Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.6% → 83.0%

Time: 25.5s

Alternatives: 17

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \left(0.5 \cdot \frac{M}{d}\right)\\ \mathbf{if}\;h \leq -1.5 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D\_m \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(0.5 \cdot {\left(t\_0 \cdot \sqrt{\frac{h}{\ell}}\right)}^{2} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(\left(1 - h \cdot \left({t\_0}^{2} \cdot \frac{0.5}{\ell}\right)\right) \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* D_m (* 0.5 (/ M d)))))
   (if (<= h -1.5e+118)
     (*
      (sqrt (/ d l))
      (*
       (/ (sqrt (- d)) (sqrt (- h)))
       (+ 1.0 (* (/ h l) (* (pow (* D_m (/ (/ M 2.0) d)) 2.0) -0.5)))))
     (if (<= h -5e-310)
       (*
        (* d (pow (* h l) -0.5))
        (+ (* 0.5 (pow (* t_0 (sqrt (/ h l))) 2.0)) -1.0))
       (*
        d
        (*
         (- 1.0 (* h (* (pow t_0 2.0) (/ 0.5 l))))
         (* (pow l -0.5) (pow 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 t_0 = D_m * (0.5 * (M / d));
	double tmp;
	if (h <= -1.5e+118) {
		tmp = sqrt((d / l)) * ((sqrt(-d) / sqrt(-h)) * (1.0 + ((h / l) * (pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))));
	} else if (h <= -5e-310) {
		tmp = (d * pow((h * l), -0.5)) * ((0.5 * pow((t_0 * sqrt((h / l))), 2.0)) + -1.0);
	} else {
		tmp = d * ((1.0 - (h * (pow(t_0, 2.0) * (0.5 / l)))) * (pow(l, -0.5) * pow(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) :: t_0
    real(8) :: tmp
    t_0 = d_m * (0.5d0 * (m / d))
    if (h <= (-1.5d+118)) then
        tmp = sqrt((d / l)) * ((sqrt(-d) / sqrt(-h)) * (1.0d0 + ((h / l) * (((d_m * ((m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))))
    else if (h <= (-5d-310)) then
        tmp = (d * ((h * l) ** (-0.5d0))) * ((0.5d0 * ((t_0 * sqrt((h / l))) ** 2.0d0)) + (-1.0d0))
    else
        tmp = d * ((1.0d0 - (h * ((t_0 ** 2.0d0) * (0.5d0 / l)))) * ((l ** (-0.5d0)) * (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 t_0 = D_m * (0.5 * (M / d));
	double tmp;
	if (h <= -1.5e+118) {
		tmp = Math.sqrt((d / l)) * ((Math.sqrt(-d) / Math.sqrt(-h)) * (1.0 + ((h / l) * (Math.pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))));
	} else if (h <= -5e-310) {
		tmp = (d * Math.pow((h * l), -0.5)) * ((0.5 * Math.pow((t_0 * Math.sqrt((h / l))), 2.0)) + -1.0);
	} else {
		tmp = d * ((1.0 - (h * (Math.pow(t_0, 2.0) * (0.5 / l)))) * (Math.pow(l, -0.5) * Math.pow(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):
	t_0 = D_m * (0.5 * (M / d))
	tmp = 0
	if h <= -1.5e+118:
		tmp = math.sqrt((d / l)) * ((math.sqrt(-d) / math.sqrt(-h)) * (1.0 + ((h / l) * (math.pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))))
	elif h <= -5e-310:
		tmp = (d * math.pow((h * l), -0.5)) * ((0.5 * math.pow((t_0 * math.sqrt((h / l))), 2.0)) + -1.0)
	else:
		tmp = d * ((1.0 - (h * (math.pow(t_0, 2.0) * (0.5 / l)))) * (math.pow(l, -0.5) * math.pow(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)
	t_0 = Float64(D_m * Float64(0.5 * Float64(M / d)))
	tmp = 0.0
	if (h <= -1.5e+118)
		tmp = Float64(sqrt(Float64(d / l)) * 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 (h <= -5e-310)
		tmp = Float64(Float64(d * (Float64(h * l) ^ -0.5)) * Float64(Float64(0.5 * (Float64(t_0 * sqrt(Float64(h / l))) ^ 2.0)) + -1.0));
	else
		tmp = Float64(d * Float64(Float64(1.0 - Float64(h * Float64((t_0 ^ 2.0) * Float64(0.5 / l)))) * Float64((l ^ -0.5) * (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)
	t_0 = D_m * (0.5 * (M / d));
	tmp = 0.0;
	if (h <= -1.5e+118)
		tmp = sqrt((d / l)) * ((sqrt(-d) / sqrt(-h)) * (1.0 + ((h / l) * (((D_m * ((M / 2.0) / d)) ^ 2.0) * -0.5))));
	elseif (h <= -5e-310)
		tmp = (d * ((h * l) ^ -0.5)) * ((0.5 * ((t_0 * sqrt((h / l))) ^ 2.0)) + -1.0);
	else
		tmp = d * ((1.0 - (h * ((t_0 ^ 2.0) * (0.5 / l)))) * ((l ^ -0.5) * (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_] := Block[{t$95$0 = N[(D$95$m * N[(0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -1.5e+118], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * 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[h, -5e-310], N[(N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[Power[N[(t$95$0 * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 - N[(h * N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.5e118

   1. Initial program 45.1%

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

   3. Simplified 45.1%

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

      1. frac-2neg 45.1%

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

      2. sqrt-div 68.8%

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

   6. Applied egg-rr 68.8%

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

   if -1.5e118 < h < -4.999999999999985e-310

   1. Initial program 57.1%

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

   3. Simplified 56.1%

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

      1. add-sqr-sqrt 56.1%

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

      2. pow2 56.1%

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

      3. sqrt-prod 56.1%

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

      4. sqrt-pow1 59.2%

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

      5. frac-times 61.3%

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

      6. associate-/r* 61.3%

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

      7. metadata-eval 61.3%

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

      8. pow1 61.3%

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

      9. div-inv 61.3%

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

      10. *-commutative 61.3%

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

      11. associate-*r* 61.3%

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

      12. div-inv 61.3%

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

      13. associate-*r/ 60.2%

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

      14. associate-/l/ 60.2%

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

      15. *-un-lft-identity 60.2%

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

      16. *-commutative 60.2%

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

      17. times-frac 60.2%

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

      18. metadata-eval 60.2%

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

   6. Applied egg-rr 60.2%

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

   7. Taylor expanded in d around 0 1.7%

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

      1. associate-/r* 1.7%

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

   9. Simplified 1.7%

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

   10. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 79.3%

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

       4. mul-1-neg 79.3%

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

       5. rem-exp-log 76.6%

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

       6. exp-neg 76.6%

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

       7. unpow1/2 76.6%

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

       8. exp-prod 76.6%

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

       9. distribute-lft-neg-out 76.6%

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

       10. distribute-rgt-neg-in 76.6%

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

       11. metadata-eval 76.6%

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

       12. exp-to-pow 79.3%

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

   12. Simplified 79.3%

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

   if -4.999999999999985e-310 < h

   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 64.7%

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

   5. Taylor expanded in d around 0 69.5%

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

      1. pow1 69.5%

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

      2. associate-*l* 70.4%

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

      3. pow1/2 70.4%

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

      4. inv-pow 70.4%

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

      5. pow-pow 70.4%

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

      6. metadata-eval 70.4%

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

      7. associate-*r* 70.4%

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

      8. div-inv 70.4%

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

      9. metadata-eval 70.4%

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

   7. Applied egg-rr 70.4%

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

   9. Simplified 80.2%

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

       1. *-commutative 80.2%

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

       2. unpow-prod-down 92.2%

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

   11. Applied egg-rr 92.2%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.5 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(\left(1 - h \cdot \left({\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right) \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := h \cdot \left({\left(D\_m \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right)\\ \mathbf{if}\;\ell \leq -1.75 \cdot 10^{+248}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + -0.5 \cdot {\left(\left(D\_m \cdot \sqrt{\frac{h}{\ell}}\right) \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-311}:\\ \;\;\;\;d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(t\_0 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(\left(1 - t\_0\right) \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* h (* (pow (* D_m (* 0.5 (/ M d))) 2.0) (/ 0.5 l)))))
   (if (<= l -1.75e+248)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (+ 1.0 (* -0.5 (pow (* (* D_m (sqrt (/ h l))) (* M (/ 0.5 d))) 2.0))))
     (if (<= l -1e-311)
       (* d (* (sqrt (/ 1.0 (* h l))) (+ t_0 -1.0)))
       (* d (* (- 1.0 t_0) (* (pow l -0.5) (pow 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 t_0 = h * (pow((D_m * (0.5 * (M / d))), 2.0) * (0.5 / l));
	double tmp;
	if (l <= -1.75e+248) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 + (-0.5 * pow(((D_m * sqrt((h / l))) * (M * (0.5 / d))), 2.0)));
	} else if (l <= -1e-311) {
		tmp = d * (sqrt((1.0 / (h * l))) * (t_0 + -1.0));
	} else {
		tmp = d * ((1.0 - t_0) * (pow(l, -0.5) * pow(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) :: t_0
    real(8) :: tmp
    t_0 = h * (((d_m * (0.5d0 * (m / d))) ** 2.0d0) * (0.5d0 / l))
    if (l <= (-1.75d+248)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 + ((-0.5d0) * (((d_m * sqrt((h / l))) * (m * (0.5d0 / d))) ** 2.0d0)))
    else if (l <= (-1d-311)) then
        tmp = d * (sqrt((1.0d0 / (h * l))) * (t_0 + (-1.0d0)))
    else
        tmp = d * ((1.0d0 - t_0) * ((l ** (-0.5d0)) * (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 t_0 = h * (Math.pow((D_m * (0.5 * (M / d))), 2.0) * (0.5 / l));
	double tmp;
	if (l <= -1.75e+248) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 + (-0.5 * Math.pow(((D_m * Math.sqrt((h / l))) * (M * (0.5 / d))), 2.0)));
	} else if (l <= -1e-311) {
		tmp = d * (Math.sqrt((1.0 / (h * l))) * (t_0 + -1.0));
	} else {
		tmp = d * ((1.0 - t_0) * (Math.pow(l, -0.5) * Math.pow(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):
	t_0 = h * (math.pow((D_m * (0.5 * (M / d))), 2.0) * (0.5 / l))
	tmp = 0
	if l <= -1.75e+248:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 + (-0.5 * math.pow(((D_m * math.sqrt((h / l))) * (M * (0.5 / d))), 2.0)))
	elif l <= -1e-311:
		tmp = d * (math.sqrt((1.0 / (h * l))) * (t_0 + -1.0))
	else:
		tmp = d * ((1.0 - t_0) * (math.pow(l, -0.5) * math.pow(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)
	t_0 = Float64(h * Float64((Float64(D_m * Float64(0.5 * Float64(M / d))) ^ 2.0) * Float64(0.5 / l)))
	tmp = 0.0
	if (l <= -1.75e+248)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 + Float64(-0.5 * (Float64(Float64(D_m * sqrt(Float64(h / l))) * Float64(M * Float64(0.5 / d))) ^ 2.0))));
	elseif (l <= -1e-311)
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(t_0 + -1.0)));
	else
		tmp = Float64(d * Float64(Float64(1.0 - t_0) * Float64((l ^ -0.5) * (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)
	t_0 = h * (((D_m * (0.5 * (M / d))) ^ 2.0) * (0.5 / l));
	tmp = 0.0;
	if (l <= -1.75e+248)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 + (-0.5 * (((D_m * sqrt((h / l))) * (M * (0.5 / d))) ^ 2.0)));
	elseif (l <= -1e-311)
		tmp = d * (sqrt((1.0 / (h * l))) * (t_0 + -1.0));
	else
		tmp = d * ((1.0 - t_0) * ((l ^ -0.5) * (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_] := Block[{t$95$0 = N[(h * N[(N[Power[N[(D$95$m * N[(0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.75e+248], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[Power[N[(N[(D$95$m * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-311], N[(d * N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 - t$95$0), $MachinePrecision] * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.75000000000000011e248

