Henrywood and Agarwal, Equation (12)

Percentage Accurate: 65.8% → 81.2%

Time: 19.4s

Alternatives: 20

Speedup: 1.5×

• 23.4% of points produce a very large (infinite) output. You may want to add a precondition.

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: 65.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 81.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{t\_0}{\sqrt{-\ell}} \cdot \left(\frac{t\_0}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D\_m \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{-51}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D\_m \cdot \left(0.5 \cdot \frac{M\_m}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(1 + \frac{-0.5 \cdot \left(h \cdot {\left(\frac{\frac{d}{M\_m}}{D\_m \cdot 0.5}\right)}^{-2}\right)}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (- d))))
   (if (<= d -2e-310)
     (*
      (/ t_0 (sqrt (- l)))
      (*
       (/ t_0 (sqrt (- h)))
       (+ 1.0 (* (/ h l) (* (pow (* D_m (/ (/ M_m 2.0) d)) 2.0) -0.5)))))
     (if (<= d 1.8e-51)
       (*
        (* d (sqrt (/ 1.0 (* l h))))
        (- 1.0 (* 0.5 (pow (* (* D_m (* 0.5 (/ M_m d))) (sqrt (/ h l))) 2.0))))
       (*
        (sqrt (/ d l))
        (*
         (/ (sqrt d) (sqrt h))
         (+
          1.0
          (/ (* -0.5 (* h (pow (/ (/ d M_m) (* D_m 0.5)) -2.0))) l))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt(-d);
	double tmp;
	if (d <= -2e-310) {
		tmp = (t_0 / sqrt(-l)) * ((t_0 / sqrt(-h)) * (1.0 + ((h / l) * (pow((D_m * ((M_m / 2.0) / d)), 2.0) * -0.5))));
	} else if (d <= 1.8e-51) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (1.0 - (0.5 * pow(((D_m * (0.5 * (M_m / d))) * sqrt((h / l))), 2.0)));
	} else {
		tmp = sqrt((d / l)) * ((sqrt(d) / sqrt(h)) * (1.0 + ((-0.5 * (h * pow(((d / M_m) / (D_m * 0.5)), -2.0))) / l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(-d)
    if (d <= (-2d-310)) then
        tmp = (t_0 / sqrt(-l)) * ((t_0 / sqrt(-h)) * (1.0d0 + ((h / l) * (((d_m * ((m_m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))))
    else if (d <= 1.8d-51) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * (1.0d0 - (0.5d0 * (((d_m * (0.5d0 * (m_m / d))) * sqrt((h / l))) ** 2.0d0)))
    else
        tmp = sqrt((d / l)) * ((sqrt(d) / sqrt(h)) * (1.0d0 + (((-0.5d0) * (h * (((d / m_m) / (d_m * 0.5d0)) ** (-2.0d0)))) / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt(-d);
	double tmp;
	if (d <= -2e-310) {
		tmp = (t_0 / Math.sqrt(-l)) * ((t_0 / Math.sqrt(-h)) * (1.0 + ((h / l) * (Math.pow((D_m * ((M_m / 2.0) / d)), 2.0) * -0.5))));
	} else if (d <= 1.8e-51) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (1.0 - (0.5 * Math.pow(((D_m * (0.5 * (M_m / d))) * Math.sqrt((h / l))), 2.0)));
	} else {
		tmp = Math.sqrt((d / l)) * ((Math.sqrt(d) / Math.sqrt(h)) * (1.0 + ((-0.5 * (h * Math.pow(((d / M_m) / (D_m * 0.5)), -2.0))) / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt(-d)
	tmp = 0
	if d <= -2e-310:
		tmp = (t_0 / math.sqrt(-l)) * ((t_0 / math.sqrt(-h)) * (1.0 + ((h / l) * (math.pow((D_m * ((M_m / 2.0) / d)), 2.0) * -0.5))))
	elif d <= 1.8e-51:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (1.0 - (0.5 * math.pow(((D_m * (0.5 * (M_m / d))) * math.sqrt((h / l))), 2.0)))
	else:
		tmp = math.sqrt((d / l)) * ((math.sqrt(d) / math.sqrt(h)) * (1.0 + ((-0.5 * (h * math.pow(((d / M_m) / (D_m * 0.5)), -2.0))) / l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(-d))
	tmp = 0.0
	if (d <= -2e-310)
		tmp = Float64(Float64(t_0 / sqrt(Float64(-l))) * Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D_m * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5)))));
	elseif (d <= 1.8e-51)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(1.0 - Float64(0.5 * (Float64(Float64(D_m * Float64(0.5 * Float64(M_m / d))) * sqrt(Float64(h / l))) ^ 2.0))));
	else
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(sqrt(d) / sqrt(h)) * Float64(1.0 + Float64(Float64(-0.5 * Float64(h * (Float64(Float64(d / M_m) / Float64(D_m * 0.5)) ^ -2.0))) / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt(-d);
	tmp = 0.0;
	if (d <= -2e-310)
		tmp = (t_0 / sqrt(-l)) * ((t_0 / sqrt(-h)) * (1.0 + ((h / l) * (((D_m * ((M_m / 2.0) / d)) ^ 2.0) * -0.5))));
	elseif (d <= 1.8e-51)
		tmp = (d * sqrt((1.0 / (l * h)))) * (1.0 - (0.5 * (((D_m * (0.5 * (M_m / d))) * sqrt((h / l))) ^ 2.0)));
	else
		tmp = sqrt((d / l)) * ((sqrt(d) / sqrt(h)) * (1.0 + ((-0.5 * (h * (((d / M_m) / (D_m * 0.5)) ^ -2.0))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[d, -2e-310], N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D$95$m * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.8e-51], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[Power[N[(N[(D$95$m * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(-0.5 * N[(h * N[Power[N[(N[(d / M$95$m), $MachinePrecision] / N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.999999999999994e-310

