Henrywood and Agarwal, Equation (12)

Percentage Accurate: 67.3% → 80.4%

Time: 22.3s

Alternatives: 13

Speedup: 1.0×

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: 67.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := 1 + \frac{h}{\ell} \cdot \left({\left(D\_m \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -1.75 \cdot 10^{-284}:\\ \;\;\;\;\frac{t\_2}{\sqrt{-\ell}} \cdot \left(\frac{t\_2}{\sqrt{-h}} \cdot t\_0\right)\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{-176}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot t\_1\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D\_m}{d \cdot \frac{2}{M}}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(t\_0 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (/ h l) (* (pow (* D_m (/ (/ M 2.0) d)) 2.0) -0.5))))
        (t_1 (sqrt (/ d l)))
        (t_2 (sqrt (- d))))
   (if (<= l -1.75e-284)
     (* (/ t_2 (sqrt (- l))) (* (/ t_2 (sqrt (- h))) t_0))
     (if (<= l 1.8e-176)
       (*
        (* (sqrt (/ d h)) t_1)
        (- 1.0 (* 0.5 (/ (* h (pow (/ D_m (* d (/ 2.0 M))) 2.0)) l))))
       (* t_1 (* t_0 (/ (sqrt d) (sqrt h))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 1.0 + ((h / l) * (pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5));
	double t_1 = sqrt((d / l));
	double t_2 = sqrt(-d);
	double tmp;
	if (l <= -1.75e-284) {
		tmp = (t_2 / sqrt(-l)) * ((t_2 / sqrt(-h)) * t_0);
	} else if (l <= 1.8e-176) {
		tmp = (sqrt((d / h)) * t_1) * (1.0 - (0.5 * ((h * pow((D_m / (d * (2.0 / M))), 2.0)) / l)));
	} else {
		tmp = t_1 * (t_0 * (sqrt(d) / sqrt(h)));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 + ((h / l) * (((d_m * ((m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))
    t_1 = sqrt((d / l))
    t_2 = sqrt(-d)
    if (l <= (-1.75d-284)) then
        tmp = (t_2 / sqrt(-l)) * ((t_2 / sqrt(-h)) * t_0)
    else if (l <= 1.8d-176) then
        tmp = (sqrt((d / h)) * t_1) * (1.0d0 - (0.5d0 * ((h * ((d_m / (d * (2.0d0 / m))) ** 2.0d0)) / l)))
    else
        tmp = t_1 * (t_0 * (sqrt(d) / sqrt(h)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 1.0 + ((h / l) * (Math.pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5));
	double t_1 = Math.sqrt((d / l));
	double t_2 = Math.sqrt(-d);
	double tmp;
	if (l <= -1.75e-284) {
		tmp = (t_2 / Math.sqrt(-l)) * ((t_2 / Math.sqrt(-h)) * t_0);
	} else if (l <= 1.8e-176) {
		tmp = (Math.sqrt((d / h)) * t_1) * (1.0 - (0.5 * ((h * Math.pow((D_m / (d * (2.0 / M))), 2.0)) / l)));
	} else {
		tmp = t_1 * (t_0 * (Math.sqrt(d) / Math.sqrt(h)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = 1.0 + ((h / l) * (math.pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))
	t_1 = math.sqrt((d / l))
	t_2 = math.sqrt(-d)
	tmp = 0
	if l <= -1.75e-284:
		tmp = (t_2 / math.sqrt(-l)) * ((t_2 / math.sqrt(-h)) * t_0)
	elif l <= 1.8e-176:
		tmp = (math.sqrt((d / h)) * t_1) * (1.0 - (0.5 * ((h * math.pow((D_m / (d * (2.0 / M))), 2.0)) / l)))
	else:
		tmp = t_1 * (t_0 * (math.sqrt(d) / math.sqrt(h)))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D_m * Float64(Float64(M / 2.0) / d)) ^ 2.0) * -0.5)))
	t_1 = sqrt(Float64(d / l))
	t_2 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -1.75e-284)
		tmp = Float64(Float64(t_2 / sqrt(Float64(-l))) * Float64(Float64(t_2 / sqrt(Float64(-h))) * t_0));
	elseif (l <= 1.8e-176)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * t_1) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(D_m / Float64(d * Float64(2.0 / M))) ^ 2.0)) / l))));
	else
		tmp = Float64(t_1 * Float64(t_0 * Float64(sqrt(d) / sqrt(h))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = 1.0 + ((h / l) * (((D_m * ((M / 2.0) / d)) ^ 2.0) * -0.5));
	t_1 = sqrt((d / l));
	t_2 = sqrt(-d);
	tmp = 0.0;
	if (l <= -1.75e-284)
		tmp = (t_2 / sqrt(-l)) * ((t_2 / sqrt(-h)) * t_0);
	elseif (l <= 1.8e-176)
		tmp = (sqrt((d / h)) * t_1) * (1.0 - (0.5 * ((h * ((D_m / (d * (2.0 / M))) ^ 2.0)) / l)));
	else
		tmp = t_1 * (t_0 * (sqrt(d) / sqrt(h)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D$95$m * N[(N[(M / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -1.75e-284], N[(N[(t$95$2 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$2 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.8e-176], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(D$95$m / N[(d * N[(2.0 / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.74999999999999988e-284

   1. Initial program 65.0%

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

   3. Simplified 63.3%

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

      1. frac-2neg 63.3%

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

      2. sqrt-div 68.1%

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

   6. Applied egg-rr 68.1%

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

      1. frac-2neg 68.1%

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

      2. sqrt-div 81.8%

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

   8. Applied egg-rr 81.8%

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

   if -1.74999999999999988e-284 < l < 1.8000000000000001e-176

   1. Initial program 65.8%

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

   3. Simplified 69.1%

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

      1. associate-*r/ 86.7%

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

      2. frac-times 83.4%

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

      3. *-commutative 83.4%

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

      4. *-un-lft-identity 83.4%

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

      5. times-frac 86.6%

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

      6. *-commutative 86.6%

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

      7. associate-/l/ 86.6%

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

      8. times-frac 83.4%

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

      9. *-un-lft-identity 83.4%

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

      10. associate-*r/ 86.6%

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

      11. clear-num 86.7%

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

      12. un-div-inv 86.8%

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

      13. div-inv 86.8%

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

      14. clear-num 86.8%

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

   6. Applied egg-rr 86.8%

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

   if 1.8000000000000001e-176 < l

   1. Initial program 68.6%

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

   3. Simplified 67.7%

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

      1. sqrt-div 83.0%

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

      2. div-inv 82.9%

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

   6. Applied egg-rr 82.9%

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

      1. associate-*r/ 83.0%

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

      2. *-rgt-identity 83.0%

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

   8. Simplified 83.0%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.75 \cdot 10^{-284}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{-176}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d \cdot \frac{2}{M}}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 70.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D\_m \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-118}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t\_0 \cdot \left(h \cdot \left(-0.125 \cdot \frac{{\left(\frac{D\_m \cdot M}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(t\_0 \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(t\_0 \cdot \left(h \cdot \left(-0.125 \cdot \frac{{\left(M \cdot \frac{D\_m}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* D_m M) (* d 2.0)) 2.0)))))))
   (if (<= t_1 -2e-118)
     (*
      (sqrt (/ d h))
      (* t_0 (* h (* -0.125 (/ (pow (/ (* D_m M) d) 2.0) l)))))
     (if (<= t_1 0.0)
       (* d (sqrt (/ 1.0 (* l h))))
       (if (<= t_1 INFINITY)
         (*
          (* t_0 (/ 1.0 (sqrt (/ h d))))
          (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M 2.0) (/ D_m d)) 2.0)))))
         (*
          (/ (sqrt (- d)) (sqrt (- h)))
          (* t_0 (* h (* -0.125 (/ (pow (* M (/ D_m d)) 2.0) l))))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt((d / l));
	double t_1 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((D_m * M) / (d * 2.0)), 2.0))));
	double tmp;
	if (t_1 <= -2e-118) {
		tmp = sqrt((d / h)) * (t_0 * (h * (-0.125 * (pow(((D_m * M) / d), 2.0) / l))));
	} else if (t_1 <= 0.0) {
		tmp = d * sqrt((1.0 / (l * h)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_0 * (1.0 / sqrt((h / d)))) * (1.0 - (0.5 * ((h / l) * pow(((M / 2.0) * (D_m / d)), 2.0))));
	} else {
		tmp = (sqrt(-d) / sqrt(-h)) * (t_0 * (h * (-0.125 * (pow((M * (D_m / d)), 2.0) / l))));
	}
	return tmp;
}