   1. Initial program 54.7%

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

   3. Simplified 54.7%

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

      1. add-sqr-sqrt 54.7%

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

      2. pow2 54.7%

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

      3. sqrt-prod 54.7%

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

      4. sqrt-pow1 66.5%

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

      5. frac-times 60.5%

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

      6. associate-/r* 60.5%

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

      7. metadata-eval 60.5%

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

      8. pow1 60.5%

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

      9. div-inv 60.5%

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

      10. *-commutative 60.5%

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

      11. associate-*r* 60.5%

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

      12. div-inv 60.5%

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

      13. associate-*r/ 66.3%

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

      14. associate-/l/ 66.3%

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

      15. *-un-lft-identity 66.3%

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

      16. *-commutative 66.3%

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

      17. times-frac 66.3%

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

      18. metadata-eval 66.3%

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

   6. Applied egg-rr 66.3%

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

   7. Taylor expanded in D around 0 60.5%

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

      1. associate-*r* 60.5%

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

      2. associate-/l* 66.3%

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

      3. associate-*r* 66.3%

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

      4. *-commutative 66.3%

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

      5. associate-*r* 66.3%

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

      6. associate-/l* 66.3%

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

      7. associate-*l* 66.3%

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

      8. *-commutative 66.3%

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

      9. *-commutative 66.3%

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

      10. associate-/l* 66.3%

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

   9. Simplified 66.3%

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

       1. cancel-sign-sub-inv 66.3%

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

       2. metadata-eval 66.3%

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

       3. associate-*r* 66.4%

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

   11. Applied egg-rr 66.4%

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

   if -1.75000000000000011e248 < l < -9.99999999999948e-312

   1. Initial program 53.2%

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

   3. Simplified 52.4%

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

   5. Taylor expanded in d around 0 2.3%

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

      1. pow1 2.3%

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

      2. associate-*l* 2.3%

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

      3. pow1/2 2.3%

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

      4. inv-pow 2.3%

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

      5. pow-pow 2.3%

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

      6. metadata-eval 2.3%

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

      7. associate-*r* 2.3%

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

      8. div-inv 2.3%

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

      9. metadata-eval 2.3%

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

   7. Applied egg-rr 2.3%

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

   9. Simplified 3.1%

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

   10. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 74.1%

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

       4. mul-1-neg 74.1%

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

   12. Simplified 74.1%

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

   if -9.99999999999948e-312 < l

   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 64.7%

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

   5. Taylor expanded in d around 0 69.5%

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

      1. pow1 69.5%

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

      2. associate-*l* 70.4%

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

      3. pow1/2 70.4%

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

      4. inv-pow 70.4%

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

      5. pow-pow 70.4%

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

      6. metadata-eval 70.4%

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

      7. associate-*r* 70.4%

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

      8. div-inv 70.4%

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

      9. metadata-eval 70.4%

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

   7. Applied egg-rr 70.4%

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

   9. Simplified 80.2%

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

       1. *-commutative 80.2%

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

       2. unpow-prod-down 92.2%

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

   11. Applied egg-rr 92.2%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.75 \cdot 10^{+248}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + -0.5 \cdot {\left(\left(D \cdot \sqrt{\frac{h}{\ell}}\right) \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-311}:\\ \;\;\;\;d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(h \cdot \left({\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(\left(1 - h \cdot \left({\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right) \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \left(0.5 \cdot \frac{M}{d}\right)\\ t_1 := h \cdot \left({t\_0}^{2} \cdot \frac{0.5}{\ell}\right)\\ \mathbf{if}\;h \leq -2.95 \cdot 10^{+273}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq -1.25 \cdot 10^{+35}:\\ \;\;\;\;d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(t\_1 + -1\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(0.5 \cdot {\left(t\_0 \cdot \sqrt{\frac{h}{\ell}}\right)}^{2} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(\left(1 - t\_1\right) \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* D_m (* 0.5 (/ M d)))) (t_1 (* h (* (pow t_0 2.0) (/ 0.5 l)))))
   (if (<= h -2.95e+273)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M 2.0) (/ D_m d)) 2.0)))))
     (if (<= h -1.25e+35)
       (* d (* (sqrt (/ 1.0 (* h l))) (+ t_1 -1.0)))
       (if (<= h -5e-310)
         (*
          (* d (pow (* h l) -0.5))
          (+ (* 0.5 (pow (* t_0 (sqrt (/ h l))) 2.0)) -1.0))
         (* d (* (- 1.0 t_1) (* (pow l -0.5) (pow 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 t_0 = D_m * (0.5 * (M / d));
	double t_1 = h * (pow(t_0, 2.0) * (0.5 / l));
	double tmp;
	if (h <= -2.95e+273) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * ((h / l) * pow(((M / 2.0) * (D_m / d)), 2.0))));
	} else if (h <= -1.25e+35) {
		tmp = d * (sqrt((1.0 / (h * l))) * (t_1 + -1.0));
	} else if (h <= -5e-310) {
		tmp = (d * pow((h * l), -0.5)) * ((0.5 * pow((t_0 * sqrt((h / l))), 2.0)) + -1.0);
	} else {
		tmp = d * ((1.0 - t_1) * (pow(l, -0.5) * pow(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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = d_m * (0.5d0 * (m / d))
    t_1 = h * ((t_0 ** 2.0d0) * (0.5d0 / l))
    if (h <= (-2.95d+273)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (0.5d0 * ((h / l) * (((m / 2.0d0) * (d_m / d)) ** 2.0d0))))
    else if (h <= (-1.25d+35)) then
        tmp = d * (sqrt((1.0d0 / (h * l))) * (t_1 + (-1.0d0)))
    else if (h <= (-5d-310)) then
        tmp = (d * ((h * l) ** (-0.5d0))) * ((0.5d0 * ((t_0 * sqrt((h / l))) ** 2.0d0)) + (-1.0d0))
    else
        tmp = d * ((1.0d0 - t_1) * ((l ** (-0.5d0)) * (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 t_0 = D_m * (0.5 * (M / d));
	double t_1 = h * (Math.pow(t_0, 2.0) * (0.5 / l));
	double tmp;
	if (h <= -2.95e+273) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (0.5 * ((h / l) * Math.pow(((M / 2.0) * (D_m / d)), 2.0))));
	} else if (h <= -1.25e+35) {
		tmp = d * (Math.sqrt((1.0 / (h * l))) * (t_1 + -1.0));
	} else if (h <= -5e-310) {
		tmp = (d * Math.pow((h * l), -0.5)) * ((0.5 * Math.pow((t_0 * Math.sqrt((h / l))), 2.0)) + -1.0);
	} else {
		tmp = d * ((1.0 - t_1) * (Math.pow(l, -0.5) * Math.pow(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):
	t_0 = D_m * (0.5 * (M / d))
	t_1 = h * (math.pow(t_0, 2.0) * (0.5 / l))
	tmp = 0
	if h <= -2.95e+273:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (0.5 * ((h / l) * math.pow(((M / 2.0) * (D_m / d)), 2.0))))
	elif h <= -1.25e+35:
		tmp = d * (math.sqrt((1.0 / (h * l))) * (t_1 + -1.0))
	elif h <= -5e-310:
		tmp = (d * math.pow((h * l), -0.5)) * ((0.5 * math.pow((t_0 * math.sqrt((h / l))), 2.0)) + -1.0)
	else:
		tmp = d * ((1.0 - t_1) * (math.pow(l, -0.5) * math.pow(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)
	t_0 = Float64(D_m * Float64(0.5 * Float64(M / d)))
	t_1 = Float64(h * Float64((t_0 ^ 2.0) * Float64(0.5 / l)))
	tmp = 0.0
	if (h <= -2.95e+273)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M / 2.0) * Float64(D_m / d)) ^ 2.0)))));
	elseif (h <= -1.25e+35)
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(t_1 + -1.0)));
	elseif (h <= -5e-310)
		tmp = Float64(Float64(d * (Float64(h * l) ^ -0.5)) * Float64(Float64(0.5 * (Float64(t_0 * sqrt(Float64(h / l))) ^ 2.0)) + -1.0));
	else
		tmp = Float64(d * Float64(Float64(1.0 - t_1) * Float64((l ^ -0.5) * (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)
	t_0 = D_m * (0.5 * (M / d));
	t_1 = h * ((t_0 ^ 2.0) * (0.5 / l));
	tmp = 0.0;
	if (h <= -2.95e+273)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * ((h / l) * (((M / 2.0) * (D_m / d)) ^ 2.0))));
	elseif (h <= -1.25e+35)
		tmp = d * (sqrt((1.0 / (h * l))) * (t_1 + -1.0));
	elseif (h <= -5e-310)
		tmp = (d * ((h * l) ^ -0.5)) * ((0.5 * ((t_0 * sqrt((h / l))) ^ 2.0)) + -1.0);
	else
		tmp = d * ((1.0 - t_1) * ((l ^ -0.5) * (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_] := Block[{t$95$0 = N[(D$95$m * N[(0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(h * N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -2.95e+273], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - 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]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1.25e+35], N[(d * N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-310], N[(N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[Power[N[(t$95$0 * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 - t$95$1), $MachinePrecision] * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -2.94999999999999989e273

