Henrywood and Agarwal, Equation (12)

Percentage Accurate: 67.2% → 83.9%

Time: 27.0s

Alternatives: 30

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: 67.2% 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: 83.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D\_m \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2}\right)\\ t_1 := \sqrt{-d}\\ t_2 := \frac{t\_1}{\sqrt{-\ell}}\\ \mathbf{if}\;\ell \leq -3.8 \cdot 10^{-149}:\\ \;\;\;\;t\_2 \cdot \left(\frac{t\_1}{\sqrt{-h}} \cdot t\_0\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_2 \cdot \left(\left(1 + \frac{\left(h \cdot -0.5\right) \cdot {\left(D\_m \cdot \left(0.5 \cdot \frac{M\_m}{d}\right)\right)}^{2}}{\ell}\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (/ h l) (* -0.5 (pow (* D_m (/ (/ M_m 2.0) d)) 2.0)))))
        (t_1 (sqrt (- d)))
        (t_2 (/ t_1 (sqrt (- l)))))
   (if (<= l -3.8e-149)
     (* t_2 (* (/ t_1 (sqrt (- h))) t_0))
     (if (<= l -5e-310)
       (*
        t_2
        (*
         (+ 1.0 (/ (* (* h -0.5) (pow (* D_m (* 0.5 (/ M_m d))) 2.0)) l))
         (sqrt (/ d h))))
       (* (/ (sqrt d) (sqrt l)) (* (/ (sqrt d) (sqrt h)) t_0))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 1.0 + ((h / l) * (-0.5 * pow((D_m * ((M_m / 2.0) / d)), 2.0)));
	double t_1 = sqrt(-d);
	double t_2 = t_1 / sqrt(-l);
	double tmp;
	if (l <= -3.8e-149) {
		tmp = t_2 * ((t_1 / sqrt(-h)) * t_0);
	} else if (l <= -5e-310) {
		tmp = t_2 * ((1.0 + (((h * -0.5) * pow((D_m * (0.5 * (M_m / d))), 2.0)) / l)) * sqrt((d / h)));
	} else {
		tmp = (sqrt(d) / sqrt(l)) * ((sqrt(d) / sqrt(h)) * t_0);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 + ((h / l) * ((-0.5d0) * ((d_m * ((m_m / 2.0d0) / d)) ** 2.0d0)))
    t_1 = sqrt(-d)
    t_2 = t_1 / sqrt(-l)
    if (l <= (-3.8d-149)) then
        tmp = t_2 * ((t_1 / sqrt(-h)) * t_0)
    else if (l <= (-5d-310)) then
        tmp = t_2 * ((1.0d0 + (((h * (-0.5d0)) * ((d_m * (0.5d0 * (m_m / d))) ** 2.0d0)) / l)) * sqrt((d / h)))
    else
        tmp = (sqrt(d) / sqrt(l)) * ((sqrt(d) / sqrt(h)) * t_0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 1.0 + ((h / l) * (-0.5 * Math.pow((D_m * ((M_m / 2.0) / d)), 2.0)));
	double t_1 = Math.sqrt(-d);
	double t_2 = t_1 / Math.sqrt(-l);
	double tmp;
	if (l <= -3.8e-149) {
		tmp = t_2 * ((t_1 / Math.sqrt(-h)) * t_0);
	} else if (l <= -5e-310) {
		tmp = t_2 * ((1.0 + (((h * -0.5) * Math.pow((D_m * (0.5 * (M_m / d))), 2.0)) / l)) * Math.sqrt((d / h)));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(l)) * ((Math.sqrt(d) / Math.sqrt(h)) * t_0);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = 1.0 + ((h / l) * (-0.5 * math.pow((D_m * ((M_m / 2.0) / d)), 2.0)))
	t_1 = math.sqrt(-d)
	t_2 = t_1 / math.sqrt(-l)
	tmp = 0
	if l <= -3.8e-149:
		tmp = t_2 * ((t_1 / math.sqrt(-h)) * t_0)
	elif l <= -5e-310:
		tmp = t_2 * ((1.0 + (((h * -0.5) * math.pow((D_m * (0.5 * (M_m / d))), 2.0)) / l)) * math.sqrt((d / h)))
	else:
		tmp = (math.sqrt(d) / math.sqrt(l)) * ((math.sqrt(d) / math.sqrt(h)) * t_0)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(D_m * Float64(Float64(M_m / 2.0) / d)) ^ 2.0))))
	t_1 = sqrt(Float64(-d))
	t_2 = Float64(t_1 / sqrt(Float64(-l)))
	tmp = 0.0
	if (l <= -3.8e-149)
		tmp = Float64(t_2 * Float64(Float64(t_1 / sqrt(Float64(-h))) * t_0));
	elseif (l <= -5e-310)
		tmp = Float64(t_2 * Float64(Float64(1.0 + Float64(Float64(Float64(h * -0.5) * (Float64(D_m * Float64(0.5 * Float64(M_m / d))) ^ 2.0)) / l)) * sqrt(Float64(d / h))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(l)) * Float64(Float64(sqrt(d) / sqrt(h)) * t_0));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = 1.0 + ((h / l) * (-0.5 * ((D_m * ((M_m / 2.0) / d)) ^ 2.0)));
	t_1 = sqrt(-d);
	t_2 = t_1 / sqrt(-l);
	tmp = 0.0;
	if (l <= -3.8e-149)
		tmp = t_2 * ((t_1 / sqrt(-h)) * t_0);
	elseif (l <= -5e-310)
		tmp = t_2 * ((1.0 + (((h * -0.5) * ((D_m * (0.5 * (M_m / d))) ^ 2.0)) / l)) * sqrt((d / h)));
	else
		tmp = (sqrt(d) / sqrt(l)) * ((sqrt(d) / sqrt(h)) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D$95$m * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.8e-149], N[(t$95$2 * N[(N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(t$95$2 * N[(N[(1.0 + N[(N[(N[(h * -0.5), $MachinePrecision] * N[Power[N[(D$95$m * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.80000000000000005e-149

   1. Initial program 78.3%

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

   3. Simplified 78.3%

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

      1. frac-2neg 84.1%

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

      2. sqrt-div 87.8%

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

   6. Applied egg-rr 80.0%

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

      1. frac-2neg 76.7%

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

      2. sqrt-div 84.1%

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

   8. Applied egg-rr 89.5%

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

   if -3.80000000000000005e-149 < l < -4.999999999999985e-310

   1. Initial program 69.4%

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

   3. Simplified 64.0%

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

      1. associate-*l/ 73.1%

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

   6. Applied egg-rr 73.1%

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

      1. frac-2neg 73.0%

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

      2. sqrt-div 86.1%

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

   8. Applied egg-rr 86.2%

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

   if -4.999999999999985e-310 < l

   1. Initial program 65.7%

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

   3. Simplified 64.2%

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

      1. sqrt-div 75.9%

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

      2. div-inv 75.9%

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

   6. Applied egg-rr 75.9%

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

      1. associate-*r/ 75.9%

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

      2. *-rgt-identity 75.9%

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

   8. Simplified 75.9%

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

      1. sqrt-div 80.4%

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

      2. div-inv 80.3%

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

   10. Applied egg-rr 80.3%

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

       1. associate-*r/ 80.4%

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

       2. *-rgt-identity 80.4%

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

   12. Simplified 80.4%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.8 \cdot 10^{-149}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\left(1 + \frac{\left(h \cdot -0.5\right) \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.2%

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

   3. Simplified 86.3%

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

      1. add-sqr-sqrt 86.3%

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

      2. pow2 86.3%

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

      3. sqrt-prod 86.3%

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

      4. sqrt-pow1 87.2%

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

      5. *-commutative 87.2%

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

      6. metadata-eval 87.2%

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

      7. pow1 87.2%

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

      8. div-inv 87.2%

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

      9. associate-*l* 87.3%

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

      10. times-frac 87.3%

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

      11. *-commutative 87.3%

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

      12. times-frac 87.3%

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

      13. metadata-eval 87.3%

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

   6. Applied egg-rr 87.3%

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

   7. Taylor expanded in D around 0 89.1%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. pow2 0.0%

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

      3. sqrt-prod 0.0%

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

      4. sqrt-pow1 4.5%

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

      5. *-commutative 4.5%

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

      6. metadata-eval 4.5%

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

      7. pow1 4.5%

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

      8. div-inv 4.5%

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

      9. associate-*l* 2.5%

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

      10. times-frac 2.5%

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

      11. *-commutative 2.5%

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

      12. times-frac 2.5%

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

      13. metadata-eval 2.5%

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

   6. Applied egg-rr 2.5%

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

      1. sqrt-div 8.0%

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

   8. Applied egg-rr 8.0%

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

   9. Taylor expanded in d around 0 31.2%

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

       1. *-commutative 31.2%

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

   11. Simplified 31.2%

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

4. Final simplification 78.0%

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

Alternative 3: 72.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.2%

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

   3. Simplified 86.5%

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

      1. associate-/l/ 86.5%

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

      2. *-un-lft-identity 86.5%

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

      3. times-frac 86.4%

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

      4. associate-*l* 86.3%

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

      5. div-inv 86.3%

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

      6. *-commutative 86.3%

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

      7. associate-*r/ 88.2%

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

      8. div-inv 88.2%

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

      9. metadata-eval 88.2%

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

   6. Applied egg-rr 88.2%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. pow2 0.0%

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

      3. sqrt-prod 0.0%

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

      4. sqrt-pow1 4.5%

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

      5. *-commutative 4.5%

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

      6. metadata-eval 4.5%

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

      7. pow1 4.5%

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

      8. div-inv 4.5%

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

      9. associate-*l* 2.5%

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

      10. times-frac 2.5%

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

      11. *-commutative 2.5%

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

      12. times-frac 2.5%

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

      13. metadata-eval 2.5%

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

   6. Applied egg-rr 2.5%

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

      1. sqrt-div 8.0%

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

   8. Applied egg-rr 8.0%

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

   9. Taylor expanded in d around 0 31.2%

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

       1. *-commutative 31.2%

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

   11. Simplified 31.2%

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

4. Final simplification 77.3%

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

Alternative 4: 80.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D\_m \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2}\right)\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{-147}:\\ \;\;\;\;\left(\frac{t\_1}{\sqrt{-h}} \cdot t\_0\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{t\_1}{\sqrt{-\ell}} \cdot \left(\left(1 + \frac{\left(h \cdot -0.5\right) \cdot {\left(D\_m \cdot \left(0.5 \cdot \frac{M\_m}{d}\right)\right)}^{2}}{\ell}\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (/ h l) (* -0.5 (pow (* D_m (/ (/ M_m 2.0) d)) 2.0)))))
        (t_1 (sqrt (- d))))
   (if (<= l -4e-147)
     (* (* (/ t_1 (sqrt (- h))) t_0) (sqrt (/ d l)))
     (if (<= l -5e-310)
       (*
        (/ t_1 (sqrt (- l)))
        (*
         (+ 1.0 (/ (* (* h -0.5) (pow (* D_m (* 0.5 (/ M_m d))) 2.0)) l))
         (sqrt (/ d h))))
       (* (/ (sqrt d) (sqrt l)) (* (/ (sqrt d) (sqrt h)) t_0))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 1.0 + ((h / l) * (-0.5 * pow((D_m * ((M_m / 2.0) / d)), 2.0)));
	double t_1 = sqrt(-d);
	double tmp;
	if (l <= -4e-147) {
		tmp = ((t_1 / sqrt(-h)) * t_0) * sqrt((d / l));
	} else if (l <= -5e-310) {
		tmp = (t_1 / sqrt(-l)) * ((1.0 + (((h * -0.5) * pow((D_m * (0.5 * (M_m / d))), 2.0)) / l)) * sqrt((d / h)));
	} else {
		tmp = (sqrt(d) / sqrt(l)) * ((sqrt(d) / sqrt(h)) * t_0);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + ((h / l) * ((-0.5d0) * ((d_m * ((m_m / 2.0d0) / d)) ** 2.0d0)))
    t_1 = sqrt(-d)
    if (l <= (-4d-147)) then
        tmp = ((t_1 / sqrt(-h)) * t_0) * sqrt((d / l))
    else if (l <= (-5d-310)) then
        tmp = (t_1 / sqrt(-l)) * ((1.0d0 + (((h * (-0.5d0)) * ((d_m * (0.5d0 * (m_m / d))) ** 2.0d0)) / l)) * sqrt((d / h)))
    else
        tmp = (sqrt(d) / sqrt(l)) * ((sqrt(d) / sqrt(h)) * t_0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 1.0 + ((h / l) * (-0.5 * Math.pow((D_m * ((M_m / 2.0) / d)), 2.0)));
	double t_1 = Math.sqrt(-d);
	double tmp;
	if (l <= -4e-147) {
		tmp = ((t_1 / Math.sqrt(-h)) * t_0) * Math.sqrt((d / l));
	} else if (l <= -5e-310) {
		tmp = (t_1 / Math.sqrt(-l)) * ((1.0 + (((h * -0.5) * Math.pow((D_m * (0.5 * (M_m / d))), 2.0)) / l)) * Math.sqrt((d / h)));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(l)) * ((Math.sqrt(d) / Math.sqrt(h)) * t_0);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = 1.0 + ((h / l) * (-0.5 * math.pow((D_m * ((M_m / 2.0) / d)), 2.0)))
	t_1 = math.sqrt(-d)
	tmp = 0
	if l <= -4e-147:
		tmp = ((t_1 / math.sqrt(-h)) * t_0) * math.sqrt((d / l))
	elif l <= -5e-310:
		tmp = (t_1 / math.sqrt(-l)) * ((1.0 + (((h * -0.5) * math.pow((D_m * (0.5 * (M_m / d))), 2.0)) / l)) * math.sqrt((d / h)))
	else:
		tmp = (math.sqrt(d) / math.sqrt(l)) * ((math.sqrt(d) / math.sqrt(h)) * t_0)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(D_m * Float64(Float64(M_m / 2.0) / d)) ^ 2.0))))
	t_1 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -4e-147)
		tmp = Float64(Float64(Float64(t_1 / sqrt(Float64(-h))) * t_0) * sqrt(Float64(d / l)));
	elseif (l <= -5e-310)
		tmp = Float64(Float64(t_1 / sqrt(Float64(-l))) * Float64(Float64(1.0 + Float64(Float64(Float64(h * -0.5) * (Float64(D_m * Float64(0.5 * Float64(M_m / d))) ^ 2.0)) / l)) * sqrt(Float64(d / h))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(l)) * Float64(Float64(sqrt(d) / sqrt(h)) * t_0));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = 1.0 + ((h / l) * (-0.5 * ((D_m * ((M_m / 2.0) / d)) ^ 2.0)));
	t_1 = sqrt(-d);
	tmp = 0.0;
	if (l <= -4e-147)
		tmp = ((t_1 / sqrt(-h)) * t_0) * sqrt((d / l));
	elseif (l <= -5e-310)
		tmp = (t_1 / sqrt(-l)) * ((1.0 + (((h * -0.5) * ((D_m * (0.5 * (M_m / d))) ^ 2.0)) / l)) * sqrt((d / h)));
	else
		tmp = (sqrt(d) / sqrt(l)) * ((sqrt(d) / sqrt(h)) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D$95$m * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -4e-147], N[(N[(N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(N[(h * -0.5), $MachinePrecision] * N[Power[N[(D$95$m * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.9999999999999999e-147

   1. Initial program 78.3%

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

   3. Simplified 78.3%

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

      1. frac-2neg 76.7%

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

      2. sqrt-div 84.1%

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

   6. Applied egg-rr 85.7%

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

   if -3.9999999999999999e-147 < l < -4.999999999999985e-310

   1. Initial program 69.4%

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

   3. Simplified 64.0%

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

      1. associate-*l/ 73.1%

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

   6. Applied egg-rr 73.1%

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

      1. frac-2neg 73.0%

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

      2. sqrt-div 86.1%

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

   8. Applied egg-rr 86.2%

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

   if -4.999999999999985e-310 < l

   1. Initial program 65.7%

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

   3. Simplified 64.2%

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

      1. sqrt-div 75.9%

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

      2. div-inv 75.9%

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

   6. Applied egg-rr 75.9%

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

      1. associate-*r/ 75.9%

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

      2. *-rgt-identity 75.9%

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

   8. Simplified 75.9%

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

      1. sqrt-div 80.4%

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

      2. div-inv 80.3%

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

   10. Applied egg-rr 80.3%

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

       1. associate-*r/ 80.4%

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

       2. *-rgt-identity 80.4%

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

   12. Simplified 80.4%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{-147}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}\right)\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\left(1 + \frac{\left(h \cdot -0.5\right) \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 76.0%

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

   3. Simplified 74.6%

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

      1. associate-*l/ 75.8%

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

   6. Applied egg-rr 75.8%

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

      1. frac-2neg 75.8%

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

      2. sqrt-div 81.3%

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

   8. Applied egg-rr 81.3%

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

      1. frac-2neg 81.3%

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

      2. sqrt-div 87.4%

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

   10. Applied egg-rr 87.4%

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

   if -4.999999999999985e-310 < l

   1. Initial program 65.7%

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

   3. Simplified 64.2%

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

      1. sqrt-div 75.9%

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

      2. div-inv 75.9%

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

   6. Applied egg-rr 75.9%

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

      1. associate-*r/ 75.9%

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

      2. *-rgt-identity 75.9%

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

   8. Simplified 75.9%

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

      1. sqrt-div 80.4%

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

      2. div-inv 80.3%

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

   10. Applied egg-rr 80.3%

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

       1. associate-*r/ 80.4%

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

       2. *-rgt-identity 80.4%

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

   12. Simplified 80.4%

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

4. Final simplification 84.2%

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

Alternative 6: 79.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -1.25 \cdot 10^{-147}:\\ \;\;\;\;\left(\frac{t\_0}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D\_m \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2}\right)\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{t\_0}{\sqrt{-\ell}} \cdot \left(\left(1 + \frac{\left(h \cdot -0.5\right) \cdot {\left(D\_m \cdot \left(0.5 \cdot \frac{M\_m}{d}\right)\right)}^{2}}{\ell}\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot \left(D\_m \cdot 0.5\right)}{d}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (- d))))
   (if (<= l -1.25e-147)
     (*
      (*
       (/ t_0 (sqrt (- h)))
       (+ 1.0 (* (/ h l) (* -0.5 (pow (* D_m (/ (/ M_m 2.0) d)) 2.0)))))
      (sqrt (/ d l)))
     (if (<= l -5e-310)
       (*
        (/ t_0 (sqrt (- l)))
        (*
         (+ 1.0 (/ (* (* h -0.5) (pow (* D_m (* 0.5 (/ M_m d))) 2.0)) l))
         (sqrt (/ d h))))
       (*
        (/ d (* (sqrt l) (sqrt h)))
        (- 1.0 (* 0.5 (* (/ h l) (pow (/ (* M_m (* D_m 0.5)) d) 2.0)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt(-d);
	double tmp;
	if (l <= -1.25e-147) {
		tmp = ((t_0 / sqrt(-h)) * (1.0 + ((h / l) * (-0.5 * pow((D_m * ((M_m / 2.0) / d)), 2.0))))) * sqrt((d / l));
	} else if (l <= -5e-310) {
		tmp = (t_0 / sqrt(-l)) * ((1.0 + (((h * -0.5) * pow((D_m * (0.5 * (M_m / d))), 2.0)) / l)) * sqrt((d / h)));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (0.5 * ((h / l) * pow(((M_m * (D_m * 0.5)) / d), 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(-d)
    if (l <= (-1.25d-147)) then
        tmp = ((t_0 / sqrt(-h)) * (1.0d0 + ((h / l) * ((-0.5d0) * ((d_m * ((m_m / 2.0d0) / d)) ** 2.0d0))))) * sqrt((d / l))
    else if (l <= (-5d-310)) then
        tmp = (t_0 / sqrt(-l)) * ((1.0d0 + (((h * (-0.5d0)) * ((d_m * (0.5d0 * (m_m / d))) ** 2.0d0)) / l)) * sqrt((d / h)))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - (0.5d0 * ((h / l) * (((m_m * (d_m * 0.5d0)) / d) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt(-d);
	double tmp;
	if (l <= -1.25e-147) {
		tmp = ((t_0 / Math.sqrt(-h)) * (1.0 + ((h / l) * (-0.5 * Math.pow((D_m * ((M_m / 2.0) / d)), 2.0))))) * Math.sqrt((d / l));
	} else if (l <= -5e-310) {
		tmp = (t_0 / Math.sqrt(-l)) * ((1.0 + (((h * -0.5) * Math.pow((D_m * (0.5 * (M_m / d))), 2.0)) / l)) * Math.sqrt((d / h)));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - (0.5 * ((h / l) * Math.pow(((M_m * (D_m * 0.5)) / d), 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt(-d)
	tmp = 0
	if l <= -1.25e-147:
		tmp = ((t_0 / math.sqrt(-h)) * (1.0 + ((h / l) * (-0.5 * math.pow((D_m * ((M_m / 2.0) / d)), 2.0))))) * math.sqrt((d / l))
	elif l <= -5e-310:
		tmp = (t_0 / math.sqrt(-l)) * ((1.0 + (((h * -0.5) * math.pow((D_m * (0.5 * (M_m / d))), 2.0)) / l)) * math.sqrt((d / h)))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - (0.5 * ((h / l) * math.pow(((M_m * (D_m * 0.5)) / d), 2.0))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -1.25e-147)
		tmp = Float64(Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(D_m * Float64(Float64(M_m / 2.0) / d)) ^ 2.0))))) * sqrt(Float64(d / l)));
	elseif (l <= -5e-310)
		tmp = Float64(Float64(t_0 / sqrt(Float64(-l))) * Float64(Float64(1.0 + Float64(Float64(Float64(h * -0.5) * (Float64(D_m * Float64(0.5 * Float64(M_m / d))) ^ 2.0)) / l)) * sqrt(Float64(d / h))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m * Float64(D_m * 0.5)) / d) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt(-d);
	tmp = 0.0;
	if (l <= -1.25e-147)
		tmp = ((t_0 / sqrt(-h)) * (1.0 + ((h / l) * (-0.5 * ((D_m * ((M_m / 2.0) / d)) ^ 2.0))))) * sqrt((d / l));
	elseif (l <= -5e-310)
		tmp = (t_0 / sqrt(-l)) * ((1.0 + (((h * -0.5) * ((D_m * (0.5 * (M_m / d))) ^ 2.0)) / l)) * sqrt((d / h)));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (0.5 * ((h / l) * (((M_m * (D_m * 0.5)) / d) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -1.25e-147], N[(N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D$95$m * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(N[(h * -0.5), $MachinePrecision] * N[Power[N[(D$95$m * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $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[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.25000000000000003e-147