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = (Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * Math.pow(((D_m * M) / (d * 2.0)), 2.0))));
	double tmp;
	if (t_1 <= -2e-118) {
		tmp = Math.sqrt((d / h)) * (t_0 * (h * (-0.125 * (Math.pow(((D_m * M) / d), 2.0) / l))));
	} else if (t_1 <= 0.0) {
		tmp = d * Math.sqrt((1.0 / (l * h)));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 * (1.0 / Math.sqrt((h / d)))) * (1.0 - (0.5 * ((h / l) * Math.pow(((M / 2.0) * (D_m / d)), 2.0))));
	} else {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * (t_0 * (h * (-0.125 * (Math.pow((M * (D_m / d)), 2.0) / l))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.sqrt((d / l))
	t_1 = (math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * math.pow(((D_m * M) / (d * 2.0)), 2.0))))
	tmp = 0
	if t_1 <= -2e-118:
		tmp = math.sqrt((d / h)) * (t_0 * (h * (-0.125 * (math.pow(((D_m * M) / d), 2.0) / l))))
	elif t_1 <= 0.0:
		tmp = d * math.sqrt((1.0 / (l * h)))
	elif t_1 <= math.inf:
		tmp = (t_0 * (1.0 / math.sqrt((h / d)))) * (1.0 - (0.5 * ((h / l) * math.pow(((M / 2.0) * (D_m / d)), 2.0))))
	else:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * (t_0 * (h * (-0.125 * (math.pow((M * (D_m / d)), 2.0) / l))))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(d / l))
	t_1 = 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) / Float64(d * 2.0)) ^ 2.0)))))
	tmp = 0.0
	if (t_1 <= -2e-118)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(t_0 * Float64(h * Float64(-0.125 * Float64((Float64(Float64(D_m * M) / d) ^ 2.0) / l)))));
	elseif (t_1 <= 0.0)
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(l * h))));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_0 * Float64(1.0 / sqrt(Float64(h / d)))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M / 2.0) * Float64(D_m / d)) ^ 2.0)))));
	else
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(t_0 * Float64(h * Float64(-0.125 * Float64((Float64(M * Float64(D_m / d)) ^ 2.0) / l)))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = sqrt((d / l));
	t_1 = (((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((h / l) * (0.5 * (((D_m * M) / (d * 2.0)) ^ 2.0))));
	tmp = 0.0;
	if (t_1 <= -2e-118)
		tmp = sqrt((d / h)) * (t_0 * (h * (-0.125 * ((((D_m * M) / d) ^ 2.0) / l))));
	elseif (t_1 <= 0.0)
		tmp = d * sqrt((1.0 / (l * h)));
	elseif (t_1 <= Inf)
		tmp = (t_0 * (1.0 / sqrt((h / d)))) * (1.0 - (0.5 * ((h / l) * (((M / 2.0) * (D_m / d)) ^ 2.0))));
	else
		tmp = (sqrt(-d) / sqrt(-h)) * (t_0 * (h * (-0.125 * (((M * (D_m / d)) ^ 2.0) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = 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), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-118], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(h * N[(-0.125 * N[(N[Power[N[(N[(D$95$m * M), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(t$95$0 * N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(h * N[(-0.125 * N[(N[Power[N[(M * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 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)))) < -1.99999999999999997e-118

   1. Initial program 79.1%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in M around inf 51.2%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

   7. Simplified 79.0%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(h \cdot \left(-0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. associate-*r/ 82.0%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(h \cdot \left(-0.125 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}}{\ell}\right)\right)\right) \]

   9. Applied egg-rr 82.0%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(h \cdot \left(-0.125 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}}{\ell}\right)\right)\right) \]

   if -1.99999999999999997e-118 < (*.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)))) < 0.0

   1. Initial program 27.3%

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

   3. Simplified 19.7%

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

      1. associate-*r/ 19.1%

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

      2. clear-num 19.1%

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

      3. frac-times 26.4%

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

      4. *-commutative 26.4%

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

      5. *-un-lft-identity 26.4%

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

      6. times-frac 25.8%

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

      7. *-commutative 25.8%

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

      8. associate-/l/ 25.8%

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

      9. times-frac 26.4%

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

      10. *-un-lft-identity 26.4%

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

      11. associate-*r/ 25.8%

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

      12. clear-num 25.8%

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

      13. un-div-inv 25.8%

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

      14. div-inv 25.8%

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

      15. clear-num 25.8%

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

   6. Applied egg-rr 25.8%

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

      1. associate-/r/ 25.8%

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

      2. *-commutative 25.8%

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

      3. associate-*r/ 25.8%

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

   8. Simplified 25.8%

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

   9. Taylor expanded in d around inf 63.2%

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

   if 0.0 < (*.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)))) < +inf.0

   1. Initial program 80.5%

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

   3. Simplified 81.4%

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

      1. clear-num 81.4%

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

      2. sqrt-div 83.7%

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

      3. metadata-eval 83.7%

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

   6. Applied egg-rr 83.7%

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

   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%

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

   5. Taylor expanded in M around inf 0.2%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

   7. Simplified 0.8%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(h \cdot \left(-0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. frac-2neg 0.2%