   1. Initial program 66.8%

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

   3. Simplified 66.8%

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

   if -2.94999999999999989e273 < h < -1.25000000000000005e35

   1. Initial program 40.0%

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

   3. Simplified 40.0%

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

   5. Taylor expanded in d around 0 7.2%

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

      1. pow1 7.2%

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

      2. associate-*l* 7.2%

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

      3. pow1/2 7.2%

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

      4. inv-pow 7.2%

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

      5. pow-pow 7.2%

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

      6. metadata-eval 7.2%

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

      7. associate-*r* 7.2%

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

      8. div-inv 7.2%

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

      9. metadata-eval 7.2%

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

   7. Applied egg-rr 7.2%

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

   9. Simplified 8.9%

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

   10. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 68.3%

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

       4. mul-1-neg 68.3%

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

   12. Simplified 68.3%

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

   if -1.25000000000000005e35 < h < -4.999999999999985e-310

   1. Initial program 63.5%

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

   3. Simplified 62.0%

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

      1. add-sqr-sqrt 62.0%

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

      2. pow2 62.0%

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

      3. sqrt-prod 62.0%

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

      4. sqrt-pow1 65.1%

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

      5. frac-times 68.0%

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

      6. associate-/r* 68.0%

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

      7. metadata-eval 68.0%

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

      8. pow1 68.0%

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

      9. div-inv 68.0%

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

      10. *-commutative 68.0%

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

      11. associate-*r* 68.0%

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

      12. div-inv 68.0%

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

      13. associate-*r/ 66.4%

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

      14. associate-/l/ 66.4%

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

      15. *-un-lft-identity 66.4%

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

      16. *-commutative 66.4%

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

      17. times-frac 66.4%

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

      18. metadata-eval 66.4%

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

   6. Applied egg-rr 66.4%

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

   7. Taylor expanded in d around 0 0.6%

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

      1. associate-/r* 0.6%

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

   9. Simplified 0.6%

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

   10. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 82.4%

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

       4. mul-1-neg 82.4%

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

       5. rem-exp-log 79.8%

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

       6. exp-neg 79.8%

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

       7. unpow1/2 79.8%

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

       8. exp-prod 79.8%

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

       9. distribute-lft-neg-out 79.8%

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

       10. distribute-rgt-neg-in 79.8%

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

       11. metadata-eval 79.8%

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

       12. exp-to-pow 82.5%

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

   12. Simplified 82.5%

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

   if -4.999999999999985e-310 < h

   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 64.7%

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

   5. Taylor expanded in d around 0 69.5%

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

      1. pow1 69.5%

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

      2. associate-*l* 70.4%

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

      3. pow1/2 70.4%

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

      4. inv-pow 70.4%

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

      5. pow-pow 70.4%

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

      6. metadata-eval 70.4%

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

      7. associate-*r* 70.4%

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

      8. div-inv 70.4%

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

      9. metadata-eval 70.4%

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

   7. Applied egg-rr 70.4%

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

   9. Simplified 80.2%

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

       1. *-commutative 80.2%

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

       2. unpow-prod-down 92.2%

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

   11. Applied egg-rr 92.2%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.95 \cdot 10^{+273}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq -1.25 \cdot 10^{+35}:\\ \;\;\;\;d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(h \cdot \left({\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right) + -1\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(\left(1 - h \cdot \left({\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right) \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \left(0.5 \cdot \frac{M}{d}\right)\\ t_1 := h \cdot \left({t\_0}^{2} \cdot \frac{0.5}{\ell}\right)\\ \mathbf{if}\;h \leq -2.6 \cdot 10^{+274}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D\_m \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;h \leq -6.4 \cdot 10^{+35}:\\ \;\;\;\;d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(t\_1 + -1\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(0.5 \cdot {\left(t\_0 \cdot \sqrt{\frac{h}{\ell}}\right)}^{2} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(\left(1 - t\_1\right) \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* D_m (* 0.5 (/ M d)))) (t_1 (* h (* (pow t_0 2.0) (/ 0.5 l)))))
   (if (<= h -2.6e+274)
     (*
      (sqrt (/ d l))
      (*
       (+ 1.0 (* (/ h l) (* (pow (* D_m (/ (/ M 2.0) d)) 2.0) -0.5)))
       (sqrt (/ d h))))
     (if (<= h -6.4e+35)
       (* d (* (sqrt (/ 1.0 (* h l))) (+ t_1 -1.0)))
       (if (<= h -5e-310)
         (*
          (* d (pow (* h l) -0.5))
          (+ (* 0.5 (pow (* t_0 (sqrt (/ h l))) 2.0)) -1.0))
         (* d (* (- 1.0 t_1) (* (pow l -0.5) (pow 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 t_0 = D_m * (0.5 * (M / d));
	double t_1 = h * (pow(t_0, 2.0) * (0.5 / l));
	double tmp;
	if (h <= -2.6e+274) {
		tmp = sqrt((d / l)) * ((1.0 + ((h / l) * (pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))) * sqrt((d / h)));
	} else if (h <= -6.4e+35) {
		tmp = d * (sqrt((1.0 / (h * l))) * (t_1 + -1.0));
	} else if (h <= -5e-310) {
		tmp = (d * pow((h * l), -0.5)) * ((0.5 * pow((t_0 * sqrt((h / l))), 2.0)) + -1.0);
	} else {
		tmp = d * ((1.0 - t_1) * (pow(l, -0.5) * pow(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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = d_m * (0.5d0 * (m / d))
    t_1 = h * ((t_0 ** 2.0d0) * (0.5d0 / l))
    if (h <= (-2.6d+274)) then
        tmp = sqrt((d / l)) * ((1.0d0 + ((h / l) * (((d_m * ((m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))) * sqrt((d / h)))
    else if (h <= (-6.4d+35)) then
        tmp = d * (sqrt((1.0d0 / (h * l))) * (t_1 + (-1.0d0)))
    else if (h <= (-5d-310)) then
        tmp = (d * ((h * l) ** (-0.5d0))) * ((0.5d0 * ((t_0 * sqrt((h / l))) ** 2.0d0)) + (-1.0d0))
    else
        tmp = d * ((1.0d0 - t_1) * ((l ** (-0.5d0)) * (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 t_0 = D_m * (0.5 * (M / d));
	double t_1 = h * (Math.pow(t_0, 2.0) * (0.5 / l));
	double tmp;
	if (h <= -2.6e+274) {
		tmp = Math.sqrt((d / l)) * ((1.0 + ((h / l) * (Math.pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))) * Math.sqrt((d / h)));
	} else if (h <= -6.4e+35) {
		tmp = d * (Math.sqrt((1.0 / (h * l))) * (t_1 + -1.0));
	} else if (h <= -5e-310) {
		tmp = (d * Math.pow((h * l), -0.5)) * ((0.5 * Math.pow((t_0 * Math.sqrt((h / l))), 2.0)) + -1.0);
	} else {
		tmp = d * ((1.0 - t_1) * (Math.pow(l, -0.5) * Math.pow(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):
	t_0 = D_m * (0.5 * (M / d))
	t_1 = h * (math.pow(t_0, 2.0) * (0.5 / l))
	tmp = 0
	if h <= -2.6e+274:
		tmp = math.sqrt((d / l)) * ((1.0 + ((h / l) * (math.pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))) * math.sqrt((d / h)))
	elif h <= -6.4e+35:
		tmp = d * (math.sqrt((1.0 / (h * l))) * (t_1 + -1.0))
	elif h <= -5e-310:
		tmp = (d * math.pow((h * l), -0.5)) * ((0.5 * math.pow((t_0 * math.sqrt((h / l))), 2.0)) + -1.0)
	else:
		tmp = d * ((1.0 - t_1) * (math.pow(l, -0.5) * math.pow(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)
	t_0 = Float64(D_m * Float64(0.5 * Float64(M / d)))
	t_1 = Float64(h * Float64((t_0 ^ 2.0) * Float64(0.5 / l)))
	tmp = 0.0
	if (h <= -2.6e+274)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D_m * Float64(Float64(M / 2.0) / d)) ^ 2.0) * -0.5))) * sqrt(Float64(d / h))));
	elseif (h <= -6.4e+35)
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(t_1 + -1.0)));
	elseif (h <= -5e-310)
		tmp = Float64(Float64(d * (Float64(h * l) ^ -0.5)) * Float64(Float64(0.5 * (Float64(t_0 * sqrt(Float64(h / l))) ^ 2.0)) + -1.0));
	else
		tmp = Float64(d * Float64(Float64(1.0 - t_1) * Float64((l ^ -0.5) * (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)
	t_0 = D_m * (0.5 * (M / d));
	t_1 = h * ((t_0 ^ 2.0) * (0.5 / l));
	tmp = 0.0;
	if (h <= -2.6e+274)
		tmp = sqrt((d / l)) * ((1.0 + ((h / l) * (((D_m * ((M / 2.0) / d)) ^ 2.0) * -0.5))) * sqrt((d / h)));
	elseif (h <= -6.4e+35)
		tmp = d * (sqrt((1.0 / (h * l))) * (t_1 + -1.0));
	elseif (h <= -5e-310)
		tmp = (d * ((h * l) ^ -0.5)) * ((0.5 * ((t_0 * sqrt((h / l))) ^ 2.0)) + -1.0);
	else
		tmp = d * ((1.0 - t_1) * ((l ^ -0.5) * (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_] := Block[{t$95$0 = N[(D$95$m * N[(0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(h * N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -2.6e+274], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(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] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -6.4e+35], N[(d * N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-310], N[(N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[Power[N[(t$95$0 * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 - t$95$1), $MachinePrecision] * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -2.5999999999999998e274