   1. Initial program 78.3%

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

   3. Simplified 78.3%

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

      1. frac-2neg 76.7%

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

      2. sqrt-div 84.1%

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

   6. Applied egg-rr 85.7%

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

   if -1.25000000000000003e-147 < l < -4.999999999999985e-310

   1. Initial program 69.4%

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

   3. Simplified 64.0%

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

      1. associate-*l/ 73.1%

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

   6. Applied egg-rr 73.1%

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

      1. frac-2neg 73.0%

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

      2. sqrt-div 86.1%

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

   8. Applied egg-rr 86.2%

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

   if -4.999999999999985e-310 < l

   1. Initial program 65.7%

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

   3. Simplified 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. *-commutative 64.8%

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

      2. sqrt-div 67.6%

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

      3. sqrt-div 78.4%

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

      4. frac-times 78.3%

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

      5. add-sqr-sqrt 78.5%

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

   6. Applied egg-rr 78.5%

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

      1. associate-*r/ 79.3%

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

      2. div-inv 79.3%

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

      3. metadata-eval 79.3%

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

      4. associate-*l* 79.3%

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

   8. Applied egg-rr 79.3%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.25 \cdot 10^{-147}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}\right)\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\left(1 + \frac{\left(h \cdot -0.5\right) \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \left(D \cdot 0.5\right)}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \left(0.5 \cdot \frac{M\_m}{d}\right)\\ \mathbf{if}\;d \leq -8.2 \cdot 10^{-292}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{\left(h \cdot -0.5\right) \cdot {t\_0}^{2}}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{-17}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.5 \cdot {\left(t\_0 \cdot \frac{\sqrt{h}}{\sqrt{\ell}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* D_m (* 0.5 (/ M_m d)))))
   (if (<= d -8.2e-292)
     (*
      (*
       (/ (sqrt (- d)) (sqrt (- h)))
       (+ 1.0 (/ (* (* h -0.5) (pow t_0 2.0)) l)))
      (sqrt (/ d l)))
     (if (<= d 4.6e-17)
       (*
        (* d (sqrt (/ 1.0 (* l h))))
        (- 1.0 (* 0.5 (pow (* t_0 (/ (sqrt h) (sqrt l))) 2.0))))
       (*
        (* d (/ (sqrt (/ 1.0 l)) (sqrt h)))
        (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = D_m * (0.5 * (M_m / d));
	double tmp;
	if (d <= -8.2e-292) {
		tmp = ((sqrt(-d) / sqrt(-h)) * (1.0 + (((h * -0.5) * pow(t_0, 2.0)) / l))) * sqrt((d / l));
	} else if (d <= 4.6e-17) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (1.0 - (0.5 * pow((t_0 * (sqrt(h) / sqrt(l))), 2.0)));
	} else {
		tmp = (d * (sqrt((1.0 / l)) / sqrt(h))) * (1.0 - (0.5 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d_m * (0.5d0 * (m_m / d))
    if (d <= (-8.2d-292)) then
        tmp = ((sqrt(-d) / sqrt(-h)) * (1.0d0 + (((h * (-0.5d0)) * (t_0 ** 2.0d0)) / l))) * sqrt((d / l))
    else if (d <= 4.6d-17) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * (1.0d0 - (0.5d0 * ((t_0 * (sqrt(h) / sqrt(l))) ** 2.0d0)))
    else
        tmp = (d * (sqrt((1.0d0 / l)) / sqrt(h))) * (1.0d0 - (0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_m / d)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = D_m * (0.5 * (M_m / d));
	double tmp;
	if (d <= -8.2e-292) {
		tmp = ((Math.sqrt(-d) / Math.sqrt(-h)) * (1.0 + (((h * -0.5) * Math.pow(t_0, 2.0)) / l))) * Math.sqrt((d / l));
	} else if (d <= 4.6e-17) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (1.0 - (0.5 * Math.pow((t_0 * (Math.sqrt(h) / Math.sqrt(l))), 2.0)));
	} else {
		tmp = (d * (Math.sqrt((1.0 / l)) / Math.sqrt(h))) * (1.0 - (0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D_m / d)), 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = D_m * (0.5 * (M_m / d))
	tmp = 0
	if d <= -8.2e-292:
		tmp = ((math.sqrt(-d) / math.sqrt(-h)) * (1.0 + (((h * -0.5) * math.pow(t_0, 2.0)) / l))) * math.sqrt((d / l))
	elif d <= 4.6e-17:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (1.0 - (0.5 * math.pow((t_0 * (math.sqrt(h) / math.sqrt(l))), 2.0)))
	else:
		tmp = (d * (math.sqrt((1.0 / l)) / math.sqrt(h))) * (1.0 - (0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D_m / d)), 2.0))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(D_m * Float64(0.5 * Float64(M_m / d)))
	tmp = 0.0
	if (d <= -8.2e-292)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(Float64(h * -0.5) * (t_0 ^ 2.0)) / l))) * sqrt(Float64(d / l)));
	elseif (d <= 4.6e-17)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(1.0 - Float64(0.5 * (Float64(t_0 * Float64(sqrt(h) / sqrt(l))) ^ 2.0))));
	else
		tmp = Float64(Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = D_m * (0.5 * (M_m / d));
	tmp = 0.0;
	if (d <= -8.2e-292)
		tmp = ((sqrt(-d) / sqrt(-h)) * (1.0 + (((h * -0.5) * (t_0 ^ 2.0)) / l))) * sqrt((d / l));
	elseif (d <= 4.6e-17)
		tmp = (d * sqrt((1.0 / (l * h)))) * (1.0 - (0.5 * ((t_0 * (sqrt(h) / sqrt(l))) ^ 2.0)));
	else
		tmp = (d * (sqrt((1.0 / l)) / sqrt(h))) * (1.0 - (0.5 * ((h / l) * (((M_m / 2.0) * (D_m / d)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(D$95$m * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -8.2e-292], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[(h * -0.5), $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.6e-17], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[Power[N[(t$95$0 * N[(N[Sqrt[h], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -8.20000000000000087e-292

   1. Initial program 76.6%

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

   3. Simplified 75.1%

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

      1. associate-*l/ 76.3%

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

   6. Applied egg-rr 76.3%

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

      1. frac-2neg 76.3%

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

      2. sqrt-div 81.9%

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

   8. Applied egg-rr 81.9%

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

   if -8.20000000000000087e-292 < d < 4.60000000000000018e-17

   1. Initial program 53.1%

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

   3. Simplified 51.5%

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

      1. add-sqr-sqrt 51.5%

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

      2. pow2 51.5%

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

      3. sqrt-prod 51.4%

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

      4. sqrt-pow1 54.8%

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

      5. *-commutative 54.8%

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

      6. metadata-eval 54.8%

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

      7. pow1 54.8%

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

      8. div-inv 54.8%

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

      9. associate-*l* 53.0%

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

      10. times-frac 53.0%

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

      11. *-commutative 53.0%

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

      12. times-frac 53.0%

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

      13. metadata-eval 53.0%

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

   6. Applied egg-rr 53.0%

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

      1. sqrt-div 55.6%

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

   8. Applied egg-rr 55.6%

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

   9. Taylor expanded in d around 0 66.8%

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

       1. *-commutative 66.8%

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

   11. Simplified 66.8%

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

   if 4.60000000000000018e-17 < d

   1. Initial program 78.7%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in d around 0 71.4%

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

      1. *-commutative 79.2%

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

   7. Simplified 71.4%

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

      1. associate-/r* 71.5%

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

      2. sqrt-div 90.6%

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

   9. Applied egg-rr 90.6%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.2 \cdot 10^{-292}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{\left(h \cdot -0.5\right) \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{-17}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \frac{\sqrt{h}}{\sqrt{\ell}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -3.9 \cdot 10^{-152}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D\_m \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2}\right)\right)\right) \cdot t\_0\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-308}:\\ \;\;\;\;t\_0 \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(h \cdot -0.5\right) \cdot {\left(0.5 \cdot \frac{D\_m \cdot M\_m}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot \left(D\_m \cdot 0.5\right)}{d}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))))
   (if (<= l -3.9e-152)
     (*
      (*
       (/ (sqrt (- d)) (sqrt (- h)))
       (+ 1.0 (* (/ h l) (* -0.5 (pow (* D_m (/ (/ M_m 2.0) d)) 2.0)))))
      t_0)
     (if (<= l 2e-308)
       (*
        t_0
        (*
         (sqrt (/ d h))
         (+ 1.0 (/ (* (* h -0.5) (pow (* 0.5 (/ (* D_m M_m) d)) 2.0)) l))))
       (*
        (/ d (* (sqrt l) (sqrt h)))
        (- 1.0 (* 0.5 (* (/ h l) (pow (/ (* M_m (* D_m 0.5)) d) 2.0)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / l));
	double tmp;
	if (l <= -3.9e-152) {
		tmp = ((sqrt(-d) / sqrt(-h)) * (1.0 + ((h / l) * (-0.5 * pow((D_m * ((M_m / 2.0) / d)), 2.0))))) * t_0;
	} else if (l <= 2e-308) {
		tmp = t_0 * (sqrt((d / h)) * (1.0 + (((h * -0.5) * pow((0.5 * ((D_m * M_m) / d)), 2.0)) / l)));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (0.5 * ((h / l) * pow(((M_m * (D_m * 0.5)) / d), 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((d / l))
    if (l <= (-3.9d-152)) then
        tmp = ((sqrt(-d) / sqrt(-h)) * (1.0d0 + ((h / l) * ((-0.5d0) * ((d_m * ((m_m / 2.0d0) / d)) ** 2.0d0))))) * t_0
    else if (l <= 2d-308) then
        tmp = t_0 * (sqrt((d / h)) * (1.0d0 + (((h * (-0.5d0)) * ((0.5d0 * ((d_m * m_m) / d)) ** 2.0d0)) / l)))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - (0.5d0 * ((h / l) * (((m_m * (d_m * 0.5d0)) / d) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((d / l));
	double tmp;
	if (l <= -3.9e-152) {
		tmp = ((Math.sqrt(-d) / Math.sqrt(-h)) * (1.0 + ((h / l) * (-0.5 * Math.pow((D_m * ((M_m / 2.0) / d)), 2.0))))) * t_0;
	} else if (l <= 2e-308) {
		tmp = t_0 * (Math.sqrt((d / h)) * (1.0 + (((h * -0.5) * Math.pow((0.5 * ((D_m * M_m) / d)), 2.0)) / l)));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - (0.5 * ((h / l) * Math.pow(((M_m * (D_m * 0.5)) / d), 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((d / l))
	tmp = 0
	if l <= -3.9e-152:
		tmp = ((math.sqrt(-d) / math.sqrt(-h)) * (1.0 + ((h / l) * (-0.5 * math.pow((D_m * ((M_m / 2.0) / d)), 2.0))))) * t_0
	elif l <= 2e-308:
		tmp = t_0 * (math.sqrt((d / h)) * (1.0 + (((h * -0.5) * math.pow((0.5 * ((D_m * M_m) / d)), 2.0)) / l)))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - (0.5 * ((h / l) * math.pow(((M_m * (D_m * 0.5)) / d), 2.0))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -3.9e-152)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(D_m * Float64(Float64(M_m / 2.0) / d)) ^ 2.0))))) * t_0);
	elseif (l <= 2e-308)
		tmp = Float64(t_0 * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(Float64(h * -0.5) * (Float64(0.5 * Float64(Float64(D_m * M_m) / d)) ^ 2.0)) / l))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m * Float64(D_m * 0.5)) / d) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((d / l));
	tmp = 0.0;
	if (l <= -3.9e-152)
		tmp = ((sqrt(-d) / sqrt(-h)) * (1.0 + ((h / l) * (-0.5 * ((D_m * ((M_m / 2.0) / d)) ^ 2.0))))) * t_0;
	elseif (l <= 2e-308)
		tmp = t_0 * (sqrt((d / h)) * (1.0 + (((h * -0.5) * ((0.5 * ((D_m * M_m) / d)) ^ 2.0)) / l)));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (0.5 * ((h / l) * (((M_m * (D_m * 0.5)) / d) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -3.9e-152], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D$95$m * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[l, 2e-308], N[(t$95$0 * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(N[(h * -0.5), $MachinePrecision] * N[Power[N[(0.5 * N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $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[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.9000000000000004e-152

   1. Initial program 78.3%

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

   3. Simplified 78.3%

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

      1. frac-2neg 76.7%

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

      2. sqrt-div 84.1%

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

   6. Applied egg-rr 85.7%

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

   if -3.9000000000000004e-152 < l < 1.9999999999999998e-308

   1. Initial program 69.4%

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

   3. Simplified 64.0%

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

      1. associate-*l/ 73.1%

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

   6. Applied egg-rr 73.1%

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

   7. Taylor expanded in D around 0 78.5%

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

   if 1.9999999999999998e-308 < l

   1. Initial program 65.7%

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

   3. Simplified 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. *-commutative 64.8%

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

      2. sqrt-div 67.6%

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

      3. sqrt-div 78.4%

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

      4. frac-times 78.3%

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

      5. add-sqr-sqrt 78.5%

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

   6. Applied egg-rr 78.5%

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

      1. associate-*r/ 79.3%

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

      2. div-inv 79.3%

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

      3. metadata-eval 79.3%

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

      4. associate-*l* 79.3%

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

   8. Applied egg-rr 79.3%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.9 \cdot 10^{-152}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}\right)\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(h \cdot -0.5\right) \cdot {\left(0.5 \cdot \frac{D \cdot M}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \left(D \cdot 0.5\right)}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -9.5 \cdot 10^{+176}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot -0.5\right) \cdot \frac{{\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(0.5 \cdot {\left(\left(0.5 \cdot \frac{D\_m \cdot M\_m}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot \left(D\_m \cdot 0.5\right)}{d}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= h -9.5e+176)
   (*
    (sqrt (/ d l))
    (*
     (sqrt (/ d h))
     (+ 1.0 (* (* h -0.5) (/ (pow (* M_m (* 0.5 (/ D_m d))) 2.0) l)))))
   (if (<= h -1e-310)
     (*
      (* d (sqrt (/ (/ 1.0 l) h)))
      (+ (* 0.5 (pow (* (* 0.5 (/ (* D_m M_m) d)) (sqrt (/ h l))) 2.0)) -1.0))
     (*
      (/ d (* (sqrt l) (sqrt h)))
      (- 1.0 (* 0.5 (* (/ h l) (pow (/ (* M_m (* D_m 0.5)) d) 2.0))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= -9.5e+176) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h * -0.5) * (pow((M_m * (0.5 * (D_m / d))), 2.0) / l))));
	} else if (h <= -1e-310) {
		tmp = (d * sqrt(((1.0 / l) / h))) * ((0.5 * pow(((0.5 * ((D_m * M_m) / d)) * sqrt((h / l))), 2.0)) + -1.0);
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (0.5 * ((h / l) * pow(((M_m * (D_m * 0.5)) / d), 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (h <= (-9.5d+176)) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + ((h * (-0.5d0)) * (((m_m * (0.5d0 * (d_m / d))) ** 2.0d0) / l))))
    else if (h <= (-1d-310)) then
        tmp = (d * sqrt(((1.0d0 / l) / h))) * ((0.5d0 * (((0.5d0 * ((d_m * m_m) / d)) * sqrt((h / l))) ** 2.0d0)) + (-1.0d0))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - (0.5d0 * ((h / l) * (((m_m * (d_m * 0.5d0)) / d) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= -9.5e+176) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + ((h * -0.5) * (Math.pow((M_m * (0.5 * (D_m / d))), 2.0) / l))));
	} else if (h <= -1e-310) {
		tmp = (d * Math.sqrt(((1.0 / l) / h))) * ((0.5 * Math.pow(((0.5 * ((D_m * M_m) / d)) * Math.sqrt((h / l))), 2.0)) + -1.0);
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - (0.5 * ((h / l) * Math.pow(((M_m * (D_m * 0.5)) / d), 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if h <= -9.5e+176:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + ((h * -0.5) * (math.pow((M_m * (0.5 * (D_m / d))), 2.0) / l))))
	elif h <= -1e-310:
		tmp = (d * math.sqrt(((1.0 / l) / h))) * ((0.5 * math.pow(((0.5 * ((D_m * M_m) / d)) * math.sqrt((h / l))), 2.0)) + -1.0)
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - (0.5 * ((h / l) * math.pow(((M_m * (D_m * 0.5)) / d), 2.0))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (h <= -9.5e+176)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h * -0.5) * Float64((Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0) / l)))));
	elseif (h <= -1e-310)
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / l) / h))) * Float64(Float64(0.5 * (Float64(Float64(0.5 * Float64(Float64(D_m * M_m) / d)) * sqrt(Float64(h / l))) ^ 2.0)) + -1.0));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m * Float64(D_m * 0.5)) / d) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (h <= -9.5e+176)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h * -0.5) * (((M_m * (0.5 * (D_m / d))) ^ 2.0) / l))));
	elseif (h <= -1e-310)
		tmp = (d * sqrt(((1.0 / l) / h))) * ((0.5 * (((0.5 * ((D_m * M_m) / d)) * sqrt((h / l))) ^ 2.0)) + -1.0);
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (0.5 * ((h / l) * (((M_m * (D_m * 0.5)) / d) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[h, -9.5e+176], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h * -0.5), $MachinePrecision] * N[(N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1e-310], N[(N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[Power[N[(N[(0.5 * N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -9.4999999999999995e176

   1. Initial program 63.7%

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

   3. Simplified 60.9%

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

      1. associate-*l/ 62.2%

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

   6. Applied egg-rr 62.2%

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

      1. associate-/l* 69.7%

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

      2. associate-*r/ 69.7%

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

      3. *-commutative 69.7%

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

      4. associate-/l* 72.6%

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

      5. associate-*l/ 72.5%

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

      6. *-commutative 72.5%

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

      7. associate-*l* 72.5%

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

   8. Simplified 72.5%

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

   if -9.4999999999999995e176 < h < -9.999999999999969e-311

   1. Initial program 80.3%

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

   3. Simplified 77.5%

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

      1. add-sqr-sqrt 77.5%

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

      2. pow2 77.5%

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

      3. sqrt-prod 77.5%

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

      4. sqrt-pow1 78.4%

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

      5. *-commutative 78.4%

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

      6. metadata-eval 78.4%

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

      7. pow1 78.4%

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

      8. div-inv 78.4%

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

      9. associate-*l* 80.2%

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

      10. times-frac 80.2%

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

      11. *-commutative 80.2%

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

      12. times-frac 80.2%

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

      13. metadata-eval 80.2%

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

   6. Applied egg-rr 80.2%

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

   7. Taylor expanded in D around 0 81.2%

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

   8. Taylor expanded in h around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-/r* 0.0%

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

      4. *-commutative 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 83.3%

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

      7. neg-mul-1 83.3%

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

   10. Simplified 83.3%

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

   if -9.999999999999969e-311 < h

   1. Initial program 65.7%

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

   3. Simplified 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. *-commutative 64.8%

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

      2. sqrt-div 67.6%

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

      3. sqrt-div 78.4%

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

      4. frac-times 78.3%

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

      5. add-sqr-sqrt 78.5%

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

   6. Applied egg-rr 78.5%

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

      1. associate-*r/ 79.3%

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

      2. div-inv 79.3%

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

      3. metadata-eval 79.3%

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

      4. associate-*l* 79.3%

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

   8. Applied egg-rr 79.3%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -9.5 \cdot 10^{+176}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot -0.5\right) \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(0.5 \cdot {\left(\left(0.5 \cdot \frac{D \cdot M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \left(D \cdot 0.5\right)}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -3.6 \cdot 10^{+183}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{d} \cdot \left(D\_m \cdot 0.5\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot \left(D\_m \cdot 0.5\right)}{d}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= h -3.6e+183)
   (*
    (sqrt (* (/ d h) (/ d l)))
    (+ 1.0 (* -0.5 (* (/ h l) (pow (* (/ M_m d) (* D_m 0.5)) 2.0)))))
   (if (<= h -1e-310)
     (*
      (* d (sqrt (/ (/ 1.0 l) h)))
      (+ (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0))) -1.0))
     (*
      (/ d (* (sqrt l) (sqrt h)))
      (- 1.0 (* 0.5 (* (/ h l) (pow (/ (* M_m (* D_m 0.5)) d) 2.0))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= -3.6e+183) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * pow(((M_m / d) * (D_m * 0.5)), 2.0))));
	} else if (h <= -1e-310) {
		tmp = (d * sqrt(((1.0 / l) / h))) * ((0.5 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0))) + -1.0);
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (0.5 * ((h / l) * pow(((M_m * (D_m * 0.5)) / d), 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (h <= (-3.6d+183)) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 + ((-0.5d0) * ((h / l) * (((m_m / d) * (d_m * 0.5d0)) ** 2.0d0))))
    else if (h <= (-1d-310)) then
        tmp = (d * sqrt(((1.0d0 / l) / h))) * ((0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_m / d)) ** 2.0d0))) + (-1.0d0))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - (0.5d0 * ((h / l) * (((m_m * (d_m * 0.5d0)) / d) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= -3.6e+183) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * Math.pow(((M_m / d) * (D_m * 0.5)), 2.0))));
	} else if (h <= -1e-310) {
		tmp = (d * Math.sqrt(((1.0 / l) / h))) * ((0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D_m / d)), 2.0))) + -1.0);
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - (0.5 * ((h / l) * Math.pow(((M_m * (D_m * 0.5)) / d), 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if h <= -3.6e+183:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * math.pow(((M_m / d) * (D_m * 0.5)), 2.0))))
	elif h <= -1e-310:
		tmp = (d * math.sqrt(((1.0 / l) / h))) * ((0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D_m / d)), 2.0))) + -1.0)
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - (0.5 * ((h / l) * math.pow(((M_m * (D_m * 0.5)) / d), 2.0))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (h <= -3.6e+183)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / d) * Float64(D_m * 0.5)) ^ 2.0)))));
	elseif (h <= -1e-310)
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / l) / h))) * Float64(Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0))) + -1.0));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m * Float64(D_m * 0.5)) / d) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (h <= -3.6e+183)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * (((M_m / d) * (D_m * 0.5)) ^ 2.0))));
	elseif (h <= -1e-310)
		tmp = (d * sqrt(((1.0 / l) / h))) * ((0.5 * ((h / l) * (((M_m / 2.0) * (D_m / d)) ^ 2.0))) + -1.0);
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (0.5 * ((h / l) * (((M_m * (D_m * 0.5)) / d) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[h, -3.6e+183], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / d), $MachinePrecision] * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1e-310], N[(N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -3.60000000000000023e183