   1. Initial program 66.2%

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

   3. Simplified 65.3%

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

      1. frac-2neg 65.3%

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

      2. sqrt-div 73.3%

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

   6. Applied egg-rr 73.3%

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

      1. frac-2neg 73.3%

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

      2. sqrt-div 80.4%

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

   8. Applied egg-rr 80.4%

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

   if -1.999999999999994e-310 < d < 1.8e-51

   1. Initial program 44.5%

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

   3. Simplified 44.5%

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

   5. Taylor expanded in d around 0 61.4%

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

      1. add-sqr-sqrt 61.4%

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

      2. pow2 61.4%

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

      3. sqrt-prod 61.4%

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

      4. sqrt-pow1 64.7%

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

      5. frac-times 68.4%

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

      6. associate-/r* 68.4%

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

      7. metadata-eval 68.4%

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

      8. pow1 68.4%

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

      9. div-inv 68.4%

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

      10. *-commutative 68.4%

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

      11. associate-*r* 68.4%

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

      12. div-inv 68.4%

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

      13. associate-*r/ 62.9%

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

      14. associate-/l/ 62.9%

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

      15. *-un-lft-identity 62.9%

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

      16. *-commutative 62.9%

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

      17. times-frac 62.9%

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

      18. metadata-eval 62.9%

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

   7. Applied egg-rr 62.9%

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

   if 1.8e-51 < d

   1. Initial program 86.8%

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

   3. Simplified 86.7%

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

      1. associate-*r/ 86.8%

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

      2. div-inv 86.8%

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

      3. associate-*r* 86.8%

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

      4. *-commutative 86.8%

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

      5. div-inv 86.8%

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

      6. associate-/r* 86.8%

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

      7. frac-times 86.8%

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

      8. associate-*r/ 86.8%

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

      9. div-inv 86.8%

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

      10. metadata-eval 86.8%

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

   6. Applied egg-rr 86.8%

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

      1. metadata-eval 86.8%

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

      2. div-inv 86.8%

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

      3. *-commutative 86.8%

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

      4. associate-*r/ 86.7%

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

      5. expm1-log1p-u 52.5%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)}\right)\right) \]

      6. log1p-define 52.5%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)}\right)\right)\right) \]

      7. expm1-undefine 52.5%

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

   8. Applied egg-rr 86.8%

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

   10. Simplified 89.2%

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

       1. sqrt-div 92.5%

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

       2. div-inv 92.5%

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

   12. Applied egg-rr 92.5%

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

       1. associate-*r/ 92.5%

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

       2. *-rgt-identity 92.5%

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

   14. Simplified 92.5%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{-51}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(1 + \frac{-0.5 \cdot \left(h \cdot {\left(\frac{\frac{d}{M}}{D \cdot 0.5}\right)}^{-2}\right)}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 1 + \frac{-0.5 \cdot \left(h \cdot {\left(\frac{\frac{d}{M\_m}}{D\_m \cdot 0.5}\right)}^{-2}\right)}{\ell}\\ \mathbf{if}\;d \leq -1.7 \cdot 10^{-304}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(t\_0 \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{-51}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D\_m \cdot \left(0.5 \cdot \frac{M\_m}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (+ 1.0 (/ (* -0.5 (* h (pow (/ (/ d M_m) (* D_m 0.5)) -2.0))) l))))
   (if (<= d -1.7e-304)
     (* (/ (sqrt (- d)) (sqrt (- l))) (* t_0 (sqrt (/ d h))))
     (if (<= d 1.8e-51)
       (*
        (* d (sqrt (/ 1.0 (* l h))))
        (- 1.0 (* 0.5 (pow (* (* D_m (* 0.5 (/ M_m d))) (sqrt (/ h l))) 2.0))))
       (* (sqrt (/ d l)) (* (/ (sqrt d) (sqrt h)) t_0))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 1.0 + ((-0.5 * (h * pow(((d / M_m) / (D_m * 0.5)), -2.0))) / l);
	double tmp;
	if (d <= -1.7e-304) {
		tmp = (sqrt(-d) / sqrt(-l)) * (t_0 * sqrt((d / h)));
	} else if (d <= 1.8e-51) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (1.0 - (0.5 * pow(((D_m * (0.5 * (M_m / d))) * sqrt((h / l))), 2.0)));
	} else {
		tmp = sqrt((d / l)) * ((sqrt(d) / sqrt(h)) * t_0);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (((-0.5d0) * (h * (((d / m_m) / (d_m * 0.5d0)) ** (-2.0d0)))) / l)
    if (d <= (-1.7d-304)) then
        tmp = (sqrt(-d) / sqrt(-l)) * (t_0 * sqrt((d / h)))
    else if (d <= 1.8d-51) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * (1.0d0 - (0.5d0 * (((d_m * (0.5d0 * (m_m / d))) * sqrt((h / l))) ** 2.0d0)))
    else
        tmp = sqrt((d / l)) * ((sqrt(d) / sqrt(h)) * t_0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 1.0 + ((-0.5 * (h * Math.pow(((d / M_m) / (D_m * 0.5)), -2.0))) / l);
	double tmp;
	if (d <= -1.7e-304) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-l)) * (t_0 * Math.sqrt((d / h)));
	} else if (d <= 1.8e-51) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (1.0 - (0.5 * Math.pow(((D_m * (0.5 * (M_m / d))) * Math.sqrt((h / l))), 2.0)));
	} else {
		tmp = Math.sqrt((d / l)) * ((Math.sqrt(d) / Math.sqrt(h)) * t_0);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = 1.0 + ((-0.5 * (h * math.pow(((d / M_m) / (D_m * 0.5)), -2.0))) / l)
	tmp = 0
	if d <= -1.7e-304:
		tmp = (math.sqrt(-d) / math.sqrt(-l)) * (t_0 * math.sqrt((d / h)))
	elif d <= 1.8e-51:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (1.0 - (0.5 * math.pow(((D_m * (0.5 * (M_m / d))) * math.sqrt((h / l))), 2.0)))
	else:
		tmp = math.sqrt((d / l)) * ((math.sqrt(d) / math.sqrt(h)) * t_0)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(1.0 + Float64(Float64(-0.5 * Float64(h * (Float64(Float64(d / M_m) / Float64(D_m * 0.5)) ^ -2.0))) / l))
	tmp = 0.0
	if (d <= -1.7e-304)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * Float64(t_0 * sqrt(Float64(d / h))));
	elseif (d <= 1.8e-51)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(1.0 - Float64(0.5 * (Float64(Float64(D_m * Float64(0.5 * Float64(M_m / d))) * sqrt(Float64(h / l))) ^ 2.0))));
	else
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(sqrt(d) / sqrt(h)) * t_0));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = 1.0 + ((-0.5 * (h * (((d / M_m) / (D_m * 0.5)) ^ -2.0))) / l);
	tmp = 0.0;
	if (d <= -1.7e-304)
		tmp = (sqrt(-d) / sqrt(-l)) * (t_0 * sqrt((d / h)));
	elseif (d <= 1.8e-51)
		tmp = (d * sqrt((1.0 / (l * h)))) * (1.0 - (0.5 * (((D_m * (0.5 * (M_m / d))) * sqrt((h / l))) ^ 2.0)));
	else
		tmp = sqrt((d / l)) * ((sqrt(d) / sqrt(h)) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(1.0 + N[(N[(-0.5 * N[(h * N[Power[N[(N[(d / M$95$m), $MachinePrecision] / N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.7e-304], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.8e-51], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[Power[N[(N[(D$95$m * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.6999999999999999e-304