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

      2. sqrt-div 32.7%

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

   9. Applied egg-rr 29.8%

      \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(h \cdot \left(-0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq -2 \cdot 10^{-118}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(h \cdot \left(-0.125 \cdot \frac{{\left(\frac{D \cdot M}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq 0:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq \infty:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(h \cdot \left(-0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D\_m \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-118}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t\_0 \cdot \left(h \cdot \left(-0.125 \cdot \frac{{\left(\frac{D\_m \cdot M}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{1}{\ell} \cdot \left(h \cdot {\left(\frac{D\_m}{\frac{d \cdot 2}{M}}\right)}^{2}\right)\right)\right) \cdot \left(t\_0 \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* D_m M) (* d 2.0)) 2.0)))))))
   (if (<= t_1 -2e-118)
     (*
      (sqrt (/ d h))
      (* t_0 (* h (* -0.125 (/ (pow (/ (* D_m M) d) 2.0) l)))))
     (if (<= t_1 0.0)
       (* d (sqrt (/ 1.0 (* l h))))
       (*
        (- 1.0 (* 0.5 (* (/ 1.0 l) (* h (pow (/ D_m (/ (* d 2.0) M)) 2.0)))))
        (* t_0 (/ 1.0 (sqrt (/ h d)))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt((d / l));
	double t_1 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((D_m * M) / (d * 2.0)), 2.0))));
	double tmp;
	if (t_1 <= -2e-118) {
		tmp = sqrt((d / h)) * (t_0 * (h * (-0.125 * (pow(((D_m * M) / d), 2.0) / l))));
	} else if (t_1 <= 0.0) {
		tmp = d * sqrt((1.0 / (l * h)));
	} else {
		tmp = (1.0 - (0.5 * ((1.0 / l) * (h * pow((D_m / ((d * 2.0) / M)), 2.0))))) * (t_0 * (1.0 / sqrt((h / d))));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = (((d / h) ** 0.5d0) * ((d / l) ** 0.5d0)) * (1.0d0 - ((h / l) * (0.5d0 * (((d_m * m) / (d * 2.0d0)) ** 2.0d0))))
    if (t_1 <= (-2d-118)) then
        tmp = sqrt((d / h)) * (t_0 * (h * ((-0.125d0) * ((((d_m * m) / d) ** 2.0d0) / l))))
    else if (t_1 <= 0.0d0) then
        tmp = d * sqrt((1.0d0 / (l * h)))
    else
        tmp = (1.0d0 - (0.5d0 * ((1.0d0 / l) * (h * ((d_m / ((d * 2.0d0) / m)) ** 2.0d0))))) * (t_0 * (1.0d0 / sqrt((h / d))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = (Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * Math.pow(((D_m * M) / (d * 2.0)), 2.0))));
	double tmp;
	if (t_1 <= -2e-118) {
		tmp = Math.sqrt((d / h)) * (t_0 * (h * (-0.125 * (Math.pow(((D_m * M) / d), 2.0) / l))));
	} else if (t_1 <= 0.0) {
		tmp = d * Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = (1.0 - (0.5 * ((1.0 / l) * (h * Math.pow((D_m / ((d * 2.0) / M)), 2.0))))) * (t_0 * (1.0 / Math.sqrt((h / d))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.sqrt((d / l))
	t_1 = (math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * math.pow(((D_m * M) / (d * 2.0)), 2.0))))
	tmp = 0
	if t_1 <= -2e-118:
		tmp = math.sqrt((d / h)) * (t_0 * (h * (-0.125 * (math.pow(((D_m * M) / d), 2.0) / l))))
	elif t_1 <= 0.0:
		tmp = d * math.sqrt((1.0 / (l * h)))
	else:
		tmp = (1.0 - (0.5 * ((1.0 / l) * (h * math.pow((D_m / ((d * 2.0) / M)), 2.0))))) * (t_0 * (1.0 / math.sqrt((h / d))))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(d / l))
	t_1 = 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) / Float64(d * 2.0)) ^ 2.0)))))
	tmp = 0.0
	if (t_1 <= -2e-118)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(t_0 * Float64(h * Float64(-0.125 * Float64((Float64(Float64(D_m * M) / d) ^ 2.0) / l)))));
	elseif (t_1 <= 0.0)
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(l * h))));
	else
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(1.0 / l) * Float64(h * (Float64(D_m / Float64(Float64(d * 2.0) / M)) ^ 2.0))))) * Float64(t_0 * Float64(1.0 / sqrt(Float64(h / d)))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = sqrt((d / l));
	t_1 = (((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((h / l) * (0.5 * (((D_m * M) / (d * 2.0)) ^ 2.0))));
	tmp = 0.0;
	if (t_1 <= -2e-118)
		tmp = sqrt((d / h)) * (t_0 * (h * (-0.125 * ((((D_m * M) / d) ^ 2.0) / l))));
	elseif (t_1 <= 0.0)
		tmp = d * sqrt((1.0 / (l * h)));
	else
		tmp = (1.0 - (0.5 * ((1.0 / l) * (h * ((D_m / ((d * 2.0) / M)) ^ 2.0))))) * (t_0 * (1.0 / sqrt((h / d))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = 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), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-118], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(h * N[(-0.125 * N[(N[Power[N[(N[(D$95$m * M), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(0.5 * N[(N[(1.0 / l), $MachinePrecision] * N[(h * N[Power[N[(D$95$m / N[(N[(d * 2.0), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.99999999999999997e-118

   1. Initial program 79.1%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in M around inf 51.2%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

   7. Simplified 79.0%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(h \cdot \left(-0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. associate-*r/ 82.0%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(h \cdot \left(-0.125 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}}{\ell}\right)\right)\right) \]

   9. Applied egg-rr 82.0%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(h \cdot \left(-0.125 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}}{\ell}\right)\right)\right) \]

   if -1.99999999999999997e-118 < (*.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)))) < 0.0

   1. Initial program 27.3%

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

   3. Simplified 19.7%

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

      1. associate-*r/ 19.1%

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

      2. clear-num 19.1%

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

      3. frac-times 26.4%

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

      4. *-commutative 26.4%

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

      5. *-un-lft-identity 26.4%

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

      6. times-frac 25.8%

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

      7. *-commutative 25.8%

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

      8. associate-/l/ 25.8%

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

      9. times-frac 26.4%

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

      10. *-un-lft-identity 26.4%

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

      11. associate-*r/ 25.8%

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

      12. clear-num 25.8%

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

      13. un-div-inv 25.8%

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

      14. div-inv 25.8%

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

      15. clear-num 25.8%

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

   6. Applied egg-rr 25.8%

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

      1. associate-/r/ 25.8%

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

      2. *-commutative 25.8%

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

      3. associate-*r/ 25.8%

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

   8. Simplified 25.8%

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

   9. Taylor expanded in d around inf 63.2%

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

   if 0.0 < (*.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 61.8%

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

   3. Simplified 62.4%

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

      1. associate-*r/ 65.5%

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

      2. clear-num 65.5%

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

      3. frac-times 64.9%

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

      4. *-commutative 64.9%

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

      5. *-un-lft-identity 64.9%

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

      6. times-frac 65.5%

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

      7. *-commutative 65.5%

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

      8. associate-/l/ 65.5%

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

      9. times-frac 64.9%

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

      10. *-un-lft-identity 64.9%

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

      11. associate-*r/ 65.5%

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

      12. clear-num 65.5%

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

      13. un-div-inv 65.5%

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

      14. div-inv 65.5%

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

      15. clear-num 65.5%

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

   6. Applied egg-rr 65.5%

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

      1. associate-/r/ 65.5%

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

      2. *-commutative 65.5%

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

      3. associate-*r/ 65.5%

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

   8. Simplified 65.5%

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

      1. clear-num 65.5%

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

      2. sqrt-div 67.3%

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

      3. metadata-eval 67.3%

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

   10. Applied egg-rr 67.3%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq -2 \cdot 10^{-118}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(h \cdot \left(-0.125 \cdot \frac{{\left(\frac{D \cdot M}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq 0:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{1}{\ell} \cdot \left(h \cdot {\left(\frac{D}{\frac{d \cdot 2}{M}}\right)}^{2}\right)\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := 1 + \frac{h}{\ell} \cdot \left({\left(D\_m \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;d \leq -3.45 \cdot 10^{+246}:\\ \;\;\;\;\frac{t\_2}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \frac{-0.5 \cdot {\left(D\_m \cdot \frac{M \cdot 0.5}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(\frac{t\_2}{\sqrt{-h}} \cdot t\_1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1 (+ 1.0 (* (/ h l) (* (pow (* D_m (/ (/ M 2.0) d)) 2.0) -0.5))))
        (t_2 (sqrt (- d))))
   (if (<= d -3.45e+246)
     (*
      (/ t_2 (sqrt (- l)))
      (*
       (sqrt (/ d h))
       (+ 1.0 (* h (/ (* -0.5 (pow (* D_m (/ (* M 0.5) d)) 2.0)) l)))))
     (if (<= d -5e-311)
       (* (* (/ t_2 (sqrt (- h))) t_1) t_0)
       (* t_0 (* t_1 (/ (sqrt d) (sqrt h))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt((d / l));
	double t_1 = 1.0 + ((h / l) * (pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5));
	double t_2 = sqrt(-d);
	double tmp;
	if (d <= -3.45e+246) {
		tmp = (t_2 / sqrt(-l)) * (sqrt((d / h)) * (1.0 + (h * ((-0.5 * pow((D_m * ((M * 0.5) / d)), 2.0)) / l))));
	} else if (d <= -5e-311) {
		tmp = ((t_2 / sqrt(-h)) * t_1) * t_0;
	} else {
		tmp = t_0 * (t_1 * (sqrt(d) / sqrt(h)));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = 1.0d0 + ((h / l) * (((d_m * ((m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))
    t_2 = sqrt(-d)
    if (d <= (-3.45d+246)) then
        tmp = (t_2 / sqrt(-l)) * (sqrt((d / h)) * (1.0d0 + (h * (((-0.5d0) * ((d_m * ((m * 0.5d0) / d)) ** 2.0d0)) / l))))
    else if (d <= (-5d-311)) then
        tmp = ((t_2 / sqrt(-h)) * t_1) * t_0
    else
        tmp = t_0 * (t_1 * (sqrt(d) / sqrt(h)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = 1.0 + ((h / l) * (Math.pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5));
	double t_2 = Math.sqrt(-d);
	double tmp;
	if (d <= -3.45e+246) {
		tmp = (t_2 / Math.sqrt(-l)) * (Math.sqrt((d / h)) * (1.0 + (h * ((-0.5 * Math.pow((D_m * ((M * 0.5) / d)), 2.0)) / l))));
	} else if (d <= -5e-311) {
		tmp = ((t_2 / Math.sqrt(-h)) * t_1) * t_0;
	} else {
		tmp = t_0 * (t_1 * (Math.sqrt(d) / Math.sqrt(h)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.sqrt((d / l))
	t_1 = 1.0 + ((h / l) * (math.pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))
	t_2 = math.sqrt(-d)
	tmp = 0
	if d <= -3.45e+246:
		tmp = (t_2 / math.sqrt(-l)) * (math.sqrt((d / h)) * (1.0 + (h * ((-0.5 * math.pow((D_m * ((M * 0.5) / d)), 2.0)) / l))))
	elif d <= -5e-311:
		tmp = ((t_2 / math.sqrt(-h)) * t_1) * t_0
	else:
		tmp = t_0 * (t_1 * (math.sqrt(d) / math.sqrt(h)))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D_m * Float64(Float64(M / 2.0) / d)) ^ 2.0) * -0.5)))
	t_2 = sqrt(Float64(-d))
	tmp = 0.0
	if (d <= -3.45e+246)
		tmp = Float64(Float64(t_2 / sqrt(Float64(-l))) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(h * Float64(Float64(-0.5 * (Float64(D_m * Float64(Float64(M * 0.5) / d)) ^ 2.0)) / l)))));
	elseif (d <= -5e-311)
		tmp = Float64(Float64(Float64(t_2 / sqrt(Float64(-h))) * t_1) * t_0);
	else
		tmp = Float64(t_0 * Float64(t_1 * Float64(sqrt(d) / sqrt(h))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = sqrt((d / l));
	t_1 = 1.0 + ((h / l) * (((D_m * ((M / 2.0) / d)) ^ 2.0) * -0.5));
	t_2 = sqrt(-d);
	tmp = 0.0;
	if (d <= -3.45e+246)
		tmp = (t_2 / sqrt(-l)) * (sqrt((d / h)) * (1.0 + (h * ((-0.5 * ((D_m * ((M * 0.5) / d)) ^ 2.0)) / l))));
	elseif (d <= -5e-311)
		tmp = ((t_2 / sqrt(-h)) * t_1) * t_0;
	else
		tmp = t_0 * (t_1 * (sqrt(d) / sqrt(h)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D$95$m * N[(N[(M / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[d, -3.45e+246], N[(N[(t$95$2 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h * N[(N[(-0.5 * N[Power[N[(D$95$m * N[(N[(M * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-311], N[(N[(N[(t$95$2 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 * N[(t$95$1 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.45e246