   1. Initial program 66.8%

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

   3. Simplified 66.7%

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

   if -2.5999999999999998e274 < h < -6.39999999999999965e35

   1. Initial program 40.0%

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

   3. Simplified 40.0%

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

   5. Taylor expanded in d around 0 7.2%

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

      1. pow1 7.2%

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

      2. associate-*l* 7.2%

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

      3. pow1/2 7.2%

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

      4. inv-pow 7.2%

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

      5. pow-pow 7.2%

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

      6. metadata-eval 7.2%

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

      7. associate-*r* 7.2%

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

      8. div-inv 7.2%

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

      9. metadata-eval 7.2%

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

   7. Applied egg-rr 7.2%

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

   9. Simplified 8.9%

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

   10. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 68.3%

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

       4. mul-1-neg 68.3%

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

   12. Simplified 68.3%

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

   if -6.39999999999999965e35 < h < -4.999999999999985e-310

   1. Initial program 63.5%

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

   3. Simplified 62.0%

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

      1. add-sqr-sqrt 62.0%

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

      2. pow2 62.0%

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

      3. sqrt-prod 62.0%

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

      4. sqrt-pow1 65.1%

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

      5. frac-times 68.0%

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

      6. associate-/r* 68.0%

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

      7. metadata-eval 68.0%

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

      8. pow1 68.0%

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

      9. div-inv 68.0%

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

      10. *-commutative 68.0%

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

      11. associate-*r* 68.0%

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

      12. div-inv 68.0%

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

      13. associate-*r/ 66.4%

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

      14. associate-/l/ 66.4%

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

      15. *-un-lft-identity 66.4%

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

      16. *-commutative 66.4%

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

      17. times-frac 66.4%

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

      18. metadata-eval 66.4%

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

   6. Applied egg-rr 66.4%

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

   7. Taylor expanded in d around 0 0.6%

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

      1. associate-/r* 0.6%

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

   9. Simplified 0.6%

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

   10. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 82.4%

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

       4. mul-1-neg 82.4%

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

       5. rem-exp-log 79.8%

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

       6. exp-neg 79.8%

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

       7. unpow1/2 79.8%

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

       8. exp-prod 79.8%

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

       9. distribute-lft-neg-out 79.8%

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

       10. distribute-rgt-neg-in 79.8%

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

       11. metadata-eval 79.8%

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

       12. exp-to-pow 82.5%

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

   12. Simplified 82.5%

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

   if -4.999999999999985e-310 < h

   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 64.7%

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

   5. Taylor expanded in d around 0 69.5%

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

      1. pow1 69.5%

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

      2. associate-*l* 70.4%

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

      3. pow1/2 70.4%

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

      4. inv-pow 70.4%

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

      5. pow-pow 70.4%

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

      6. metadata-eval 70.4%

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

      7. associate-*r* 70.4%

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

      8. div-inv 70.4%

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

      9. metadata-eval 70.4%

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

   7. Applied egg-rr 70.4%

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

   9. Simplified 80.2%

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

       1. *-commutative 80.2%

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

       2. unpow-prod-down 92.2%

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

   11. Applied egg-rr 92.2%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.6 \cdot 10^{+274}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;h \leq -6.4 \cdot 10^{+35}:\\ \;\;\;\;d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(h \cdot \left({\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right) + -1\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(\left(1 - h \cdot \left({\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right) \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := h \cdot \left({\left(D\_m \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right)\\ \mathbf{if}\;h \leq -1.52 \cdot 10^{+274}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D\_m \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(t\_0 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(\left(1 - t\_0\right) \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* h (* (pow (* D_m (* 0.5 (/ M d))) 2.0) (/ 0.5 l)))))
   (if (<= h -1.52e+274)
     (*
      (sqrt (/ d l))
      (*
       (+ 1.0 (* (/ h l) (* (pow (* D_m (/ (/ M 2.0) d)) 2.0) -0.5)))
       (sqrt (/ d h))))
     (if (<= h -5e-310)
       (* d (* (sqrt (/ 1.0 (* h l))) (+ t_0 -1.0)))
       (* d (* (- 1.0 t_0) (* (pow l -0.5) (pow 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 t_0 = h * (pow((D_m * (0.5 * (M / d))), 2.0) * (0.5 / l));
	double tmp;
	if (h <= -1.52e+274) {
		tmp = sqrt((d / l)) * ((1.0 + ((h / l) * (pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))) * sqrt((d / h)));
	} else if (h <= -5e-310) {
		tmp = d * (sqrt((1.0 / (h * l))) * (t_0 + -1.0));
	} else {
		tmp = d * ((1.0 - t_0) * (pow(l, -0.5) * pow(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) :: t_0
    real(8) :: tmp
    t_0 = h * (((d_m * (0.5d0 * (m / d))) ** 2.0d0) * (0.5d0 / l))
    if (h <= (-1.52d+274)) then
        tmp = sqrt((d / l)) * ((1.0d0 + ((h / l) * (((d_m * ((m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))) * sqrt((d / h)))
    else if (h <= (-5d-310)) then
        tmp = d * (sqrt((1.0d0 / (h * l))) * (t_0 + (-1.0d0)))
    else
        tmp = d * ((1.0d0 - t_0) * ((l ** (-0.5d0)) * (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 t_0 = h * (Math.pow((D_m * (0.5 * (M / d))), 2.0) * (0.5 / l));
	double tmp;
	if (h <= -1.52e+274) {
		tmp = Math.sqrt((d / l)) * ((1.0 + ((h / l) * (Math.pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))) * Math.sqrt((d / h)));
	} else if (h <= -5e-310) {
		tmp = d * (Math.sqrt((1.0 / (h * l))) * (t_0 + -1.0));
	} else {
		tmp = d * ((1.0 - t_0) * (Math.pow(l, -0.5) * Math.pow(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):
	t_0 = h * (math.pow((D_m * (0.5 * (M / d))), 2.0) * (0.5 / l))
	tmp = 0
	if h <= -1.52e+274:
		tmp = math.sqrt((d / l)) * ((1.0 + ((h / l) * (math.pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))) * math.sqrt((d / h)))
	elif h <= -5e-310:
		tmp = d * (math.sqrt((1.0 / (h * l))) * (t_0 + -1.0))
	else:
		tmp = d * ((1.0 - t_0) * (math.pow(l, -0.5) * math.pow(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)
	t_0 = Float64(h * Float64((Float64(D_m * Float64(0.5 * Float64(M / d))) ^ 2.0) * Float64(0.5 / l)))
	tmp = 0.0
	if (h <= -1.52e+274)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D_m * Float64(Float64(M / 2.0) / d)) ^ 2.0) * -0.5))) * sqrt(Float64(d / h))));
	elseif (h <= -5e-310)
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(t_0 + -1.0)));
	else
		tmp = Float64(d * Float64(Float64(1.0 - t_0) * Float64((l ^ -0.5) * (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)
	t_0 = h * (((D_m * (0.5 * (M / d))) ^ 2.0) * (0.5 / l));
	tmp = 0.0;
	if (h <= -1.52e+274)
		tmp = sqrt((d / l)) * ((1.0 + ((h / l) * (((D_m * ((M / 2.0) / d)) ^ 2.0) * -0.5))) * sqrt((d / h)));
	elseif (h <= -5e-310)
		tmp = d * (sqrt((1.0 / (h * l))) * (t_0 + -1.0));
	else
		tmp = d * ((1.0 - t_0) * ((l ^ -0.5) * (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_] := Block[{t$95$0 = N[(h * N[(N[Power[N[(D$95$m * N[(0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -1.52e+274], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(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] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-310], N[(d * N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 - t$95$0), $MachinePrecision] * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.52e274

   1. Initial program 66.8%

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

   3. Simplified 66.7%

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

   if -1.52e274 < h < -4.999999999999985e-310

   1. Initial program 52.4%

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

   3. Simplified 51.7%

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

   5. Taylor expanded in d around 0 3.7%

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

      1. pow1 3.7%

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

      2. associate-*l* 3.7%

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

      3. pow1/2 3.7%

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

      4. inv-pow 3.7%

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

      5. pow-pow 3.7%

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

      6. metadata-eval 3.7%

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

      7. associate-*r* 3.7%

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

      8. div-inv 3.7%

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

      9. metadata-eval 3.7%

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

   7. Applied egg-rr 3.7%

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

   9. Simplified 4.6%

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

   10. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 73.3%

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

       4. mul-1-neg 73.3%

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

   12. Simplified 73.3%

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

   if -4.999999999999985e-310 < h

   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 64.7%

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

   5. Taylor expanded in d around 0 69.5%

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

      1. pow1 69.5%

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

      2. associate-*l* 70.4%

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

      3. pow1/2 70.4%

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

      4. inv-pow 70.4%

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

      5. pow-pow 70.4%

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

      6. metadata-eval 70.4%

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

      7. associate-*r* 70.4%

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

      8. div-inv 70.4%

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

      9. metadata-eval 70.4%

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

   7. Applied egg-rr 70.4%

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

   9. Simplified 80.2%

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

       1. *-commutative 80.2%

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

       2. unpow-prod-down 92.2%

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

   11. Applied egg-rr 92.2%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.52 \cdot 10^{+274}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(h \cdot \left({\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(\left(1 - h \cdot \left({\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right) \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := {\left(D\_m \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{-307}:\\ \;\;\;\;d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(h \cdot \left(t\_0 \cdot \frac{0.5}{\ell}\right) + -1\right)\right)\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+49}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 + -0.5 \cdot \frac{h \cdot {\left(M \cdot \frac{0.5}{\frac{d}{D\_m}}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \frac{t\_0}{\frac{\ell}{h}}\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (pow (* D_m (* 0.5 (/ M d))) 2.0)))
   (if (<= l -1e-307)
     (* d (* (sqrt (/ 1.0 (* h l))) (+ (* h (* t_0 (/ 0.5 l))) -1.0)))
     (if (<= l 1.3e+49)
       (*
        d
        (*
         (pow (* h l) -0.5)
         (+ 1.0 (* -0.5 (/ (* h (pow (* M (/ 0.5 (/ d D_m))) 2.0)) l)))))
       (* (/ d (* (sqrt l) (sqrt h))) (- 1.0 (* 0.5 (/ t_0 (/ l 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 = pow((D_m * (0.5 * (M / d))), 2.0);
	double tmp;
	if (l <= -1e-307) {
		tmp = d * (sqrt((1.0 / (h * l))) * ((h * (t_0 * (0.5 / l))) + -1.0));
	} else if (l <= 1.3e+49) {
		tmp = d * (pow((h * l), -0.5) * (1.0 + (-0.5 * ((h * pow((M * (0.5 / (d / D_m))), 2.0)) / l))));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (0.5 * (t_0 / (l / 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 = (d_m * (0.5d0 * (m / d))) ** 2.0d0
    if (l <= (-1d-307)) then
        tmp = d * (sqrt((1.0d0 / (h * l))) * ((h * (t_0 * (0.5d0 / l))) + (-1.0d0)))
    else if (l <= 1.3d+49) then
        tmp = d * (((h * l) ** (-0.5d0)) * (1.0d0 + ((-0.5d0) * ((h * ((m * (0.5d0 / (d / d_m))) ** 2.0d0)) / l))))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - (0.5d0 * (t_0 / (l / 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 = Math.pow((D_m * (0.5 * (M / d))), 2.0);
	double tmp;
	if (l <= -1e-307) {
		tmp = d * (Math.sqrt((1.0 / (h * l))) * ((h * (t_0 * (0.5 / l))) + -1.0));
	} else if (l <= 1.3e+49) {
		tmp = d * (Math.pow((h * l), -0.5) * (1.0 + (-0.5 * ((h * Math.pow((M * (0.5 / (d / D_m))), 2.0)) / l))));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - (0.5 * (t_0 / (l / 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 = math.pow((D_m * (0.5 * (M / d))), 2.0)
	tmp = 0
	if l <= -1e-307:
		tmp = d * (math.sqrt((1.0 / (h * l))) * ((h * (t_0 * (0.5 / l))) + -1.0))
	elif l <= 1.3e+49:
		tmp = d * (math.pow((h * l), -0.5) * (1.0 + (-0.5 * ((h * math.pow((M * (0.5 / (d / D_m))), 2.0)) / l))))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - (0.5 * (t_0 / (l / 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(D_m * Float64(0.5 * Float64(M / d))) ^ 2.0
	tmp = 0.0
	if (l <= -1e-307)
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(Float64(h * Float64(t_0 * Float64(0.5 / l))) + -1.0)));
	elseif (l <= 1.3e+49)
		tmp = Float64(d * Float64((Float64(h * l) ^ -0.5) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h * (Float64(M * Float64(0.5 / Float64(d / D_m))) ^ 2.0)) / l)))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - Float64(0.5 * Float64(t_0 / Float64(l / 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 = (D_m * (0.5 * (M / d))) ^ 2.0;
	tmp = 0.0;
	if (l <= -1e-307)
		tmp = d * (sqrt((1.0 / (h * l))) * ((h * (t_0 * (0.5 / l))) + -1.0));
	elseif (l <= 1.3e+49)
		tmp = d * (((h * l) ^ -0.5) * (1.0 + (-0.5 * ((h * ((M * (0.5 / (d / D_m))) ^ 2.0)) / l))));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (0.5 * (t_0 / (l / 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[Power[N[(D$95$m * N[(0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[l, -1e-307], N[(d * N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(h * N[(t$95$0 * N[(0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.3e+49], N[(d * N[(N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h * N[Power[N[(M * N[(0.5 / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(t$95$0 / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.99999999999999909e-308