   1. Initial program 66.7%

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

   3. Simplified 65.9%

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

      1. associate-*r/ 66.7%

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

      2. div-inv 66.7%

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

      3. associate-*r* 66.7%

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

      4. *-commutative 66.7%

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

      5. div-inv 66.7%

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

      6. associate-/r* 66.7%

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

      7. frac-times 65.9%

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

      8. associate-*r/ 66.7%

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

      9. div-inv 66.7%

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

      10. metadata-eval 66.7%

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

   6. Applied egg-rr 66.7%

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

      1. metadata-eval 66.7%

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

      2. div-inv 66.7%

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

      3. *-commutative 66.7%

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

      4. associate-*r/ 65.9%

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

      5. expm1-log1p-u 37.3%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)}\right)\right) \]

      6. log1p-define 37.3%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)}\right)\right)\right) \]

      7. expm1-undefine 37.3%

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

   8. Applied egg-rr 65.9%

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

   10. Simplified 69.4%

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

       1. frac-2neg 73.9%

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

       2. sqrt-div 80.2%

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

   12. Applied egg-rr 77.9%

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

   if -1.6999999999999999e-304 < d < 1.8e-51

   1. Initial program 43.7%

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

   3. Simplified 43.7%

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

   5. Taylor expanded in d around 0 60.3%

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

      1. add-sqr-sqrt 60.3%

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

      2. pow2 60.3%

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

      3. sqrt-prod 60.3%

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

      4. sqrt-pow1 63.5%

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

      5. frac-times 67.1%

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

      6. associate-/r* 67.1%

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

      7. metadata-eval 67.1%

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

      8. pow1 67.1%

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

      9. div-inv 67.1%

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

      10. *-commutative 67.1%

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

      11. associate-*r* 67.1%

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

      12. div-inv 67.1%

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

      13. associate-*r/ 61.7%

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

      14. associate-/l/ 61.7%

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

      15. *-un-lft-identity 61.7%

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

      16. *-commutative 61.7%

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

      17. times-frac 61.7%

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

      18. metadata-eval 61.7%

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

   7. Applied egg-rr 61.7%

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

   if 1.8e-51 < d

   1. Initial program 86.8%

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

   3. Simplified 86.7%

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

      1. associate-*r/ 86.8%

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

      2. div-inv 86.8%

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

      3. associate-*r* 86.8%

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

      4. *-commutative 86.8%

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

      5. div-inv 86.8%

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

      6. associate-/r* 86.8%

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

      7. frac-times 86.8%

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

      8. associate-*r/ 86.8%

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

      9. div-inv 86.8%

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

      10. metadata-eval 86.8%

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

   6. Applied egg-rr 86.8%

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

      1. metadata-eval 86.8%

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

      2. div-inv 86.8%

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

      3. *-commutative 86.8%

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

      4. associate-*r/ 86.7%

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

      5. expm1-log1p-u 52.5%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)}\right)\right) \]

      6. log1p-define 52.5%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)}\right)\right)\right) \]