   1. Initial program 69.1%

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

   3. Simplified 69.1%

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

      1. frac-2neg 69.1%

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

      2. sqrt-div 78.8%

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

   6. Applied egg-rr 78.8%

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

   8. Applied egg-rr 0.0%

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

      1. prod-exp 0.0%

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

      2. *-commutative 0.0%

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

      3. exp-to-pow 5.3%

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

      4. rem-exp-log 78.8%

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

      5. associate-/l* 78.8%

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

      6. associate-*r* 78.8%

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

      7. metadata-eval 78.8%

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

      8. distribute-rgt-neg-in 78.8%

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

      9. *-commutative 78.8%

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

      10. associate-/l* 99.8%

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

      11. distribute-lft-neg-in 99.8%

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

      12. distribute-lft-neg-in 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/l* 99.8%

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

   10. Simplified 99.8%

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

   if -3.45e246 < d < -5.00000000000023e-311

   1. Initial program 63.6%

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

   3. Simplified 61.8%

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

      1. frac-2neg 65.4%

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

      2. sqrt-div 81.2%

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

   6. Applied egg-rr 74.7%

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

   if -5.00000000000023e-311 < d

   1. Initial program 68.6%

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

   3. Simplified 68.6%

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

      1. sqrt-div 81.7%

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

      2. div-inv 81.6%

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

   6. Applied egg-rr 81.6%

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

      1. associate-*r/ 81.7%

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

      2. *-rgt-identity 81.7%

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

   8. Simplified 81.7%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.45 \cdot 10^{+246}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \frac{-0.5 \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -1.08 \cdot 10^{+248}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(t\_0 \cdot \left(h \cdot \left(-0.125 \cdot \frac{{\left(M \cdot \frac{D\_m}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{1}{\ell} \cdot \left(h \cdot {\left(\frac{D\_m}{\frac{d \cdot 2}{M}}\right)}^{2}\right)\right)\right) \cdot \left(t\_0 \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D\_m \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))))
   (if (<= h -1.08e+248)
     (*
      (/ (sqrt (- d)) (sqrt (- h)))
      (* t_0 (* h (* -0.125 (/ (pow (* M (/ D_m d)) 2.0) l)))))
     (if (<= h -5e-311)
       (*
        (- 1.0 (* 0.5 (* (/ 1.0 l) (* h (pow (/ D_m (/ (* d 2.0) M)) 2.0)))))
        (* t_0 (/ 1.0 (sqrt (/ h d)))))
       (*
        t_0
        (*
         (+ 1.0 (* (/ h l) (* (pow (* D_m (/ (/ M 2.0) d)) 2.0) -0.5)))
         (/ (sqrt d) (sqrt h))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt((d / l));
	double tmp;
	if (h <= -1.08e+248) {
		tmp = (sqrt(-d) / sqrt(-h)) * (t_0 * (h * (-0.125 * (pow((M * (D_m / d)), 2.0) / l))));
	} else if (h <= -5e-311) {
		tmp = (1.0 - (0.5 * ((1.0 / l) * (h * pow((D_m / ((d * 2.0) / M)), 2.0))))) * (t_0 * (1.0 / sqrt((h / d))));
	} else {
		tmp = t_0 * ((1.0 + ((h / l) * (pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))) * (sqrt(d) / sqrt(h)));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((d / l))
    if (h <= (-1.08d+248)) then
        tmp = (sqrt(-d) / sqrt(-h)) * (t_0 * (h * ((-0.125d0) * (((m * (d_m / d)) ** 2.0d0) / l))))
    else if (h <= (-5d-311)) then
        tmp = (1.0d0 - (0.5d0 * ((1.0d0 / l) * (h * ((d_m / ((d * 2.0d0) / m)) ** 2.0d0))))) * (t_0 * (1.0d0 / sqrt((h / d))))
    else
        tmp = t_0 * ((1.0d0 + ((h / l) * (((d_m * ((m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))) * (sqrt(d) / sqrt(h)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.sqrt((d / l));
	double tmp;
	if (h <= -1.08e+248) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * (t_0 * (h * (-0.125 * (Math.pow((M * (D_m / d)), 2.0) / l))));
	} else if (h <= -5e-311) {
		tmp = (1.0 - (0.5 * ((1.0 / l) * (h * Math.pow((D_m / ((d * 2.0) / M)), 2.0))))) * (t_0 * (1.0 / Math.sqrt((h / d))));
	} else {
		tmp = t_0 * ((1.0 + ((h / l) * (Math.pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))) * (Math.sqrt(d) / Math.sqrt(h)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.sqrt((d / l))
	tmp = 0
	if h <= -1.08e+248:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * (t_0 * (h * (-0.125 * (math.pow((M * (D_m / d)), 2.0) / l))))
	elif h <= -5e-311:
		tmp = (1.0 - (0.5 * ((1.0 / l) * (h * math.pow((D_m / ((d * 2.0) / M)), 2.0))))) * (t_0 * (1.0 / math.sqrt((h / d))))
	else:
		tmp = t_0 * ((1.0 + ((h / l) * (math.pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))) * (math.sqrt(d) / math.sqrt(h)))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(d / l))
	tmp = 0.0
	if (h <= -1.08e+248)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(t_0 * Float64(h * Float64(-0.125 * Float64((Float64(M * Float64(D_m / d)) ^ 2.0) / l)))));
	elseif (h <= -5e-311)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(1.0 / l) * Float64(h * (Float64(D_m / Float64(Float64(d * 2.0) / M)) ^ 2.0))))) * Float64(t_0 * Float64(1.0 / sqrt(Float64(h / d)))));
	else
		tmp = Float64(t_0 * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D_m * Float64(Float64(M / 2.0) / d)) ^ 2.0) * -0.5))) * Float64(sqrt(d) / sqrt(h))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = sqrt((d / l));
	tmp = 0.0;
	if (h <= -1.08e+248)
		tmp = (sqrt(-d) / sqrt(-h)) * (t_0 * (h * (-0.125 * (((M * (D_m / d)) ^ 2.0) / l))));
	elseif (h <= -5e-311)
		tmp = (1.0 - (0.5 * ((1.0 / l) * (h * ((D_m / ((d * 2.0) / M)) ^ 2.0))))) * (t_0 * (1.0 / sqrt((h / d))));
	else
		tmp = t_0 * ((1.0 + ((h / l) * (((D_m * ((M / 2.0) / d)) ^ 2.0) * -0.5))) * (sqrt(d) / sqrt(h)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -1.08e+248], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(h * N[(-0.125 * N[(N[Power[N[(M * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-311], N[(N[(1.0 - N[(0.5 * N[(N[(1.0 / l), $MachinePrecision] * N[(h * N[Power[N[(D$95$m / N[(N[(d * 2.0), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D$95$m * N[(N[(M / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.08e248

