / l))) * (d * ((l * h) ** (-0.5d0)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= -1.8e-141) {
		tmp = -d * Math.sqrt(((1.0 / h) / l));
	} else if (l <= -2.8e-289) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = (1.0 - (0.5 * ((h * Math.pow((M * (0.5 * (D_m / d))), 2.0)) / l))) * (d * Math.pow((l * h), -0.5));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if l <= -1.8e-141:
		tmp = -d * math.sqrt(((1.0 / h) / l))
	elif l <= -2.8e-289:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = (1.0 - (0.5 * ((h * math.pow((M * (0.5 * (D_m / d))), 2.0)) / l))) * (d * math.pow((l * h), -0.5))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (l <= -1.8e-141)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	elseif (l <= -2.8e-289)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(M * Float64(0.5 * Float64(D_m / d))) ^ 2.0)) / l))) * Float64(d * (Float64(l * h) ^ -0.5)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (l <= -1.8e-141)
		tmp = -d * sqrt(((1.0 / h) / l));
	elseif (l <= -2.8e-289)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = (1.0 - (0.5 * ((h * ((M * (0.5 * (D_m / d))) ^ 2.0)) / l))) * (d * ((l * h) ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[l, -1.8e-141], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2.8e-289], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(M * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.80000000000000007e-141

   1. Initial program 59.2%

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

   3. Simplified 59.2%

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

      1. add-sqr-sqrt 59.2%

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

      2. pow2 59.2%

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

      3. sqrt-prod 59.2%

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

      4. sqrt-pow1 61.2%

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

      5. metadata-eval 61.2%

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

      6. pow1 61.2%

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

      7. div-inv 61.2%

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

      8. metadata-eval 61.2%

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

      9. associate-*l* 61.2%

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

   6. Applied egg-rr 61.2%

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

   7. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 51.3%

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

      6. neg-mul-1 51.3%

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

   9. Simplified 51.3%

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

   if -1.80000000000000007e-141 < l < -2.79999999999999985e-289

   1. Initial program 82.6%

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

   3. Simplified 82.6%

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

      1. add-sqr-sqrt 82.5%

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

      2. pow2 82.5%

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

      3. sqrt-prod 82.5%

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

      4. sqrt-pow1 82.5%

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

      5. metadata-eval 82.5%

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

      6. pow1 82.5%

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

      7. div-inv 82.5%

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

      8. metadata-eval 82.5%

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

      9. associate-*l* 82.5%

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

   6. Applied egg-rr 82.5%

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

   7. Taylor expanded in M around 0 52.4%

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

   if -2.79999999999999985e-289 < l

   1. Initial program 61.8%

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

   3. Simplified 61.7%

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

      1. *-commutative 61.7%

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

      2. unpow2 61.7%

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

      3. associate-*r* 63.1%

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

      4. *-commutative 63.1%

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

      5. associate-*l/ 63.1%

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

      6. associate-*r/ 63.1%

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

      7. *-commutative 63.1%

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

      8. associate-*l/ 64.8%

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

      9. associate-*r/ 64.8%

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

      10. associate-*r* 61.8%

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

      11. unpow2 61.8%

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

      12. associate-*l/ 63.5%

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

   6. Applied egg-rr 63.5%

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

      1. add-sqr-sqrt 63.3%

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

      2. pow2 63.3%

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

      3. pow1/2 63.3%

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

      4. sqrt-pow1 63.4%

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

      5. metadata-eval 63.4%

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

   8. Applied egg-rr 63.4%

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

      1. pow-pow 63.5%

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

      2. metadata-eval 63.5%

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

      3. pow1/2 63.5%

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

      4. *-commutative 63.5%

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

      5. clear-num 63.4%

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

      6. sqrt-div 63.4%

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

      7. metadata-eval 63.4%

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

      8. div-inv 63.4%

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

      9. clear-num 63.4%

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

   10. Applied egg-rr 63.4%

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

   11. Taylor expanded in h around 0 77.3%

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

       1. rem-exp-log 75.1%

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

       2. exp-neg 75.1%

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

       3. unpow1/2 75.1%

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

       4. exp-prod 75.1%

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

       5. distribute-lft-neg-out 75.1%

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

       6. distribute-rgt-neg-in 75.1%

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

       7. metadata-eval 75.1%

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

       8. exp-to-pow 77.4%

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

   13. Simplified 77.4%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{-141}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq -2.8 \cdot 10^{-289}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * ((h * ((m * (0.5d0 * (d_m / d))) ** 2.0d0)) / l)
    t_1 = d * ((l * h) ** (-0.5d0))
    if (h <= (-5d-310)) then
        tmp = t_1 * (t_0 + (-1.0d0))
    else
        tmp = (1.0d0 - t_0) * t_1
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 0.5 * ((h * Math.pow((M * (0.5 * (D_m / d))), 2.0)) / l);
	double t_1 = d * Math.pow((l * h), -0.5);
	double tmp;
	if (h <= -5e-310) {
		tmp = t_1 * (t_0 + -1.0);
	} else {
		tmp = (1.0 - t_0) * t_1;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = 0.5 * ((h * math.pow((M * (0.5 * (D_m / d))), 2.0)) / l)
	t_1 = d * math.pow((l * h), -0.5)
	tmp = 0
	if h <= -5e-310:
		tmp = t_1 * (t_0 + -1.0)
	else:
		tmp = (1.0 - t_0) * t_1
	return tmp