   1. Initial program 63.7%

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

   3. Simplified 63.7%

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

      1. add-sqr-sqrt 63.7%

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

      2. pow2 63.7%

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

      3. sqrt-prod 63.7%

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

      4. sqrt-pow1 63.7%

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

      5. *-commutative 63.7%

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

      6. metadata-eval 63.7%

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

      7. pow1 63.7%

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

      8. div-inv 63.7%

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

      9. associate-*l* 61.2%

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

      10. times-frac 61.2%

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

      11. *-commutative 61.2%

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

      12. times-frac 61.2%

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

      13. metadata-eval 61.2%

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

   6. Applied egg-rr 61.2%

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

      1. pow1 61.2%

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

      2. sqrt-unprod 51.8%

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

      3. cancel-sign-sub-inv 51.8%

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

      4. metadata-eval 51.8%

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

      5. *-commutative 51.8%

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

      6. unpow-prod-down 51.6%

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

      7. pow2 51.6%

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

      8. add-sqr-sqrt 51.6%

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

      9. associate-*r* 51.6%

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

   8. Applied egg-rr 51.6%

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

      1. unpow1 51.6%

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

      2. *-lft-identity 51.6%

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

      3. *-lft-identity 51.6%

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

      4. *-commutative 51.6%

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

   10. Simplified 51.6%

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

   if -3.60000000000000023e183 < h < -9.999999999999969e-311

   1. Initial program 80.3%

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

   3. Simplified 77.5%

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

   5. Taylor expanded in h around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-/r* 0.0%

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

      4. *-commutative 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 83.3%

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

      7. neg-mul-1 83.3%

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

   7. Simplified 81.4%

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

   if -9.999999999999969e-311 < h

   1. Initial program 65.7%

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

   3. Simplified 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. *-commutative 64.8%

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

      2. sqrt-div 67.6%

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

      3. sqrt-div 78.4%

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

      4. frac-times 78.3%

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

      5. add-sqr-sqrt 78.5%

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

   6. Applied egg-rr 78.5%

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

      1. associate-*r/ 79.3%

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

      2. div-inv 79.3%

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

      3. metadata-eval 79.3%

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

      4. associate-*l* 79.3%

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

   8. Applied egg-rr 79.3%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -3.6 \cdot 10^{+183}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \left(D \cdot 0.5\right)}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\\ \mathbf{if}\;h \leq -6.2 \cdot 10^{+181}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{d} \cdot \left(D\_m \cdot 0.5\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(t\_0 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - t\_0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0)))))
   (if (<= h -6.2e+181)
     (*
      (sqrt (* (/ d h) (/ d l)))
      (+ 1.0 (* -0.5 (* (/ h l) (pow (* (/ M_m d) (* D_m 0.5)) 2.0)))))
     (if (<= h -1e-310)
       (* (* d (sqrt (/ (/ 1.0 l) h))) (+ t_0 -1.0))
       (* (/ d (* (sqrt l) (sqrt h))) (- 1.0 t_0))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 0.5 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0));
	double tmp;
	if (h <= -6.2e+181) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * pow(((M_m / d) * (D_m * 0.5)), 2.0))));
	} else if (h <= -1e-310) {
		tmp = (d * sqrt(((1.0 / l) / h))) * (t_0 + -1.0);
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - t_0);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_m / d)) ** 2.0d0))
    if (h <= (-6.2d+181)) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 + ((-0.5d0) * ((h / l) * (((m_m / d) * (d_m * 0.5d0)) ** 2.0d0))))
    else if (h <= (-1d-310)) then
        tmp = (d * sqrt(((1.0d0 / l) / h))) * (t_0 + (-1.0d0))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - t_0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D_m / d)), 2.0));
	double tmp;
	if (h <= -6.2e+181) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * Math.pow(((M_m / d) * (D_m * 0.5)), 2.0))));
	} else if (h <= -1e-310) {
		tmp = (d * Math.sqrt(((1.0 / l) / h))) * (t_0 + -1.0);
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - t_0);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = 0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D_m / d)), 2.0))
	tmp = 0
	if h <= -6.2e+181:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * math.pow(((M_m / d) * (D_m * 0.5)), 2.0))))
	elif h <= -1e-310:
		tmp = (d * math.sqrt(((1.0 / l) / h))) * (t_0 + -1.0)
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - t_0)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0)))
	tmp = 0.0
	if (h <= -6.2e+181)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / d) * Float64(D_m * 0.5)) ^ 2.0)))));
	elseif (h <= -1e-310)
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / l) / h))) * Float64(t_0 + -1.0));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - t_0));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = 0.5 * ((h / l) * (((M_m / 2.0) * (D_m / d)) ^ 2.0));
	tmp = 0.0;
	if (h <= -6.2e+181)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * (((M_m / d) * (D_m * 0.5)) ^ 2.0))));
	elseif (h <= -1e-310)
		tmp = (d * sqrt(((1.0 / l) / h))) * (t_0 + -1.0);
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - t_0);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -6.2e+181], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / d), $MachinePrecision] * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1e-310], N[(N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -6.19999999999999978e181

   1. Initial program 63.7%

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

   3. Simplified 63.7%

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

      1. add-sqr-sqrt 63.7%

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

      2. pow2 63.7%

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

      3. sqrt-prod 63.7%

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

      4. sqrt-pow1 63.7%

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

      5. *-commutative 63.7%

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

      6. metadata-eval 63.7%

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

      7. pow1 63.7%

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

      8. div-inv 63.7%

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

      9. associate-*l* 61.2%

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

      10. times-frac 61.2%

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

      11. *-commutative 61.2%

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

      12. times-frac 61.2%

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

      13. metadata-eval 61.2%

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

   6. Applied egg-rr 61.2%

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

      1. pow1 61.2%

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

      2. sqrt-unprod 51.8%

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

      3. cancel-sign-sub-inv 51.8%

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

      4. metadata-eval 51.8%

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

      5. *-commutative 51.8%

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

      6. unpow-prod-down 51.6%

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

      7. pow2 51.6%

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

      8. add-sqr-sqrt 51.6%

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

      9. associate-*r* 51.6%

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

   8. Applied egg-rr 51.6%

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

      1. unpow1 51.6%

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

      2. *-lft-identity 51.6%

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

      3. *-lft-identity 51.6%

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

      4. *-commutative 51.6%

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

   10. Simplified 51.6%

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

   if -6.19999999999999978e181 < h < -9.999999999999969e-311

   1. Initial program 80.3%

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

   3. Simplified 77.5%

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

   5. Taylor expanded in h around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-/r* 0.0%

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

      4. *-commutative 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 83.3%

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

      7. neg-mul-1 83.3%

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

   7. Simplified 81.4%

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

   if -9.999999999999969e-311 < h

   1. Initial program 65.7%

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

   3. Simplified 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. *-commutative 64.8%

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

      2. sqrt-div 67.6%

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

      3. sqrt-div 78.4%

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

      4. frac-times 78.3%

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

      5. add-sqr-sqrt 78.5%

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

   6. Applied egg-rr 78.5%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -6.2 \cdot 10^{+181}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -8.2 \cdot 10^{-292}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(h \cdot -0.5\right) \cdot {\left(0.5 \cdot \frac{D\_m \cdot M\_m}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot \left(D\_m \cdot 0.5\right)}{d}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -8.2e-292)
   (*
    (sqrt (/ d l))
    (*
     (sqrt (/ d h))
     (+ 1.0 (/ (* (* h -0.5) (pow (* 0.5 (/ (* D_m M_m) d)) 2.0)) l))))
   (*
    (/ d (* (sqrt l) (sqrt h)))
    (- 1.0 (* 0.5 (* (/ h l) (pow (/ (* M_m (* D_m 0.5)) d) 2.0)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -8.2e-292) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + (((h * -0.5) * pow((0.5 * ((D_m * M_m) / d)), 2.0)) / l)));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (0.5 * ((h / l) * pow(((M_m * (D_m * 0.5)) / d), 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-8.2d-292)) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + (((h * (-0.5d0)) * ((0.5d0 * ((d_m * m_m) / d)) ** 2.0d0)) / l)))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - (0.5d0 * ((h / l) * (((m_m * (d_m * 0.5d0)) / d) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -8.2e-292) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + (((h * -0.5) * Math.pow((0.5 * ((D_m * M_m) / d)), 2.0)) / l)));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - (0.5 * ((h / l) * Math.pow(((M_m * (D_m * 0.5)) / d), 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -8.2e-292:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + (((h * -0.5) * math.pow((0.5 * ((D_m * M_m) / d)), 2.0)) / l)))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - (0.5 * ((h / l) * math.pow(((M_m * (D_m * 0.5)) / d), 2.0))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -8.2e-292)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(Float64(h * -0.5) * (Float64(0.5 * Float64(Float64(D_m * M_m) / d)) ^ 2.0)) / l))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m * Float64(D_m * 0.5)) / d) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -8.2e-292)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + (((h * -0.5) * ((0.5 * ((D_m * M_m) / d)) ^ 2.0)) / l)));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (0.5 * ((h / l) * (((M_m * (D_m * 0.5)) / d) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -8.2e-292], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(N[(h * -0.5), $MachinePrecision] * N[Power[N[(0.5 * N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $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[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -8.20000000000000087e-292

   1. Initial program 76.6%

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

   3. Simplified 75.1%

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

      1. associate-*l/ 76.3%

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

   6. Applied egg-rr 76.3%

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

   7. Taylor expanded in D around 0 77.7%

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

   if -8.20000000000000087e-292 < d

   1. Initial program 65.1%

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

   3. Simplified 64.3%

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

      1. *-commutative 64.3%

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

      2. sqrt-div 67.0%

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

      3. sqrt-div 77.7%

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

      4. frac-times 77.7%

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

      5. add-sqr-sqrt 77.8%

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

   6. Applied egg-rr 77.8%

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

      1. associate-*r/ 78.6%

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

      2. div-inv 78.6%

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

      3. metadata-eval 78.6%

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

      4. associate-*l* 78.6%

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

   8. Applied egg-rr 78.6%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.2 \cdot 10^{-292}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{\left(h \cdot -0.5\right) \cdot {\left(0.5 \cdot \frac{D \cdot M}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \left(D \cdot 0.5\right)}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -8.2 \cdot 10^{-292}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot -0.5\right) \cdot \frac{{\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot \left(D\_m \cdot 0.5\right)}{d}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -8.2e-292)
   (*
    (sqrt (/ d l))
    (*
     (sqrt (/ d h))
     (+ 1.0 (* (* h -0.5) (/ (pow (* M_m (* 0.5 (/ D_m d))) 2.0) l)))))
   (*
    (/ d (* (sqrt l) (sqrt h)))
    (- 1.0 (* 0.5 (* (/ h l) (pow (/ (* M_m (* D_m 0.5)) d) 2.0)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -8.2e-292) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h * -0.5) * (pow((M_m * (0.5 * (D_m / d))), 2.0) / l))));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (0.5 * ((h / l) * pow(((M_m * (D_m * 0.5)) / d), 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-8.2d-292)) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + ((h * (-0.5d0)) * (((m_m * (0.5d0 * (d_m / d))) ** 2.0d0) / l))))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - (0.5d0 * ((h / l) * (((m_m * (d_m * 0.5d0)) / d) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -8.2e-292) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + ((h * -0.5) * (Math.pow((M_m * (0.5 * (D_m / d))), 2.0) / l))));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - (0.5 * ((h / l) * Math.pow(((M_m * (D_m * 0.5)) / d), 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -8.2e-292:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + ((h * -0.5) * (math.pow((M_m * (0.5 * (D_m / d))), 2.0) / l))))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - (0.5 * ((h / l) * math.pow(((M_m * (D_m * 0.5)) / d), 2.0))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -8.2e-292)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h * -0.5) * Float64((Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0) / l)))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m * Float64(D_m * 0.5)) / d) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -8.2e-292)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h * -0.5) * (((M_m * (0.5 * (D_m / d))) ^ 2.0) / l))));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (0.5 * ((h / l) * (((M_m * (D_m * 0.5)) / d) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -8.2e-292], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h * -0.5), $MachinePrecision] * N[(N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $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[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -8.20000000000000087e-292

   1. Initial program 76.6%

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

   3. Simplified 75.1%

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

      1. associate-*l/ 76.3%

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

   6. Applied egg-rr 76.3%

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

      1. associate-/l* 77.6%

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

      2. associate-*r/ 77.6%

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

      3. *-commutative 77.6%

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

      4. associate-/l* 78.4%

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

      5. associate-*l/ 76.9%

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

      6. *-commutative 76.9%

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

      7. associate-*l* 76.9%

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

   8. Simplified 76.9%

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

   if -8.20000000000000087e-292 < d

   1. Initial program 65.1%

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

   3. Simplified 64.3%

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

      1. *-commutative 64.3%

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

      2. sqrt-div 67.0%

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

      3. sqrt-div 77.7%

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

      4. frac-times 77.7%

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

      5. add-sqr-sqrt 77.8%

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

   6. Applied egg-rr 77.8%

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

      1. associate-*r/ 78.6%

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

      2. div-inv 78.6%

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

      3. metadata-eval 78.6%

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

      4. associate-*l* 78.6%

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

   8. Applied egg-rr 78.6%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.2 \cdot 10^{-292}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \left(h \cdot -0.5\right) \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \left(D \cdot 0.5\right)}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 73.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{+157}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+126}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 + h \cdot \left(-0.25 \cdot \left({\left(\frac{D\_m \cdot M\_m}{d}\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{h}^{-0.5} \cdot \frac{d}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -4e+157)
   (* (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ d l)))
   (if (<= l -5e-310)
     (*
      (* d (sqrt (/ (/ 1.0 l) h)))
      (+ (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0))) -1.0))
     (if (<= l 5.8e+126)
       (*
        (* d (sqrt (/ 1.0 (* l h))))
        (+ 1.0 (* h (* -0.25 (* (pow (/ (* D_m M_m) d) 2.0) (/ 0.5 l))))))
       (* (pow h -0.5) (/ d (sqrt l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -4e+157) {
		tmp = (sqrt(-d) / sqrt(-h)) * sqrt((d / l));
	} else if (l <= -5e-310) {
		tmp = (d * sqrt(((1.0 / l) / h))) * ((0.5 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0))) + -1.0);
	} else if (l <= 5.8e+126) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (1.0 + (h * (-0.25 * (pow(((D_m * M_m) / d), 2.0) * (0.5 / l)))));
	} else {
		tmp = pow(h, -0.5) * (d / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-4d+157)) then
        tmp = (sqrt(-d) / sqrt(-h)) * sqrt((d / l))
    else if (l <= (-5d-310)) then
        tmp = (d * sqrt(((1.0d0 / l) / h))) * ((0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_m / d)) ** 2.0d0))) + (-1.0d0))
    else if (l <= 5.8d+126) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * (1.0d0 + (h * ((-0.25d0) * ((((d_m * m_m) / d) ** 2.0d0) * (0.5d0 / l)))))
    else
        tmp = (h ** (-0.5d0)) * (d / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -4e+157) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * Math.sqrt((d / l));
	} else if (l <= -5e-310) {
		tmp = (d * Math.sqrt(((1.0 / l) / h))) * ((0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D_m / d)), 2.0))) + -1.0);
	} else if (l <= 5.8e+126) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (1.0 + (h * (-0.25 * (Math.pow(((D_m * M_m) / d), 2.0) * (0.5 / l)))));
	} else {
		tmp = Math.pow(h, -0.5) * (d / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -4e+157:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * math.sqrt((d / l))
	elif l <= -5e-310:
		tmp = (d * math.sqrt(((1.0 / l) / h))) * ((0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D_m / d)), 2.0))) + -1.0)
	elif l <= 5.8e+126:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (1.0 + (h * (-0.25 * (math.pow(((D_m * M_m) / d), 2.0) * (0.5 / l)))))
	else:
		tmp = math.pow(h, -0.5) * (d / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -4e+157)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * sqrt(Float64(d / l)));
	elseif (l <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / l) / h))) * Float64(Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0))) + -1.0));
	elseif (l <= 5.8e+126)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(1.0 + Float64(h * Float64(-0.25 * Float64((Float64(Float64(D_m * M_m) / d) ^ 2.0) * Float64(0.5 / l))))));
	else
		tmp = Float64((h ^ -0.5) * Float64(d / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -4e+157)
		tmp = (sqrt(-d) / sqrt(-h)) * sqrt((d / l));
	elseif (l <= -5e-310)
		tmp = (d * sqrt(((1.0 / l) / h))) * ((0.5 * ((h / l) * (((M_m / 2.0) * (D_m / d)) ^ 2.0))) + -1.0);
	elseif (l <= 5.8e+126)
		tmp = (d * sqrt((1.0 / (l * h)))) * (1.0 + (h * (-0.25 * ((((D_m * M_m) / d) ^ 2.0) * (0.5 / l)))));
	else
		tmp = (h ^ -0.5) * (d / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -4e+157], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.8e+126], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(h * N[(-0.25 * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[h, -0.5], $MachinePrecision] * N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -3.99999999999999993e157

   1. Initial program 70.8%

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

   3. Simplified 68.0%

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

      1. add-sqr-sqrt 68.0%

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

      2. pow2 68.0%

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

      3. sqrt-prod 68.1%

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

      4. sqrt-pow1 68.2%

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

      5. *-commutative 68.2%

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

      6. metadata-eval 68.2%

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

      7. pow1 68.2%

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

      8. div-inv 68.2%

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

      9. associate-*l* 70.9%

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

      10. times-frac 70.9%

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

      11. *-commutative 70.9%

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

      12. times-frac 70.9%

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

      13. metadata-eval 70.9%

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

   6. Applied egg-rr 70.9%

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

   7. Taylor expanded in D around 0 59.5%

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

      1. frac-2neg 68.4%

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

      2. sqrt-div 73.9%

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

   9. Applied egg-rr 65.1%

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

   if -3.99999999999999993e157 < l < -4.999999999999985e-310

   1. Initial program 77.7%

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

   3. Simplified 75.9%

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

   5. Taylor expanded in h around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-/r* 0.0%

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

      4. *-commutative 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 77.0%

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

      7. neg-mul-1 77.0%

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

   7. Simplified 75.9%

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

   if -4.999999999999985e-310 < l < 5.79999999999999971e126

   1. Initial program 71.2%

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

   3. Simplified 70.1%

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

   5. Taylor expanded in d around 0 78.4%

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

      1. *-commutative 86.0%

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

   7. Simplified 78.4%

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

      1. add-sqr-sqrt 78.4%

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

      2. pow2 78.4%

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

      3. *-commutative 78.4%

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

      4. sqrt-prod 78.3%

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

      5. *-commutative 78.3%

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

      6. sqrt-prod 78.4%

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

      7. sqrt-pow1 80.8%

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

      8. metadata-eval 80.8%

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

      9. pow1 80.8%

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

      10. associate-*r/ 80.8%

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

      11. div-inv 80.8%

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

      12. metadata-eval 80.8%

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

   9. Applied egg-rr 80.8%

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

   10. Taylor expanded in h around inf 63.6%

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

       1. sub-neg 63.6%

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

       2. distribute-lft-in 63.6%

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

       3. rgt-mult-inverse 63.6%

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

       4. distribute-lft-neg-in 63.6%

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

       5. metadata-eval 63.6%

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

       6. associate-*r* 63.6%

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

       7. unpow2 63.6%

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

       8. rem-square-sqrt 63.7%

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

       9. times-frac 62.8%

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

   12. Simplified 83.3%

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

   if 5.79999999999999971e126 < l

   1. Initial program 52.2%

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

   3. Simplified 52.2%

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

      1. add-sqr-sqrt 52.2%

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

      2. pow2 52.2%

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

      3. sqrt-prod 52.2%

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

      4. sqrt-pow1 55.6%

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

      5. *-commutative 55.6%

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

      6. metadata-eval 55.6%

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

      7. pow1 55.6%

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

      8. div-inv 55.6%

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

      9. associate-*l* 49.6%

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

      10. times-frac 49.6%

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

      11. *-commutative 49.6%

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

      12. times-frac 49.6%

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

      13. metadata-eval 49.6%

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

   6. Applied egg-rr 49.6%

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

   7. Taylor expanded in d around inf 36.9%

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

      1. unpow-1 36.9%

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

      2. metadata-eval 36.9%

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

      3. pow-sqr 36.9%

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

      4. rem-sqrt-square 36.9%

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

      5. rem-square-sqrt 36.8%

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

      6. fabs-sqr 36.8%

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

      7. rem-square-sqrt 36.9%

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

   9. Simplified 36.9%

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

   10. Taylor expanded in d around 0 36.9%

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

       1. *-commutative 36.9%

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

       2. unpow1/2 36.9%

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

       3. rem-exp-log 35.4%

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

       4. rec-exp 35.4%

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

       5. exp-prod 35.4%

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

       6. distribute-lft-neg-out 35.4%

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

       7. exp-neg 35.4%

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

       8. exp-to-pow 36.8%

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

       9. unpow1/2 36.8%

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

       10. associate-*l/ 36.8%

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

       11. *-lft-identity 36.8%

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

   12. Simplified 36.8%

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

       1. *-un-lft-identity 36.8%

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

       2. sqrt-prod 59.3%

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

       3. times-frac 59.3%

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

       4. add-sqr-sqrt 59.3%

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

       5. sqrt-unprod 25.2%

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

       6. sqr-neg 25.2%

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

       7. sqrt-unprod 0.0%

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

       8. add-sqr-sqrt 0.0%

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

       9. pow1/2 0.0%

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

       10. pow-flip 0.0%

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

       11. add-sqr-sqrt 0.0%

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

       12. sqrt-unprod 25.3%

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

       13. sqr-neg 25.3%

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

       14. sqrt-unprod 59.4%

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

       15. add-sqr-sqrt 59.7%

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

       16. metadata-eval 59.7%

           \[\leadsto {h}^{\color{blue}{-0.5}} \cdot \frac{d}{\sqrt{\ell}} \]

   14. Applied egg-rr 59.7%

       \[\leadsto \color{blue}{{h}^{-0.5} \cdot \frac{d}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{+157}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+126}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 + h \cdot \left(-0.25 \cdot \left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{h}^{-0.5} \cdot \frac{d}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 72.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.15 \cdot 10^{+151}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 2.55 \cdot 10^{+126}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 + h \cdot \left(-0.25 \cdot \left({\left(\frac{D\_m \cdot M\_m}{d}\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{h}^{-0.5} \cdot \frac{d}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -1.15e+151)
   (/ (sqrt (/ d l)) (sqrt (/ h d)))
   (if (<= l -5e-310)
     (*
      (* d (pow (* l h) -0.5))
      (+ (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0))) -1.0))
     (if (<= l 2.55e+126)
       (*
        (* d (sqrt (/ 1.0 (* l h))))
        (+ 1.0 (* h (* -0.25 (* (pow (/ (* D_m M_m) d) 2.0) (/ 0.5 l))))))
       (* (pow h -0.5) (/ d (sqrt l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1.15e+151) {
		tmp = sqrt((d / l)) / sqrt((h / d));
	} else if (l <= -5e-310) {
		tmp = (d * pow((l * h), -0.5)) * ((0.5 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0))) + -1.0);
	} else if (l <= 2.55e+126) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (1.0 + (h * (-0.25 * (pow(((D_m * M_m) / d), 2.0) * (0.5 / l)))));
	} else {
		tmp = pow(h, -0.5) * (d / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-1.15d+151)) then
        tmp = sqrt((d / l)) / sqrt((h / d))
    else if (l <= (-5d-310)) then
        tmp = (d * ((l * h) ** (-0.5d0))) * ((0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_m / d)) ** 2.0d0))) + (-1.0d0))
    else if (l <= 2.55d+126) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * (1.0d0 + (h * ((-0.25d0) * ((((d_m * m_m) / d) ** 2.0d0) * (0.5d0 / l)))))
    else
        tmp = (h ** (-0.5d0)) * (d / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1.15e+151) {
		tmp = Math.sqrt((d / l)) / Math.sqrt((h / d));
	} else if (l <= -5e-310) {
		tmp = (d * Math.pow((l * h), -0.5)) * ((0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D_m / d)), 2.0))) + -1.0);
	} else if (l <= 2.55e+126) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (1.0 + (h * (-0.25 * (Math.pow(((D_m * M_m) / d), 2.0) * (0.5 / l)))));
	} else {
		tmp = Math.pow(h, -0.5) * (d / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -1.15e+151:
		tmp = math.sqrt((d / l)) / math.sqrt((h / d))
	elif l <= -5e-310:
		tmp = (d * math.pow((l * h), -0.5)) * ((0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D_m / d)), 2.0))) + -1.0)
	elif l <= 2.55e+126:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (1.0 + (h * (-0.25 * (math.pow(((D_m * M_m) / d), 2.0) * (0.5 / l)))))
	else:
		tmp = math.pow(h, -0.5) * (d / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -1.15e+151)
		tmp = Float64(sqrt(Float64(d / l)) / sqrt(Float64(h / d)));
	elseif (l <= -5e-310)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0))) + -1.0));
	elseif (l <= 2.55e+126)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(1.0 + Float64(h * Float64(-0.25 * Float64((Float64(Float64(D_m * M_m) / d) ^ 2.0) * Float64(0.5 / l))))));
	else
		tmp = Float64((h ^ -0.5) * Float64(d / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -1.15e+151)
		tmp = sqrt((d / l)) / sqrt((h / d));
	elseif (l <= -5e-310)
		tmp = (d * ((l * h) ^ -0.5)) * ((0.5 * ((h / l) * (((M_m / 2.0) * (D_m / d)) ^ 2.0))) + -1.0);
	elseif (l <= 2.55e+126)
		tmp = (d * sqrt((1.0 / (l * h)))) * (1.0 + (h * (-0.25 * ((((D_m * M_m) / d) ^ 2.0) * (0.5 / l)))));
	else
		tmp = (h ^ -0.5) * (d / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -1.15e+151], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.55e+126], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(h * N[(-0.25 * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[h, -0.5], $MachinePrecision] * N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.15e151