   1. Initial program 8.6%

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

   3. Simplified 8.6%

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

   5. Taylor expanded in M around inf 0.8%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

   7. Simplified 9.4%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(h \cdot \left(-0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. frac-2neg 8.6%

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

      2. sqrt-div 62.0%

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

   9. Applied egg-rr 55.0%

      \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(h \cdot \left(-0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right) \]

   if -1.08e248 < h < -5.00000000000023e-311

   1. Initial program 71.1%

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

   3. Simplified 71.0%

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

      1. associate-*r/ 72.9%

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

      2. clear-num 72.9%

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

      3. frac-times 73.8%

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

      4. *-commutative 73.8%

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

      5. *-un-lft-identity 73.8%

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

      6. times-frac 72.9%

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

      7. *-commutative 72.9%

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

      8. associate-/l/ 72.9%

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

      9. times-frac 73.8%

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

      10. *-un-lft-identity 73.8%

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

      11. associate-*r/ 72.9%

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

      12. clear-num 72.9%

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

      13. un-div-inv 72.9%

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

      14. div-inv 72.9%

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

      15. clear-num 72.9%

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

   6. Applied egg-rr 72.9%

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

      1. associate-/r/ 72.9%

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

      2. *-commutative 72.9%

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

      3. associate-*r/ 72.9%

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

   8. Simplified 72.9%

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

      1. clear-num 72.9%

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

      2. sqrt-div 73.9%

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

      3. metadata-eval 73.9%

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

   10. Applied egg-rr 73.9%

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

   if -5.00000000000023e-311 < h

   1. Initial program 68.6%

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

   3. Simplified 68.6%

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

      1. sqrt-div 81.7%

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

      2. div-inv 81.6%

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

   6. Applied egg-rr 81.6%

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

      1. associate-*r/ 81.7%

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

      2. *-rgt-identity 81.7%

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

   8. Simplified 81.7%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.08 \cdot 10^{+248}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(h \cdot \left(-0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{1}{\ell} \cdot \left(h \cdot {\left(\frac{D}{\frac{d \cdot 2}{M}}\right)}^{2}\right)\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := 1 + \frac{h}{\ell} \cdot \left({\left(D\_m \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -1.15 \cdot 10^{-292}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t\_0\right) \cdot t\_1\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{-177}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot t\_1\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D\_m}{d \cdot \frac{2}{M}}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(t\_0 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (/ h l) (* (pow (* D_m (/ (/ M 2.0) d)) 2.0) -0.5))))
        (t_1 (sqrt (/ d l))))
   (if (<= l -1.15e-292)
     (* (* (/ (sqrt (- d)) (sqrt (- h))) t_0) t_1)
     (if (<= l 7.8e-177)
       (*
        (* (sqrt (/ d h)) t_1)
        (- 1.0 (* 0.5 (/ (* h (pow (/ D_m (* d (/ 2.0 M))) 2.0)) l))))
       (* t_1 (* t_0 (/ (sqrt d) (sqrt h))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 1.0 + ((h / l) * (pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5));
	double t_1 = sqrt((d / l));
	double tmp;
	if (l <= -1.15e-292) {
		tmp = ((sqrt(-d) / sqrt(-h)) * t_0) * t_1;
	} else if (l <= 7.8e-177) {
		tmp = (sqrt((d / h)) * t_1) * (1.0 - (0.5 * ((h * pow((D_m / (d * (2.0 / M))), 2.0)) / l)));
	} else {
		tmp = t_1 * (t_0 * (sqrt(d) / sqrt(h)));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + ((h / l) * (((d_m * ((m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))
    t_1 = sqrt((d / l))
    if (l <= (-1.15d-292)) then
        tmp = ((sqrt(-d) / sqrt(-h)) * t_0) * t_1
    else if (l <= 7.8d-177) then
        tmp = (sqrt((d / h)) * t_1) * (1.0d0 - (0.5d0 * ((h * ((d_m / (d * (2.0d0 / m))) ** 2.0d0)) / l)))
    else
        tmp = t_1 * (t_0 * (sqrt(d) / sqrt(h)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 1.0 + ((h / l) * (Math.pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5));
	double t_1 = Math.sqrt((d / l));
	double tmp;
	if (l <= -1.15e-292) {
		tmp = ((Math.sqrt(-d) / Math.sqrt(-h)) * t_0) * t_1;
	} else if (l <= 7.8e-177) {
		tmp = (Math.sqrt((d / h)) * t_1) * (1.0 - (0.5 * ((h * Math.pow((D_m / (d * (2.0 / M))), 2.0)) / l)));
	} else {
		tmp = t_1 * (t_0 * (Math.sqrt(d) / Math.sqrt(h)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = 1.0 + ((h / l) * (math.pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))
	t_1 = math.sqrt((d / l))
	tmp = 0
	if l <= -1.15e-292:
		tmp = ((math.sqrt(-d) / math.sqrt(-h)) * t_0) * t_1
	elif l <= 7.8e-177:
		tmp = (math.sqrt((d / h)) * t_1) * (1.0 - (0.5 * ((h * math.pow((D_m / (d * (2.0 / M))), 2.0)) / l)))
	else:
		tmp = t_1 * (t_0 * (math.sqrt(d) / math.sqrt(h)))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D_m * Float64(Float64(M / 2.0) / d)) ^ 2.0) * -0.5)))
	t_1 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -1.15e-292)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_0) * t_1);
	elseif (l <= 7.8e-177)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * t_1) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(D_m / Float64(d * Float64(2.0 / M))) ^ 2.0)) / l))));
	else
		tmp = Float64(t_1 * Float64(t_0 * Float64(sqrt(d) / sqrt(h))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = 1.0 + ((h / l) * (((D_m * ((M / 2.0) / d)) ^ 2.0) * -0.5));
	t_1 = sqrt((d / l));
	tmp = 0.0;
	if (l <= -1.15e-292)
		tmp = ((sqrt(-d) / sqrt(-h)) * t_0) * t_1;
	elseif (l <= 7.8e-177)
		tmp = (sqrt((d / h)) * t_1) * (1.0 - (0.5 * ((h * ((D_m / (d * (2.0 / M))) ^ 2.0)) / l)));
	else
		tmp = t_1 * (t_0 * (sqrt(d) / sqrt(h)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D$95$m * N[(N[(M / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.15e-292], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[l, 7.8e-177], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(D$95$m / N[(d * N[(2.0 / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.1499999999999999e-292