   1. Initial program 53.8%

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

   3. Simplified 53.0%

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

   5. Taylor expanded in d around 0 3.7%

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

      1. pow1 3.7%

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

      2. associate-*l* 3.7%

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

      3. pow1/2 3.7%

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

      4. inv-pow 3.7%

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

      5. pow-pow 3.7%

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

      6. metadata-eval 3.7%

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

      7. associate-*r* 3.7%

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

      8. div-inv 3.7%

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

      9. metadata-eval 3.7%

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

   7. Applied egg-rr 3.7%

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

   9. Simplified 3.8%

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

   10. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 70.0%

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

       4. mul-1-neg 70.0%

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

   12. Simplified 70.0%

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

   if -9.99999999999999909e-308 < l < 1.29999999999999994e49

   1. Initial program 71.9%

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

   3. Simplified 69.7%

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

      1. add-sqr-sqrt 69.7%

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

      2. pow2 69.7%

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

      3. sqrt-prod 69.7%

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

      4. sqrt-pow1 71.0%

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

      5. frac-times 73.2%

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

      6. associate-/r* 73.2%

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

      7. metadata-eval 73.2%

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

      8. pow1 73.2%

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

      9. div-inv 73.2%

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

      10. *-commutative 73.2%

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

      11. associate-*r* 73.2%

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

      12. div-inv 73.2%

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

      13. associate-*r/ 71.9%

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

      14. associate-/l/ 71.9%

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

      15. *-un-lft-identity 71.9%

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

      16. *-commutative 71.9%

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

      17. times-frac 71.9%

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

      18. metadata-eval 71.9%

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

   6. Applied egg-rr 71.9%

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

   7. Taylor expanded in d around 0 78.9%

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

      1. associate-/r* 78.9%

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

   9. Simplified 78.9%

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

       1. pow1 78.9%

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

   11. Applied egg-rr 78.9%

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

       1. unpow1 78.9%

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

       2. *-commutative 78.9%

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

       3. associate-*r/ 78.9%

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

       4. *-commutative 78.9%

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

       5. associate-/l* 78.0%

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

       6. associate-*l/ 78.0%

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

       7. associate-*r/ 88.8%

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

       8. associate-/r/ 88.8%

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

   13. Simplified 88.8%

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

   if 1.29999999999999994e49 < l

   1. Initial program 52.4%

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

   3. Simplified 54.7%

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

      1. clear-num 52.5%

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

      2. un-div-inv 52.5%

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

   6. Applied egg-rr 54.7%

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

      1. *-commutative 54.7%

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

      2. sqrt-div 63.0%

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

      3. sqrt-div 76.9%

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

      4. frac-times 77.0%

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

      5. add-sqr-sqrt 77.2%

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

   8. Applied egg-rr 77.2%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-307}:\\ \;\;\;\;d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(h \cdot \left({\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right) + -1\right)\right)\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+49}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 + -0.5 \cdot \frac{h \cdot {\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\frac{\ell}{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-307}:\\ \;\;\;\;d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(h \cdot \left({\left(D\_m \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right) + -1\right)\right)\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{+16}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 + -0.5 \cdot \frac{h \cdot {\left(M \cdot \frac{0.5}{\frac{d}{D\_m}}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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
 (if (<= l -1e-307)
   (*
    d
    (*
     (sqrt (/ 1.0 (* h l)))
     (+ (* h (* (pow (* D_m (* 0.5 (/ M d))) 2.0) (/ 0.5 l))) -1.0)))
   (if (<= l 4.8e+16)
     (*
      d
      (*
       (pow (* h l) -0.5)
       (+ 1.0 (* -0.5 (/ (* h (pow (* M (/ 0.5 (/ d D_m))) 2.0)) l)))))
     (*
      (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M 2.0) (/ D_m d)) 2.0))))
      (/ 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 tmp;
	if (l <= -1e-307) {
		tmp = d * (sqrt((1.0 / (h * l))) * ((h * (pow((D_m * (0.5 * (M / d))), 2.0) * (0.5 / l))) + -1.0));
	} else if (l <= 4.8e+16) {
		tmp = d * (pow((h * l), -0.5) * (1.0 + (-0.5 * ((h * pow((M * (0.5 / (d / D_m))), 2.0)) / l))));
	} else {
		tmp = (1.0 - (0.5 * ((h / l) * pow(((M / 2.0) * (D_m / d)), 2.0)))) * (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) :: tmp
    if (l <= (-1d-307)) then
        tmp = d * (sqrt((1.0d0 / (h * l))) * ((h * (((d_m * (0.5d0 * (m / d))) ** 2.0d0) * (0.5d0 / l))) + (-1.0d0)))
    else if (l <= 4.8d+16) then
        tmp = d * (((h * l) ** (-0.5d0)) * (1.0d0 + ((-0.5d0) * ((h * ((m * (0.5d0 / (d / d_m))) ** 2.0d0)) / l))))
    else
        tmp = (1.0d0 - (0.5d0 * ((h / l) * (((m / 2.0d0) * (d_m / d)) ** 2.0d0)))) * (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 tmp;
	if (l <= -1e-307) {
		tmp = d * (Math.sqrt((1.0 / (h * l))) * ((h * (Math.pow((D_m * (0.5 * (M / d))), 2.0) * (0.5 / l))) + -1.0));
	} else if (l <= 4.8e+16) {
		tmp = d * (Math.pow((h * l), -0.5) * (1.0 + (-0.5 * ((h * Math.pow((M * (0.5 / (d / D_m))), 2.0)) / l))));
	} else {
		tmp = (1.0 - (0.5 * ((h / l) * Math.pow(((M / 2.0) * (D_m / d)), 2.0)))) * (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):
	tmp = 0
	if l <= -1e-307:
		tmp = d * (math.sqrt((1.0 / (h * l))) * ((h * (math.pow((D_m * (0.5 * (M / d))), 2.0) * (0.5 / l))) + -1.0))
	elif l <= 4.8e+16:
		tmp = d * (math.pow((h * l), -0.5) * (1.0 + (-0.5 * ((h * math.pow((M * (0.5 / (d / D_m))), 2.0)) / l))))
	else:
		tmp = (1.0 - (0.5 * ((h / l) * math.pow(((M / 2.0) * (D_m / d)), 2.0)))) * (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)
	tmp = 0.0
	if (l <= -1e-307)
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(Float64(h * Float64((Float64(D_m * Float64(0.5 * Float64(M / d))) ^ 2.0) * Float64(0.5 / l))) + -1.0)));
	elseif (l <= 4.8e+16)
		tmp = Float64(d * Float64((Float64(h * l) ^ -0.5) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h * (Float64(M * Float64(0.5 / Float64(d / D_m))) ^ 2.0)) / l)))));
	else
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M / 2.0) * Float64(D_m / d)) ^ 2.0)))) * 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)
	tmp = 0.0;
	if (l <= -1e-307)
		tmp = d * (sqrt((1.0 / (h * l))) * ((h * (((D_m * (0.5 * (M / d))) ^ 2.0) * (0.5 / l))) + -1.0));
	elseif (l <= 4.8e+16)
		tmp = d * (((h * l) ^ -0.5) * (1.0 + (-0.5 * ((h * ((M * (0.5 / (d / D_m))) ^ 2.0)) / l))));
	else
		tmp = (1.0 - (0.5 * ((h / l) * (((M / 2.0) * (D_m / d)) ^ 2.0)))) * (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_] := If[LessEqual[l, -1e-307], N[(d * N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(h * N[(N[Power[N[(D$95$m * N[(0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.8e+16], N[(d * N[(N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h * N[Power[N[(M * N[(0.5 / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - 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]), $MachinePrecision] * N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.99999999999999909e-308

   1. Initial program 53.8%

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

   3. Simplified 53.0%

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

   5. Taylor expanded in d around 0 3.7%

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

      1. pow1 3.7%

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

      2. associate-*l* 3.7%

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

      3. pow1/2 3.7%

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

      4. inv-pow 3.7%

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

      5. pow-pow 3.7%

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

      6. metadata-eval 3.7%

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

      7. associate-*r* 3.7%

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

      8. div-inv 3.7%

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

      9. metadata-eval 3.7%

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

   7. Applied egg-rr 3.7%

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

   9. Simplified 3.8%

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

   10. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 70.0%

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

       4. mul-1-neg 70.0%

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

   12. Simplified 70.0%

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

   if -9.99999999999999909e-308 < l < 4.8e16

   1. Initial program 68.5%

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

   3. Simplified 66.8%

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

      1. add-sqr-sqrt 66.9%

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

      2. pow2 66.9%

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

      3. sqrt-prod 66.8%

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

      4. sqrt-pow1 68.4%

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

      5. frac-times 70.1%

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

      6. associate-/r* 70.1%

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

      7. metadata-eval 70.1%

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

      8. pow1 70.1%

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

      9. div-inv 70.1%

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

      10. *-commutative 70.1%

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

      11. associate-*r* 70.1%

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

      12. div-inv 70.1%

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

      13. associate-*r/ 68.6%

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

      14. associate-/l/ 68.6%

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

      15. *-un-lft-identity 68.6%

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

      16. *-commutative 68.6%

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

      17. times-frac 68.6%

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

      18. metadata-eval 68.6%

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

   6. Applied egg-rr 68.6%

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

   7. Taylor expanded in d around 0 75.3%

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

      1. associate-/r* 75.3%

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

   9. Simplified 75.3%

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

       1. pow1 75.3%

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

   11. Applied egg-rr 75.3%

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

       1. unpow1 75.3%

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

       2. *-commutative 75.3%

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

       3. associate-*r/ 75.3%

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

       4. *-commutative 75.3%

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

       5. associate-/l* 75.1%

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

       6. associate-*l/ 75.1%

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

       7. associate-*r/ 87.7%

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

       8. associate-/r/ 87.7%

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

   13. Simplified 87.7%

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

   if 4.8e16 < l

   1. Initial program 60.1%

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

   3. Simplified 61.0%

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

      1. *-commutative 62.0%

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

      2. sqrt-div 68.5%

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

      3. sqrt-div 81.3%

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

      4. frac-times 81.6%

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

      5. add-sqr-sqrt 81.7%

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

   6. Applied egg-rr 80.8%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-307}:\\ \;\;\;\;d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(h \cdot \left({\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right) + -1\right)\right)\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{+16}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 + -0.5 \cdot \frac{h \cdot {\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := h \cdot \left({\left(D\_m \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right)\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{-311}:\\ \;\;\;\;d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(t\_0 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(\left(1 - t\_0\right) \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* h (* (pow (* D_m (* 0.5 (/ M d))) 2.0) (/ 0.5 l)))))
   (if (<= l -1e-311)
     (* d (* (sqrt (/ 1.0 (* h l))) (+ t_0 -1.0)))
     (* d (* (- 1.0 t_0) (* (pow l -0.5) (pow 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 t_0 = h * (pow((D_m * (0.5 * (M / d))), 2.0) * (0.5 / l));
	double tmp;
	if (l <= -1e-311) {
		tmp = d * (sqrt((1.0 / (h * l))) * (t_0 + -1.0));
	} else {
		tmp = d * ((1.0 - t_0) * (pow(l, -0.5) * pow(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) :: t_0
    real(8) :: tmp
    t_0 = h * (((d_m * (0.5d0 * (m / d))) ** 2.0d0) * (0.5d0 / l))
    if (l <= (-1d-311)) then
        tmp = d * (sqrt((1.0d0 / (h * l))) * (t_0 + (-1.0d0)))
    else
        tmp = d * ((1.0d0 - t_0) * ((l ** (-0.5d0)) * (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 t_0 = h * (Math.pow((D_m * (0.5 * (M / d))), 2.0) * (0.5 / l));
	double tmp;
	if (l <= -1e-311) {
		tmp = d * (Math.sqrt((1.0 / (h * l))) * (t_0 + -1.0));
	} else {
		tmp = d * ((1.0 - t_0) * (Math.pow(l, -0.5) * Math.pow(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):
	t_0 = h * (math.pow((D_m * (0.5 * (M / d))), 2.0) * (0.5 / l))
	tmp = 0
	if l <= -1e-311:
		tmp = d * (math.sqrt((1.0 / (h * l))) * (t_0 + -1.0))
	else:
		tmp = d * ((1.0 - t_0) * (math.pow(l, -0.5) * math.pow(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)
	t_0 = Float64(h * Float64((Float64(D_m * Float64(0.5 * Float64(M / d))) ^ 2.0) * Float64(0.5 / l)))
	tmp = 0.0
	if (l <= -1e-311)
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(t_0 + -1.0)));
	else
		tmp = Float64(d * Float64(Float64(1.0 - t_0) * Float64((l ^ -0.5) * (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)
	t_0 = h * (((D_m * (0.5 * (M / d))) ^ 2.0) * (0.5 / l));
	tmp = 0.0;
	if (l <= -1e-311)
		tmp = d * (sqrt((1.0 / (h * l))) * (t_0 + -1.0));
	else
		tmp = d * ((1.0 - t_0) * ((l ^ -0.5) * (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_] := Block[{t$95$0 = N[(h * N[(N[Power[N[(D$95$m * N[(0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1e-311], N[(d * N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 - t$95$0), $MachinePrecision] * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -9.99999999999948e-312