   1. Initial program 70.8%

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

   3. Simplified 68.0%

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

      1. add-sqr-sqrt 68.0%

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

      2. pow2 68.0%

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

      3. sqrt-prod 68.1%

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

      4. sqrt-pow1 68.2%

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

      5. *-commutative 68.2%

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

      6. metadata-eval 68.2%

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

      7. pow1 68.2%

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

      8. div-inv 68.2%

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

      9. associate-*l* 70.9%

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

      10. times-frac 70.9%

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

      11. *-commutative 70.9%

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

      12. times-frac 70.9%

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

      13. metadata-eval 70.9%

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

   6. Applied egg-rr 70.9%

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

   7. Taylor expanded in D around 0 59.5%

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

      1. *-rgt-identity 59.5%

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

      2. *-commutative 59.5%

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

      3. frac-2neg 59.5%

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

      4. sqrt-undiv 65.1%

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

      5. clear-num 65.0%

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

      6. un-div-inv 65.0%

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

      7. sqrt-undiv 61.6%

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

      8. frac-2neg 61.6%

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

   9. Applied egg-rr 61.6%

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

   if -1.15e151 < l < -4.999999999999985e-310

   1. Initial program 77.7%

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

   3. Simplified 75.9%

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

   5. Taylor expanded in d around 0 0.4%

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

      1. *-commutative 0.0%

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

   7. Simplified 0.4%

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

   8. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 75.0%

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

      4. unpow-1 75.0%

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

      5. metadata-eval 75.0%

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

      6. pow-sqr 75.1%

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

      7. rem-sqrt-square 75.1%

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

      8. rem-square-sqrt 74.9%

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

      9. fabs-sqr 74.9%

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

      10. rem-square-sqrt 75.1%

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

   10. Simplified 75.1%

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

   if -4.999999999999985e-310 < l < 2.5500000000000001e126

   1. Initial program 71.2%

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

   3. Simplified 70.1%

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

   5. Taylor expanded in d around 0 78.4%

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

      1. *-commutative 86.0%

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

   7. Simplified 78.4%

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

      1. add-sqr-sqrt 78.4%

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

      2. pow2 78.4%

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

      3. *-commutative 78.4%

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

      4. sqrt-prod 78.3%

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

      5. *-commutative 78.3%

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

      6. sqrt-prod 78.4%

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

      7. sqrt-pow1 80.8%

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

      8. metadata-eval 80.8%

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

      9. pow1 80.8%

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

      10. associate-*r/ 80.8%

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

      11. div-inv 80.8%

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

      12. metadata-eval 80.8%

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

   9. Applied egg-rr 80.8%

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

   10. Taylor expanded in h around inf 63.6%

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

       1. sub-neg 63.6%

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

       2. distribute-lft-in 63.6%

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

       3. rgt-mult-inverse 63.6%

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

       4. distribute-lft-neg-in 63.6%

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

       5. metadata-eval 63.6%

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

       6. associate-*r* 63.6%

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

       7. unpow2 63.6%

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

       8. rem-square-sqrt 63.7%

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

       9. times-frac 62.8%

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

   12. Simplified 83.3%

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

   if 2.5500000000000001e126 < l

   1. Initial program 52.2%

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

   3. Simplified 52.2%

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

      1. add-sqr-sqrt 52.2%

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

      2. pow2 52.2%

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

      3. sqrt-prod 52.2%

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

      4. sqrt-pow1 55.6%

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

      5. *-commutative 55.6%

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

      6. metadata-eval 55.6%

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

      7. pow1 55.6%

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

      8. div-inv 55.6%

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

      9. associate-*l* 49.6%

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

      10. times-frac 49.6%

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

      11. *-commutative 49.6%

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

      12. times-frac 49.6%

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

      13. metadata-eval 49.6%

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

   6. Applied egg-rr 49.6%

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

   7. Taylor expanded in d around inf 36.9%

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

      1. unpow-1 36.9%

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

      2. metadata-eval 36.9%

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

      3. pow-sqr 36.9%

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

      4. rem-sqrt-square 36.9%

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

      5. rem-square-sqrt 36.8%

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

      6. fabs-sqr 36.8%

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

      7. rem-square-sqrt 36.9%

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

   9. Simplified 36.9%

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

   10. Taylor expanded in d around 0 36.9%

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

       1. *-commutative 36.9%

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

       2. unpow1/2 36.9%

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

       3. rem-exp-log 35.4%

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

       4. rec-exp 35.4%

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

       5. exp-prod 35.4%

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

       6. distribute-lft-neg-out 35.4%

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

       7. exp-neg 35.4%

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

       8. exp-to-pow 36.8%

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

       9. unpow1/2 36.8%

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

       10. associate-*l/ 36.8%

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

       11. *-lft-identity 36.8%

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

   12. Simplified 36.8%

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

       1. *-un-lft-identity 36.8%

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

       2. sqrt-prod 59.3%

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

       3. times-frac 59.3%

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

       4. add-sqr-sqrt 59.3%

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

       5. sqrt-unprod 25.2%

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

       6. sqr-neg 25.2%

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

       7. sqrt-unprod 0.0%

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

       8. add-sqr-sqrt 0.0%

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

       9. pow1/2 0.0%

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

       10. pow-flip 0.0%

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

       11. add-sqr-sqrt 0.0%

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

       12. sqrt-unprod 25.3%

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

       13. sqr-neg 25.3%

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

       14. sqrt-unprod 59.4%

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

       15. add-sqr-sqrt 59.7%

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

       16. metadata-eval 59.7%

           \[\leadsto {h}^{\color{blue}{-0.5}} \cdot \frac{d}{\sqrt{\ell}} \]

   14. Applied egg-rr 59.7%

       \[\leadsto \color{blue}{{h}^{-0.5} \cdot \frac{d}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.15 \cdot 10^{+151}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 2.55 \cdot 10^{+126}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 + h \cdot \left(-0.25 \cdot \left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{h}^{-0.5} \cdot \frac{d}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 57.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-181}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;\ell \leq 8.4 \cdot 10^{-305}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{1.5}}\\ \mathbf{elif}\;\ell \leq 1.52 \cdot 10^{+119}:\\ \;\;\;\;\frac{d \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \left(0.5 \cdot M\_m\right)\right)}^{2}\right)\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;{h}^{-0.5} \cdot \frac{d}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -1e-181)
   (/ (sqrt (/ d l)) (sqrt (/ h d)))
   (if (<= l 8.4e-305)
     (* d (cbrt (pow (/ (/ 1.0 h) l) 1.5)))
     (if (<= l 1.52e+119)
       (/
        (* d (+ 1.0 (* -0.5 (* (/ h l) (pow (* (/ D_m d) (* 0.5 M_m)) 2.0)))))
        (sqrt (* l h)))
       (* (pow h -0.5) (/ d (sqrt l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1e-181) {
		tmp = sqrt((d / l)) / sqrt((h / d));
	} else if (l <= 8.4e-305) {
		tmp = d * cbrt(pow(((1.0 / h) / l), 1.5));
	} else if (l <= 1.52e+119) {
		tmp = (d * (1.0 + (-0.5 * ((h / l) * pow(((D_m / d) * (0.5 * M_m)), 2.0))))) / sqrt((l * h));
	} else {
		tmp = pow(h, -0.5) * (d / sqrt(l));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1e-181) {
		tmp = Math.sqrt((d / l)) / Math.sqrt((h / d));
	} else if (l <= 8.4e-305) {
		tmp = d * Math.cbrt(Math.pow(((1.0 / h) / l), 1.5));
	} else if (l <= 1.52e+119) {
		tmp = (d * (1.0 + (-0.5 * ((h / l) * Math.pow(((D_m / d) * (0.5 * M_m)), 2.0))))) / Math.sqrt((l * h));
	} else {
		tmp = Math.pow(h, -0.5) * (d / Math.sqrt(l));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -1e-181)
		tmp = Float64(sqrt(Float64(d / l)) / sqrt(Float64(h / d)));
	elseif (l <= 8.4e-305)
		tmp = Float64(d * cbrt((Float64(Float64(1.0 / h) / l) ^ 1.5)));
	elseif (l <= 1.52e+119)
		tmp = Float64(Float64(d * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(0.5 * M_m)) ^ 2.0))))) / sqrt(Float64(l * h)));
	else
		tmp = Float64((h ^ -0.5) * Float64(d / sqrt(l)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -1e-181], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8.4e-305], N[(d * N[Power[N[Power[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.52e+119], N[(N[(d * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(0.5 * M$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[h, -0.5], $MachinePrecision] * N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.00000000000000005e-181

   1. Initial program 75.2%

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

   3. Simplified 73.4%

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

      1. add-sqr-sqrt 73.4%

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

      2. pow2 73.4%

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

      3. sqrt-prod 73.4%

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

      4. sqrt-pow1 74.3%

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

      5. *-commutative 74.3%

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

      6. metadata-eval 74.3%

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

      7. pow1 74.3%

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

      8. div-inv 74.3%

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

      9. associate-*l* 76.1%

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

      10. times-frac 76.1%

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

      11. *-commutative 76.1%

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

      12. times-frac 76.1%

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

      13. metadata-eval 76.1%

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

   6. Applied egg-rr 76.1%

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

   7. Taylor expanded in D around 0 45.8%

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

      1. *-rgt-identity 45.8%

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

      2. *-commutative 45.8%

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

      3. frac-2neg 45.8%

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

      4. sqrt-undiv 49.9%

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

      5. clear-num 49.8%

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

      6. un-div-inv 49.8%

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

      7. sqrt-undiv 46.9%

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

      8. frac-2neg 46.9%

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

   9. Applied egg-rr 46.9%

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

   if -1.00000000000000005e-181 < l < 8.3999999999999999e-305

   1. Initial program 74.1%

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

   3. Simplified 71.2%

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

      1. add-sqr-sqrt 71.2%

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

      2. pow2 71.2%

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

      3. sqrt-prod 71.2%

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

      4. sqrt-pow1 71.2%

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

      5. *-commutative 71.2%

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

      6. metadata-eval 71.2%

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

      7. pow1 71.2%

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

      8. div-inv 71.1%

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

      9. associate-*l* 67.9%

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

      10. times-frac 67.9%

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

      11. *-commutative 67.9%

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

      12. times-frac 67.9%

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

      13. 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 31.1%

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

      1. unpow-1 31.1%

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

      2. metadata-eval 31.1%

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

      3. pow-sqr 31.1%

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

      4. rem-sqrt-square 31.1%

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

      5. rem-square-sqrt 31.1%

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

      6. fabs-sqr 31.1%

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

      7. rem-square-sqrt 31.1%

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

   9. Simplified 31.1%

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

       1. *-commutative 31.1%

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

       2. metadata-eval 31.1%

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

       3. pow-pow 31.1%

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

       4. inv-pow 31.1%

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

       5. pow1/2 31.1%

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

       6. add-cbrt-cube 37.4%

          \[\leadsto d \cdot \color{blue}{\sqrt[3]{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}}} \]

       7. add-sqr-sqrt 37.4%

          \[\leadsto d \cdot \sqrt[3]{\color{blue}{\frac{1}{\ell \cdot h}} \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

       8. pow1 37.4%

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

       9. pow1/2 37.4%

          \[\leadsto d \cdot \sqrt[3]{{\left(\frac{1}{\ell \cdot h}\right)}^{1} \cdot \color{blue}{{\left(\frac{1}{\ell \cdot h}\right)}^{0.5}}} \]

       10. pow-prod-up 37.4%

           \[\leadsto d \cdot \sqrt[3]{\color{blue}{{\left(\frac{1}{\ell \cdot h}\right)}^{\left(1 + 0.5\right)}}} \]

       11. *-commutative 37.4%

           \[\leadsto d \cdot \sqrt[3]{{\left(\frac{1}{\color{blue}{h \cdot \ell}}\right)}^{\left(1 + 0.5\right)}} \]

       12. metadata-eval 37.4%

           \[\leadsto d \cdot \sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{1.5}}} \]

   11. Applied egg-rr 37.4%

       \[\leadsto d \cdot \color{blue}{\sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{1.5}}} \]
   12. Step-by-step derivation

       1. associate-/r* 37.4%

          \[\leadsto d \cdot \sqrt[3]{{\color{blue}{\left(\frac{\frac{1}{h}}{\ell}\right)}}^{1.5}} \]

   13. Simplified 37.4%

       \[\leadsto d \cdot \color{blue}{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{1.5}}} \]

   if 8.3999999999999999e-305 < l < 1.52000000000000007e119

   1. Initial program 73.2%

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

   3. Simplified 71.9%

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

   5. Taylor expanded in d around 0 79.6%

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

      1. *-commutative 86.3%

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

   7. Simplified 79.6%

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

      1. add-sqr-sqrt 79.6%

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

      2. pow2 79.6%

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

      3. *-commutative 79.6%

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

      4. sqrt-prod 79.5%

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

      5. *-commutative 79.5%

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

      6. sqrt-prod 79.5%

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

      7. sqrt-pow1 82.1%

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

      8. metadata-eval 82.1%

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

      9. pow1 82.1%

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

      10. associate-*r/ 82.1%

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

      11. div-inv 82.1%

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

      12. metadata-eval 82.1%

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

   9. Applied egg-rr 82.1%

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

   11. Applied egg-rr 80.9%

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

   if 1.52000000000000007e119 < l

   1. Initial program 53.5%

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

   3. Simplified 53.5%

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

      1. add-sqr-sqrt 53.5%

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

      2. pow2 53.5%

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

      3. sqrt-prod 53.5%

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

      4. sqrt-pow1 56.6%

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

      5. *-commutative 56.6%

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

      6. metadata-eval 56.6%

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

      7. pow1 56.6%

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

      8. div-inv 56.6%

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

      9. associate-*l* 51.1%

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

      10. times-frac 51.1%

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

      11. *-commutative 51.1%

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

      12. times-frac 51.1%

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

      13. metadata-eval 51.1%

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

   6. Applied egg-rr 51.1%

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

   7. Taylor expanded in d around inf 39.4%

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

      1. unpow-1 39.4%

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

      2. metadata-eval 39.4%

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

      3. pow-sqr 39.4%

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

      4. rem-sqrt-square 39.4%

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

      5. rem-square-sqrt 39.2%

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

      6. fabs-sqr 39.2%

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

      7. rem-square-sqrt 39.4%

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

   9. Simplified 39.4%

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

   10. Taylor expanded in d around 0 39.4%

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

       1. *-commutative 39.4%

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

       2. unpow1/2 39.4%

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

       3. rem-exp-log 37.5%

          \[\leadsto {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot d \]

       4. rec-exp 37.5%

          \[\leadsto {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot d \]

       5. exp-prod 37.5%

          \[\leadsto \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot d \]

       6. distribute-lft-neg-out 37.5%

          \[\leadsto e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot d \]

       7. exp-neg 37.5%

          \[\leadsto \color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \cdot d \]

       8. exp-to-pow 39.2%

          \[\leadsto \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot d \]

       9. unpow1/2 39.2%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot d \]

       10. associate-*l/ 39.2%

           \[\leadsto \color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}} \]

       11. *-lft-identity 39.2%

           \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   12. Simplified 39.2%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   13. Step-by-step derivation

       1. *-un-lft-identity 39.2%

          \[\leadsto \frac{\color{blue}{1 \cdot d}}{\sqrt{h \cdot \ell}} \]

       2. sqrt-prod 59.9%

          \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       3. times-frac 60.0%

          \[\leadsto \color{blue}{\frac{1}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}} \]

       4. add-sqr-sqrt 60.0%

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\sqrt{h} \cdot \sqrt{h}}}} \cdot \frac{d}{\sqrt{\ell}} \]

       5. sqrt-unprod 26.0%

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\sqrt{h \cdot h}}}} \cdot \frac{d}{\sqrt{\ell}} \]

       6. sqr-neg 26.0%

          \[\leadsto \frac{1}{\sqrt{\sqrt{\color{blue}{\left(-h\right) \cdot \left(-h\right)}}}} \cdot \frac{d}{\sqrt{\ell}} \]

       7. sqrt-unprod 0.0%

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\sqrt{-h} \cdot \sqrt{-h}}}} \cdot \frac{d}{\sqrt{\ell}} \]

       8. add-sqr-sqrt 0.0%

          \[\leadsto \frac{1}{\sqrt{\color{blue}{-h}}} \cdot \frac{d}{\sqrt{\ell}} \]

       9. pow1/2 0.0%

          \[\leadsto \frac{1}{\color{blue}{{\left(-h\right)}^{0.5}}} \cdot \frac{d}{\sqrt{\ell}} \]

       10. pow-flip 0.0%

           \[\leadsto \color{blue}{{\left(-h\right)}^{\left(-0.5\right)}} \cdot \frac{d}{\sqrt{\ell}} \]

       11. add-sqr-sqrt 0.0%

           \[\leadsto {\color{blue}{\left(\sqrt{-h} \cdot \sqrt{-h}\right)}}^{\left(-0.5\right)} \cdot \frac{d}{\sqrt{\ell}} \]

       12. sqrt-unprod 26.1%

           \[\leadsto {\color{blue}{\left(\sqrt{\left(-h\right) \cdot \left(-h\right)}\right)}}^{\left(-0.5\right)} \cdot \frac{d}{\sqrt{\ell}} \]

       13. sqr-neg 26.1%

           \[\leadsto {\left(\sqrt{\color{blue}{h \cdot h}}\right)}^{\left(-0.5\right)} \cdot \frac{d}{\sqrt{\ell}} \]

       14. sqrt-unprod 60.1%

           \[\leadsto {\color{blue}{\left(\sqrt{h} \cdot \sqrt{h}\right)}}^{\left(-0.5\right)} \cdot \frac{d}{\sqrt{\ell}} \]

       15. add-sqr-sqrt 60.3%

           \[\leadsto {\color{blue}{h}}^{\left(-0.5\right)} \cdot \frac{d}{\sqrt{\ell}} \]

       16. metadata-eval 60.3%

           \[\leadsto {h}^{\color{blue}{-0.5}} \cdot \frac{d}{\sqrt{\ell}} \]

   14. Applied egg-rr 60.3%

       \[\leadsto \color{blue}{{h}^{-0.5} \cdot \frac{d}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-181}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;\ell \leq 8.4 \cdot 10^{-305}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{1.5}}\\ \mathbf{elif}\;\ell \leq 1.52 \cdot 10^{+119}:\\ \;\;\;\;\frac{d \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2}\right)\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;{h}^{-0.5} \cdot \frac{d}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 48.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.38 \cdot 10^{-142}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-23}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-0.25 \cdot \left({\left(\frac{D\_m \cdot M\_m}{d}\right)}^{2} \cdot \left(h \cdot \frac{0.5}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -1.38e-142)
   (/ (sqrt (/ d l)) (sqrt (/ h d)))
   (if (<= d -4e-310)
     (/ d (cbrt (pow (* l h) 1.5)))
     (if (<= d 1.4e-23)
       (*
        (* d (sqrt (/ 1.0 (* l h))))
        (* -0.25 (* (pow (/ (* D_m M_m) d) 2.0) (* h (/ 0.5 l)))))
       (* d (* (pow h -0.5) (pow l -0.5)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1.38e-142) {
		tmp = sqrt((d / l)) / sqrt((h / d));
	} else if (d <= -4e-310) {
		tmp = d / cbrt(pow((l * h), 1.5));
	} else if (d <= 1.4e-23) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-0.25 * (pow(((D_m * M_m) / d), 2.0) * (h * (0.5 / l))));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1.38e-142) {
		tmp = Math.sqrt((d / l)) / Math.sqrt((h / d));
	} else if (d <= -4e-310) {
		tmp = d / Math.cbrt(Math.pow((l * h), 1.5));
	} else if (d <= 1.4e-23) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (-0.25 * (Math.pow(((D_m * M_m) / d), 2.0) * (h * (0.5 / l))));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -1.38e-142)
		tmp = Float64(sqrt(Float64(d / l)) / sqrt(Float64(h / d)));
	elseif (d <= -4e-310)
		tmp = Float64(d / cbrt((Float64(l * h) ^ 1.5)));
	elseif (d <= 1.4e-23)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-0.25 * Float64((Float64(Float64(D_m * M_m) / d) ^ 2.0) * Float64(h * Float64(0.5 / l)))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -1.38e-142], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4e-310], N[(d / N[Power[N[Power[N[(l * h), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.4e-23], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.25 * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * N[(0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.37999999999999998e-142

   1. Initial program 80.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 78.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 78.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 78.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 78.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. metadata-eval 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. pow1 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(D \cdot \frac{1}{d}\right)} \cdot \frac{M}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. associate-*l* 80.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(\frac{1}{d} \cdot \frac{M}{2}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. times-frac 80.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{1 \cdot M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. *-commutative 80.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. times-frac 80.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. metadata-eval 80.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 80.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in D around 0 49.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{1} \]
   8. Step-by-step derivation

      1. *-rgt-identity 49.2%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

      2. *-commutative 49.2%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}} \]

      3. frac-2neg 49.2%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{-d}{-h}}} \]

      4. sqrt-undiv 53.6%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \]

      5. clear-num 53.5%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\frac{1}{\frac{\sqrt{-h}}{\sqrt{-d}}}} \]

      6. un-div-inv 53.5%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\frac{\sqrt{-h}}{\sqrt{-d}}}} \]

      7. sqrt-undiv 50.4%

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\color{blue}{\sqrt{\frac{-h}{-d}}}} \]

      8. frac-2neg 50.4%

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\color{blue}{\frac{h}{d}}}} \]

   9. Applied egg-rr 50.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}} \]

   if -1.37999999999999998e-142 < d < -3.999999999999988e-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 60.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 60.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 60.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 60.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 60.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 60.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. metadata-eval 60.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. pow1 60.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 60.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(D \cdot \frac{1}{d}\right)} \cdot \frac{M}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. associate-*l* 60.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(\frac{1}{d} \cdot \frac{M}{2}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. times-frac 60.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{1 \cdot M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. *-commutative 60.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. times-frac 60.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. metadata-eval 60.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 60.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 24.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. unpow-1 24.9%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. metadata-eval 24.9%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      3. pow-sqr 24.9%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      4. rem-sqrt-square 24.9%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      5. rem-square-sqrt 24.9%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

      6. fabs-sqr 24.9%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      7. rem-square-sqrt 24.9%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 24.9%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Taylor expanded in d around 0 24.9%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. *-commutative 24.9%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

       2. unpow1/2 24.9%

          \[\leadsto \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot d \]

       3. rem-exp-log 24.9%

          \[\leadsto {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot d \]

       4. rec-exp 24.9%

          \[\leadsto {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot d \]

       5. exp-prod 24.9%

          \[\leadsto \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot d \]

       6. distribute-lft-neg-out 24.9%

          \[\leadsto e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot d \]

       7. exp-neg 24.9%

          \[\leadsto \color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \cdot d \]

       8. exp-to-pow 24.9%

          \[\leadsto \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot d \]

       9. unpow1/2 24.9%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot d \]

       10. associate-*l/ 24.9%

           \[\leadsto \color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}} \]

       11. *-lft-identity 24.9%

           \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   12. Simplified 24.9%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   13. Step-by-step derivation

       1. add-cbrt-cube 30.5%

          \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{\left(\sqrt{h \cdot \ell} \cdot \sqrt{h \cdot \ell}\right) \cdot \sqrt{h \cdot \ell}}}} \]

       2. pow1/3 30.5%

          \[\leadsto \frac{d}{\color{blue}{{\left(\left(\sqrt{h \cdot \ell} \cdot \sqrt{h \cdot \ell}\right) \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}}} \]