      7. expm1-undefine 52.5%

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

   8. Applied egg-rr 86.8%

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

   10. Simplified 89.2%

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

       1. sqrt-div 92.5%

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

       2. div-inv 92.5%

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

   12. Applied egg-rr 92.5%

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

       1. associate-*r/ 92.5%

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

       2. *-rgt-identity 92.5%

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

   14. Simplified 92.5%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.7 \cdot 10^{-304}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\left(1 + \frac{-0.5 \cdot \left(h \cdot {\left(\frac{\frac{d}{M}}{D \cdot 0.5}\right)}^{-2}\right)}{\ell}\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{-51}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(1 + \frac{-0.5 \cdot \left(h \cdot {\left(\frac{\frac{d}{M}}{D \cdot 0.5}\right)}^{-2}\right)}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -1.7 \cdot 10^{-304}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D\_m \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \cdot t\_0\\ \mathbf{elif}\;d \leq 3.05 \cdot 10^{-50}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D\_m \cdot \left(0.5 \cdot \frac{M\_m}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(1 + \frac{-0.5 \cdot \left(h \cdot {\left(\frac{\frac{d}{M\_m}}{D\_m \cdot 0.5}\right)}^{-2}\right)}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))))
   (if (<= d -1.7e-304)
     (*
      (*
       (/ (sqrt (- d)) (sqrt (- h)))
       (+ 1.0 (* (/ h l) (* (pow (* D_m (/ (/ M_m 2.0) d)) 2.0) -0.5))))
      t_0)
     (if (<= d 3.05e-50)
       (*
        (* d (sqrt (/ 1.0 (* l h))))
        (- 1.0 (* 0.5 (pow (* (* D_m (* 0.5 (/ M_m d))) (sqrt (/ h l))) 2.0))))
       (*
        t_0
        (*
         (/ (sqrt d) (sqrt h))
         (+
          1.0
          (/ (* -0.5 (* h (pow (/ (/ d M_m) (* D_m 0.5)) -2.0))) l))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / l));
	double tmp;
	if (d <= -1.7e-304) {
		tmp = ((sqrt(-d) / sqrt(-h)) * (1.0 + ((h / l) * (pow((D_m * ((M_m / 2.0) / d)), 2.0) * -0.5)))) * t_0;
	} else if (d <= 3.05e-50) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (1.0 - (0.5 * pow(((D_m * (0.5 * (M_m / d))) * sqrt((h / l))), 2.0)));
	} else {
		tmp = t_0 * ((sqrt(d) / sqrt(h)) * (1.0 + ((-0.5 * (h * pow(((d / M_m) / (D_m * 0.5)), -2.0))) / l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((d / l))
    if (d <= (-1.7d-304)) then
        tmp = ((sqrt(-d) / sqrt(-h)) * (1.0d0 + ((h / l) * (((d_m * ((m_m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0))))) * t_0
    else if (d <= 3.05d-50) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * (1.0d0 - (0.5d0 * (((d_m * (0.5d0 * (m_m / d))) * sqrt((h / l))) ** 2.0d0)))
    else
        tmp = t_0 * ((sqrt(d) / sqrt(h)) * (1.0d0 + (((-0.5d0) * (h * (((d / m_m) / (d_m * 0.5d0)) ** (-2.0d0)))) / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((d / l));
	double tmp;
	if (d <= -1.7e-304) {
		tmp = ((Math.sqrt(-d) / Math.sqrt(-h)) * (1.0 + ((h / l) * (Math.pow((D_m * ((M_m / 2.0) / d)), 2.0) * -0.5)))) * t_0;
	} else if (d <= 3.05e-50) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (1.0 - (0.5 * Math.pow(((D_m * (0.5 * (M_m / d))) * Math.sqrt((h / l))), 2.0)));
	} else {
		tmp = t_0 * ((Math.sqrt(d) / Math.sqrt(h)) * (1.0 + ((-0.5 * (h * Math.pow(((d / M_m) / (D_m * 0.5)), -2.0))) / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((d / l))
	tmp = 0
	if d <= -1.7e-304:
		tmp = ((math.sqrt(-d) / math.sqrt(-h)) * (1.0 + ((h / l) * (math.pow((D_m * ((M_m / 2.0) / d)), 2.0) * -0.5)))) * t_0
	elif d <= 3.05e-50:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (1.0 - (0.5 * math.pow(((D_m * (0.5 * (M_m / d))) * math.sqrt((h / l))), 2.0)))
	else:
		tmp = t_0 * ((math.sqrt(d) / math.sqrt(h)) * (1.0 + ((-0.5 * (h * math.pow(((d / M_m) / (D_m * 0.5)), -2.0))) / l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / l))
	tmp = 0.0
	if (d <= -1.7e-304)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D_m * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5)))) * t_0);
	elseif (d <= 3.05e-50)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(1.0 - Float64(0.5 * (Float64(Float64(D_m * Float64(0.5 * Float64(M_m / d))) * sqrt(Float64(h / l))) ^ 2.0))));
	else
		tmp = Float64(t_0 * Float64(Float64(sqrt(d) / sqrt(h)) * Float64(1.0 + Float64(Float64(-0.5 * Float64(h * (Float64(Float64(d / M_m) / Float64(D_m * 0.5)) ^ -2.0))) / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((d / l));
	tmp = 0.0;
	if (d <= -1.7e-304)
		tmp = ((sqrt(-d) / sqrt(-h)) * (1.0 + ((h / l) * (((D_m * ((M_m / 2.0) / d)) ^ 2.0) * -0.5)))) * t_0;
	elseif (d <= 3.05e-50)
		tmp = (d * sqrt((1.0 / (l * h)))) * (1.0 - (0.5 * (((D_m * (0.5 * (M_m / d))) * sqrt((h / l))) ^ 2.0)));
	else
		tmp = t_0 * ((sqrt(d) / sqrt(h)) * (1.0 + ((-0.5 * (h * (((d / M_m) / (D_m * 0.5)) ^ -2.0))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.7e-304], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D$95$m * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[d, 3.05e-50], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[Power[N[(N[(D$95$m * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(-0.5 * N[(h * N[Power[N[(N[(d / M$95$m), $MachinePrecision] / N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.6999999999999999e-304