Julia

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

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = 0.5 * ((h * ((M * (0.5 * (D_m / d))) ^ 2.0)) / l);
	t_1 = d * ((l * h) ^ -0.5);
	tmp = 0.0;
	if (h <= -5e-310)
		tmp = t_1 * (t_0 + -1.0);
	else
		tmp = (1.0 - t_0) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 63.6%

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

   3. Simplified 63.6%

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

      1. *-commutative 63.6%

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

      2. unpow2 63.6%

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

      3. associate-*r* 65.1%

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

      4. *-commutative 65.1%

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

      5. associate-*l/ 64.4%

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

      6. associate-*r/ 64.3%

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

      7. *-commutative 64.3%

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

      8. associate-*l/ 64.3%

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

      9. associate-*r/ 65.0%

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

      10. associate-*r* 62.8%

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

      11. unpow2 62.8%

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

      12. associate-*l/ 64.4%

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

   6. Applied egg-rr 64.7%

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

      1. add-sqr-sqrt 64.5%

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

      2. pow2 64.5%

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

      3. pow1/2 64.5%

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

      4. sqrt-pow1 64.6%

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

      5. metadata-eval 64.6%

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

   8. Applied egg-rr 64.6%

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

      1. pow-pow 64.7%

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

      2. metadata-eval 64.7%

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

      3. pow1/2 64.7%

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

      4. *-commutative 64.7%

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

      5. clear-num 64.7%

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

      6. sqrt-div 65.6%

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

      7. metadata-eval 65.6%

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

      8. div-inv 65.6%

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

      9. clear-num 65.7%

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

   10. Applied egg-rr 65.7%

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

   11. Taylor expanded in d around -inf 71.3%

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

       1. associate-*r* 71.3%

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

       2. rem-exp-log 68.9%

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

       3. exp-neg 68.9%

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

       4. unpow1/2 68.9%

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

       5. exp-prod 68.9%

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

       6. distribute-lft-neg-out 68.9%

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

       7. distribute-rgt-neg-in 68.9%

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

       8. metadata-eval 68.9%

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

       9. exp-to-pow 71.3%

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

       10. neg-mul-1 71.3%

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

       11. distribute-lft-neg-in 71.3%

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

       12. distribute-rgt-neg-in 71.3%

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

   13. Simplified 71.3%

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

   if -4.999999999999985e-310 < h

   1. Initial program 63.3%

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

   3. Simplified 63.2%

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

      1. *-commutative 63.2%

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

      2. unpow2 63.2%

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

      3. associate-*r* 64.6%

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

      4. *-commutative 64.6%

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

      5. associate-*l/ 64.7%

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

      6. associate-*r/ 64.7%

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

      7. *-commutative 64.7%

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

      8. associate-*l/ 65.6%

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

      9. associate-*r/ 65.6%

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

      10. associate-*r* 63.3%

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

      11. unpow2 63.3%

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

      12. associate-*l/ 65.1%

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

   6. Applied egg-rr 65.0%

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

      1. add-sqr-sqrt 64.8%

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

      2. pow2 64.8%

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

      3. pow1/2 64.8%

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

      4. sqrt-pow1 64.9%

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

      5. metadata-eval 64.9%

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

   8. Applied egg-rr 64.9%

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

      1. pow-pow 65.0%

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

      2. metadata-eval 65.0%

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

      3. pow1/2 65.0%

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

      4. *-commutative 65.0%

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

      5. clear-num 64.9%

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

      6. sqrt-div 64.9%

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

      7. metadata-eval 64.9%

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

      8. div-inv 64.9%

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

      9. clear-num 64.9%

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

   10. Applied egg-rr 64.9%

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

   11. Taylor expanded in h around 0 78.5%

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

       1. rem-exp-log 76.2%

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

       2. exp-neg 76.2%

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

       3. unpow1/2 76.2%

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

       4. exp-prod 76.2%

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

       5. distribute-lft-neg-out 76.2%

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

       6. distribute-rgt-neg-in 76.2%

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

       7. metadata-eval 76.2%

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

       8. exp-to-pow 78.5%