       3. add-sqr-sqrt 30.5%

          \[\leadsto \frac{d}{{\left(\color{blue}{\left(h \cdot \ell\right)} \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}} \]

       4. pow1 30.5%

          \[\leadsto \frac{d}{{\left(\color{blue}{{\left(h \cdot \ell\right)}^{1}} \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}} \]

       5. pow1/2 30.5%

          \[\leadsto \frac{d}{{\left({\left(h \cdot \ell\right)}^{1} \cdot \color{blue}{{\left(h \cdot \ell\right)}^{0.5}}\right)}^{0.3333333333333333}} \]

       6. pow-prod-up 30.5%

          \[\leadsto \frac{d}{{\color{blue}{\left({\left(h \cdot \ell\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}} \]

       7. metadata-eval 30.5%

          \[\leadsto \frac{d}{{\left({\left(h \cdot \ell\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \]

   14. Applied egg-rr 30.5%

       \[\leadsto \frac{d}{\color{blue}{{\left({\left(h \cdot \ell\right)}^{1.5}\right)}^{0.3333333333333333}}} \]
   15. Step-by-step derivation

       1. unpow1/3 30.5%

          \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}} \]

   16. Simplified 30.5%

       \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}} \]

   if -3.999999999999988e-310 < d < 1.3999999999999999e-23

   1. Initial program 54.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 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 61.7%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 66.8%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \frac{\sqrt{h}}{\sqrt{\ell}}\right)}^{2}\right) \]

   7. Simplified 61.7%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 61.7%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \color{blue}{\sqrt{0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)} \cdot \sqrt{0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)}}\right) \]

      2. pow2 61.7%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \color{blue}{{\left(\sqrt{0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)}^{2}}\right) \]

      3. *-commutative 61.7%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\sqrt{\color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.5}}\right)}^{2}\right) \]

      4. sqrt-prod 61.6%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{0.5}\right)}}^{2}\right) \]

      5. *-commutative 61.6%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\sqrt{\color{blue}{\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}}} \cdot \sqrt{0.5}\right)}^{2}\right) \]

      6. sqrt-prod 61.6%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\color{blue}{\left(\sqrt{\frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}}\right)} \cdot \sqrt{0.5}\right)}^{2}\right) \]

      7. sqrt-pow1 66.7%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\left(\sqrt{\frac{h}{\ell}} \cdot \color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}}\right) \cdot \sqrt{0.5}\right)}^{2}\right) \]

      8. metadata-eval 66.7%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\left(\sqrt{\frac{h}{\ell}} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}}\right) \cdot \sqrt{0.5}\right)}^{2}\right) \]

      9. pow1 66.7%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\left(\sqrt{\frac{h}{\ell}} \cdot \color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}\right) \cdot \sqrt{0.5}\right)}^{2}\right) \]

      10. associate-*r/ 68.4%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\left(\sqrt{\frac{h}{\ell}} \cdot \color{blue}{\frac{\frac{M}{2} \cdot D}{d}}\right) \cdot \sqrt{0.5}\right)}^{2}\right) \]

      11. div-inv 68.4%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\left(\sqrt{\frac{h}{\ell}} \cdot \frac{\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot D}{d}\right) \cdot \sqrt{0.5}\right)}^{2}\right) \]

      12. metadata-eval 68.4%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\left(\sqrt{\frac{h}{\ell}} \cdot \frac{\left(M \cdot \color{blue}{0.5}\right) \cdot D}{d}\right) \cdot \sqrt{0.5}\right)}^{2}\right) \]

   9. Applied egg-rr 68.4%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \color{blue}{{\left(\left(\sqrt{\frac{h}{\ell}} \cdot \frac{\left(M \cdot 0.5\right) \cdot D}{d}\right) \cdot \sqrt{0.5}\right)}^{2}}\right) \]

   10. Taylor expanded in h around inf 39.6%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \color{blue}{\left(-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot {\left(\sqrt{0.5}\right)}^{2}\right)\right)}{{d}^{2} \cdot \ell}\right)} \]
   11. Step-by-step derivation

       1. *-commutative 39.6%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot {\left(\sqrt{0.5}\right)}^{2}\right)\right)}{{d}^{2} \cdot \ell} \cdot -0.25\right)} \]

       2. associate-*r* 38.0%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(h \cdot {\left(\sqrt{0.5}\right)}^{2}\right)}}{{d}^{2} \cdot \ell} \cdot -0.25\right) \]

       3. times-frac 37.8%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h \cdot {\left(\sqrt{0.5}\right)}^{2}}{\ell}\right)} \cdot -0.25\right) \]

       4. associate-/l* 36.2%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\left(\color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{{d}^{2}}\right)} \cdot \frac{h \cdot {\left(\sqrt{0.5}\right)}^{2}}{\ell}\right) \cdot -0.25\right) \]

       5. unpow2 36.2%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\left(\left(\color{blue}{\left(D \cdot D\right)} \cdot \frac{{M}^{2}}{{d}^{2}}\right) \cdot \frac{h \cdot {\left(\sqrt{0.5}\right)}^{2}}{\ell}\right) \cdot -0.25\right) \]

       6. unpow2 36.2%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\left(\left(\left(D \cdot D\right) \cdot \frac{\color{blue}{M \cdot M}}{{d}^{2}}\right) \cdot \frac{h \cdot {\left(\sqrt{0.5}\right)}^{2}}{\ell}\right) \cdot -0.25\right) \]

       7. unpow2 36.2%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\left(\left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{\color{blue}{d \cdot d}}\right) \cdot \frac{h \cdot {\left(\sqrt{0.5}\right)}^{2}}{\ell}\right) \cdot -0.25\right) \]

       8. times-frac 45.0%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{M}{d}\right)}\right) \cdot \frac{h \cdot {\left(\sqrt{0.5}\right)}^{2}}{\ell}\right) \cdot -0.25\right) \]

       9. swap-sqr 51.7%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\left(\color{blue}{\left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)} \cdot \frac{h \cdot {\left(\sqrt{0.5}\right)}^{2}}{\ell}\right) \cdot -0.25\right) \]

       10. unpow2 51.7%

           \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\left(\color{blue}{{\left(D \cdot \frac{M}{d}\right)}^{2}} \cdot \frac{h \cdot {\left(\sqrt{0.5}\right)}^{2}}{\ell}\right) \cdot -0.25\right) \]

       11. associate-*r/ 53.4%

           \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\left({\color{blue}{\left(\frac{D \cdot M}{d}\right)}}^{2} \cdot \frac{h \cdot {\left(\sqrt{0.5}\right)}^{2}}{\ell}\right) \cdot -0.25\right) \]

       12. unpow2 53.4%

           \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot \frac{h \cdot \color{blue}{\left(\sqrt{0.5} \cdot \sqrt{0.5}\right)}}{\ell}\right) \cdot -0.25\right) \]

       13. rem-square-sqrt 53.5%

           \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot \frac{h \cdot \color{blue}{0.5}}{\ell}\right) \cdot -0.25\right) \]

       14. associate-/l* 53.5%

           \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot \color{blue}{\left(h \cdot \frac{0.5}{\ell}\right)}\right) \cdot -0.25\right) \]

   12. Simplified 53.5%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \color{blue}{\left(\left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot \left(h \cdot \frac{0.5}{\ell}\right)\right) \cdot -0.25\right)} \]

   if 1.3999999999999999e-23 < d

   1. Initial program 77.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 77.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 77.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 77.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 77.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 79.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 79.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. metadata-eval 79.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. pow1 79.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 79.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(D \cdot \frac{1}{d}\right)} \cdot \frac{M}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. associate-*l* 78.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(\frac{1}{d} \cdot \frac{M}{2}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. times-frac 78.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{1 \cdot M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. *-commutative 78.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. times-frac 78.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. metadata-eval 78.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 78.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 53.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. unpow-1 53.2%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. metadata-eval 53.2%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      3. pow-sqr 53.2%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      4. rem-sqrt-square 53.2%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      5. rem-square-sqrt 53.1%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

      6. fabs-sqr 53.1%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      7. rem-square-sqrt 53.2%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 53.2%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. *-commutative 53.2%

          \[\leadsto d \cdot {\color{blue}{\left(\ell \cdot h\right)}}^{-0.5} \]

       2. unpow-prod-down 70.0%

          \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]

   11. Applied egg-rr 70.0%

       \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.38 \cdot 10^{-142}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-23}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-0.25 \cdot \left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot \left(h \cdot \frac{0.5}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 73.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 4.1 \cdot 10^{+125}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 + h \cdot \left(-0.25 \cdot \left({\left(\frac{D\_m \cdot M\_m}{d}\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{h}^{-0.5} \cdot \frac{d}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -5e-310)
   (*
    (* d (sqrt (/ (/ 1.0 l) h)))
    (+ (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0))) -1.0))
   (if (<= l 4.1e+125)
     (*
      (* d (sqrt (/ 1.0 (* l h))))
      (+ 1.0 (* h (* -0.25 (* (pow (/ (* D_m M_m) d) 2.0) (/ 0.5 l))))))
     (* (pow h -0.5) (/ d (sqrt l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -5e-310) {
		tmp = (d * sqrt(((1.0 / l) / h))) * ((0.5 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0))) + -1.0);
	} else if (l <= 4.1e+125) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (1.0 + (h * (-0.25 * (pow(((D_m * M_m) / d), 2.0) * (0.5 / l)))));
	} else {
		tmp = pow(h, -0.5) * (d / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-5d-310)) then
        tmp = (d * sqrt(((1.0d0 / l) / h))) * ((0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_m / d)) ** 2.0d0))) + (-1.0d0))
    else if (l <= 4.1d+125) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * (1.0d0 + (h * ((-0.25d0) * ((((d_m * m_m) / d) ** 2.0d0) * (0.5d0 / l)))))
    else
        tmp = (h ** (-0.5d0)) * (d / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -5e-310) {
		tmp = (d * Math.sqrt(((1.0 / l) / h))) * ((0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D_m / d)), 2.0))) + -1.0);
	} else if (l <= 4.1e+125) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (1.0 + (h * (-0.25 * (Math.pow(((D_m * M_m) / d), 2.0) * (0.5 / l)))));
	} else {
		tmp = Math.pow(h, -0.5) * (d / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -5e-310:
		tmp = (d * math.sqrt(((1.0 / l) / h))) * ((0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D_m / d)), 2.0))) + -1.0)
	elif l <= 4.1e+125:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (1.0 + (h * (-0.25 * (math.pow(((D_m * M_m) / d), 2.0) * (0.5 / l)))))
	else:
		tmp = math.pow(h, -0.5) * (d / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / l) / h))) * Float64(Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0))) + -1.0));
	elseif (l <= 4.1e+125)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(1.0 + Float64(h * Float64(-0.25 * Float64((Float64(Float64(D_m * M_m) / d) ^ 2.0) * Float64(0.5 / l))))));
	else
		tmp = Float64((h ^ -0.5) * Float64(d / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -5e-310)
		tmp = (d * sqrt(((1.0 / l) / h))) * ((0.5 * ((h / l) * (((M_m / 2.0) * (D_m / d)) ^ 2.0))) + -1.0);
	elseif (l <= 4.1e+125)
		tmp = (d * sqrt((1.0 / (l * h)))) * (1.0 + (h * (-0.25 * ((((D_m * M_m) / d) ^ 2.0) * (0.5 / l)))));
	else
		tmp = (h ^ -0.5) * (d / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -5e-310], N[(N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.1e+125], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(h * N[(-0.25 * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[h, -0.5], $MachinePrecision] * N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 76.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 74.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \cdot \left(1 - 0.5 \cdot {\left(\left(0.5 \cdot \frac{D \cdot M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. *-commutative 0.0%

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot \left(1 - 0.5 \cdot {\left(\left(0.5 \cdot \frac{D \cdot M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. associate-/r* 0.0%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot \left(1 - 0.5 \cdot {\left(\left(0.5 \cdot \frac{D \cdot M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. *-commutative 0.0%

         \[\leadsto \left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(0.5 \cdot \frac{D \cdot M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. unpow2 0.0%

         \[\leadsto \left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right)\right) \cdot \left(1 - 0.5 \cdot {\left(\left(0.5 \cdot \frac{D \cdot M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. rem-square-sqrt 71.6%

         \[\leadsto \left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(\color{blue}{-1} \cdot d\right)\right) \cdot \left(1 - 0.5 \cdot {\left(\left(0.5 \cdot \frac{D \cdot M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. neg-mul-1 71.6%

         \[\leadsto \left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \color{blue}{\left(-d\right)}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(0.5 \cdot \frac{D \cdot M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   7. Simplified 70.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   if -4.999999999999985e-310 < l < 4.09999999999999992e125

   1. Initial program 71.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 70.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around 0 78.4%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 86.0%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \frac{\sqrt{h}}{\sqrt{\ell}}\right)}^{2}\right) \]

   7. Simplified 78.4%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 78.4%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \color{blue}{\sqrt{0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)} \cdot \sqrt{0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)}}\right) \]

      2. pow2 78.4%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \color{blue}{{\left(\sqrt{0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)}^{2}}\right) \]

      3. *-commutative 78.4%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\sqrt{\color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.5}}\right)}^{2}\right) \]

      4. sqrt-prod 78.3%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{0.5}\right)}}^{2}\right) \]

      5. *-commutative 78.3%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\sqrt{\color{blue}{\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}}} \cdot \sqrt{0.5}\right)}^{2}\right) \]

      6. sqrt-prod 78.4%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\color{blue}{\left(\sqrt{\frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}}\right)} \cdot \sqrt{0.5}\right)}^{2}\right) \]

      7. sqrt-pow1 80.8%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\left(\sqrt{\frac{h}{\ell}} \cdot \color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}}\right) \cdot \sqrt{0.5}\right)}^{2}\right) \]

      8. metadata-eval 80.8%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\left(\sqrt{\frac{h}{\ell}} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}}\right) \cdot \sqrt{0.5}\right)}^{2}\right) \]

      9. pow1 80.8%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\left(\sqrt{\frac{h}{\ell}} \cdot \color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}\right) \cdot \sqrt{0.5}\right)}^{2}\right) \]

      10. associate-*r/ 80.8%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\left(\sqrt{\frac{h}{\ell}} \cdot \color{blue}{\frac{\frac{M}{2} \cdot D}{d}}\right) \cdot \sqrt{0.5}\right)}^{2}\right) \]

      11. div-inv 80.8%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\left(\sqrt{\frac{h}{\ell}} \cdot \frac{\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot D}{d}\right) \cdot \sqrt{0.5}\right)}^{2}\right) \]

      12. metadata-eval 80.8%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\left(\sqrt{\frac{h}{\ell}} \cdot \frac{\left(M \cdot \color{blue}{0.5}\right) \cdot D}{d}\right) \cdot \sqrt{0.5}\right)}^{2}\right) \]

   9. Applied egg-rr 80.8%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \color{blue}{{\left(\left(\sqrt{\frac{h}{\ell}} \cdot \frac{\left(M \cdot 0.5\right) \cdot D}{d}\right) \cdot \sqrt{0.5}\right)}^{2}}\right) \]

   10. Taylor expanded in h around inf 63.6%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \color{blue}{\left(h \cdot \left(\frac{1}{h} - 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{0.5}\right)}^{2}\right)}{{d}^{2} \cdot \ell}\right)\right)} \]
   11. Step-by-step derivation

       1. sub-neg 63.6%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(h \cdot \color{blue}{\left(\frac{1}{h} + \left(-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{0.5}\right)}^{2}\right)}{{d}^{2} \cdot \ell}\right)\right)}\right) \]

       2. distribute-lft-in 63.6%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \color{blue}{\left(h \cdot \frac{1}{h} + h \cdot \left(-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{0.5}\right)}^{2}\right)}{{d}^{2} \cdot \ell}\right)\right)} \]

       3. rgt-mult-inverse 63.6%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\color{blue}{1} + h \cdot \left(-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{0.5}\right)}^{2}\right)}{{d}^{2} \cdot \ell}\right)\right) \]

       4. distribute-lft-neg-in 63.6%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 + h \cdot \color{blue}{\left(\left(-0.25\right) \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{0.5}\right)}^{2}\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

       5. metadata-eval 63.6%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 + h \cdot \left(\color{blue}{-0.25} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{0.5}\right)}^{2}\right)}{{d}^{2} \cdot \ell}\right)\right) \]

       6. associate-*r* 63.6%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 + h \cdot \left(-0.25 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot {\left(\sqrt{0.5}\right)}^{2}}}{{d}^{2} \cdot \ell}\right)\right) \]

       7. unpow2 63.6%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 + h \cdot \left(-0.25 \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \color{blue}{\left(\sqrt{0.5} \cdot \sqrt{0.5}\right)}}{{d}^{2} \cdot \ell}\right)\right) \]

       8. rem-square-sqrt 63.7%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 + h \cdot \left(-0.25 \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \color{blue}{0.5}}{{d}^{2} \cdot \ell}\right)\right) \]

       9. times-frac 62.8%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 + h \cdot \left(-0.25 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{0.5}{\ell}\right)}\right)\right) \]

   12. Simplified 83.3%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \color{blue}{\left(1 + h \cdot \left(-0.25 \cdot \left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right)\right)} \]

   if 4.09999999999999992e125 < l

   1. Initial program 52.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 52.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 52.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 52.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 52.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 55.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 55.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. metadata-eval 55.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. pow1 55.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 55.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(D \cdot \frac{1}{d}\right)} \cdot \frac{M}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. associate-*l* 49.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(\frac{1}{d} \cdot \frac{M}{2}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. times-frac 49.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{1 \cdot M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. *-commutative 49.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. times-frac 49.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. metadata-eval 49.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 49.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 36.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. unpow-1 36.9%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. metadata-eval 36.9%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      3. pow-sqr 36.9%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      4. rem-sqrt-square 36.9%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      5. rem-square-sqrt 36.8%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

      6. fabs-sqr 36.8%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      7. rem-square-sqrt 36.9%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 36.9%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Taylor expanded in d around 0 36.9%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. *-commutative 36.9%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

       2. unpow1/2 36.9%

          \[\leadsto \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot d \]

       3. rem-exp-log 35.4%

          \[\leadsto {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot d \]

       4. rec-exp 35.4%

          \[\leadsto {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot d \]

       5. exp-prod 35.4%

          \[\leadsto \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot d \]

       6. distribute-lft-neg-out 35.4%

          \[\leadsto e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot d \]

       7. exp-neg 35.4%

          \[\leadsto \color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \cdot d \]

       8. exp-to-pow 36.8%

          \[\leadsto \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot d \]

       9. unpow1/2 36.8%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot d \]

       10. associate-*l/ 36.8%

           \[\leadsto \color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}} \]

       11. *-lft-identity 36.8%

           \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   12. Simplified 36.8%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   13. Step-by-step derivation

       1. *-un-lft-identity 36.8%

          \[\leadsto \frac{\color{blue}{1 \cdot d}}{\sqrt{h \cdot \ell}} \]

       2. sqrt-prod 59.3%

          \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       3. times-frac 59.3%

          \[\leadsto \color{blue}{\frac{1}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}} \]

       4. add-sqr-sqrt 59.3%

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\sqrt{h} \cdot \sqrt{h}}}} \cdot \frac{d}{\sqrt{\ell}} \]

       5. sqrt-unprod 25.2%

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\sqrt{h \cdot h}}}} \cdot \frac{d}{\sqrt{\ell}} \]

       6. sqr-neg 25.2%

          \[\leadsto \frac{1}{\sqrt{\sqrt{\color{blue}{\left(-h\right) \cdot \left(-h\right)}}}} \cdot \frac{d}{\sqrt{\ell}} \]

       7. sqrt-unprod 0.0%

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\sqrt{-h} \cdot \sqrt{-h}}}} \cdot \frac{d}{\sqrt{\ell}} \]

       8. add-sqr-sqrt 0.0%

          \[\leadsto \frac{1}{\sqrt{\color{blue}{-h}}} \cdot \frac{d}{\sqrt{\ell}} \]

       9. pow1/2 0.0%

          \[\leadsto \frac{1}{\color{blue}{{\left(-h\right)}^{0.5}}} \cdot \frac{d}{\sqrt{\ell}} \]

       10. pow-flip 0.0%

           \[\leadsto \color{blue}{{\left(-h\right)}^{\left(-0.5\right)}} \cdot \frac{d}{\sqrt{\ell}} \]

       11. add-sqr-sqrt 0.0%

           \[\leadsto {\color{blue}{\left(\sqrt{-h} \cdot \sqrt{-h}\right)}}^{\left(-0.5\right)} \cdot \frac{d}{\sqrt{\ell}} \]

       12. sqrt-unprod 25.3%

           \[\leadsto {\color{blue}{\left(\sqrt{\left(-h\right) \cdot \left(-h\right)}\right)}}^{\left(-0.5\right)} \cdot \frac{d}{\sqrt{\ell}} \]

       13. sqr-neg 25.3%

           \[\leadsto {\left(\sqrt{\color{blue}{h \cdot h}}\right)}^{\left(-0.5\right)} \cdot \frac{d}{\sqrt{\ell}} \]

       14. sqrt-unprod 59.4%

           \[\leadsto {\color{blue}{\left(\sqrt{h} \cdot \sqrt{h}\right)}}^{\left(-0.5\right)} \cdot \frac{d}{\sqrt{\ell}} \]

       15. add-sqr-sqrt 59.7%

           \[\leadsto {\color{blue}{h}}^{\left(-0.5\right)} \cdot \frac{d}{\sqrt{\ell}} \]

       16. metadata-eval 59.7%

           \[\leadsto {h}^{\color{blue}{-0.5}} \cdot \frac{d}{\sqrt{\ell}} \]

   14. Applied egg-rr 59.7%

       \[\leadsto \color{blue}{{h}^{-0.5} \cdot \frac{d}{\sqrt{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 4.1 \cdot 10^{+125}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 + h \cdot \left(-0.25 \cdot \left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{h}^{-0.5} \cdot \frac{d}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 66.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{d} \cdot \left(D\_m \cdot 0.5\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{+126}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 + h \cdot \left(-0.25 \cdot \left({\left(\frac{D\_m \cdot M\_m}{d}\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{h}^{-0.5} \cdot \frac{d}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -5e-310)
   (*
    (sqrt (* (/ d h) (/ d l)))
    (+ 1.0 (* -0.5 (* (/ h l) (pow (* (/ M_m d) (* D_m 0.5)) 2.0)))))
   (if (<= l 5.5e+126)
     (*
      (* d (sqrt (/ 1.0 (* l h))))
      (+ 1.0 (* h (* -0.25 (* (pow (/ (* D_m M_m) d) 2.0) (/ 0.5 l))))))
     (* (pow h -0.5) (/ d (sqrt l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -5e-310) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * pow(((M_m / d) * (D_m * 0.5)), 2.0))));
	} else if (l <= 5.5e+126) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (1.0 + (h * (-0.25 * (pow(((D_m * M_m) / d), 2.0) * (0.5 / l)))));
	} else {
		tmp = pow(h, -0.5) * (d / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-5d-310)) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 + ((-0.5d0) * ((h / l) * (((m_m / d) * (d_m * 0.5d0)) ** 2.0d0))))
    else if (l <= 5.5d+126) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * (1.0d0 + (h * ((-0.25d0) * ((((d_m * m_m) / d) ** 2.0d0) * (0.5d0 / l)))))
    else
        tmp = (h ** (-0.5d0)) * (d / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -5e-310) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * Math.pow(((M_m / d) * (D_m * 0.5)), 2.0))));
	} else if (l <= 5.5e+126) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (1.0 + (h * (-0.25 * (Math.pow(((D_m * M_m) / d), 2.0) * (0.5 / l)))));
	} else {
		tmp = Math.pow(h, -0.5) * (d / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -5e-310:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * math.pow(((M_m / d) * (D_m * 0.5)), 2.0))))
	elif l <= 5.5e+126:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (1.0 + (h * (-0.25 * (math.pow(((D_m * M_m) / d), 2.0) * (0.5 / l)))))
	else:
		tmp = math.pow(h, -0.5) * (d / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / d) * Float64(D_m * 0.5)) ^ 2.0)))));
	elseif (l <= 5.5e+126)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(1.0 + Float64(h * Float64(-0.25 * Float64((Float64(Float64(D_m * M_m) / d) ^ 2.0) * Float64(0.5 / l))))));
	else
		tmp = Float64((h ^ -0.5) * Float64(d / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -5e-310)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * (((M_m / d) * (D_m * 0.5)) ^ 2.0))));
	elseif (l <= 5.5e+126)
		tmp = (d * sqrt((1.0 / (l * h)))) * (1.0 + (h * (-0.25 * ((((D_m * M_m) / d) ^ 2.0) * (0.5 / l)))));
	else
		tmp = (h ^ -0.5) * (d / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -5e-310], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / d), $MachinePrecision] * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.5e+126], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(h * N[(-0.25 * N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[h, -0.5], $MachinePrecision] * N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 76.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 74.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 74.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 74.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 74.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. metadata-eval 74.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. pow1 74.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 74.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(D \cdot \frac{1}{d}\right)} \cdot \frac{M}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. associate-*l* 75.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(\frac{1}{d} \cdot \frac{M}{2}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. times-frac 75.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{1 \cdot M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. *-commutative 75.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. times-frac 75.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. metadata-eval 75.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 75.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. pow1 75.3%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)}^{1}} \]