   1. Initial program 66.7%

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

   3. Simplified 65.9%

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

      1. frac-2neg 65.9%

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

      2. sqrt-div 73.9%

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

   6. Applied egg-rr 73.9%

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

   if -1.6999999999999999e-304 < d < 3.0499999999999998e-50

   1. Initial program 43.7%

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

   3. Simplified 43.7%

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

   5. Taylor expanded in d around 0 60.3%

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

      1. add-sqr-sqrt 60.3%

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

      2. pow2 60.3%

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

      3. sqrt-prod 60.3%

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

      4. sqrt-pow1 63.5%

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

      5. frac-times 67.1%

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

      6. associate-/r* 67.1%

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

      7. metadata-eval 67.1%

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

      8. pow1 67.1%

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

      9. div-inv 67.1%

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

      10. *-commutative 67.1%

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

      11. associate-*r* 67.1%

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

      12. div-inv 67.1%

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

      13. associate-*r/ 61.7%

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

      14. associate-/l/ 61.7%

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

      15. *-un-lft-identity 61.7%

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

      16. *-commutative 61.7%

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

      17. times-frac 61.7%

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

      18. metadata-eval 61.7%

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

   7. Applied egg-rr 61.7%

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

   if 3.0499999999999998e-50 < d

   1. Initial program 86.8%

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

   3. Simplified 86.7%

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

      1. associate-*r/ 86.8%

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

      2. div-inv 86.8%

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

      3. associate-*r* 86.8%

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

      4. *-commutative 86.8%

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

      5. div-inv 86.8%

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

      6. associate-/r* 86.8%

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

      7. frac-times 86.8%

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

      8. associate-*r/ 86.8%

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

      9. div-inv 86.8%

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

      10. metadata-eval 86.8%

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

   6. Applied egg-rr 86.8%

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

      1. metadata-eval 86.8%

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

      2. div-inv 86.8%

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

      3. *-commutative 86.8%

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

      4. associate-*r/ 86.7%

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

      5. expm1-log1p-u 52.5%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)}\right)\right) \]

      6. log1p-define 52.5%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)}\right)\right)\right) \]

      7. expm1-undefine 52.5%

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

   8. Applied egg-rr 86.8%

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

   10. Simplified 89.2%

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

       1. sqrt-div 92.5%

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

       2. div-inv 92.5%

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

   12. Applied egg-rr 92.5%

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

       1. associate-*r/ 92.5%

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

       2. *-rgt-identity 92.5%

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

   14. Simplified 92.5%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.7 \cdot 10^{-304}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 3.05 \cdot 10^{-50}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(1 + \frac{-0.5 \cdot \left(h \cdot {\left(\frac{\frac{d}{M}}{D \cdot 0.5}\right)}^{-2}\right)}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 1 - 0.5 \cdot {\left(\left(D\_m \cdot \left(0.5 \cdot \frac{M\_m}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -1.7 \cdot 10^{-304}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{-47}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(1 + \frac{-0.5 \cdot \left(h \cdot {\left(\frac{\frac{d}{M\_m}}{D\_m \cdot 0.5}\right)}^{-2}\right)}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (-
          1.0
          (* 0.5 (pow (* (* D_m (* 0.5 (/ M_m d))) (sqrt (/ h l))) 2.0))))
        (t_1 (sqrt (/ d l))))
   (if (<= d -1.7e-304)
     (* t_0 (* t_1 (sqrt (/ d h))))
     (if (<= d 4.5e-47)
       (* (* d (sqrt (/ 1.0 (* l h)))) t_0)
       (*
        t_1
        (*
         (/ (sqrt d) (sqrt h))
         (+
          1.0
          (/ (* -0.5 (* h (pow (/ (/ d M_m) (* D_m 0.5)) -2.0))) l))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 1.0 - (0.5 * pow(((D_m * (0.5 * (M_m / d))) * sqrt((h / l))), 2.0));
	double t_1 = sqrt((d / l));
	double tmp;
	if (d <= -1.7e-304) {
		tmp = t_0 * (t_1 * sqrt((d / h)));
	} else if (d <= 4.5e-47) {
		tmp = (d * sqrt((1.0 / (l * h)))) * t_0;
	} else {
		tmp = t_1 * ((sqrt(d) / sqrt(h)) * (1.0 + ((-0.5 * (h * pow(((d / M_m) / (D_m * 0.5)), -2.0))) / l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (0.5d0 * (((d_m * (0.5d0 * (m_m / d))) * sqrt((h / l))) ** 2.0d0))
    t_1 = sqrt((d / l))
    if (d <= (-1.7d-304)) then
        tmp = t_0 * (t_1 * sqrt((d / h)))
    else if (d <= 4.5d-47) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * t_0
    else
        tmp = t_1 * ((sqrt(d) / sqrt(h)) * (1.0d0 + (((-0.5d0) * (h * (((d / m_m) / (d_m * 0.5d0)) ** (-2.0d0)))) / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 1.0 - (0.5 * Math.pow(((D_m * (0.5 * (M_m / d))) * Math.sqrt((h / l))), 2.0));
	double t_1 = Math.sqrt((d / l));
	double tmp;
	if (d <= -1.7e-304) {
		tmp = t_0 * (t_1 * Math.sqrt((d / h)));
	} else if (d <= 4.5e-47) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * t_0;
	} else {
		tmp = t_1 * ((Math.sqrt(d) / Math.sqrt(h)) * (1.0 + ((-0.5 * (h * Math.pow(((d / M_m) / (D_m * 0.5)), -2.0))) / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = 1.0 - (0.5 * math.pow(((D_m * (0.5 * (M_m / d))) * math.sqrt((h / l))), 2.0))
	t_1 = math.sqrt((d / l))
	tmp = 0
	if d <= -1.7e-304:
		tmp = t_0 * (t_1 * math.sqrt((d / h)))
	elif d <= 4.5e-47:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * t_0
	else:
		tmp = t_1 * ((math.sqrt(d) / math.sqrt(h)) * (1.0 + ((-0.5 * (h * math.pow(((d / M_m) / (D_m * 0.5)), -2.0))) / l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(1.0 - Float64(0.5 * (Float64(Float64(D_m * Float64(0.5 * Float64(M_m / d))) * sqrt(Float64(h / l))) ^ 2.0)))
	t_1 = sqrt(Float64(d / l))
	tmp = 0.0
	if (d <= -1.7e-304)
		tmp = Float64(t_0 * Float64(t_1 * sqrt(Float64(d / h))));
	elseif (d <= 4.5e-47)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * t_0);
	else
		tmp = Float64(t_1 * Float64(Float64(sqrt(d) / sqrt(h)) * Float64(1.0 + Float64(Float64(-0.5 * Float64(h * (Float64(Float64(d / M_m) / Float64(D_m * 0.5)) ^ -2.0))) / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = 1.0 - (0.5 * (((D_m * (0.5 * (M_m / d))) * sqrt((h / l))) ^ 2.0));
	t_1 = sqrt((d / l));
	tmp = 0.0;
	if (d <= -1.7e-304)
		tmp = t_0 * (t_1 * sqrt((d / h)));
	elseif (d <= 4.5e-47)
		tmp = (d * sqrt((1.0 / (l * h)))) * t_0;
	else
		tmp = t_1 * ((sqrt(d) / sqrt(h)) * (1.0 + ((-0.5 * (h * (((d / M_m) / (D_m * 0.5)) ^ -2.0))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(1.0 - N[(0.5 * N[Power[N[(N[(D$95$m * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.7e-304], N[(t$95$0 * N[(t$95$1 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.5e-47], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$1 * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(-0.5 * N[(h * N[Power[N[(N[(d / M$95$m), $MachinePrecision] / N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.6999999999999999e-304