   1. Initial program 53.4%

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

   3. Simplified 52.7%

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

   5. Taylor expanded in d around 0 3.7%

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

      1. pow1 3.7%

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

      2. associate-*l* 3.7%

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

      3. pow1/2 3.7%

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

      4. inv-pow 3.7%

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

      5. pow-pow 3.7%

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

      6. metadata-eval 3.7%

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

      7. associate-*r* 3.7%

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

      8. div-inv 3.7%

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

      9. metadata-eval 3.7%

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

   7. Applied egg-rr 3.7%

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

   9. Simplified 4.5%

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

   10. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 69.5%

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

       4. mul-1-neg 69.5%

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

   12. Simplified 69.5%

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

   if -9.99999999999948e-312 < l

   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 64.7%

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

   5. Taylor expanded in d around 0 69.5%

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

      1. pow1 69.5%

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

      2. associate-*l* 70.4%

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

      3. pow1/2 70.4%

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

      4. inv-pow 70.4%

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

      5. pow-pow 70.4%

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

      6. metadata-eval 70.4%

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

      7. associate-*r* 70.4%

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

      8. div-inv 70.4%

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

      9. metadata-eval 70.4%

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

   7. Applied egg-rr 70.4%

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

   9. Simplified 80.2%

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

       1. *-commutative 80.2%

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

       2. unpow-prod-down 92.2%

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

   11. Applied egg-rr 92.2%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-311}:\\ \;\;\;\;d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(h \cdot \left({\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(\left(1 - h \cdot \left({\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right) \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := 1 - h \cdot \left({\left(D\_m \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right)\\ \mathbf{if}\;\ell \leq -1.15 \cdot 10^{-75}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -4.7 \cdot 10^{-300}:\\ \;\;\;\;t\_0 \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(t\_0 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (- 1.0 (* h (* (pow (* D_m (* 0.5 (/ M d))) 2.0) (/ 0.5 l))))))
   (if (<= l -1.15e-75)
     (* (- d) (sqrt (/ (/ 1.0 l) h)))
     (if (<= l -4.7e-300)
       (* t_0 (sqrt (* (/ d l) (/ d h))))
       (* d (* t_0 (sqrt (/ 1.0 (* h l)))))))))

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 = 1.0 - (h * (pow((D_m * (0.5 * (M / d))), 2.0) * (0.5 / l)));
	double tmp;
	if (l <= -1.15e-75) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else if (l <= -4.7e-300) {
		tmp = t_0 * sqrt(((d / l) * (d / h)));
	} else {
		tmp = d * (t_0 * sqrt((1.0 / (h * l))));
	}
	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 = 1.0d0 - (h * (((d_m * (0.5d0 * (m / d))) ** 2.0d0) * (0.5d0 / l)))
    if (l <= (-1.15d-75)) then
        tmp = -d * sqrt(((1.0d0 / l) / h))
    else if (l <= (-4.7d-300)) then
        tmp = t_0 * sqrt(((d / l) * (d / h)))
    else
        tmp = d * (t_0 * sqrt((1.0d0 / (h * l))))
    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 = 1.0 - (h * (Math.pow((D_m * (0.5 * (M / d))), 2.0) * (0.5 / l)));
	double tmp;
	if (l <= -1.15e-75) {
		tmp = -d * Math.sqrt(((1.0 / l) / h));
	} else if (l <= -4.7e-300) {
		tmp = t_0 * Math.sqrt(((d / l) * (d / h)));
	} else {
		tmp = d * (t_0 * Math.sqrt((1.0 / (h * l))));
	}
	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 = 1.0 - (h * (math.pow((D_m * (0.5 * (M / d))), 2.0) * (0.5 / l)))
	tmp = 0
	if l <= -1.15e-75:
		tmp = -d * math.sqrt(((1.0 / l) / h))
	elif l <= -4.7e-300:
		tmp = t_0 * math.sqrt(((d / l) * (d / h)))
	else:
		tmp = d * (t_0 * math.sqrt((1.0 / (h * l))))
	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(1.0 - Float64(h * Float64((Float64(D_m * Float64(0.5 * Float64(M / d))) ^ 2.0) * Float64(0.5 / l))))
	tmp = 0.0
	if (l <= -1.15e-75)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	elseif (l <= -4.7e-300)
		tmp = Float64(t_0 * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	else
		tmp = Float64(d * Float64(t_0 * sqrt(Float64(1.0 / Float64(h * l)))));
	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 = 1.0 - (h * (((D_m * (0.5 * (M / d))) ^ 2.0) * (0.5 / l)));
	tmp = 0.0;
	if (l <= -1.15e-75)
		tmp = -d * sqrt(((1.0 / l) / h));
	elseif (l <= -4.7e-300)
		tmp = t_0 * sqrt(((d / l) * (d / h)));
	else
		tmp = d * (t_0 * sqrt((1.0 / (h * l))));
	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[(1.0 - N[(h * N[(N[Power[N[(D$95$m * N[(0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.15e-75], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -4.7e-300], N[(t$95$0 * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(t$95$0 * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.15e-75

   1. Initial program 49.0%

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

   3. Simplified 47.8%

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

      1. add-sqr-sqrt 47.8%

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

      2. pow2 47.8%

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

      3. sqrt-prod 47.8%

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

      4. sqrt-pow1 52.4%

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

      5. frac-times 53.6%

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

      6. associate-/r* 53.6%

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

      7. metadata-eval 53.6%

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

      8. pow1 53.6%

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

      9. div-inv 53.6%

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

      10. *-commutative 53.6%

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

      11. associate-*r* 53.6%

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

      12. div-inv 53.6%

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

      13. associate-*r/ 53.5%

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

      14. associate-/l/ 53.5%

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

      15. *-un-lft-identity 53.5%

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

      16. *-commutative 53.5%

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

      17. times-frac 53.5%

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

      18. metadata-eval 53.5%

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

   6. Applied egg-rr 53.5%

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

   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. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 48.8%

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

      4. mul-1-neg 48.8%

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

      5. *-commutative 48.8%

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

      6. associate-/r* 49.5%

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

   9. Simplified 49.5%

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

   if -1.15e-75 < l < -4.7e-300

   1. Initial program 64.8%

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

   3. Simplified 64.8%

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

      1. add-sqr-sqrt 64.8%

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

      2. pow2 64.8%

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

      3. sqrt-prod 64.8%

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

      4. sqrt-pow1 64.8%

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

      5. frac-times 64.8%

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

      6. associate-/r* 64.8%

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

      7. metadata-eval 64.8%

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

      8. pow1 64.8%

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

      9. div-inv 64.8%

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

      10. *-commutative 64.8%

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

      11. associate-*r* 64.8%

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

      12. div-inv 64.8%

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

      13. associate-*r/ 64.8%

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

      14. associate-/l/ 64.8%

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

      15. *-un-lft-identity 64.8%

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

      16. *-commutative 64.8%

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

      17. times-frac 64.8%

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

      18. metadata-eval 64.8%

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

   6. Applied egg-rr 64.8%

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

      1. cancel-sign-sub-inv 64.8%

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

      2. unpow-prod-down 64.8%

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

      3. pow2 64.8%

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

      4. add-sqr-sqrt 64.8%

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

      5. clear-num 64.9%

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

      6. div-inv 66.2%

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

      7. cancel-sign-sub-inv 66.2%

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

      8. pow1 66.2%

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

   8. Applied egg-rr 61.7%

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

   10. Simplified 64.1%

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

   if -4.7e-300 < l

   1. Initial program 63.5%

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

   3. Simplified 63.0%

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

   5. Taylor expanded in d around 0 67.8%

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

      1. pow1 67.8%

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

      2. associate-*l* 68.6%

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

      3. pow1/2 68.6%

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

      4. inv-pow 68.6%

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

      5. pow-pow 68.6%

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

      6. metadata-eval 68.6%

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

      7. associate-*r* 68.6%

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

      8. div-inv 68.6%

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

      9. metadata-eval 68.6%

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

   7. Applied egg-rr 68.6%

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

   9. Simplified 79.1%

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

   10. Taylor expanded in h around 0 79.1%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.15 \cdot 10^{-75}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -4.7 \cdot 10^{-300}:\\ \;\;\;\;\left(1 - h \cdot \left({\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(\left(1 - h \cdot \left({\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := h \cdot \left({\left(D\_m \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right)\\ t_1 := \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{-307}:\\ \;\;\;\;d \cdot \left(t\_1 \cdot \left(t\_0 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(\left(1 - t\_0\right) \cdot t\_1\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* h (* (pow (* D_m (* 0.5 (/ M d))) 2.0) (/ 0.5 l))))
        (t_1 (sqrt (/ 1.0 (* h l)))))
   (if (<= l -1e-307) (* d (* t_1 (+ t_0 -1.0))) (* d (* (- 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 = h * (pow((D_m * (0.5 * (M / d))), 2.0) * (0.5 / l));
	double t_1 = sqrt((1.0 / (h * l)));
	double tmp;
	if (l <= -1e-307) {
		tmp = d * (t_1 * (t_0 + -1.0));
	} else {
		tmp = d * ((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 = h * (((d_m * (0.5d0 * (m / d))) ** 2.0d0) * (0.5d0 / l))
    t_1 = sqrt((1.0d0 / (h * l)))
    if (l <= (-1d-307)) then
        tmp = d * (t_1 * (t_0 + (-1.0d0)))
    else
        tmp = d * ((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 = h * (Math.pow((D_m * (0.5 * (M / d))), 2.0) * (0.5 / l));
	double t_1 = Math.sqrt((1.0 / (h * l)));
	double tmp;
	if (l <= -1e-307) {
		tmp = d * (t_1 * (t_0 + -1.0));
	} else {
		tmp = d * ((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 = h * (math.pow((D_m * (0.5 * (M / d))), 2.0) * (0.5 / l))
	t_1 = math.sqrt((1.0 / (h * l)))
	tmp = 0
	if l <= -1e-307:
		tmp = d * (t_1 * (t_0 + -1.0))
	else:
		tmp = d * ((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(h * Float64((Float64(D_m * Float64(0.5 * Float64(M / d))) ^ 2.0) * Float64(0.5 / l)))
	t_1 = sqrt(Float64(1.0 / Float64(h * l)))
	tmp = 0.0
	if (l <= -1e-307)
		tmp = Float64(d * Float64(t_1 * Float64(t_0 + -1.0)));
	else
		tmp = Float64(d * 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 = h * (((D_m * (0.5 * (M / d))) ^ 2.0) * (0.5 / l));
	t_1 = sqrt((1.0 / (h * l)));
	tmp = 0.0;
	if (l <= -1e-307)
		tmp = d * (t_1 * (t_0 + -1.0));
	else
		tmp = d * ((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[(h * N[(N[Power[N[(D$95$m * N[(0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1e-307], N[(d * N[(t$95$1 * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 - t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if l < -9.99999999999999909e-308