      2. sqrt-unprod 63.3%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)}^{1} \]

      3. cancel-sign-sub-inv 63.3%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-0.5\right) \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)}\right)}^{1} \]

      4. metadata-eval 63.3%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \color{blue}{-0.5} \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)}^{1} \]

      5. *-commutative 63.3%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot {\color{blue}{\left(\sqrt{\frac{h}{\ell}} \cdot \left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)\right)}}^{2}\right)\right)}^{1} \]

      6. unpow-prod-down 62.6%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \color{blue}{\left({\left(\sqrt{\frac{h}{\ell}}\right)}^{2} \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}\right)}\right)\right)}^{1} \]

      7. pow2 62.6%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\color{blue}{\left(\sqrt{\frac{h}{\ell}} \cdot \sqrt{\frac{h}{\ell}}\right)} \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}\right)\right)\right)}^{1} \]

      8. add-sqr-sqrt 62.7%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\color{blue}{\frac{h}{\ell}} \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}\right)\right)\right)}^{1} \]

      9. associate-*r* 62.7%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\color{blue}{\left(\left(D \cdot 0.5\right) \cdot \frac{M}{d}\right)}}^{2}\right)\right)\right)}^{1} \]

   8. Applied egg-rr 62.7%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(D \cdot 0.5\right) \cdot \frac{M}{d}\right)}^{2}\right)\right)\right)}^{1}} \]
   9. Step-by-step derivation

      1. unpow1 62.7%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(D \cdot 0.5\right) \cdot \frac{M}{d}\right)}^{2}\right)\right)} \]

      2. *-lft-identity 62.7%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(D \cdot 0.5\right) \cdot \frac{M}{d}\right)}^{2}\right)\right)\right)} \]

      3. *-lft-identity 62.7%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(D \cdot 0.5\right) \cdot \frac{M}{d}\right)}^{2}\right)\right)} \]

      4. *-commutative 62.7%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \color{blue}{\left(\frac{h}{\ell} \cdot {\left(\left(D \cdot 0.5\right) \cdot \frac{M}{d}\right)}^{2}\right) \cdot -0.5}\right) \]

   10. Simplified 62.7%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot {\left(\left(D \cdot 0.5\right) \cdot \frac{M}{d}\right)}^{2}\right) \cdot -0.5\right)} \]

   if -4.999999999999985e-310 < l < 5.5000000000000004e126

   1. Initial program 71.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 70.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around 0 78.4%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 86.0%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \frac{\sqrt{h}}{\sqrt{\ell}}\right)}^{2}\right) \]

   7. Simplified 78.4%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 78.4%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \color{blue}{\sqrt{0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)} \cdot \sqrt{0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)}}\right) \]

      2. pow2 78.4%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \color{blue}{{\left(\sqrt{0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)}^{2}}\right) \]

      3. *-commutative 78.4%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\sqrt{\color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.5}}\right)}^{2}\right) \]

      4. sqrt-prod 78.3%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{0.5}\right)}}^{2}\right) \]

      5. *-commutative 78.3%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\sqrt{\color{blue}{\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}}} \cdot \sqrt{0.5}\right)}^{2}\right) \]

      6. sqrt-prod 78.4%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\color{blue}{\left(\sqrt{\frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}}\right)} \cdot \sqrt{0.5}\right)}^{2}\right) \]

      7. sqrt-pow1 80.8%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\left(\sqrt{\frac{h}{\ell}} \cdot \color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}}\right) \cdot \sqrt{0.5}\right)}^{2}\right) \]

      8. metadata-eval 80.8%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\left(\sqrt{\frac{h}{\ell}} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}}\right) \cdot \sqrt{0.5}\right)}^{2}\right) \]

      9. pow1 80.8%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\left(\sqrt{\frac{h}{\ell}} \cdot \color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}\right) \cdot \sqrt{0.5}\right)}^{2}\right) \]

      10. associate-*r/ 80.8%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\left(\sqrt{\frac{h}{\ell}} \cdot \color{blue}{\frac{\frac{M}{2} \cdot D}{d}}\right) \cdot \sqrt{0.5}\right)}^{2}\right) \]

      11. div-inv 80.8%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\left(\sqrt{\frac{h}{\ell}} \cdot \frac{\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot D}{d}\right) \cdot \sqrt{0.5}\right)}^{2}\right) \]

      12. metadata-eval 80.8%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\left(\sqrt{\frac{h}{\ell}} \cdot \frac{\left(M \cdot \color{blue}{0.5}\right) \cdot D}{d}\right) \cdot \sqrt{0.5}\right)}^{2}\right) \]

   9. Applied egg-rr 80.8%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \color{blue}{{\left(\left(\sqrt{\frac{h}{\ell}} \cdot \frac{\left(M \cdot 0.5\right) \cdot D}{d}\right) \cdot \sqrt{0.5}\right)}^{2}}\right) \]

   10. Taylor expanded in h around inf 63.6%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \color{blue}{\left(h \cdot \left(\frac{1}{h} - 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{0.5}\right)}^{2}\right)}{{d}^{2} \cdot \ell}\right)\right)} \]
   11. Step-by-step derivation

       1. sub-neg 63.6%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(h \cdot \color{blue}{\left(\frac{1}{h} + \left(-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{0.5}\right)}^{2}\right)}{{d}^{2} \cdot \ell}\right)\right)}\right) \]

       2. distribute-lft-in 63.6%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \color{blue}{\left(h \cdot \frac{1}{h} + h \cdot \left(-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{0.5}\right)}^{2}\right)}{{d}^{2} \cdot \ell}\right)\right)} \]

       3. rgt-mult-inverse 63.6%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\color{blue}{1} + h \cdot \left(-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{0.5}\right)}^{2}\right)}{{d}^{2} \cdot \ell}\right)\right) \]

       4. distribute-lft-neg-in 63.6%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 + h \cdot \color{blue}{\left(\left(-0.25\right) \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{0.5}\right)}^{2}\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

       5. metadata-eval 63.6%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 + h \cdot \left(\color{blue}{-0.25} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{0.5}\right)}^{2}\right)}{{d}^{2} \cdot \ell}\right)\right) \]

       6. associate-*r* 63.6%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 + h \cdot \left(-0.25 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot {\left(\sqrt{0.5}\right)}^{2}}}{{d}^{2} \cdot \ell}\right)\right) \]

       7. unpow2 63.6%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 + h \cdot \left(-0.25 \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \color{blue}{\left(\sqrt{0.5} \cdot \sqrt{0.5}\right)}}{{d}^{2} \cdot \ell}\right)\right) \]

       8. rem-square-sqrt 63.7%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 + h \cdot \left(-0.25 \cdot \frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \color{blue}{0.5}}{{d}^{2} \cdot \ell}\right)\right) \]

       9. times-frac 62.8%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 + h \cdot \left(-0.25 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{0.5}{\ell}\right)}\right)\right) \]

   12. Simplified 83.3%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \color{blue}{\left(1 + h \cdot \left(-0.25 \cdot \left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right)\right)} \]

   if 5.5000000000000004e126 < l

   1. Initial program 52.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 52.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 52.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 52.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 52.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 55.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 55.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. metadata-eval 55.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. pow1 55.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 55.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(D \cdot \frac{1}{d}\right)} \cdot \frac{M}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. associate-*l* 49.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(\frac{1}{d} \cdot \frac{M}{2}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. times-frac 49.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{1 \cdot M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. *-commutative 49.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. times-frac 49.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. metadata-eval 49.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 49.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 36.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. unpow-1 36.9%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. metadata-eval 36.9%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      3. pow-sqr 36.9%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      4. rem-sqrt-square 36.9%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      5. rem-square-sqrt 36.8%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

      6. fabs-sqr 36.8%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      7. rem-square-sqrt 36.9%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 36.9%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Taylor expanded in d around 0 36.9%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. *-commutative 36.9%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

       2. unpow1/2 36.9%

          \[\leadsto \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot d \]

       3. rem-exp-log 35.4%

          \[\leadsto {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot d \]

       4. rec-exp 35.4%

          \[\leadsto {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot d \]

       5. exp-prod 35.4%

          \[\leadsto \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot d \]

       6. distribute-lft-neg-out 35.4%

          \[\leadsto e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot d \]

       7. exp-neg 35.4%

          \[\leadsto \color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \cdot d \]

       8. exp-to-pow 36.8%

          \[\leadsto \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot d \]

       9. unpow1/2 36.8%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot d \]

       10. associate-*l/ 36.8%

           \[\leadsto \color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}} \]

       11. *-lft-identity 36.8%

           \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   12. Simplified 36.8%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   13. Step-by-step derivation

       1. *-un-lft-identity 36.8%

          \[\leadsto \frac{\color{blue}{1 \cdot d}}{\sqrt{h \cdot \ell}} \]

       2. sqrt-prod 59.3%

          \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       3. times-frac 59.3%

          \[\leadsto \color{blue}{\frac{1}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}} \]

       4. add-sqr-sqrt 59.3%

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\sqrt{h} \cdot \sqrt{h}}}} \cdot \frac{d}{\sqrt{\ell}} \]

       5. sqrt-unprod 25.2%

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\sqrt{h \cdot h}}}} \cdot \frac{d}{\sqrt{\ell}} \]

       6. sqr-neg 25.2%

          \[\leadsto \frac{1}{\sqrt{\sqrt{\color{blue}{\left(-h\right) \cdot \left(-h\right)}}}} \cdot \frac{d}{\sqrt{\ell}} \]

       7. sqrt-unprod 0.0%

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\sqrt{-h} \cdot \sqrt{-h}}}} \cdot \frac{d}{\sqrt{\ell}} \]

       8. add-sqr-sqrt 0.0%

          \[\leadsto \frac{1}{\sqrt{\color{blue}{-h}}} \cdot \frac{d}{\sqrt{\ell}} \]

       9. pow1/2 0.0%

          \[\leadsto \frac{1}{\color{blue}{{\left(-h\right)}^{0.5}}} \cdot \frac{d}{\sqrt{\ell}} \]

       10. pow-flip 0.0%

           \[\leadsto \color{blue}{{\left(-h\right)}^{\left(-0.5\right)}} \cdot \frac{d}{\sqrt{\ell}} \]

       11. add-sqr-sqrt 0.0%

           \[\leadsto {\color{blue}{\left(\sqrt{-h} \cdot \sqrt{-h}\right)}}^{\left(-0.5\right)} \cdot \frac{d}{\sqrt{\ell}} \]

       12. sqrt-unprod 25.3%

           \[\leadsto {\color{blue}{\left(\sqrt{\left(-h\right) \cdot \left(-h\right)}\right)}}^{\left(-0.5\right)} \cdot \frac{d}{\sqrt{\ell}} \]

       13. sqr-neg 25.3%

           \[\leadsto {\left(\sqrt{\color{blue}{h \cdot h}}\right)}^{\left(-0.5\right)} \cdot \frac{d}{\sqrt{\ell}} \]

       14. sqrt-unprod 59.4%

           \[\leadsto {\color{blue}{\left(\sqrt{h} \cdot \sqrt{h}\right)}}^{\left(-0.5\right)} \cdot \frac{d}{\sqrt{\ell}} \]

       15. add-sqr-sqrt 59.7%

           \[\leadsto {\color{blue}{h}}^{\left(-0.5\right)} \cdot \frac{d}{\sqrt{\ell}} \]

       16. metadata-eval 59.7%

           \[\leadsto {h}^{\color{blue}{-0.5}} \cdot \frac{d}{\sqrt{\ell}} \]

   14. Applied egg-rr 59.7%

       \[\leadsto \color{blue}{{h}^{-0.5} \cdot \frac{d}{\sqrt{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{+126}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 + h \cdot \left(-0.25 \cdot \left({\left(\frac{D \cdot M}{d}\right)}^{2} \cdot \frac{0.5}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{h}^{-0.5} \cdot \frac{d}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 64.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{d} \cdot \left(D\_m \cdot 0.5\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+116}:\\ \;\;\;\;\frac{d \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \left(0.5 \cdot M\_m\right)\right)}^{2}\right)\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;{h}^{-0.5} \cdot \frac{d}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -5e-310)
   (*
    (sqrt (* (/ d h) (/ d l)))
    (+ 1.0 (* -0.5 (* (/ h l) (pow (* (/ M_m d) (* D_m 0.5)) 2.0)))))
   (if (<= l 1.5e+116)
     (/
      (* d (+ 1.0 (* -0.5 (* (/ h l) (pow (* (/ D_m d) (* 0.5 M_m)) 2.0)))))
      (sqrt (* l h)))
     (* (pow h -0.5) (/ d (sqrt l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -5e-310) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * pow(((M_m / d) * (D_m * 0.5)), 2.0))));
	} else if (l <= 1.5e+116) {
		tmp = (d * (1.0 + (-0.5 * ((h / l) * pow(((D_m / d) * (0.5 * M_m)), 2.0))))) / sqrt((l * h));
	} else {
		tmp = pow(h, -0.5) * (d / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-5d-310)) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 + ((-0.5d0) * ((h / l) * (((m_m / d) * (d_m * 0.5d0)) ** 2.0d0))))
    else if (l <= 1.5d+116) then
        tmp = (d * (1.0d0 + ((-0.5d0) * ((h / l) * (((d_m / d) * (0.5d0 * m_m)) ** 2.0d0))))) / sqrt((l * h))
    else
        tmp = (h ** (-0.5d0)) * (d / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -5e-310) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * Math.pow(((M_m / d) * (D_m * 0.5)), 2.0))));
	} else if (l <= 1.5e+116) {
		tmp = (d * (1.0 + (-0.5 * ((h / l) * Math.pow(((D_m / d) * (0.5 * M_m)), 2.0))))) / Math.sqrt((l * h));
	} else {
		tmp = Math.pow(h, -0.5) * (d / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -5e-310:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * math.pow(((M_m / d) * (D_m * 0.5)), 2.0))))
	elif l <= 1.5e+116:
		tmp = (d * (1.0 + (-0.5 * ((h / l) * math.pow(((D_m / d) * (0.5 * M_m)), 2.0))))) / math.sqrt((l * h))
	else:
		tmp = math.pow(h, -0.5) * (d / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / d) * Float64(D_m * 0.5)) ^ 2.0)))));
	elseif (l <= 1.5e+116)
		tmp = Float64(Float64(d * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(0.5 * M_m)) ^ 2.0))))) / sqrt(Float64(l * h)));
	else
		tmp = Float64((h ^ -0.5) * Float64(d / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -5e-310)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * (((M_m / d) * (D_m * 0.5)) ^ 2.0))));
	elseif (l <= 1.5e+116)
		tmp = (d * (1.0 + (-0.5 * ((h / l) * (((D_m / d) * (0.5 * M_m)) ^ 2.0))))) / sqrt((l * h));
	else
		tmp = (h ^ -0.5) * (d / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -5e-310], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / d), $MachinePrecision] * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.5e+116], N[(N[(d * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(0.5 * M$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[h, -0.5], $MachinePrecision] * N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 76.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 74.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 74.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 74.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 74.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. metadata-eval 74.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. pow1 74.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 74.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(D \cdot \frac{1}{d}\right)} \cdot \frac{M}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. associate-*l* 75.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(\frac{1}{d} \cdot \frac{M}{2}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. times-frac 75.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{1 \cdot M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. *-commutative 75.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. times-frac 75.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. metadata-eval 75.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 75.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. pow1 75.3%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)}^{1}} \]

      2. sqrt-unprod 63.3%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)}^{1} \]

      3. cancel-sign-sub-inv 63.3%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-0.5\right) \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)}\right)}^{1} \]

      4. metadata-eval 63.3%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \color{blue}{-0.5} \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)}^{1} \]

      5. *-commutative 63.3%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot {\color{blue}{\left(\sqrt{\frac{h}{\ell}} \cdot \left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)\right)}}^{2}\right)\right)}^{1} \]

      6. unpow-prod-down 62.6%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \color{blue}{\left({\left(\sqrt{\frac{h}{\ell}}\right)}^{2} \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}\right)}\right)\right)}^{1} \]

      7. pow2 62.6%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\color{blue}{\left(\sqrt{\frac{h}{\ell}} \cdot \sqrt{\frac{h}{\ell}}\right)} \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}\right)\right)\right)}^{1} \]

      8. add-sqr-sqrt 62.7%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\color{blue}{\frac{h}{\ell}} \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}\right)\right)\right)}^{1} \]

      9. associate-*r* 62.7%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\color{blue}{\left(\left(D \cdot 0.5\right) \cdot \frac{M}{d}\right)}}^{2}\right)\right)\right)}^{1} \]

   8. Applied egg-rr 62.7%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(D \cdot 0.5\right) \cdot \frac{M}{d}\right)}^{2}\right)\right)\right)}^{1}} \]
   9. Step-by-step derivation

      1. unpow1 62.7%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(D \cdot 0.5\right) \cdot \frac{M}{d}\right)}^{2}\right)\right)} \]

      2. *-lft-identity 62.7%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(D \cdot 0.5\right) \cdot \frac{M}{d}\right)}^{2}\right)\right)\right)} \]

      3. *-lft-identity 62.7%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(D \cdot 0.5\right) \cdot \frac{M}{d}\right)}^{2}\right)\right)} \]

      4. *-commutative 62.7%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \color{blue}{\left(\frac{h}{\ell} \cdot {\left(\left(D \cdot 0.5\right) \cdot \frac{M}{d}\right)}^{2}\right) \cdot -0.5}\right) \]

   10. Simplified 62.7%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot {\left(\left(D \cdot 0.5\right) \cdot \frac{M}{d}\right)}^{2}\right) \cdot -0.5\right)} \]

   if -4.999999999999985e-310 < l < 1.4999999999999999e116

   1. Initial program 71.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 70.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around 0 78.8%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 86.7%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \frac{\sqrt{h}}{\sqrt{\ell}}\right)}^{2}\right) \]

   7. Simplified 78.8%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 78.8%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \color{blue}{\sqrt{0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)} \cdot \sqrt{0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)}}\right) \]

      2. pow2 78.8%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \color{blue}{{\left(\sqrt{0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)}^{2}}\right) \]

      3. *-commutative 78.8%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\sqrt{\color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.5}}\right)}^{2}\right) \]

      4. sqrt-prod 78.7%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{0.5}\right)}}^{2}\right) \]

      5. *-commutative 78.7%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\sqrt{\color{blue}{\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}}} \cdot \sqrt{0.5}\right)}^{2}\right) \]

      6. sqrt-prod 78.7%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\color{blue}{\left(\sqrt{\frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}}\right)} \cdot \sqrt{0.5}\right)}^{2}\right) \]

      7. sqrt-pow1 81.3%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\left(\sqrt{\frac{h}{\ell}} \cdot \color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}}\right) \cdot \sqrt{0.5}\right)}^{2}\right) \]

      8. metadata-eval 81.3%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\left(\sqrt{\frac{h}{\ell}} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}}\right) \cdot \sqrt{0.5}\right)}^{2}\right) \]

      9. pow1 81.3%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\left(\sqrt{\frac{h}{\ell}} \cdot \color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}\right) \cdot \sqrt{0.5}\right)}^{2}\right) \]

      10. associate-*r/ 81.2%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\left(\sqrt{\frac{h}{\ell}} \cdot \color{blue}{\frac{\frac{M}{2} \cdot D}{d}}\right) \cdot \sqrt{0.5}\right)}^{2}\right) \]

      11. div-inv 81.2%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\left(\sqrt{\frac{h}{\ell}} \cdot \frac{\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot D}{d}\right) \cdot \sqrt{0.5}\right)}^{2}\right) \]

      12. metadata-eval 81.2%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - {\left(\left(\sqrt{\frac{h}{\ell}} \cdot \frac{\left(M \cdot \color{blue}{0.5}\right) \cdot D}{d}\right) \cdot \sqrt{0.5}\right)}^{2}\right) \]

   9. Applied egg-rr 81.2%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \color{blue}{{\left(\left(\sqrt{\frac{h}{\ell}} \cdot \frac{\left(M \cdot 0.5\right) \cdot D}{d}\right) \cdot \sqrt{0.5}\right)}^{2}}\right) \]

   11. Applied egg-rr 80.1%

       \[\leadsto \color{blue}{\frac{d \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)}{\sqrt{h \cdot \ell}}} \]

   if 1.4999999999999999e116 < l

   1. Initial program 53.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 53.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 53.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 53.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 53.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 56.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 56.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. metadata-eval 56.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. pow1 56.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 56.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(D \cdot \frac{1}{d}\right)} \cdot \frac{M}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. associate-*l* 51.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(\frac{1}{d} \cdot \frac{M}{2}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. times-frac 51.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{1 \cdot M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. *-commutative 51.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. times-frac 51.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. metadata-eval 51.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 51.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 39.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. unpow-1 39.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. metadata-eval 39.4%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      3. pow-sqr 39.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      4. rem-sqrt-square 39.4%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      5. rem-square-sqrt 39.2%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

      6. fabs-sqr 39.2%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      7. rem-square-sqrt 39.4%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 39.4%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Taylor expanded in d around 0 39.4%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. *-commutative 39.4%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

       2. unpow1/2 39.4%

          \[\leadsto \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot d \]

       3. rem-exp-log 37.5%

          \[\leadsto {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot d \]

       4. rec-exp 37.5%

          \[\leadsto {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot d \]

       5. exp-prod 37.5%

          \[\leadsto \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot d \]

       6. distribute-lft-neg-out 37.5%

          \[\leadsto e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot d \]

       7. exp-neg 37.5%

          \[\leadsto \color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \cdot d \]

       8. exp-to-pow 39.2%

          \[\leadsto \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot d \]

       9. unpow1/2 39.2%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot d \]

       10. associate-*l/ 39.2%

           \[\leadsto \color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}} \]

       11. *-lft-identity 39.2%

           \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   12. Simplified 39.2%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   13. Step-by-step derivation

       1. *-un-lft-identity 39.2%

          \[\leadsto \frac{\color{blue}{1 \cdot d}}{\sqrt{h \cdot \ell}} \]