   1. Initial program 66.7%

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

   3. Simplified 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. add-sqr-sqrt 0.9%

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

      2. pow2 0.9%

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

      3. sqrt-prod 0.9%

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

      4. sqrt-pow1 0.9%

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

      5. frac-times 1.1%

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

      6. associate-/r* 1.1%

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

      7. metadata-eval 1.1%

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

      8. pow1 1.1%

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

      9. div-inv 1.1%

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

      10. *-commutative 1.1%

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

      11. associate-*r* 1.1%

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

      12. div-inv 1.1%

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

      13. associate-*r/ 1.1%

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

      14. associate-/l/ 1.1%

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

      15. *-un-lft-identity 1.1%

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

      16. *-commutative 1.1%

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

      17. times-frac 1.1%

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

      18. metadata-eval 1.1%

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

   6. Applied egg-rr 70.7%

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

   if -1.6999999999999999e-304 < d < 4.5e-47

   1. Initial program 43.7%

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

   3. Simplified 43.7%

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

   5. Taylor expanded in d around 0 60.3%

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

      1. add-sqr-sqrt 60.3%

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

      2. pow2 60.3%

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

      3. sqrt-prod 60.3%

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

      4. sqrt-pow1 63.5%

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

      5. frac-times 67.1%

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

      6. associate-/r* 67.1%

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

      7. metadata-eval 67.1%

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

      8. pow1 67.1%

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

      9. div-inv 67.1%

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

      10. *-commutative 67.1%

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

      11. associate-*r* 67.1%

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

      12. div-inv 67.1%

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

      13. associate-*r/ 61.7%

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

      14. associate-/l/ 61.7%

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

      15. *-un-lft-identity 61.7%

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

      16. *-commutative 61.7%

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

      17. times-frac 61.7%

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

      18. metadata-eval 61.7%

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

   7. Applied egg-rr 61.7%

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

   if 4.5e-47 < d

   1. Initial program 86.8%

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

   3. Simplified 86.7%

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

      1. associate-*r/ 86.8%

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

      2. div-inv 86.8%

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

      3. associate-*r* 86.8%

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

      4. *-commutative 86.8%

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

      5. div-inv 86.8%

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

      6. associate-/r* 86.8%

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

      7. frac-times 86.8%

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

      8. associate-*r/ 86.8%

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

      9. div-inv 86.8%

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

      10. metadata-eval 86.8%

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

   6. Applied egg-rr 86.8%

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

      1. metadata-eval 86.8%

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

      2. div-inv 86.8%

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

      3. *-commutative 86.8%

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

      4. associate-*r/ 86.7%

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

      5. expm1-log1p-u 52.5%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)}\right)\right) \]

      6. log1p-define 52.5%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)}\right)\right)\right) \]