   1. Initial program 53.8%

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

   3. Simplified 53.0%

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

   5. Taylor expanded in d around 0 3.7%

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

      1. pow1 3.7%

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

      2. associate-*l* 3.7%

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

      3. pow1/2 3.7%

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

      4. inv-pow 3.7%

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

      5. pow-pow 3.7%

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

      6. metadata-eval 3.7%

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

      7. associate-*r* 3.7%

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

      8. div-inv 3.7%

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

      9. metadata-eval 3.7%

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

   7. Applied egg-rr 3.7%

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

   9. Simplified 3.8%

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

   10. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 70.0%

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

       4. mul-1-neg 70.0%

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

   12. Simplified 70.0%

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

   if -9.99999999999999909e-308 < l

   1. Initial program 64.6%

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

   3. Simplified 64.1%

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

   5. Taylor expanded in d around 0 68.9%

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

      1. pow1 68.9%

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

      2. associate-*l* 69.8%

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

      3. pow1/2 69.8%

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

      4. inv-pow 69.8%

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

      5. pow-pow 69.8%

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

      6. metadata-eval 69.8%

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

      7. associate-*r* 69.8%

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

      8. div-inv 69.8%

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

      9. metadata-eval 69.8%

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

   7. Applied egg-rr 69.8%

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

   9. Simplified 80.4%

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

   10. Taylor expanded in h around 0 80.4%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-307}:\\ \;\;\;\;d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(h \cdot \left({\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(\left(1 - h \cdot \left({\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.7 \cdot 10^{-300}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(\left(1 - h \cdot \left({\left(D\_m \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \end{array} \end{array} \]

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 -4.7e-300)
   (* (- d) (sqrt (/ (/ 1.0 l) h)))
   (*
    d
    (*
     (- 1.0 (* h (* (pow (* D_m (* 0.5 (/ M d))) 2.0) (/ 0.5 l))))
     (sqrt (/ 1.0 (* h l)))))))

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 <= -4.7e-300) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else {
		tmp = d * ((1.0 - (h * (pow((D_m * (0.5 * (M / d))), 2.0) * (0.5 / l)))) * sqrt((1.0 / (h * l))));
	}
	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 <= (-4.7d-300)) then
        tmp = -d * sqrt(((1.0d0 / l) / h))
    else
        tmp = d * ((1.0d0 - (h * (((d_m * (0.5d0 * (m / d))) ** 2.0d0) * (0.5d0 / l)))) * sqrt((1.0d0 / (h * l))))
    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 <= -4.7e-300) {
		tmp = -d * Math.sqrt(((1.0 / l) / h));
	} else {
		tmp = d * ((1.0 - (h * (Math.pow((D_m * (0.5 * (M / d))), 2.0) * (0.5 / l)))) * Math.sqrt((1.0 / (h * l))));
	}
	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 <= -4.7e-300:
		tmp = -d * math.sqrt(((1.0 / l) / h))
	else:
		tmp = d * ((1.0 - (h * (math.pow((D_m * (0.5 * (M / d))), 2.0) * (0.5 / l)))) * math.sqrt((1.0 / (h * l))))
	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 <= -4.7e-300)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	else
		tmp = Float64(d * Float64(Float64(1.0 - Float64(h * Float64((Float64(D_m * Float64(0.5 * Float64(M / d))) ^ 2.0) * Float64(0.5 / l)))) * sqrt(Float64(1.0 / Float64(h * l)))));
	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 <= -4.7e-300)
		tmp = -d * sqrt(((1.0 / l) / h));
	else
		tmp = d * ((1.0 - (h * (((D_m * (0.5 * (M / d))) ^ 2.0) * (0.5 / l)))) * sqrt((1.0 / (h * l))));
	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, -4.7e-300], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 - N[(h * N[(N[Power[N[(D$95$m * N[(0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.7e-300

   1. Initial program 54.6%

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

   3. Simplified 53.8%

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

      1. add-sqr-sqrt 53.8%

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

      2. pow2 53.8%

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

      3. sqrt-prod 53.8%

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

      4. sqrt-pow1 56.8%

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

      5. frac-times 57.6%

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

      6. associate-/r* 57.6%

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

      7. metadata-eval 57.6%

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

      8. pow1 57.6%

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

      9. div-inv 57.6%

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

      10. *-commutative 57.6%

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

      11. associate-*r* 57.6%

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

      12. div-inv 57.6%

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

      13. associate-*r/ 57.5%

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

      14. associate-/l/ 57.5%

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

      15. *-un-lft-identity 57.5%

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

      16. *-commutative 57.5%

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

      17. times-frac 57.5%

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

      18. metadata-eval 57.5%

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

   6. Applied egg-rr 57.5%

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

   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. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 45.0%

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

      4. mul-1-neg 45.0%

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

      5. *-commutative 45.0%

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

      6. associate-/r* 45.5%

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

   9. Simplified 45.5%

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

   if -4.7e-300 < l

   1. Initial program 63.5%

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

   3. Simplified 63.0%

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

   5. Taylor expanded in d around 0 67.8%

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

      1. pow1 67.8%

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

      2. associate-*l* 68.6%

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

      3. pow1/2 68.6%

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

      4. inv-pow 68.6%

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

      5. pow-pow 68.6%

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

      6. metadata-eval 68.6%

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

      7. associate-*r* 68.6%

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

      8. div-inv 68.6%

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

      9. metadata-eval 68.6%

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

   7. Applied egg-rr 68.6%

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

   9. Simplified 79.1%

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

   10. Taylor expanded in h around 0 79.1%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.7 \cdot 10^{-300}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(\left(1 - h \cdot \left({\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M \cdot 0.5}{d}\\ \mathbf{if}\;\ell \leq -4.7 \cdot 10^{-300}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - h \cdot \left(\frac{0.5}{\ell} \cdot \left(t\_0 \cdot t\_0\right)\right)\right)\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* D_m (/ (* M 0.5) d))))
   (if (<= l -4.7e-300)
     (* (- d) (sqrt (/ (/ 1.0 l) h)))
     (* d (* (pow (* h l) -0.5) (- 1.0 (* h (* (/ 0.5 l) (* t_0 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_m * ((M * 0.5) / d);
	double tmp;
	if (l <= -4.7e-300) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else {
		tmp = d * (pow((h * l), -0.5) * (1.0 - (h * ((0.5 / l) * (t_0 * 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_m * ((m * 0.5d0) / d)
    if (l <= (-4.7d-300)) then
        tmp = -d * sqrt(((1.0d0 / l) / h))
    else
        tmp = d * (((h * l) ** (-0.5d0)) * (1.0d0 - (h * ((0.5d0 / l) * (t_0 * 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_m * ((M * 0.5) / d);
	double tmp;
	if (l <= -4.7e-300) {
		tmp = -d * Math.sqrt(((1.0 / l) / h));
	} else {
		tmp = d * (Math.pow((h * l), -0.5) * (1.0 - (h * ((0.5 / l) * (t_0 * 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_m * ((M * 0.5) / d)
	tmp = 0
	if l <= -4.7e-300:
		tmp = -d * math.sqrt(((1.0 / l) / h))
	else:
		tmp = d * (math.pow((h * l), -0.5) * (1.0 - (h * ((0.5 / l) * (t_0 * 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_m * Float64(Float64(M * 0.5) / d))
	tmp = 0.0
	if (l <= -4.7e-300)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	else
		tmp = Float64(d * Float64((Float64(h * l) ^ -0.5) * Float64(1.0 - Float64(h * Float64(Float64(0.5 / l) * Float64(t_0 * 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_m * ((M * 0.5) / d);
	tmp = 0.0;
	if (l <= -4.7e-300)
		tmp = -d * sqrt(((1.0 / l) / h));
	else
		tmp = d * (((h * l) ^ -0.5) * (1.0 - (h * ((0.5 / l) * (t_0 * 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$95$m * N[(N[(M * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4.7e-300], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision] * N[(1.0 - N[(h * N[(N[(0.5 / l), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -4.7e-300

   1. Initial program 54.6%

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

   3. Simplified 53.8%

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

      1. add-sqr-sqrt 53.8%

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

      2. pow2 53.8%

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

      3. sqrt-prod 53.8%

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

      4. sqrt-pow1 56.8%

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

      5. frac-times 57.6%

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

      6. associate-/r* 57.6%

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

      7. metadata-eval 57.6%

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

      8. pow1 57.6%

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

      9. div-inv 57.6%

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

      10. *-commutative 57.6%

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

      11. associate-*r* 57.6%

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

      12. div-inv 57.6%

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

      13. associate-*r/ 57.5%

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

      14. associate-/l/ 57.5%

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

      15. *-un-lft-identity 57.5%

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

      16. *-commutative 57.5%

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

      17. times-frac 57.5%

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

      18. metadata-eval 57.5%

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

   6. Applied egg-rr 57.5%

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

   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. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 45.0%

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

      4. mul-1-neg 45.0%

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

      5. *-commutative 45.0%

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

      6. associate-/r* 45.5%

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

   9. Simplified 45.5%

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

   if -4.7e-300 < l

   1. Initial program 63.5%

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

   3. Simplified 63.0%

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

   5. Taylor expanded in d around 0 67.8%

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

      1. pow1 67.8%

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

      2. associate-*l* 68.6%

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

      3. pow1/2 68.6%

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

      4. inv-pow 68.6%

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

      5. pow-pow 68.6%

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

      6. metadata-eval 68.6%

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

      7. associate-*r* 68.6%

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

      8. div-inv 68.6%

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

      9. metadata-eval 68.6%

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

   7. Applied egg-rr 68.6%

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

   9. Simplified 79.1%

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

       1. unpow2 79.1%

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

       2. associate-*r/ 79.1%

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

       3. associate-*r/ 79.1%

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

   11. Applied egg-rr 79.1%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.7 \cdot 10^{-300}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 - h \cdot \left(\frac{0.5}{\ell} \cdot \left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 42.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{if}\;\ell \leq -4.8 \cdot 10^{-229}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot t\_0\\ \end{array} \end{array} \]

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 (sqrt (/ (/ 1.0 l) h))))
   (if (<= l -4.8e-229) (* (- d) t_0) (* d 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 = sqrt(((1.0 / l) / h));
	double tmp;
	if (l <= -4.8e-229) {
		tmp = -d * t_0;
	} else {
		tmp = d * 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 = sqrt(((1.0d0 / l) / h))
    if (l <= (-4.8d-229)) then
        tmp = -d * t_0
    else
        tmp = d * 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 = Math.sqrt(((1.0 / l) / h));
	double tmp;
	if (l <= -4.8e-229) {
		tmp = -d * t_0;
	} else {
		tmp = d * 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 = math.sqrt(((1.0 / l) / h))
	tmp = 0
	if l <= -4.8e-229:
		tmp = -d * t_0
	else:
		tmp = d * 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 = sqrt(Float64(Float64(1.0 / l) / h))
	tmp = 0.0
	if (l <= -4.8e-229)
		tmp = Float64(Float64(-d) * t_0);
	else
		tmp = Float64(d * 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 = sqrt(((1.0 / l) / h));
	tmp = 0.0;
	if (l <= -4.8e-229)
		tmp = -d * t_0;
	else
		tmp = d * 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[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4.8e-229], N[((-d) * t$95$0), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -4.8e-229

   1. Initial program 51.4%

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

   3. Simplified 50.5%

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

      1. add-sqr-sqrt 50.5%

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

      2. pow2 50.5%

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

      3. sqrt-prod 50.5%

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

      4. sqrt-pow1 53.8%

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

      5. frac-times 54.7%

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

      6. associate-/r* 54.7%

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

      7. metadata-eval 54.7%

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

      8. pow1 54.7%

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

      9. div-inv 54.7%

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

      10. *-commutative 54.7%

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

      11. associate-*r* 54.7%

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

      12. div-inv 54.7%

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

      13. associate-*r/ 54.6%

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

      14. associate-/l/ 54.6%

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

      15. *-un-lft-identity 54.6%

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

      16. *-commutative 54.6%

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

      17. times-frac 54.6%

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

      18. metadata-eval 54.6%

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

   6. Applied egg-rr 54.6%

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

   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. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 47.3%