       2. sqrt-prod 59.9%

          \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       3. times-frac 60.0%

          \[\leadsto \color{blue}{\frac{1}{\sqrt{h}} \cdot \frac{d}{\sqrt{\ell}}} \]

       4. add-sqr-sqrt 60.0%

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\sqrt{h} \cdot \sqrt{h}}}} \cdot \frac{d}{\sqrt{\ell}} \]

       5. sqrt-unprod 26.0%

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\sqrt{h \cdot h}}}} \cdot \frac{d}{\sqrt{\ell}} \]

       6. sqr-neg 26.0%

          \[\leadsto \frac{1}{\sqrt{\sqrt{\color{blue}{\left(-h\right) \cdot \left(-h\right)}}}} \cdot \frac{d}{\sqrt{\ell}} \]

       7. sqrt-unprod 0.0%

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\sqrt{-h} \cdot \sqrt{-h}}}} \cdot \frac{d}{\sqrt{\ell}} \]

       8. add-sqr-sqrt 0.0%

          \[\leadsto \frac{1}{\sqrt{\color{blue}{-h}}} \cdot \frac{d}{\sqrt{\ell}} \]

       9. pow1/2 0.0%

          \[\leadsto \frac{1}{\color{blue}{{\left(-h\right)}^{0.5}}} \cdot \frac{d}{\sqrt{\ell}} \]

       10. pow-flip 0.0%

           \[\leadsto \color{blue}{{\left(-h\right)}^{\left(-0.5\right)}} \cdot \frac{d}{\sqrt{\ell}} \]

       11. add-sqr-sqrt 0.0%

           \[\leadsto {\color{blue}{\left(\sqrt{-h} \cdot \sqrt{-h}\right)}}^{\left(-0.5\right)} \cdot \frac{d}{\sqrt{\ell}} \]

       12. sqrt-unprod 26.1%

           \[\leadsto {\color{blue}{\left(\sqrt{\left(-h\right) \cdot \left(-h\right)}\right)}}^{\left(-0.5\right)} \cdot \frac{d}{\sqrt{\ell}} \]

       13. sqr-neg 26.1%

           \[\leadsto {\left(\sqrt{\color{blue}{h \cdot h}}\right)}^{\left(-0.5\right)} \cdot \frac{d}{\sqrt{\ell}} \]

       14. sqrt-unprod 60.1%

           \[\leadsto {\color{blue}{\left(\sqrt{h} \cdot \sqrt{h}\right)}}^{\left(-0.5\right)} \cdot \frac{d}{\sqrt{\ell}} \]

       15. add-sqr-sqrt 60.3%

           \[\leadsto {\color{blue}{h}}^{\left(-0.5\right)} \cdot \frac{d}{\sqrt{\ell}} \]

       16. metadata-eval 60.3%

           \[\leadsto {h}^{\color{blue}{-0.5}} \cdot \frac{d}{\sqrt{\ell}} \]

   14. Applied egg-rr 60.3%

       \[\leadsto \color{blue}{{h}^{-0.5} \cdot \frac{d}{\sqrt{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+116}:\\ \;\;\;\;\frac{d \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2}\right)\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;{h}^{-0.5} \cdot \frac{d}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 45.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -9.8 \cdot 10^{-142}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -9.8e-142)
   (/ (sqrt (/ d l)) (sqrt (/ h d)))
   (if (<= d -4e-310)
     (/ d (cbrt (pow (* l h) 1.5)))
     (* d (* (pow h -0.5) (pow l -0.5))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -9.8e-142) {
		tmp = sqrt((d / l)) / sqrt((h / d));
	} else if (d <= -4e-310) {
		tmp = d / cbrt(pow((l * h), 1.5));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -9.8e-142) {
		tmp = Math.sqrt((d / l)) / Math.sqrt((h / d));
	} else if (d <= -4e-310) {
		tmp = d / Math.cbrt(Math.pow((l * h), 1.5));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -9.8e-142)
		tmp = Float64(sqrt(Float64(d / l)) / sqrt(Float64(h / d)));
	elseif (d <= -4e-310)
		tmp = Float64(d / cbrt((Float64(l * h) ^ 1.5)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -9.8e-142], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4e-310], N[(d / N[Power[N[Power[N[(l * h), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -9.8000000000000007e-142

   1. Initial program 80.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 78.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 78.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 78.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 78.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. metadata-eval 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. pow1 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(D \cdot \frac{1}{d}\right)} \cdot \frac{M}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. associate-*l* 80.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(\frac{1}{d} \cdot \frac{M}{2}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. times-frac 80.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{1 \cdot M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. *-commutative 80.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. times-frac 80.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. metadata-eval 80.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 80.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in D around 0 49.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{1} \]
   8. Step-by-step derivation

      1. *-rgt-identity 49.2%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

      2. *-commutative 49.2%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}} \]

      3. frac-2neg 49.2%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{-d}{-h}}} \]

      4. sqrt-undiv 53.6%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \]

      5. clear-num 53.5%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\frac{1}{\frac{\sqrt{-h}}{\sqrt{-d}}}} \]

      6. un-div-inv 53.5%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\frac{\sqrt{-h}}{\sqrt{-d}}}} \]

      7. sqrt-undiv 50.4%

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\color{blue}{\sqrt{\frac{-h}{-d}}}} \]

      8. frac-2neg 50.4%

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\color{blue}{\frac{h}{d}}}} \]

   9. Applied egg-rr 50.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}} \]

   if -9.8000000000000007e-142 < d < -3.999999999999988e-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 60.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 60.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 60.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 60.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 60.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 60.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. metadata-eval 60.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. pow1 60.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 60.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(D \cdot \frac{1}{d}\right)} \cdot \frac{M}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. associate-*l* 60.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(\frac{1}{d} \cdot \frac{M}{2}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. times-frac 60.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{1 \cdot M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. *-commutative 60.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. times-frac 60.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. metadata-eval 60.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 60.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 24.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. unpow-1 24.9%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. metadata-eval 24.9%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      3. pow-sqr 24.9%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      4. rem-sqrt-square 24.9%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      5. rem-square-sqrt 24.9%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

      6. fabs-sqr 24.9%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      7. rem-square-sqrt 24.9%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 24.9%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Taylor expanded in d around 0 24.9%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. *-commutative 24.9%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

       2. unpow1/2 24.9%

          \[\leadsto \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot d \]

       3. rem-exp-log 24.9%

          \[\leadsto {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot d \]

       4. rec-exp 24.9%

          \[\leadsto {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot d \]

       5. exp-prod 24.9%

          \[\leadsto \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot d \]

       6. distribute-lft-neg-out 24.9%

          \[\leadsto e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot d \]

       7. exp-neg 24.9%

          \[\leadsto \color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \cdot d \]

       8. exp-to-pow 24.9%

          \[\leadsto \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot d \]

       9. unpow1/2 24.9%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot d \]

       10. associate-*l/ 24.9%

           \[\leadsto \color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}} \]

       11. *-lft-identity 24.9%

           \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   12. Simplified 24.9%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   13. Step-by-step derivation

       1. add-cbrt-cube 30.5%

          \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{\left(\sqrt{h \cdot \ell} \cdot \sqrt{h \cdot \ell}\right) \cdot \sqrt{h \cdot \ell}}}} \]

       2. pow1/3 30.5%

          \[\leadsto \frac{d}{\color{blue}{{\left(\left(\sqrt{h \cdot \ell} \cdot \sqrt{h \cdot \ell}\right) \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}}} \]

       3. add-sqr-sqrt 30.5%

          \[\leadsto \frac{d}{{\left(\color{blue}{\left(h \cdot \ell\right)} \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}} \]

       4. pow1 30.5%

          \[\leadsto \frac{d}{{\left(\color{blue}{{\left(h \cdot \ell\right)}^{1}} \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}} \]

       5. pow1/2 30.5%

          \[\leadsto \frac{d}{{\left({\left(h \cdot \ell\right)}^{1} \cdot \color{blue}{{\left(h \cdot \ell\right)}^{0.5}}\right)}^{0.3333333333333333}} \]

       6. pow-prod-up 30.5%

          \[\leadsto \frac{d}{{\color{blue}{\left({\left(h \cdot \ell\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}} \]

       7. metadata-eval 30.5%

          \[\leadsto \frac{d}{{\left({\left(h \cdot \ell\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \]

   14. Applied egg-rr 30.5%

       \[\leadsto \frac{d}{\color{blue}{{\left({\left(h \cdot \ell\right)}^{1.5}\right)}^{0.3333333333333333}}} \]
   15. Step-by-step derivation

       1. unpow1/3 30.5%

          \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}} \]

   16. Simplified 30.5%

       \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}} \]

   if -3.999999999999988e-310 < d

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 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 67.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 67.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. metadata-eval 67.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. pow1 67.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 67.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(D \cdot \frac{1}{d}\right)} \cdot \frac{M}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. associate-*l* 66.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(\frac{1}{d} \cdot \frac{M}{2}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. times-frac 66.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{1 \cdot M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. *-commutative 66.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. times-frac 66.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. metadata-eval 66.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 66.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 34.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. unpow-1 34.8%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. metadata-eval 34.8%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      3. pow-sqr 34.8%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      4. rem-sqrt-square 34.8%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      5. rem-square-sqrt 34.7%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

      6. fabs-sqr 34.7%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      7. rem-square-sqrt 34.8%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 34.8%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. *-commutative 34.8%

          \[\leadsto d \cdot {\color{blue}{\left(\ell \cdot h\right)}}^{-0.5} \]

       2. unpow-prod-down 45.5%

          \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]

   11. Applied egg-rr 45.5%

       \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.8 \cdot 10^{-142}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 46.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -8.2 \cdot 10^{-145}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -8.2e-145)
   (* (- d) (sqrt (/ (/ 1.0 l) h)))
   (if (<= d -4e-310)
     (/ d (cbrt (pow (* l h) 1.5)))
     (* d (* (pow h -0.5) (pow l -0.5))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -8.2e-145) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else if (d <= -4e-310) {
		tmp = d / cbrt(pow((l * h), 1.5));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -8.2e-145) {
		tmp = -d * Math.sqrt(((1.0 / l) / h));
	} else if (d <= -4e-310) {
		tmp = d / Math.cbrt(Math.pow((l * h), 1.5));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -8.2e-145)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	elseif (d <= -4e-310)
		tmp = Float64(d / cbrt((Float64(l * h) ^ 1.5)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -8.2e-145], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4e-310], N[(d / N[Power[N[Power[N[(l * h), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -8.1999999999999995e-145

   1. Initial program 79.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 77.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 77.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 77.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 77.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 78.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 78.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. metadata-eval 78.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. pow1 78.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 78.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(D \cdot \frac{1}{d}\right)} \cdot \frac{M}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. associate-*l* 79.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(\frac{1}{d} \cdot \frac{M}{2}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. times-frac 79.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{1 \cdot M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. *-commutative 79.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. times-frac 79.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. metadata-eval 79.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 79.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in D around 0 48.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{1} \]

   8. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. 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. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      3. associate-/r* 0.0%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      4. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      5. unpow2 0.0%

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      6. rem-square-sqrt 48.9%

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      7. neg-mul-1 48.9%

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \color{blue}{\left(-d\right)} \]

   10. Simplified 48.9%

       \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)} \]

   if -8.1999999999999995e-145 < d < -3.999999999999988e-310

   1. Initial program 65.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 62.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 62.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 62.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 62.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 62.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 62.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. metadata-eval 62.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. pow1 62.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 62.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(D \cdot \frac{1}{d}\right)} \cdot \frac{M}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. associate-*l* 62.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(\frac{1}{d} \cdot \frac{M}{2}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. times-frac 62.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{1 \cdot M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. *-commutative 62.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. times-frac 62.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. metadata-eval 62.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 62.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 25.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. unpow-1 25.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. metadata-eval 25.5%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      3. pow-sqr 25.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      4. rem-sqrt-square 25.6%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      5. rem-square-sqrt 25.6%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

      6. fabs-sqr 25.6%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      7. rem-square-sqrt 25.6%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 25.6%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Taylor expanded in d around 0 25.5%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. *-commutative 25.5%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

       2. unpow1/2 25.5%

          \[\leadsto \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot d \]

       3. rem-exp-log 25.5%

          \[\leadsto {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot d \]

       4. rec-exp 25.5%

          \[\leadsto {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot d \]

       5. exp-prod 25.6%

          \[\leadsto \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot d \]

       6. distribute-lft-neg-out 25.6%

          \[\leadsto e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot d \]

       7. exp-neg 25.6%

          \[\leadsto \color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \cdot d \]

       8. exp-to-pow 25.6%

          \[\leadsto \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot d \]

       9. unpow1/2 25.6%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot d \]

       10. associate-*l/ 25.6%

           \[\leadsto \color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}} \]

       11. *-lft-identity 25.6%

           \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   12. Simplified 25.6%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   13. Step-by-step derivation

       1. add-cbrt-cube 31.3%

          \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{\left(\sqrt{h \cdot \ell} \cdot \sqrt{h \cdot \ell}\right) \cdot \sqrt{h \cdot \ell}}}} \]

       2. pow1/3 31.3%

          \[\leadsto \frac{d}{\color{blue}{{\left(\left(\sqrt{h \cdot \ell} \cdot \sqrt{h \cdot \ell}\right) \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}}} \]

       3. add-sqr-sqrt 31.3%

          \[\leadsto \frac{d}{{\left(\color{blue}{\left(h \cdot \ell\right)} \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}} \]

       4. pow1 31.3%

          \[\leadsto \frac{d}{{\left(\color{blue}{{\left(h \cdot \ell\right)}^{1}} \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}} \]

       5. pow1/2 31.3%

          \[\leadsto \frac{d}{{\left({\left(h \cdot \ell\right)}^{1} \cdot \color{blue}{{\left(h \cdot \ell\right)}^{0.5}}\right)}^{0.3333333333333333}} \]

       6. pow-prod-up 31.3%

          \[\leadsto \frac{d}{{\color{blue}{\left({\left(h \cdot \ell\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}} \]

       7. metadata-eval 31.3%

          \[\leadsto \frac{d}{{\left({\left(h \cdot \ell\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \]

   14. Applied egg-rr 31.3%

       \[\leadsto \frac{d}{\color{blue}{{\left({\left(h \cdot \ell\right)}^{1.5}\right)}^{0.3333333333333333}}} \]
   15. Step-by-step derivation

       1. unpow1/3 31.3%

          \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}} \]

   16. Simplified 31.3%

       \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}} \]

   if -3.999999999999988e-310 < d

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 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 67.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 67.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. metadata-eval 67.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. pow1 67.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 67.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(D \cdot \frac{1}{d}\right)} \cdot \frac{M}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. associate-*l* 66.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(\frac{1}{d} \cdot \frac{M}{2}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. times-frac 66.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{1 \cdot M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. *-commutative 66.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. times-frac 66.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. metadata-eval 66.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 66.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 34.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. unpow-1 34.8%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. metadata-eval 34.8%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      3. pow-sqr 34.8%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      4. rem-sqrt-square 34.8%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      5. rem-square-sqrt 34.7%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

      6. fabs-sqr 34.7%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      7. rem-square-sqrt 34.8%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 34.8%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. *-commutative 34.8%

          \[\leadsto d \cdot {\color{blue}{\left(\ell \cdot h\right)}}^{-0.5} \]

       2. unpow-prod-down 45.5%

          \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]

   11. Applied egg-rr 45.5%

       \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.2 \cdot 10^{-145}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 46.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2.1 \cdot 10^{-146}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -2.1e-146)
   (* (- d) (sqrt (/ (/ 1.0 l) h)))
   (if (<= d -4e-310)
     (/ d (cbrt (pow (* l h) 1.5)))
     (/ d (* (sqrt l) (sqrt h))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -2.1e-146) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else if (d <= -4e-310) {
		tmp = d / cbrt(pow((l * h), 1.5));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -2.1e-146) {
		tmp = -d * Math.sqrt(((1.0 / l) / h));
	} else if (d <= -4e-310) {
		tmp = d / Math.cbrt(Math.pow((l * h), 1.5));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -2.1e-146)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	elseif (d <= -4e-310)
		tmp = Float64(d / cbrt((Float64(l * h) ^ 1.5)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -2.1e-146], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4e-310], N[(d / N[Power[N[Power[N[(l * h), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.0999999999999999e-146

   1. Initial program 79.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 77.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 77.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 77.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 77.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 78.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 78.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. metadata-eval 78.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. pow1 78.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 78.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(D \cdot \frac{1}{d}\right)} \cdot \frac{M}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. associate-*l* 79.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(\frac{1}{d} \cdot \frac{M}{2}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. times-frac 79.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{1 \cdot M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. *-commutative 79.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. times-frac 79.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. metadata-eval 79.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 79.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in D around 0 48.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{1} \]

   8. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. 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. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      3. associate-/r* 0.0%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      4. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      5. unpow2 0.0%

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      6. rem-square-sqrt 48.9%

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      7. neg-mul-1 48.9%

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \color{blue}{\left(-d\right)} \]

   10. Simplified 48.9%

       \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)} \]

   if -2.0999999999999999e-146 < d < -3.999999999999988e-310

   1. Initial program 65.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 62.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 62.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 62.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 62.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 62.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 62.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. metadata-eval 62.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. pow1 62.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 62.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(D \cdot \frac{1}{d}\right)} \cdot \frac{M}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. associate-*l* 62.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(\frac{1}{d} \cdot \frac{M}{2}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. times-frac 62.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{1 \cdot M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. *-commutative 62.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. times-frac 62.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. metadata-eval 62.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 62.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 25.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. unpow-1 25.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. metadata-eval 25.5%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      3. pow-sqr 25.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      4. rem-sqrt-square 25.6%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      5. rem-square-sqrt 25.6%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

      6. fabs-sqr 25.6%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      7. rem-square-sqrt 25.6%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 25.6%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Taylor expanded in d around 0 25.5%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. *-commutative 25.5%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

       2. unpow1/2 25.5%

          \[\leadsto \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot d \]

       3. rem-exp-log 25.5%

          \[\leadsto {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot d \]

       4. rec-exp 25.5%

          \[\leadsto {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot d \]

       5. exp-prod 25.6%

          \[\leadsto \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot d \]

       6. distribute-lft-neg-out 25.6%

          \[\leadsto e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot d \]

       7. exp-neg 25.6%

          \[\leadsto \color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \cdot d \]

       8. exp-to-pow 25.6%

          \[\leadsto \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot d \]

       9. unpow1/2 25.6%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot d \]

       10. associate-*l/ 25.6%

           \[\leadsto \color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}} \]

       11. *-lft-identity 25.6%

           \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   12. Simplified 25.6%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   13. Step-by-step derivation

       1. add-cbrt-cube 31.3%

          \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{\left(\sqrt{h \cdot \ell} \cdot \sqrt{h \cdot \ell}\right) \cdot \sqrt{h \cdot \ell}}}} \]

       2. pow1/3 31.3%

          \[\leadsto \frac{d}{\color{blue}{{\left(\left(\sqrt{h \cdot \ell} \cdot \sqrt{h \cdot \ell}\right) \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}}} \]

       3. add-sqr-sqrt 31.3%

          \[\leadsto \frac{d}{{\left(\color{blue}{\left(h \cdot \ell\right)} \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}} \]

       4. pow1 31.3%

          \[\leadsto \frac{d}{{\left(\color{blue}{{\left(h \cdot \ell\right)}^{1}} \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}} \]

       5. pow1/2 31.3%

          \[\leadsto \frac{d}{{\left({\left(h \cdot \ell\right)}^{1} \cdot \color{blue}{{\left(h \cdot \ell\right)}^{0.5}}\right)}^{0.3333333333333333}} \]

       6. pow-prod-up 31.3%

          \[\leadsto \frac{d}{{\color{blue}{\left({\left(h \cdot \ell\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}} \]

       7. metadata-eval 31.3%

          \[\leadsto \frac{d}{{\left({\left(h \cdot \ell\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \]

   14. Applied egg-rr 31.3%

       \[\leadsto \frac{d}{\color{blue}{{\left({\left(h \cdot \ell\right)}^{1.5}\right)}^{0.3333333333333333}}} \]
   15. Step-by-step derivation

       1. unpow1/3 31.3%

          \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}} \]

   16. Simplified 31.3%

       \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}} \]

   if -3.999999999999988e-310 < d

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 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 67.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 67.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. metadata-eval 67.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. pow1 67.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 67.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(D \cdot \frac{1}{d}\right)} \cdot \frac{M}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. associate-*l* 66.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(\frac{1}{d} \cdot \frac{M}{2}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. times-frac 66.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{1 \cdot M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. *-commutative 66.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. times-frac 66.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. metadata-eval 66.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 66.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 34.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. unpow-1 34.8%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. metadata-eval 34.8%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      3. pow-sqr 34.8%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      4. rem-sqrt-square 34.8%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      5. rem-square-sqrt 34.7%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

      6. fabs-sqr 34.7%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      7. rem-square-sqrt 34.8%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 34.8%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. *-commutative 34.8%

          \[\leadsto d \cdot {\color{blue}{\left(\ell \cdot h\right)}}^{-0.5} \]

       2. metadata-eval 34.8%

          \[\leadsto d \cdot {\left(\ell \cdot h\right)}^{\color{blue}{\left(-1 \cdot 0.5\right)}} \]

       3. pow-pow 34.8%

          \[\leadsto d \cdot \color{blue}{{\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.5}} \]

       4. inv-pow 34.8%

          \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{\ell \cdot h}\right)}}^{0.5} \]

       5. pow1/2 34.8%

          \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \]

       6. sqrt-div 34.8%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \]

       7. metadata-eval 34.8%

          \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}} \]

       8. sqrt-unprod 45.4%

          \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

       9. div-inv 45.4%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]

       10. associate-/r* 43.9%

           \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]

   11. Applied egg-rr 43.9%

       \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]
   12. Step-by-step derivation

       1. associate-/l/ 45.4%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]

   13. Simplified 45.4%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.1 \cdot 10^{-146}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 46.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 4.5 \cdot 10^{-203}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d 4.5e-203)
   (* (- d) (sqrt (/ (/ 1.0 l) h)))
   (/ d (* (sqrt l) (sqrt h)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 4.5e-203) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= 4.5d-203) then
        tmp = -d * sqrt(((1.0d0 / l) / h))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 4.5e-203) {
		tmp = -d * Math.sqrt(((1.0 / l) / h));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= 4.5e-203:
		tmp = -d * math.sqrt(((1.0 / l) / h))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= 4.5e-203)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= 4.5e-203)
		tmp = -d * sqrt(((1.0 / l) / h));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, 4.5e-203], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 4.5000000000000002e-203

   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 70.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 70.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 70.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 70.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 70.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 70.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. metadata-eval 70.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. pow1 70.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 70.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(D \cdot \frac{1}{d}\right)} \cdot \frac{M}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. associate-*l* 70.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(\frac{1}{d} \cdot \frac{M}{2}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. times-frac 70.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{1 \cdot M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. *-commutative 70.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. times-frac 70.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. metadata-eval 70.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 70.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in D around 0 35.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{1} \]

   8. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. 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. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      3. associate-/r* 0.0%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      4. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      5. unpow2 0.0%

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      6. rem-square-sqrt 38.1%

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      7. neg-mul-1 38.1%

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \color{blue}{\left(-d\right)} \]

   10. Simplified 38.1%

       \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)} \]

   if 4.5000000000000002e-203 < d

   1. Initial program 70.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 69.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 69.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 69.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 69.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 72.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 72.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. metadata-eval 72.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. pow1 72.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 72.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(D \cdot \frac{1}{d}\right)} \cdot \frac{M}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. associate-*l* 71.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(\frac{1}{d} \cdot \frac{M}{2}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. times-frac 71.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{1 \cdot M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. *-commutative 71.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. times-frac 71.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. metadata-eval 71.7%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 71.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 39.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. unpow-1 39.3%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. metadata-eval 39.3%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      3. pow-sqr 39.3%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      4. rem-sqrt-square 39.3%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      5. rem-square-sqrt 39.2%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