      7. expm1-undefine 52.5%

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

   8. Applied egg-rr 86.8%

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

   10. Simplified 89.2%

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

       1. sqrt-div 92.5%

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

       2. div-inv 92.5%

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

   12. Applied egg-rr 92.5%

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

       1. associate-*r/ 92.5%

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

       2. *-rgt-identity 92.5%

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

   14. Simplified 92.5%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.7 \cdot 10^{-304}:\\ \;\;\;\;\left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{-47}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(1 + \frac{-0.5 \cdot \left(h \cdot {\left(\frac{\frac{d}{M}}{D \cdot 0.5}\right)}^{-2}\right)}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \left(0.5 \cdot \frac{M\_m}{d}\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -1.7 \cdot 10^{-304}:\\ \;\;\;\;\left(t\_1 \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {t\_0}^{2}}{\ell}\right)\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{-50}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.5 \cdot {\left(t\_0 \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(1 + \frac{-0.5 \cdot \left(h \cdot {\left(\frac{\frac{d}{M\_m}}{D\_m \cdot 0.5}\right)}^{-2}\right)}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* D_m (* 0.5 (/ M_m d)))) (t_1 (sqrt (/ d l))))
   (if (<= d -1.7e-304)
     (* (* t_1 (sqrt (/ d h))) (- 1.0 (* 0.5 (/ (* h (pow t_0 2.0)) l))))
     (if (<= d 1.2e-50)
       (*
        (* d (sqrt (/ 1.0 (* l h))))
        (- 1.0 (* 0.5 (pow (* t_0 (sqrt (/ h l))) 2.0))))
       (*
        t_1
        (*
         (/ (sqrt d) (sqrt h))
         (+
          1.0
          (/ (* -0.5 (* h (pow (/ (/ d M_m) (* D_m 0.5)) -2.0))) l))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = D_m * (0.5 * (M_m / d));
	double t_1 = sqrt((d / l));
	double tmp;
	if (d <= -1.7e-304) {
		tmp = (t_1 * sqrt((d / h))) * (1.0 - (0.5 * ((h * pow(t_0, 2.0)) / l)));
	} else if (d <= 1.2e-50) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (1.0 - (0.5 * pow((t_0 * sqrt((h / l))), 2.0)));
	} else {
		tmp = t_1 * ((sqrt(d) / sqrt(h)) * (1.0 + ((-0.5 * (h * pow(((d / M_m) / (D_m * 0.5)), -2.0))) / l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = d_m * (0.5d0 * (m_m / d))
    t_1 = sqrt((d / l))
    if (d <= (-1.7d-304)) then
        tmp = (t_1 * sqrt((d / h))) * (1.0d0 - (0.5d0 * ((h * (t_0 ** 2.0d0)) / l)))
    else if (d <= 1.2d-50) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * (1.0d0 - (0.5d0 * ((t_0 * sqrt((h / l))) ** 2.0d0)))
    else
        tmp = t_1 * ((sqrt(d) / sqrt(h)) * (1.0d0 + (((-0.5d0) * (h * (((d / m_m) / (d_m * 0.5d0)) ** (-2.0d0)))) / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = D_m * (0.5 * (M_m / d));
	double t_1 = Math.sqrt((d / l));
	double tmp;
	if (d <= -1.7e-304) {
		tmp = (t_1 * Math.sqrt((d / h))) * (1.0 - (0.5 * ((h * Math.pow(t_0, 2.0)) / l)));
	} else if (d <= 1.2e-50) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (1.0 - (0.5 * Math.pow((t_0 * Math.sqrt((h / l))), 2.0)));
	} else {
		tmp = t_1 * ((Math.sqrt(d) / Math.sqrt(h)) * (1.0 + ((-0.5 * (h * Math.pow(((d / M_m) / (D_m * 0.5)), -2.0))) / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = D_m * (0.5 * (M_m / d))
	t_1 = math.sqrt((d / l))
	tmp = 0
	if d <= -1.7e-304:
		tmp = (t_1 * math.sqrt((d / h))) * (1.0 - (0.5 * ((h * math.pow(t_0, 2.0)) / l)))
	elif d <= 1.2e-50:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (1.0 - (0.5 * math.pow((t_0 * math.sqrt((h / l))), 2.0)))
	else:
		tmp = t_1 * ((math.sqrt(d) / math.sqrt(h)) * (1.0 + ((-0.5 * (h * math.pow(((d / M_m) / (D_m * 0.5)), -2.0))) / l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(D_m * Float64(0.5 * Float64(M_m / d)))
	t_1 = sqrt(Float64(d / l))
	tmp = 0.0
	if (d <= -1.7e-304)
		tmp = Float64(Float64(t_1 * sqrt(Float64(d / h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (t_0 ^ 2.0)) / l))));
	elseif (d <= 1.2e-50)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(1.0 - Float64(0.5 * (Float64(t_0 * sqrt(Float64(h / l))) ^ 2.0))));
	else
		tmp = Float64(t_1 * Float64(Float64(sqrt(d) / sqrt(h)) * Float64(1.0 + Float64(Float64(-0.5 * Float64(h * (Float64(Float64(d / M_m) / Float64(D_m * 0.5)) ^ -2.0))) / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = D_m * (0.5 * (M_m / d));
	t_1 = sqrt((d / l));
	tmp = 0.0;
	if (d <= -1.7e-304)
		tmp = (t_1 * sqrt((d / h))) * (1.0 - (0.5 * ((h * (t_0 ^ 2.0)) / l)));
	elseif (d <= 1.2e-50)
		tmp = (d * sqrt((1.0 / (l * h)))) * (1.0 - (0.5 * ((t_0 * sqrt((h / l))) ^ 2.0)));
	else
		tmp = t_1 * ((sqrt(d) / sqrt(h)) * (1.0 + ((-0.5 * (h * (((d / M_m) / (D_m * 0.5)) ^ -2.0))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(D$95$m * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.7e-304], N[(N[(t$95$1 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.2e-50], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[Power[N[(t$95$0 * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(-0.5 * N[(h * N[Power[N[(N[(d / M$95$m), $MachinePrecision] / N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.6999999999999999e-304