   1. Initial program 65.0%

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

   3. Simplified 63.3%

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

      1. frac-2neg 68.1%

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

      2. sqrt-div 81.8%

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

   6. Applied egg-rr 74.6%

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

   if -1.1499999999999999e-292 < l < 7.80000000000000028e-177

   1. Initial program 65.8%

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

   3. Simplified 69.1%

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

      1. associate-*r/ 86.7%

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

      2. frac-times 83.4%

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

      3. *-commutative 83.4%

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

      4. *-un-lft-identity 83.4%

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

      5. times-frac 86.6%

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

      6. *-commutative 86.6%

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

      7. associate-/l/ 86.6%

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

      8. times-frac 83.4%

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

      9. *-un-lft-identity 83.4%

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

      10. associate-*r/ 86.6%

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

      11. clear-num 86.7%

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

      12. un-div-inv 86.8%

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

      13. div-inv 86.8%

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

      14. clear-num 86.8%

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

   6. Applied egg-rr 86.8%

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

   if 7.80000000000000028e-177 < l

   1. Initial program 68.6%

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

   3. Simplified 67.7%

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

      1. sqrt-div 83.0%

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

      2. div-inv 82.9%

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

   6. Applied egg-rr 82.9%

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

      1. associate-*r/ 83.0%

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

      2. *-rgt-identity 83.0%

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

   8. Simplified 83.0%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.15 \cdot 10^{-292}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{-177}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d \cdot \frac{2}{M}}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t\_0\right) \cdot \left(1 - 0.5 \cdot \left(\frac{1}{\ell} \cdot \left(h \cdot {\left(\frac{D\_m}{\frac{d \cdot 2}{M}}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D\_m \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))))
   (if (<= l -5e-311)
     (*
      (* (/ (sqrt (- d)) (sqrt (- h))) t_0)
      (- 1.0 (* 0.5 (* (/ 1.0 l) (* h (pow (/ D_m (/ (* d 2.0) M)) 2.0))))))
     (*
      t_0
      (*
       (+ 1.0 (* (/ h l) (* (pow (* D_m (/ (/ M 2.0) d)) 2.0) -0.5)))
       (/ (sqrt d) (sqrt h)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt((d / l));
	double tmp;
	if (l <= -5e-311) {
		tmp = ((sqrt(-d) / sqrt(-h)) * t_0) * (1.0 - (0.5 * ((1.0 / l) * (h * pow((D_m / ((d * 2.0) / M)), 2.0)))));
	} else {
		tmp = t_0 * ((1.0 + ((h / l) * (pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))) * (sqrt(d) / sqrt(h)));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((d / l))
    if (l <= (-5d-311)) then
        tmp = ((sqrt(-d) / sqrt(-h)) * t_0) * (1.0d0 - (0.5d0 * ((1.0d0 / l) * (h * ((d_m / ((d * 2.0d0) / m)) ** 2.0d0)))))
    else
        tmp = t_0 * ((1.0d0 + ((h / l) * (((d_m * ((m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))) * (sqrt(d) / sqrt(h)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.sqrt((d / l));
	double tmp;
	if (l <= -5e-311) {
		tmp = ((Math.sqrt(-d) / Math.sqrt(-h)) * t_0) * (1.0 - (0.5 * ((1.0 / l) * (h * Math.pow((D_m / ((d * 2.0) / M)), 2.0)))));
	} else {
		tmp = t_0 * ((1.0 + ((h / l) * (Math.pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))) * (Math.sqrt(d) / Math.sqrt(h)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.sqrt((d / l))
	tmp = 0
	if l <= -5e-311:
		tmp = ((math.sqrt(-d) / math.sqrt(-h)) * t_0) * (1.0 - (0.5 * ((1.0 / l) * (h * math.pow((D_m / ((d * 2.0) / M)), 2.0)))))
	else:
		tmp = t_0 * ((1.0 + ((h / l) * (math.pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))) * (math.sqrt(d) / math.sqrt(h)))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -5e-311)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_0) * Float64(1.0 - Float64(0.5 * Float64(Float64(1.0 / l) * Float64(h * (Float64(D_m / Float64(Float64(d * 2.0) / M)) ^ 2.0))))));
	else
		tmp = Float64(t_0 * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D_m * Float64(Float64(M / 2.0) / d)) ^ 2.0) * -0.5))) * Float64(sqrt(d) / sqrt(h))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = sqrt((d / l));
	tmp = 0.0;
	if (l <= -5e-311)
		tmp = ((sqrt(-d) / sqrt(-h)) * t_0) * (1.0 - (0.5 * ((1.0 / l) * (h * ((D_m / ((d * 2.0) / M)) ^ 2.0)))));
	else
		tmp = t_0 * ((1.0 + ((h / l) * (((D_m * ((M / 2.0) / d)) ^ 2.0) * -0.5))) * (sqrt(d) / sqrt(h)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -5e-311], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(1.0 / l), $MachinePrecision] * N[(h * N[Power[N[(D$95$m / N[(N[(d * 2.0), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D$95$m * N[(N[(M / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -5.00000000000023e-311

   1. Initial program 64.5%

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

   3. Simplified 64.4%

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

      1. associate-*r/ 67.0%

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

      2. clear-num 67.0%

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

      3. frac-times 67.8%

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

      4. *-commutative 67.8%

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

      5. *-un-lft-identity 67.8%

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

      6. times-frac 67.0%

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

      7. *-commutative 67.0%

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

      8. associate-/l/ 67.0%

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

      9. times-frac 67.8%

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

      10. *-un-lft-identity 67.8%

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

      11. associate-*r/ 67.0%

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

      12. clear-num 67.0%

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

      13. un-div-inv 67.0%

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

      14. div-inv 66.9%

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

      15. clear-num 66.9%

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

   6. Applied egg-rr 66.9%

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

      1. associate-/r/ 66.9%

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

      2. *-commutative 66.9%

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

      3. associate-*r/ 67.0%

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

   8. Simplified 67.0%

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

      1. frac-2neg 67.5%

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

      2. sqrt-div 80.8%

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

   10. Applied egg-rr 77.8%

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

   if -5.00000000000023e-311 < l

   1. Initial program 68.6%

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

   3. Simplified 68.6%

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

      1. sqrt-div 81.7%

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

      2. div-inv 81.6%

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

   6. Applied egg-rr 81.6%

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

      1. associate-*r/ 81.7%

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

      2. *-rgt-identity 81.7%

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

   8. Simplified 81.7%

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

4. Final simplification 79.8%

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

Alternative 8: 67.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq 5.8 \cdot 10^{-289} \lor \neg \left(h \leq 1.35 \cdot 10^{-190}\right):\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D\_m}{d \cdot \frac{2}{M}}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (or (<= h 5.8e-289) (not (<= h 1.35e-190)))
   (*
    (* (sqrt (/ d h)) (sqrt (/ d l)))
    (- 1.0 (* 0.5 (/ (* h (pow (/ D_m (* d (/ 2.0 M))) 2.0)) l))))
   (* d (sqrt (/ (/ 1.0 l) h)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if ((h <= 5.8e-289) || !(h <= 1.35e-190)) {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - (0.5 * ((h * pow((D_m / (d * (2.0 / M))), 2.0)) / l)));
	} else {
		tmp = d * sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if ((h <= 5.8d-289) .or. (.not. (h <= 1.35d-190))) then
        tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0d0 - (0.5d0 * ((h * ((d_m / (d * (2.0d0 / m))) ** 2.0d0)) / l)))
    else
        tmp = d * sqrt(((1.0d0 / l) / h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if ((h <= 5.8e-289) || !(h <= 1.35e-190)) {
		tmp = (Math.sqrt((d / h)) * Math.sqrt((d / l))) * (1.0 - (0.5 * ((h * Math.pow((D_m / (d * (2.0 / M))), 2.0)) / l)));
	} else {
		tmp = d * Math.sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if (h <= 5.8e-289) or not (h <= 1.35e-190):
		tmp = (math.sqrt((d / h)) * math.sqrt((d / l))) * (1.0 - (0.5 * ((h * math.pow((D_m / (d * (2.0 / M))), 2.0)) / l)))
	else:
		tmp = d * math.sqrt(((1.0 / l) / h))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if ((h <= 5.8e-289) || !(h <= 1.35e-190))
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(D_m / Float64(d * Float64(2.0 / M))) ^ 2.0)) / l))));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) / h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if ((h <= 5.8e-289) || ~((h <= 1.35e-190)))
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - (0.5 * ((h * ((D_m / (d * (2.0 / M))) ^ 2.0)) / l)));
	else
		tmp = d * sqrt(((1.0 / l) / h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[Or[LessEqual[h, 5.8e-289], N[Not[LessEqual[h, 1.35e-190]], $MachinePrecision]], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(D$95$m / N[(d * N[(2.0 / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $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 h < 5.80000000000000012e-289 or 1.35e-190 < h