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

      4. mul-1-neg 47.3%

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

      5. *-commutative 47.3%

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

      6. associate-/r* 47.8%

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

   9. Simplified 47.8%

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

   if -4.8e-229 < l

   1. Initial program 65.6%

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

   3. Simplified 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 65.2%

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

      4. sqrt-pow1 66.7%

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

      5. frac-times 67.1%

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

      6. associate-/r* 67.1%

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

      7. metadata-eval 67.1%

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

      8. pow1 67.1%

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

      9. div-inv 67.1%

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

      10. *-commutative 67.1%

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

      11. associate-*r* 67.1%

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

      12. div-inv 67.1%

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

      13. associate-*r/ 67.9%

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

      14. associate-/l/ 67.9%

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

      15. *-un-lft-identity 67.9%

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

      16. *-commutative 67.9%

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

      17. times-frac 67.9%

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

      18. metadata-eval 67.9%

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

   6. Applied egg-rr 67.9%

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

   7. Taylor expanded in d around inf 41.8%

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

      1. *-commutative 41.8%

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

      2. associate-/r* 42.1%

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

   9. Simplified 42.1%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.8 \cdot 10^{-229}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 42.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.6 \cdot 10^{-229}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \end{array} \]

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
 (if (<= l -4.6e-229)
   (* d (- (pow (* h l) -0.5)))
   (* d (sqrt (/ (/ 1.0 l) 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 tmp;
	if (l <= -4.6e-229) {
		tmp = d * -pow((h * l), -0.5);
	} else {
		tmp = d * sqrt(((1.0 / l) / 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) :: tmp
    if (l <= (-4.6d-229)) then
        tmp = d * -((h * l) ** (-0.5d0))
    else
        tmp = d * sqrt(((1.0d0 / l) / 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 tmp;
	if (l <= -4.6e-229) {
		tmp = d * -Math.pow((h * l), -0.5);
	} else {
		tmp = d * Math.sqrt(((1.0 / l) / 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):
	tmp = 0
	if l <= -4.6e-229:
		tmp = d * -math.pow((h * l), -0.5)
	else:
		tmp = d * math.sqrt(((1.0 / l) / 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)
	tmp = 0.0
	if (l <= -4.6e-229)
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) / 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)
	tmp = 0.0;
	if (l <= -4.6e-229)
		tmp = d * -((h * l) ^ -0.5);
	else
		tmp = d * sqrt(((1.0 / l) / 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_] := If[LessEqual[l, -4.6e-229], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.59999999999999992e-229

   1. Initial program 51.4%

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

   3. Simplified 50.5%

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

      1. add-sqr-sqrt 50.5%

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

      2. pow2 50.5%

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

      3. sqrt-prod 50.5%

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

      4. sqrt-pow1 53.8%

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

      5. frac-times 54.7%

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

      6. associate-/r* 54.7%

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

      7. metadata-eval 54.7%

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

      8. pow1 54.7%

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

      9. div-inv 54.7%

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

      10. *-commutative 54.7%

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

      11. associate-*r* 54.7%

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

      12. div-inv 54.7%

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

      13. associate-*r/ 54.6%

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

      14. associate-/l/ 54.6%

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

      15. *-un-lft-identity 54.6%

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

      16. *-commutative 54.6%

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

      17. times-frac 54.6%

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

      18. metadata-eval 54.6%

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

   6. Applied egg-rr 54.6%

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

   7. Taylor expanded in D around 0 54.7%

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

      1. associate-*r* 54.7%

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

      2. associate-/l* 54.6%

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

      3. associate-*r* 54.6%

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

      4. *-commutative 54.6%

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

      5. associate-*r* 54.6%

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

      6. associate-/l* 54.6%

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

      7. associate-*l* 54.6%

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

      8. *-commutative 54.6%

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

      9. *-commutative 54.6%

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

      10. associate-/l* 54.6%

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

   9. Simplified 54.6%

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

   10. Taylor expanded in l around -inf 0.0%

       \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. 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. rem-exp-log 0.0%

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

       3. exp-neg 0.0%

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

       4. unpow1/2 0.0%

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

       5. exp-prod 0.0%

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

       6. distribute-lft-neg-out 0.0%

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

       7. distribute-rgt-neg-in 0.0%

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

       8. metadata-eval 0.0%

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

       9. exp-to-pow 0.0%

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

       10. *-commutative 0.0%

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

       11. unpow2 0.0%

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

       12. rem-square-sqrt 47.4%

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

       13. mul-1-neg 47.4%

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

   12. Simplified 47.4%

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

   if -4.59999999999999992e-229 < l

   1. Initial program 65.6%

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

   3. Simplified 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 65.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. frac-times 67.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-/r* 67.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. metadata-eval 67.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. pow1 67.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. div-inv 67.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. *-commutative 67.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. associate-*r* 67.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot \frac{1}{2}\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. div-inv 67.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \color{blue}{\frac{M}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. associate-*r/ 67.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{\frac{M}{2}}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      14. associate-/l/ 67.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      15. *-un-lft-identity 67.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{\color{blue}{1 \cdot M}}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      16. *-commutative 67.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      17. times-frac 67.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      18. metadata-eval 67.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 67.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 41.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative 41.8%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 42.1%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   9. Simplified 42.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.6 \cdot 10^{-229}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 26.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}} \end{array} \]

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 (sqrt (/ (/ 1.0 l) 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) {
	return d * sqrt(((1.0 / l) / h));
}

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 * sqrt(((1.0d0 / l) / h))
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.sqrt(((1.0 / l) / h));
}

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.sqrt(((1.0 / l) / h))

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 * sqrt(Float64(Float64(1.0 / l) / h)))
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 * sqrt(((1.0 / l) / h));
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[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.8%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 58.1%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 58.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

   2. pow2 58.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

   3. sqrt-prod 58.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

   4. sqrt-pow1 60.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   5. frac-times 61.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. associate-/r* 61.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   7. metadata-eval 61.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   8. pow1 61.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   9. div-inv 61.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   10. *-commutative 61.1%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   11. associate-*r* 61.1%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot \frac{1}{2}\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   12. div-inv 61.1%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \color{blue}{\frac{M}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   13. associate-*r/ 61.5%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{\frac{M}{2}}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   14. associate-/l/ 61.5%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   15. *-un-lft-identity 61.5%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{\color{blue}{1 \cdot M}}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   16. *-commutative 61.5%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   17. times-frac 61.5%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   18. metadata-eval 61.5%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

6. Applied egg-rr 61.5%

   \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

7. Taylor expanded in d around inf 25.2%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
8. Step-by-step derivation

   1. *-commutative 25.2%

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   2. associate-/r* 25.3%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

9. Simplified 25.3%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
10. Add Preprocessing

Alternative 16: 26.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ d \cdot \sqrt{\frac{1}{h \cdot \ell}} \end{array} \]

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 (sqrt (/ 1.0 (* h l)))))

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 * sqrt((1.0 / (h * l)));
}

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 * sqrt((1.0d0 / (h * l)))
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.sqrt((1.0 / (h * l)));
}

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.sqrt((1.0 / (h * l)))

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 * sqrt(Float64(1.0 / Float64(h * l))))
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 * sqrt((1.0 / (h * l)));
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[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.8%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 58.1%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 58.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

   2. pow2 58.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

   3. sqrt-prod 58.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

   4. sqrt-pow1 60.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   5. frac-times 61.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. associate-/r* 61.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   7. metadata-eval 61.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   8. pow1 61.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   9. div-inv 61.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   10. *-commutative 61.1%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   11. associate-*r* 61.1%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot \frac{1}{2}\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   12. div-inv 61.1%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \color{blue}{\frac{M}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   13. associate-*r/ 61.5%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{\frac{M}{2}}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   14. associate-/l/ 61.5%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   15. *-un-lft-identity 61.5%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{\color{blue}{1 \cdot M}}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   16. *-commutative 61.5%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   17. times-frac 61.5%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   18. metadata-eval 61.5%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

6. Applied egg-rr 61.5%

   \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

7. Taylor expanded in d around inf 25.2%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
8. Add Preprocessing

Alternative 17: 26.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ d \cdot {\left(h \cdot \ell\right)}^{-0.5} \end{array} \]

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 (* h l) -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((h * l), -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 * ((h * l) ** (-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((h * l), -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((h * l), -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(h * l) ^ -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 * ((h * l) ^ -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[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.8%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 58.1%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 58.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

   2. pow2 58.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

   3. sqrt-prod 58.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

   4. sqrt-pow1 60.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   5. frac-times 61.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. associate-/r* 61.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   7. metadata-eval 61.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   8. pow1 61.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   9. div-inv 61.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   10. *-commutative 61.1%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   11. associate-*r* 61.1%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot \frac{1}{2}\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   12. div-inv 61.1%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \color{blue}{\frac{M}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   13. associate-*r/ 61.5%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{\frac{M}{2}}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   14. associate-/l/ 61.5%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   15. *-un-lft-identity 61.5%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{\color{blue}{1 \cdot M}}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   16. *-commutative 61.5%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   17. times-frac 61.5%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   18. metadata-eval 61.5%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

6. Applied egg-rr 61.5%

   \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

7. Taylor expanded in D around 0 61.1%

   \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(0.5 \cdot \left(\frac{D \cdot M}{d} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]
8. Step-by-step derivation

   1. associate-*r* 61.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\left(0.5 \cdot \frac{D \cdot M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

   2. associate-/l* 61.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(0.5 \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   3. associate-*r* 61.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\left(0.5 \cdot D\right) \cdot \frac{M}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   4. *-commutative 61.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(D \cdot 0.5\right)} \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   5. associate-*r* 61.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. associate-/l* 61.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{0.5 \cdot M}{d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   7. associate-*l* 61.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(D \cdot \left(\frac{0.5 \cdot M}{d} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]

   8. *-commutative 61.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \color{blue}{\left(\sqrt{\frac{h}{\ell}} \cdot \frac{0.5 \cdot M}{d}\right)}\right)}^{2}\right) \]

   9. *-commutative 61.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\sqrt{\frac{h}{\ell}} \cdot \frac{\color{blue}{M \cdot 0.5}}{d}\right)\right)}^{2}\right) \]

   10. associate-/l* 61.1%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\sqrt{\frac{h}{\ell}} \cdot \color{blue}{\left(M \cdot \frac{0.5}{d}\right)}\right)\right)}^{2}\right) \]

9. Simplified 61.1%

   \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(D \cdot \left(\sqrt{\frac{h}{\ell}} \cdot \left(M \cdot \frac{0.5}{d}\right)\right)\right)}}^{2}\right) \]

10. Taylor expanded in d around inf 25.2%

    \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
11. Step-by-step derivation

    1. rem-exp-log 24.2%

       \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}} \]

    2. exp-neg 24.2%

       \[\leadsto d \cdot \sqrt{\color{blue}{e^{-\log \left(h \cdot \ell\right)}}} \]

    3. unpow1/2 24.2%

       \[\leadsto d \cdot \color{blue}{{\left(e^{-\log \left(h \cdot \ell\right)}\right)}^{0.5}} \]

    4. exp-prod 24.2%

       \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

    5. distribute-lft-neg-out 24.2%

       \[\leadsto d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

    6. distribute-rgt-neg-in 24.2%

       \[\leadsto d \cdot e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \]

    7. metadata-eval 24.2%

       \[\leadsto d \cdot e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \]

    8. exp-to-pow 25.1%

       \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

12. Simplified 25.1%

    \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024166 
(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)))))