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

   13. Simplified 78.5%

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

4. Final simplification 74.6%

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

Alternative 9: 72.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d * ((l * h) ** (-0.5d0))
    if (h <= (-5d-310)) then
        tmp = t_0 * ((0.5d0 * ((h / l) * (((m / 2.0d0) * (d_m / d)) ** 2.0d0))) + (-1.0d0))
    else
        tmp = (1.0d0 - (0.5d0 * ((h * ((m * (0.5d0 * (d_m / d))) ** 2.0d0)) / l))) * t_0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = d * Math.pow((l * h), -0.5);
	double tmp;
	if (h <= -5e-310) {
		tmp = t_0 * ((0.5 * ((h / l) * Math.pow(((M / 2.0) * (D_m / d)), 2.0))) + -1.0);
	} else {
		tmp = (1.0 - (0.5 * ((h * Math.pow((M * (0.5 * (D_m / d))), 2.0)) / l))) * t_0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = d * math.pow((l * h), -0.5)
	tmp = 0
	if h <= -5e-310:
		tmp = t_0 * ((0.5 * ((h / l) * math.pow(((M / 2.0) * (D_m / d)), 2.0))) + -1.0)
	else:
		tmp = (1.0 - (0.5 * ((h * math.pow((M * (0.5 * (D_m / d))), 2.0)) / l))) * t_0
	return tmp

Julia

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

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = d * ((l * h) ^ -0.5);
	tmp = 0.0;
	if (h <= -5e-310)
		tmp = t_0 * ((0.5 * ((h / l) * (((M / 2.0) * (D_m / d)) ^ 2.0))) + -1.0);
	else
		tmp = (1.0 - (0.5 * ((h * ((M * (0.5 * (D_m / d))) ^ 2.0)) / l))) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 63.6%

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

   3. Simplified 63.6%

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

      1. clear-num 63.6%

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

      2. sqrt-div 64.5%

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

      3. metadata-eval 64.5%

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

   6. Applied egg-rr 64.5%

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

   7. Taylor expanded in d around -inf 66.8%

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

      1. mul-1-neg 66.8%

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

      2. *-commutative 66.8%

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

      3. inv-pow 66.8%

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

      4. sqrt-pow1 66.8%

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

      5. metadata-eval 66.8%

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

   9. Applied egg-rr 66.8%

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

   if -4.999999999999985e-310 < h

   1. Initial program 63.3%

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

   3. Simplified 63.2%

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

      1. *-commutative 63.2%

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

      2. unpow2 63.2%

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

      3. associate-*r* 64.6%

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

      4. *-commutative 64.6%

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

      5. associate-*l/ 64.7%

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

      6. associate-*r/ 64.7%

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

      7. *-commutative 64.7%

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

      8. associate-*l/ 65.6%

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

      9. associate-*r/ 65.6%

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

      10. associate-*r* 63.3%

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

      11. unpow2 63.3%

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

      12. associate-*l/ 65.1%

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

   6. Applied egg-rr 65.0%

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

      1. add-sqr-sqrt 64.8%

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

      2. pow2 64.8%

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

      3. pow1/2 64.8%

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

      4. sqrt-pow1 64.9%

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

      5. metadata-eval 64.9%

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

   8. Applied egg-rr 64.9%

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

      1. pow-pow 65.0%

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

      2. metadata-eval 65.0%

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

      3. pow1/2 65.0%

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

      4. *-commutative 65.0%

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

      5. clear-num 64.9%

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

      6. sqrt-div 64.9%

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

      7. metadata-eval 64.9%

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

      8. div-inv 64.9%

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

      9. clear-num 64.9%

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

   10. Applied egg-rr 64.9%

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

   11. Taylor expanded in h around 0 78.5%

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

       1. rem-exp-log 76.2%

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

       2. exp-neg 76.2%

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

       3. unpow1/2 76.2%

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

       4. exp-prod 76.2%

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

       5. distribute-lft-neg-out 76.2%

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

       6. distribute-rgt-neg-in 76.2%

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

       7. metadata-eval 76.2%

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

       8. exp-to-pow 78.5%

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

   13. Simplified 78.5%

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

4. Final simplification 72.2%

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

Alternative 10: 42.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2.1 \cdot 10^{-183}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= d -2.1e-183)
   (* (- d) (sqrt (/ (/ 1.0 h) l)))
   (* d (pow (* l h) -0.5))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= -2.1e-183) {
		tmp = -d * sqrt(((1.0 / h) / l));
	} else {
		tmp = d * pow((l * h), -0.5);
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-2.1d-183)) then
        tmp = -d * sqrt(((1.0d0 / h) / l))
    else
        tmp = d * ((l * h) ** (-0.5d0))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= -2.1e-183) {
		tmp = -d * Math.sqrt(((1.0 / h) / l));
	} else {
		tmp = d * Math.pow((l * h), -0.5);
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if d <= -2.1e-183:
		tmp = -d * math.sqrt(((1.0 / h) / l))
	else:
		tmp = d * math.pow((l * h), -0.5)
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (d <= -2.1e-183)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l)));
	else
		tmp = Float64(d * (Float64(l * h) ^ -0.5));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (d <= -2.1e-183)
		tmp = -d * sqrt(((1.0 / h) / l));
	else
		tmp = d * ((l * h) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[d, -2.1e-183], N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -2.1000000000000002e-183