   1. Initial program 68.6%

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

   3. Simplified 68.1%

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

      1. associate-*r/ 70.2%

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

      2. frac-times 71.0%

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

      3. *-commutative 71.0%

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

      4. *-un-lft-identity 71.0%

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

      5. times-frac 71.0%

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

      6. *-commutative 71.0%

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

      7. associate-/l/ 71.0%

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

      8. times-frac 71.0%

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

      9. *-un-lft-identity 71.0%

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

      10. associate-*r/ 71.0%

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

      11. clear-num 71.0%

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

      12. un-div-inv 71.0%

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

      13. div-inv 71.0%

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

      14. clear-num 71.0%

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

   6. Applied egg-rr 71.0%

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

   if 5.80000000000000012e-289 < h < 1.35e-190

   1. Initial program 49.2%

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

   3. Simplified 45.5%

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

      1. associate-*r/ 45.5%

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

      2. clear-num 45.5%

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

      3. frac-times 49.2%

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

      4. *-commutative 49.2%

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

      5. *-un-lft-identity 49.2%

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

      6. times-frac 49.2%

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

      7. *-commutative 49.2%

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

      8. associate-/l/ 49.2%

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

      9. times-frac 49.2%

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

      10. *-un-lft-identity 49.2%

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

      11. associate-*r/ 49.2%

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

      12. clear-num 49.2%

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

      13. un-div-inv 49.2%

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

      14. div-inv 49.2%

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

      15. clear-num 49.2%

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

   6. Applied egg-rr 49.2%

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

      1. associate-/r/ 49.2%

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

      2. *-commutative 49.2%

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

      3. associate-*r/ 49.2%

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

   8. Simplified 49.2%

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

   9. Taylor expanded in d around inf 81.1%

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

       1. *-commutative 81.1%

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

       2. associate-/r* 81.1%

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

   11. Simplified 81.1%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 5.8 \cdot 10^{-289} \lor \neg \left(h \leq 1.35 \cdot 10^{-190}\right):\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d \cdot \frac{2}{M}}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -5.4 \cdot 10^{+244}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;d \leq 7 \cdot 10^{+183}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D\_m \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= d -5.4e+244)
   (* d (- (sqrt (/ 1.0 (* l h)))))
   (if (<= d 7e+183)
     (*
      (sqrt (/ d l))
      (*
       (+ 1.0 (* (/ h l) (* (pow (* D_m (/ (/ M 2.0) d)) 2.0) -0.5)))
       (sqrt (/ d h))))
     (* d (sqrt (/ (/ 1.0 l) h))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= -5.4e+244) {
		tmp = d * -sqrt((1.0 / (l * h)));
	} else if (d <= 7e+183) {
		tmp = sqrt((d / l)) * ((1.0 + ((h / l) * (pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))) * sqrt((d / h)));
	} else {
		tmp = d * sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-5.4d+244)) then
        tmp = d * -sqrt((1.0d0 / (l * h)))
    else if (d <= 7d+183) then
        tmp = sqrt((d / l)) * ((1.0d0 + ((h / l) * (((d_m * ((m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))) * sqrt((d / h)))
    else
        tmp = d * sqrt(((1.0d0 / l) / h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= -5.4e+244) {
		tmp = d * -Math.sqrt((1.0 / (l * h)));
	} else if (d <= 7e+183) {
		tmp = Math.sqrt((d / l)) * ((1.0 + ((h / l) * (Math.pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))) * Math.sqrt((d / h)));
	} else {
		tmp = d * Math.sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if d <= -5.4e+244:
		tmp = d * -math.sqrt((1.0 / (l * h)))
	elif d <= 7e+183:
		tmp = math.sqrt((d / l)) * ((1.0 + ((h / l) * (math.pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))) * math.sqrt((d / h)))
	else:
		tmp = d * math.sqrt(((1.0 / l) / h))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (d <= -5.4e+244)
		tmp = Float64(d * Float64(-sqrt(Float64(1.0 / Float64(l * h)))));
	elseif (d <= 7e+183)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D_m * Float64(Float64(M / 2.0) / d)) ^ 2.0) * -0.5))) * sqrt(Float64(d / h))));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) / h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (d <= -5.4e+244)
		tmp = d * -sqrt((1.0 / (l * h)));
	elseif (d <= 7e+183)
		tmp = sqrt((d / l)) * ((1.0 + ((h / l) * (((D_m * ((M / 2.0) / d)) ^ 2.0) * -0.5))) * sqrt((d / h)));
	else
		tmp = d * sqrt(((1.0 / l) / h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[d, -5.4e+244], N[(d * (-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[d, 7e+183], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D$95$m * N[(N[(M / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.39999999999999995e244

   1. Initial program 65.9%

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

   3. Simplified 65.9%

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

      1. associate-*r/ 76.4%

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

      2. clear-num 76.4%

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

      3. frac-times 76.4%

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

      4. *-commutative 76.4%

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

      5. *-un-lft-identity 76.4%

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

      6. times-frac 76.4%

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

      7. *-commutative 76.4%

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

      8. associate-/l/ 76.4%

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

      9. times-frac 76.4%

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

      10. *-un-lft-identity 76.4%

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

      11. associate-*r/ 76.4%

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

      12. clear-num 76.4%

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

      13. un-div-inv 76.4%

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

      14. div-inv 76.4%

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

      15. clear-num 76.4%

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

   6. Applied egg-rr 76.4%

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

      1. associate-/r/ 76.4%

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

      2. *-commutative 76.4%

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

      3. associate-*r/ 76.4%

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

   8. Simplified 76.4%

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

   9. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. 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%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 90.0%

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

       5. neg-mul-1 90.0%

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

   11. Simplified 90.0%

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

   if -5.39999999999999995e244 < d < 6.99999999999999974e183

   1. Initial program 67.4%

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

   3. Simplified 66.4%

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

   if 6.99999999999999974e183 < d

   1. Initial program 59.6%

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

   3. Simplified 59.6%

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

      1. associate-*r/ 60.2%

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

      2. clear-num 60.2%

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

      3. frac-times 60.2%

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

      4. *-commutative 60.2%

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

      5. *-un-lft-identity 60.2%

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

      6. times-frac 60.2%

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

      7. *-commutative 60.2%

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

      8. associate-/l/ 60.2%

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

      9. times-frac 60.2%

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

      10. *-un-lft-identity 60.2%

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

      11. associate-*r/ 60.2%

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

      12. clear-num 60.2%

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

      13. un-div-inv 60.2%

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

      14. div-inv 60.2%

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

      15. clear-num 60.2%

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

   6. Applied egg-rr 60.2%

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

      1. associate-/r/ 60.2%

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

      2. *-commutative 60.2%

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

      3. associate-*r/ 60.2%

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

   8. Simplified 60.2%

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

   9. Taylor expanded in d around inf 85.7%

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

       1. *-commutative 85.7%

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

       2. associate-/r* 85.9%

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

   11. Simplified 85.9%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.4 \cdot 10^{+244}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;d \leq 7 \cdot 10^{+183}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 49.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \cdot M \leq 4 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5 \cdot \left(h \cdot {\left(D\_m \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= (* D_m M) 4e-277)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (*
    (sqrt (* (/ d h) (/ d l)))
    (+ 1.0 (/ (* -0.5 (* h (pow (* D_m (/ M (* d 2.0))) 2.0))) l)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if ((D_m * M) <= 4e-277) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + ((-0.5 * (h * pow((D_m * (M / (d * 2.0))), 2.0))) / l));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if ((d_m * m) <= 4d-277) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 + (((-0.5d0) * (h * ((d_m * (m / (d * 2.0d0))) ** 2.0d0))) / l))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if ((D_m * M) <= 4e-277) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 + ((-0.5 * (h * Math.pow((D_m * (M / (d * 2.0))), 2.0))) / l));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if (D_m * M) <= 4e-277:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	else:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 + ((-0.5 * (h * math.pow((D_m * (M / (d * 2.0))), 2.0))) / l))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (Float64(D_m * M) <= 4e-277)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(Float64(-0.5 * Float64(h * (Float64(D_m * Float64(M / Float64(d * 2.0))) ^ 2.0))) / l)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if ((D_m * M) <= 4e-277)
		tmp = sqrt((d / h)) * sqrt((d / l));
	else
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + ((-0.5 * (h * ((D_m * (M / (d * 2.0))) ^ 2.0))) / l));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[N[(D$95$m * M), $MachinePrecision], 4e-277], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(-0.5 * N[(h * N[Power[N[(D$95$m * N[(M / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 3.99999999999999988e-277