      6. fabs-sqr 39.2%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      7. rem-square-sqrt 39.3%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 39.3%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. *-commutative 39.3%

          \[\leadsto d \cdot {\color{blue}{\left(\ell \cdot h\right)}}^{-0.5} \]

       2. metadata-eval 39.3%

          \[\leadsto d \cdot {\left(\ell \cdot h\right)}^{\color{blue}{\left(-1 \cdot 0.5\right)}} \]

       3. pow-pow 39.3%

          \[\leadsto d \cdot \color{blue}{{\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.5}} \]

       4. inv-pow 39.3%

          \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{\ell \cdot h}\right)}}^{0.5} \]

       5. pow1/2 39.3%

          \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \]

       6. sqrt-div 39.3%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \]

       7. metadata-eval 39.3%

          \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}} \]

       8. sqrt-unprod 52.4%

          \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

       9. div-inv 52.4%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]

       10. associate-/r* 50.5%

           \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]

   11. Applied egg-rr 50.5%

       \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]
   12. Step-by-step derivation

       1. associate-/l/ 52.4%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]

   13. Simplified 52.4%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 4.5 \cdot 10^{-203}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 41.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{if}\;d \leq -9.8 \cdot 10^{-147}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot t\_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ (/ 1.0 l) h))))
   (if (<= d -9.8e-147) (* (- d) t_0) (* d t_0))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt(((1.0 / l) / h));
	double tmp;
	if (d <= -9.8e-147) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((1.0d0 / l) / h))
    if (d <= (-9.8d-147)) then
        tmp = -d * t_0
    else
        tmp = d * t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt(((1.0 / l) / h));
	double tmp;
	if (d <= -9.8e-147) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt(((1.0 / l) / h))
	tmp = 0
	if d <= -9.8e-147:
		tmp = -d * t_0
	else:
		tmp = d * t_0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(Float64(1.0 / l) / h))
	tmp = 0.0
	if (d <= -9.8e-147)
		tmp = Float64(Float64(-d) * t_0);
	else
		tmp = Float64(d * t_0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt(((1.0 / l) / h));
	tmp = 0.0;
	if (d <= -9.8e-147)
		tmp = -d * t_0;
	else
		tmp = d * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -9.8e-147], N[((-d) * t$95$0), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -9.8000000000000001e-147

   1. Initial program 79.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 77.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 77.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 77.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 77.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 78.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 78.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. metadata-eval 78.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. pow1 78.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 78.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(D \cdot \frac{1}{d}\right)} \cdot \frac{M}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. associate-*l* 79.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(\frac{1}{d} \cdot \frac{M}{2}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. times-frac 79.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{1 \cdot M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. *-commutative 79.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. times-frac 79.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. metadata-eval 79.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 79.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in D around 0 48.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{1} \]

   8. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. 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. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      3. associate-/r* 0.0%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      4. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      5. unpow2 0.0%

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      6. rem-square-sqrt 48.9%

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      7. neg-mul-1 48.9%

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \color{blue}{\left(-d\right)} \]

   10. Simplified 48.9%

       \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)} \]

   if -9.8000000000000001e-147 < d

   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 64.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 64.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 64.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 64.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 66.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 66.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. metadata-eval 66.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. pow1 66.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 66.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(D \cdot \frac{1}{d}\right)} \cdot \frac{M}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. associate-*l* 65.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(\frac{1}{d} \cdot \frac{M}{2}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. times-frac 65.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{1 \cdot M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. *-commutative 65.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. times-frac 65.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. metadata-eval 65.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 65.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 32.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. unpow-1 32.7%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. metadata-eval 32.7%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      3. pow-sqr 32.7%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      4. rem-sqrt-square 32.7%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      5. rem-square-sqrt 32.6%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

      6. fabs-sqr 32.6%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      7. rem-square-sqrt 32.7%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 32.7%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Taylor expanded in d around 0 32.7%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. *-commutative 32.7%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

       2. unpow1/2 32.7%

          \[\leadsto \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot d \]

       3. rem-exp-log 31.5%

          \[\leadsto {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot d \]

       4. rec-exp 31.5%

          \[\leadsto {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot d \]

       5. exp-prod 31.5%

          \[\leadsto \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot d \]

       6. distribute-lft-neg-out 31.5%

          \[\leadsto e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot d \]

       7. exp-neg 31.5%

          \[\leadsto \color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \cdot d \]

       8. exp-to-pow 32.7%

          \[\leadsto \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot d \]

       9. unpow1/2 32.7%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot d \]

       10. associate-*l/ 32.7%

           \[\leadsto \color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}} \]

       11. *-lft-identity 32.7%

           \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   12. Simplified 32.7%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   13. Taylor expanded in d around 0 32.7%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   14. Step-by-step derivation

       1. *-commutative 32.7%

          \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

       2. associate-/r* 32.7%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   15. Simplified 32.7%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.8 \cdot 10^{-147}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 41.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.55 \cdot 10^{-143}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -1.55e-143)
   (* d (- (pow (* l h) -0.5)))
   (* d (sqrt (/ (/ 1.0 l) h)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1.55e-143) {
		tmp = d * -pow((l * h), -0.5);
	} else {
		tmp = d * sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-1.55d-143)) then
        tmp = d * -((l * h) ** (-0.5d0))
    else
        tmp = d * sqrt(((1.0d0 / l) / h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1.55e-143) {
		tmp = d * -Math.pow((l * h), -0.5);
	} else {
		tmp = d * Math.sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -1.55e-143:
		tmp = d * -math.pow((l * h), -0.5)
	else:
		tmp = d * math.sqrt(((1.0 / l) / h))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -1.55e-143)
		tmp = Float64(d * Float64(-(Float64(l * h) ^ -0.5)));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) / h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -1.55e-143)
		tmp = d * -((l * h) ^ -0.5);
	else
		tmp = d * sqrt(((1.0 / l) / h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -1.55e-143], N[(d * (-N[Power[N[(l * h), $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 d < -1.55000000000000004e-143

   1. Initial program 80.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 78.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 78.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 78.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 78.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. metadata-eval 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. pow1 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(D \cdot \frac{1}{d}\right)} \cdot \frac{M}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. associate-*l* 80.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(\frac{1}{d} \cdot \frac{M}{2}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. times-frac 80.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{1 \cdot M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. *-commutative 80.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. times-frac 80.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. metadata-eval 80.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 80.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. unpow-1 0.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      3. metadata-eval 0.0%

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      4. pow-sqr 0.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      5. rem-sqrt-square 0.0%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      6. rem-square-sqrt 0.0%

         \[\leadsto \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      7. fabs-sqr 0.0%

         \[\leadsto \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      8. rem-square-sqrt 0.0%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      9. *-commutative 0.0%

         \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      10. unpow2 0.0%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      11. rem-square-sqrt 48.2%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      12. mul-1-neg 48.2%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left(-d\right)} \]

   9. Simplified 48.2%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if -1.55000000000000004e-143 < d

   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 63.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 65.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 65.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. metadata-eval 65.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. pow1 65.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 65.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(D \cdot \frac{1}{d}\right)} \cdot \frac{M}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. associate-*l* 64.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(\frac{1}{d} \cdot \frac{M}{2}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. times-frac 64.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{1 \cdot M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. *-commutative 64.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. times-frac 64.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. 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. Taylor expanded in d around inf 32.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. unpow-1 32.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. metadata-eval 32.5%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      3. pow-sqr 32.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      4. rem-sqrt-square 32.5%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      5. rem-square-sqrt 32.4%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

      6. fabs-sqr 32.4%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      7. rem-square-sqrt 32.5%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 32.5%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Taylor expanded in d around 0 32.5%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. *-commutative 32.5%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

       2. unpow1/2 32.5%

          \[\leadsto \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot d \]

       3. rem-exp-log 31.3%

          \[\leadsto {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot d \]

       4. rec-exp 31.3%

          \[\leadsto {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot d \]

       5. exp-prod 31.3%

          \[\leadsto \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot d \]

       6. distribute-lft-neg-out 31.3%

          \[\leadsto e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot d \]

       7. exp-neg 31.3%

          \[\leadsto \color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \cdot d \]

       8. exp-to-pow 32.5%

          \[\leadsto \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot d \]

       9. unpow1/2 32.5%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot d \]

       10. associate-*l/ 32.5%

           \[\leadsto \color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}} \]

       11. *-lft-identity 32.5%

           \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   12. Simplified 32.5%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   13. Taylor expanded in d around 0 32.5%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   14. Step-by-step derivation

       1. *-commutative 32.5%

          \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

       2. associate-/r* 32.5%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   15. Simplified 32.5%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.55 \cdot 10^{-143}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 41.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.76 \cdot 10^{-143}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -1.76e-143) (/ d (- (sqrt (* l h)))) (* d (sqrt (/ (/ 1.0 l) h)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1.76e-143) {
		tmp = d / -sqrt((l * h));
	} else {
		tmp = d * sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-1.76d-143)) then
        tmp = d / -sqrt((l * h))
    else
        tmp = d * sqrt(((1.0d0 / l) / h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1.76e-143) {
		tmp = d / -Math.sqrt((l * h));
	} else {
		tmp = d * Math.sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -1.76e-143:
		tmp = d / -math.sqrt((l * h))
	else:
		tmp = d * math.sqrt(((1.0 / l) / h))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -1.76e-143)
		tmp = Float64(d / Float64(-sqrt(Float64(l * h))));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) / h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -1.76e-143)
		tmp = d / -sqrt((l * h));
	else
		tmp = d * sqrt(((1.0 / l) / h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -1.76e-143], N[(d / (-N[Sqrt[N[(l * h), $MachinePrecision]], $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 d < -1.76000000000000005e-143

   1. Initial program 80.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 78.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 78.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 78.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 78.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. metadata-eval 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. pow1 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(D \cdot \frac{1}{d}\right)} \cdot \frac{M}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. associate-*l* 80.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(\frac{1}{d} \cdot \frac{M}{2}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. times-frac 80.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{1 \cdot M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. *-commutative 80.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. times-frac 80.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. metadata-eval 80.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 80.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 6.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. unpow-1 6.3%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. metadata-eval 6.3%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      3. pow-sqr 6.3%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      4. rem-sqrt-square 6.3%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      5. rem-square-sqrt 6.3%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

      6. fabs-sqr 6.3%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      7. rem-square-sqrt 6.3%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 6.3%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Taylor expanded in h 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. 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 48.2%

          \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

       4. mul-1-neg 48.2%

          \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

       5. distribute-rgt-neg-out 48.2%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

       6. *-commutative 48.2%

          \[\leadsto -\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

       7. unpow1/2 48.2%

          \[\leadsto -\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot d \]

       8. rem-exp-log 45.3%

          \[\leadsto -{\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot d \]

       9. rec-exp 45.3%

          \[\leadsto -{\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot d \]

       10. exp-prod 45.3%

           \[\leadsto -\color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot d \]

       11. distribute-lft-neg-out 45.3%

           \[\leadsto -e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot d \]

       12. exp-neg 45.3%

           \[\leadsto -\color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \cdot d \]

       13. exp-to-pow 48.1%

           \[\leadsto -\frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot d \]

       14. unpow1/2 48.1%

           \[\leadsto -\frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot d \]

       15. associate-*l/ 48.2%

           \[\leadsto -\color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}} \]

       16. *-lft-identity 48.2%

           \[\leadsto -\frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

       17. distribute-frac-neg2 48.2%

           \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   12. Simplified 48.2%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   if -1.76000000000000005e-143 < d

   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 63.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 65.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 65.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. metadata-eval 65.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. pow1 65.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 65.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(D \cdot \frac{1}{d}\right)} \cdot \frac{M}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. associate-*l* 64.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(\frac{1}{d} \cdot \frac{M}{2}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. times-frac 64.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{1 \cdot M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. *-commutative 64.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. times-frac 64.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. 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. Taylor expanded in d around inf 32.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. unpow-1 32.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. metadata-eval 32.5%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      3. pow-sqr 32.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      4. rem-sqrt-square 32.5%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      5. rem-square-sqrt 32.4%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

      6. fabs-sqr 32.4%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      7. rem-square-sqrt 32.5%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 32.5%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Taylor expanded in d around 0 32.5%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. *-commutative 32.5%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

       2. unpow1/2 32.5%

          \[\leadsto \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot d \]

       3. rem-exp-log 31.3%

          \[\leadsto {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot d \]

       4. rec-exp 31.3%

          \[\leadsto {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot d \]

       5. exp-prod 31.3%

          \[\leadsto \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot d \]

       6. distribute-lft-neg-out 31.3%

          \[\leadsto e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot d \]

       7. exp-neg 31.3%

          \[\leadsto \color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \cdot d \]

       8. exp-to-pow 32.5%

          \[\leadsto \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot d \]

       9. unpow1/2 32.5%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot d \]

       10. associate-*l/ 32.5%

           \[\leadsto \color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}} \]

       11. *-lft-identity 32.5%

           \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   12. Simplified 32.5%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   13. Taylor expanded in d around 0 32.5%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   14. Step-by-step derivation

       1. *-commutative 32.5%

          \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

       2. associate-/r* 32.5%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   15. Simplified 32.5%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.76 \cdot 10^{-143}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 41.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.55 \cdot 10^{-143}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -1.55e-143) (/ d (- (sqrt (* l h)))) (* d (sqrt (/ (/ 1.0 h) l)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1.55e-143) {
		tmp = d / -sqrt((l * h));
	} else {
		tmp = d * sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-1.55d-143)) then
        tmp = d / -sqrt((l * h))
    else
        tmp = d * sqrt(((1.0d0 / h) / l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1.55e-143) {
		tmp = d / -Math.sqrt((l * h));
	} else {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -1.55e-143:
		tmp = d / -math.sqrt((l * h))
	else:
		tmp = d * math.sqrt(((1.0 / h) / l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -1.55e-143)
		tmp = Float64(d / Float64(-sqrt(Float64(l * h))));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -1.55e-143)
		tmp = d / -sqrt((l * h));
	else
		tmp = d * sqrt(((1.0 / h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -1.55e-143], N[(d / (-N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.55000000000000004e-143

   1. Initial program 80.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 78.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 78.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 78.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 78.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. metadata-eval 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. pow1 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(D \cdot \frac{1}{d}\right)} \cdot \frac{M}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. associate-*l* 80.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(\frac{1}{d} \cdot \frac{M}{2}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. times-frac 80.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{1 \cdot M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. *-commutative 80.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. times-frac 80.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. metadata-eval 80.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 80.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 6.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. unpow-1 6.3%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. metadata-eval 6.3%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      3. pow-sqr 6.3%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      4. rem-sqrt-square 6.3%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      5. rem-square-sqrt 6.3%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

      6. fabs-sqr 6.3%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      7. rem-square-sqrt 6.3%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 6.3%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Taylor expanded in h 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. 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 48.2%

          \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

       4. mul-1-neg 48.2%

          \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

       5. distribute-rgt-neg-out 48.2%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

       6. *-commutative 48.2%

          \[\leadsto -\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

       7. unpow1/2 48.2%

          \[\leadsto -\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot d \]

       8. rem-exp-log 45.3%

          \[\leadsto -{\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot d \]

       9. rec-exp 45.3%

          \[\leadsto -{\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot d \]

       10. exp-prod 45.3%

           \[\leadsto -\color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot d \]

       11. distribute-lft-neg-out 45.3%

           \[\leadsto -e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot d \]

       12. exp-neg 45.3%

           \[\leadsto -\color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \cdot d \]

       13. exp-to-pow 48.1%

           \[\leadsto -\frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot d \]

       14. unpow1/2 48.1%

           \[\leadsto -\frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot d \]

       15. associate-*l/ 48.2%

           \[\leadsto -\color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}} \]

       16. *-lft-identity 48.2%

           \[\leadsto -\frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

       17. distribute-frac-neg2 48.2%

           \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   12. Simplified 48.2%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   if -1.55000000000000004e-143 < d

   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 63.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 65.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 65.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. metadata-eval 65.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. pow1 65.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 65.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(D \cdot \frac{1}{d}\right)} \cdot \frac{M}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. associate-*l* 64.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(\frac{1}{d} \cdot \frac{M}{2}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. times-frac 64.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{1 \cdot M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. *-commutative 64.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. times-frac 64.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. 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. Taylor expanded in d around inf 32.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. unpow-1 32.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. metadata-eval 32.5%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      3. pow-sqr 32.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      4. rem-sqrt-square 32.5%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      5. rem-square-sqrt 32.4%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

      6. fabs-sqr 32.4%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      7. rem-square-sqrt 32.5%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 32.5%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Taylor expanded in d around 0 32.5%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. associate-/r* 32.5%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   12. Simplified 32.5%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.55 \cdot 10^{-143}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 29: 41.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;d \leq -1.55 \cdot 10^{-143}:\\ \;\;\;\;\frac{d}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t\_0}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* l h)))) (if (<= d -1.55e-143) (/ d (- t_0)) (/ d t_0))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (d <= -1.55e-143) {
		tmp = d / -t_0;
	} else {
		tmp = d / t_0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((l * h))
    if (d <= (-1.55d-143)) then
        tmp = d / -t_0
    else
        tmp = d / t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((l * h));
	double tmp;
	if (d <= -1.55e-143) {
		tmp = d / -t_0;
	} else {
		tmp = d / t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((l * h))
	tmp = 0
	if d <= -1.55e-143:
		tmp = d / -t_0
	else:
		tmp = d / t_0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (d <= -1.55e-143)
		tmp = Float64(d / Float64(-t_0));
	else
		tmp = Float64(d / t_0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((l * h));
	tmp = 0.0;
	if (d <= -1.55e-143)
		tmp = d / -t_0;
	else
		tmp = d / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.55e-143], N[(d / (-t$95$0)), $MachinePrecision], N[(d / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -1.55000000000000004e-143

   1. Initial program 80.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 78.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 78.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 78.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 78.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. metadata-eval 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. pow1 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 79.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(D \cdot \frac{1}{d}\right)} \cdot \frac{M}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. associate-*l* 80.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(\frac{1}{d} \cdot \frac{M}{2}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. times-frac 80.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{1 \cdot M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. *-commutative 80.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. times-frac 80.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. metadata-eval 80.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 80.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 6.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. unpow-1 6.3%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. metadata-eval 6.3%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      3. pow-sqr 6.3%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      4. rem-sqrt-square 6.3%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      5. rem-square-sqrt 6.3%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

      6. fabs-sqr 6.3%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      7. rem-square-sqrt 6.3%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 6.3%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Taylor expanded in h 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. 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 48.2%

          \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

       4. mul-1-neg 48.2%

          \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

       5. distribute-rgt-neg-out 48.2%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

       6. *-commutative 48.2%

          \[\leadsto -\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

       7. unpow1/2 48.2%

          \[\leadsto -\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot d \]

       8. rem-exp-log 45.3%

          \[\leadsto -{\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot d \]

       9. rec-exp 45.3%

          \[\leadsto -{\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot d \]

       10. exp-prod 45.3%

           \[\leadsto -\color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot d \]

       11. distribute-lft-neg-out 45.3%

           \[\leadsto -e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot d \]

       12. exp-neg 45.3%

           \[\leadsto -\color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \cdot d \]

       13. exp-to-pow 48.1%

           \[\leadsto -\frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot d \]

       14. unpow1/2 48.1%

           \[\leadsto -\frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot d \]

       15. associate-*l/ 48.2%

           \[\leadsto -\color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}} \]

       16. *-lft-identity 48.2%

           \[\leadsto -\frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

       17. distribute-frac-neg2 48.2%

           \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   12. Simplified 48.2%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   if -1.55000000000000004e-143 < d

   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 63.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 65.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 65.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. metadata-eval 65.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. pow1 65.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. div-inv 65.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(D \cdot \frac{1}{d}\right)} \cdot \frac{M}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. associate-*l* 64.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(\frac{1}{d} \cdot \frac{M}{2}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. times-frac 64.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{1 \cdot M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. *-commutative 64.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. times-frac 64.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. 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. Taylor expanded in d around inf 32.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. unpow-1 32.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. metadata-eval 32.5%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      3. pow-sqr 32.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      4. rem-sqrt-square 32.5%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      5. rem-square-sqrt 32.4%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

      6. fabs-sqr 32.4%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      7. rem-square-sqrt 32.5%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 32.5%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Taylor expanded in d around 0 32.5%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. *-commutative 32.5%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

       2. unpow1/2 32.5%

          \[\leadsto \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot d \]

       3. rem-exp-log 31.3%

          \[\leadsto {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot d \]

       4. rec-exp 31.3%

          \[\leadsto {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot d \]

       5. exp-prod 31.3%

          \[\leadsto \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot d \]

       6. distribute-lft-neg-out 31.3%

          \[\leadsto e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot d \]

       7. exp-neg 31.3%

          \[\leadsto \color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \cdot d \]

       8. exp-to-pow 32.5%

          \[\leadsto \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot d \]

       9. unpow1/2 32.5%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot d \]

       10. associate-*l/ 32.5%

           \[\leadsto \color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}} \]

       11. *-lft-identity 32.5%

           \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   12. Simplified 32.5%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.55 \cdot 10^{-143}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 30: 25.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (/ d (sqrt (* l h))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return d / sqrt((l * h));
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = d / sqrt((l * h))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return d / Math.sqrt((l * h));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return d / math.sqrt((l * h))

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = d / sqrt((l * h));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.3%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 69.8%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 69.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

   2. pow2 69.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

   3. sqrt-prod 69.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

   4. sqrt-pow1 71.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   5. *-commutative 71.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. metadata-eval 71.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   7. pow1 71.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   8. div-inv 71.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(D \cdot \frac{1}{d}\right)} \cdot \frac{M}{2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   9. associate-*l* 71.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(\frac{1}{d} \cdot \frac{M}{2}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   10. times-frac 71.1%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{1 \cdot M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   11. *-commutative 71.1%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   12. times-frac 71.1%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   13. metadata-eval 71.1%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

6. Applied egg-rr 71.1%

   \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

7. Taylor expanded in d around inf 21.7%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
8. Step-by-step derivation

   1. unpow-1 21.7%

      \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

   2. metadata-eval 21.7%

      \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

   3. pow-sqr 21.7%

      \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

   4. rem-sqrt-square 21.7%

      \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

   5. rem-square-sqrt 21.7%

      \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

   6. fabs-sqr 21.7%

      \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

   7. rem-square-sqrt 21.7%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

9. Simplified 21.7%

   \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

10. Taylor expanded in d around 0 21.7%

    \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
11. Step-by-step derivation

    1. *-commutative 21.7%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

    2. unpow1/2 21.7%

       \[\leadsto \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot d \]

    3. rem-exp-log 21.0%

       \[\leadsto {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot d \]

    4. rec-exp 21.1%

       \[\leadsto {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot d \]

    5. exp-prod 21.1%

       \[\leadsto \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot d \]

    6. distribute-lft-neg-out 21.1%

       \[\leadsto e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot d \]

    7. exp-neg 21.1%

       \[\leadsto \color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \cdot d \]

    8. exp-to-pow 21.7%

       \[\leadsto \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot d \]

    9. unpow1/2 21.7%

       \[\leadsto \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot d \]

    10. associate-*l/ 21.7%

        \[\leadsto \color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}} \]

    11. *-lft-identity 21.7%

        \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

12. Simplified 21.7%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

13. Final simplification 21.7%

    \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024150 
(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)))))