   1. Initial program 65.2%

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

   3. Simplified 65.2%

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

      1. add-sqr-sqrt 65.2%

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

      2. pow2 65.2%

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

      3. sqrt-prod 65.1%

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

      4. sqrt-pow1 66.9%

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

      5. metadata-eval 66.9%

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

      6. pow1 66.9%

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

      7. div-inv 66.9%

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

      8. metadata-eval 66.9%

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

      9. associate-*l* 66.9%

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

   6. Applied egg-rr 66.9%

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

   7. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 53.7%

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

      6. neg-mul-1 53.7%

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

   9. Simplified 53.7%

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

   if -2.1000000000000002e-183 < d

   1. Initial program 62.0%

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

   3. Simplified 61.9%

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

      1. add-sqr-sqrt 61.9%

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

      2. pow2 61.9%

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

      3. sqrt-prod 61.9%

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

      4. sqrt-pow1 63.2%

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

      5. metadata-eval 63.2%

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

      6. pow1 63.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. div-inv 63.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. metadata-eval 63.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. associate-*l* 63.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 63.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 40.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. associate-/r* 40.2%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   9. Simplified 40.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]

   10. Taylor expanded in h around 0 40.2%

       \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. rem-exp-log 38.5%

          \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}} \]

       2. exp-neg 38.5%

          \[\leadsto d \cdot \sqrt{\color{blue}{e^{-\log \left(h \cdot \ell\right)}}} \]

       3. unpow1/2 38.5%

          \[\leadsto d \cdot \color{blue}{{\left(e^{-\log \left(h \cdot \ell\right)}\right)}^{0.5}} \]

       4. exp-prod 38.5%

          \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

       5. distribute-lft-neg-out 38.5%

          \[\leadsto d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

       6. distribute-rgt-neg-in 38.5%

          \[\leadsto d \cdot e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \]

       7. metadata-eval 38.5%

          \[\leadsto d \cdot e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \]

       8. exp-to-pow 40.2%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   12. Simplified 40.2%

       \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.1 \cdot 10^{-183}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 27.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ d \cdot {\left(\ell \cdot h\right)}^{-0.5} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m) :precision binary64 (* d (pow (* l h) -0.5)))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	return d * pow((l * h), -0.5);
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    code = d * ((l * h) ** (-0.5d0))
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	return d * Math.pow((l * h), -0.5);
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	return d * math.pow((l * h), -0.5)

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	return Float64(d * (Float64(l * h) ^ -0.5))
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp = code(d, h, l, M, D_m)
	tmp = d * ((l * h) ^ -0.5);
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.4%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 63.4%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 63.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

   2. pow2 63.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

   3. sqrt-prod 63.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

   4. sqrt-pow1 64.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   5. metadata-eval 64.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. pow1 64.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   7. div-inv 64.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   8. metadata-eval 64.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   9. associate-*l* 64.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

6. Applied egg-rr 64.9%

   \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

7. Taylor expanded in d around inf 24.6%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
8. Step-by-step derivation

   1. associate-/r* 24.6%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

9. Simplified 24.6%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]

10. Taylor expanded in h around 0 24.6%

    \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]
11. Step-by-step derivation

    1. rem-exp-log 23.6%

       \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}} \]

    2. exp-neg 23.6%

       \[\leadsto d \cdot \sqrt{\color{blue}{e^{-\log \left(h \cdot \ell\right)}}} \]

    3. unpow1/2 23.6%

       \[\leadsto d \cdot \color{blue}{{\left(e^{-\log \left(h \cdot \ell\right)}\right)}^{0.5}} \]

    4. exp-prod 23.6%

       \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

    5. distribute-lft-neg-out 23.6%

       \[\leadsto d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

    6. distribute-rgt-neg-in 23.6%

       \[\leadsto d \cdot e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \]

    7. metadata-eval 23.6%

       \[\leadsto d \cdot e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \]

    8. exp-to-pow 24.6%

       \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

12. Simplified 24.6%

    \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

13. Final simplification 24.6%

    \[\leadsto d \cdot {\left(\ell \cdot h\right)}^{-0.5} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024111 
(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)))))