   1. Initial program 62.5%

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

   3. Simplified 61.9%

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

   5. Taylor expanded in M around 0 41.7%

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

   if 3.99999999999999988e-277 < (*.f64 M D)

   1. Initial program 73.3%

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

   3. Simplified 72.3%

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

      1. associate-*r/ 72.5%

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

      2. clear-num 72.5%

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

      3. frac-times 73.5%

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

      4. *-commutative 73.5%

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

      5. *-un-lft-identity 73.5%

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

      6. times-frac 73.4%

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

      7. *-commutative 73.4%

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

      8. associate-/l/ 73.4%

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

      9. times-frac 73.5%

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

      10. *-un-lft-identity 73.5%

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

      11. associate-*r/ 73.4%

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

      12. clear-num 73.4%

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

      13. un-div-inv 73.4%

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

      14. div-inv 73.4%

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

      15. clear-num 73.4%

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

   6. Applied egg-rr 73.4%

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

      1. associate-/r/ 73.4%

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

      2. *-commutative 73.4%

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

      3. associate-*r/ 73.4%

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

   8. Simplified 73.4%

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

      1. pow1 73.4%

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

      2. sqrt-unprod 67.7%

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

      3. cancel-sign-sub-inv 67.7%

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

      4. metadata-eval 67.7%

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

      5. associate-*l/ 67.7%

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

      6. *-un-lft-identity 67.7%

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

      7. associate-/r/ 66.7%

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

   10. Applied egg-rr 66.7%

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

   12. Simplified 67.7%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot M \leq 4 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5 \cdot \left(h \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.1 \cdot 10^{-299}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= l -3.1e-299)
   (* d (- (sqrt (/ 1.0 (* l h)))))
   (* d (sqrt (/ (/ 1.0 l) h)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= -3.1e-299) {
		tmp = d * -sqrt((1.0 / (l * h)));
	} else {
		tmp = d * sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-3.1d-299)) then
        tmp = d * -sqrt((1.0d0 / (l * h)))
    else
        tmp = d * sqrt(((1.0d0 / l) / h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= -3.1e-299) {
		tmp = d * -Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = d * Math.sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if l <= -3.1e-299:
		tmp = d * -math.sqrt((1.0 / (l * h)))
	else:
		tmp = d * math.sqrt(((1.0 / l) / h))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (l <= -3.1e-299)
		tmp = Float64(d * Float64(-sqrt(Float64(1.0 / Float64(l * h)))));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) / h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (l <= -3.1e-299)
		tmp = d * -sqrt((1.0 / (l * h)));
	else
		tmp = d * sqrt(((1.0 / l) / h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[l, -3.1e-299], N[(d * (-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $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 l < -3.1e-299

   1. Initial program 64.2%

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

   3. Simplified 64.2%

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

      1. associate-*r/ 66.7%

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

      2. clear-num 66.7%

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

      3. frac-times 67.5%

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

      4. *-commutative 67.5%

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

      5. *-un-lft-identity 67.5%

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

      6. times-frac 66.7%

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

      7. *-commutative 66.7%

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

      8. associate-/l/ 66.7%

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

      9. times-frac 67.5%

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

      10. *-un-lft-identity 67.5%

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

      11. associate-*r/ 66.7%

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

      12. clear-num 66.7%

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

      13. un-div-inv 66.7%

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

      14. div-inv 66.7%

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

      15. clear-num 66.7%

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

   6. Applied egg-rr 66.7%

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

      1. associate-/r/ 66.7%

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

      2. *-commutative 66.7%

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

      3. associate-*r/ 66.7%

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

   8. Simplified 66.7%

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

   9. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. 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%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 44.2%

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

       5. neg-mul-1 44.2%

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

   11. Simplified 44.2%

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

   if -3.1e-299 < l

   1. Initial program 68.8%

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

   3. Simplified 67.4%

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

      1. associate-*r/ 68.6%

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

      2. clear-num 68.6%

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

      3. frac-times 70.0%

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

      4. *-commutative 70.0%

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

      5. *-un-lft-identity 70.0%

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

      6. times-frac 70.7%

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

      7. *-commutative 70.7%

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

      8. associate-/l/ 70.7%

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

      9. times-frac 70.0%

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

      10. *-un-lft-identity 70.0%

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

      11. associate-*r/ 70.7%

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

      12. clear-num 70.7%

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

      13. un-div-inv 70.7%

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

      14. div-inv 70.7%

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

      15. clear-num 70.7%

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

   6. Applied egg-rr 70.7%

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

      1. associate-/r/ 70.7%

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

      2. *-commutative 70.7%

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

      3. associate-*r/ 70.7%

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

   8. Simplified 70.7%

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

   9. Taylor expanded in d around inf 47.0%

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

       1. *-commutative 47.0%

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

       2. associate-/r* 47.1%

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

   11. Simplified 47.1%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.1 \cdot 10^{-299}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 26.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ d \cdot \sqrt{\frac{1}{\ell \cdot h}} \end{array} \]

FPCore

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

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	return d * sqrt((1.0 / (l * h)));
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    code = d * sqrt((1.0d0 / (l * h)))
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	return d * Math.sqrt((1.0 / (l * h)));
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	return d * math.sqrt((1.0 / (l * h)))

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	return Float64(d * sqrt(Float64(1.0 / Float64(l * h))))
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp = code(d, h, l, M, D_m)
	tmp = d * sqrt((1.0 / (l * h)));
end

Wolfram

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

Derivation

1. Initial program 66.6%

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

3. Simplified 65.8%

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

   1. associate-*r/ 67.7%

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

   2. clear-num 67.7%

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

   3. frac-times 68.8%

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

   4. *-commutative 68.8%

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

   5. *-un-lft-identity 68.8%

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

   6. times-frac 68.8%

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

   7. *-commutative 68.8%

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

   8. associate-/l/ 68.8%

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

   9. times-frac 68.8%

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

   10. *-un-lft-identity 68.8%

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

   11. associate-*r/ 68.8%

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

   12. clear-num 68.8%

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

   13. un-div-inv 68.8%

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

   14. div-inv 68.8%

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

   15. clear-num 68.8%

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

6. Applied egg-rr 68.8%

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

   1. associate-/r/ 68.8%

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

   2. *-commutative 68.8%

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

   3. associate-*r/ 68.8%

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

8. Simplified 68.8%

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

9. Taylor expanded in d around inf 27.8%

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

10. Final simplification 27.8%

    \[\leadsto d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]
11. Add Preprocessing

Alternative 13: 27.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}} \end{array} \]

FPCore

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

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	return d * sqrt(((1.0 / l) / h));
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    code = d * sqrt(((1.0d0 / l) / h))
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	return d * Math.sqrt(((1.0 / l) / h));
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	return d * math.sqrt(((1.0 / l) / h))

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	return Float64(d * sqrt(Float64(Float64(1.0 / l) / h)))
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp = code(d, h, l, M, D_m)
	tmp = d * sqrt(((1.0 / l) / h));
end

Wolfram

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

Derivation

1. Initial program 66.6%

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

3. Simplified 65.8%

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

   1. associate-*r/ 67.7%

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

   2. clear-num 67.7%

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

   3. frac-times 68.8%

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

   4. *-commutative 68.8%

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

   5. *-un-lft-identity 68.8%

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

   6. times-frac 68.8%

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

   7. *-commutative 68.8%

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

   8. associate-/l/ 68.8%

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

   9. times-frac 68.8%

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

   10. *-un-lft-identity 68.8%

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

   11. associate-*r/ 68.8%

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

   12. clear-num 68.8%

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

   13. un-div-inv 68.8%

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

   14. div-inv 68.8%

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

   15. clear-num 68.8%

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

6. Applied egg-rr 68.8%

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

   1. associate-/r/ 68.8%

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

   2. *-commutative 68.8%

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

   3. associate-*r/ 68.8%

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

8. Simplified 68.8%

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

9. Taylor expanded in d around inf 27.8%

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

    1. *-commutative 27.8%

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

    2. associate-/r* 27.8%

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

11. Simplified 27.8%

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

12. Final simplification 27.8%

    \[\leadsto d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024081 
(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)))))