   1. Initial program 66.7%

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

   3. Simplified 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/ 69.3%

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

   6. Applied egg-rr 69.4%

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

   if -1.6999999999999999e-304 < d < 1.20000000000000001e-50

   1. Initial program 43.7%

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

   3. Simplified 43.7%

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

   5. Taylor expanded in d around 0 60.3%

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

      1. add-sqr-sqrt 60.3%

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

      2. pow2 60.3%

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

      3. sqrt-prod 60.3%

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

      4. sqrt-pow1 63.5%

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

      5. frac-times 67.1%

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

      6. associate-/r* 67.1%

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

      7. metadata-eval 67.1%

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

      8. pow1 67.1%

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

      9. div-inv 67.1%

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

      10. *-commutative 67.1%

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

      11. associate-*r* 67.1%

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

      12. div-inv 67.1%

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

      13. associate-*r/ 61.7%

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

      14. associate-/l/ 61.7%

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

      15. *-un-lft-identity 61.7%

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

      16. *-commutative 61.7%

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

      17. times-frac 61.7%

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

      18. metadata-eval 61.7%

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

   7. Applied egg-rr 61.7%

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

   if 1.20000000000000001e-50 < d

   1. Initial program 86.8%

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

   3. Simplified 86.7%

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

      1. associate-*r/ 86.8%

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

      2. div-inv 86.8%

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

      3. associate-*r* 86.8%

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

      4. *-commutative 86.8%

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

      5. div-inv 86.8%

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

      6. associate-/r* 86.8%

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

      7. frac-times 86.8%

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

      8. associate-*r/ 86.8%

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

      9. div-inv 86.8%

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

      10. metadata-eval 86.8%

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

   6. Applied egg-rr 86.8%

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

      1. metadata-eval 86.8%

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

      2. div-inv 86.8%

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

      3. *-commutative 86.8%

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

      4. associate-*r/ 86.7%

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

      5. expm1-log1p-u 52.5%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)}\right)\right) \]

      6. log1p-define 52.5%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)}\right)\right)\right) \]

      7. expm1-undefine 52.5%

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

   8. Applied egg-rr 86.8%

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

   10. Simplified 89.2%

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

       1. sqrt-div 92.5%

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

       2. div-inv 92.5%

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

   12. Applied egg-rr 92.5%

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

       1. associate-*r/ 92.5%

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

       2. *-rgt-identity 92.5%

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

   14. Simplified 92.5%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.7 \cdot 10^{-304}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{-50}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(1 + \frac{-0.5 \cdot \left(h \cdot {\left(\frac{\frac{d}{M}}{D \cdot 0.5}\right)}^{-2}\right)}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ t_1 := 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\\ \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+213}:\\ \;\;\;\;\left(1 - 0.5 \cdot {\left(\left(D\_m \cdot \left(0.5 \cdot \frac{M\_m}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 + -1\right)\\ \mathbf{elif}\;\ell \leq 10^{-203}:\\ \;\;\;\;t\_0 \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - t\_1\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* d (pow (* l h) -0.5)))
        (t_1 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0)))))
   (if (<= l -4.5e+213)
     (*
      (- 1.0 (* 0.5 (pow (* (* D_m (* 0.5 (/ M_m d))) (sqrt (/ h l))) 2.0)))
      (sqrt (* (/ d l) (/ d h))))
     (if (<= l -5e-310)
       (* t_0 (+ t_1 -1.0))
       (if (<= l 1e-203)
         (*
          t_0
          (- 1.0 (* 0.5 (/ (* h (pow (* D_m (* M_m (/ 0.5 d))) 2.0)) l))))
         (* (/ d (* (sqrt h) (sqrt l))) (- 1.0 t_1)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = d * pow((l * h), -0.5);
	double t_1 = 0.5 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0));
	double tmp;
	if (l <= -4.5e+213) {
		tmp = (1.0 - (0.5 * pow(((D_m * (0.5 * (M_m / d))) * sqrt((h / l))), 2.0))) * sqrt(((d / l) * (d / h)));
	} else if (l <= -5e-310) {
		tmp = t_0 * (t_1 + -1.0);
	} else if (l <= 1e-203) {
		tmp = t_0 * (1.0 - (0.5 * ((h * pow((D_m * (M_m * (0.5 / d))), 2.0)) / l)));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - t_1);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = d * ((l * h) ** (-0.5d0))
    t_1 = 0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_m / d)) ** 2.0d0))
    if (l <= (-4.5d+213)) then
        tmp = (1.0d0 - (0.5d0 * (((d_m * (0.5d0 * (m_m / d))) * sqrt((h / l))) ** 2.0d0))) * sqrt(((d / l) * (d / h)))
    else if (l <= (-5d-310)) then
        tmp = t_0 * (t_1 + (-1.0d0))
    else if (l <= 1d-203) then
        tmp = t_0 * (1.0d0 - (0.5d0 * ((h * ((d_m * (m_m * (0.5d0 / d))) ** 2.0d0)) / l)))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 - t_1)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = d * Math.pow((l * h), -0.5);
	double t_1 = 0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D_m / d)), 2.0));
	double tmp;
	if (l <= -4.5e+213) {
		tmp = (1.0 - (0.5 * Math.pow(((D_m * (0.5 * (M_m / d))) * Math.sqrt((h / l))), 2.0))) * Math.sqrt(((d / l) * (d / h)));
	} else if (l <= -5e-310) {
		tmp = t_0 * (t_1 + -1.0);
	} else if (l <= 1e-203) {
		tmp = t_0 * (1.0 - (0.5 * ((h * Math.pow((D_m * (M_m * (0.5 / d))), 2.0)) / l)));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 - t_1);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = d * math.pow((l * h), -0.5)
	t_1 = 0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D_m / d)), 2.0))
	tmp = 0
	if l <= -4.5e+213:
		tmp = (1.0 - (0.5 * math.pow(((D_m * (0.5 * (M_m / d))) * math.sqrt((h / l))), 2.0))) * math.sqrt(((d / l) * (d / h)))
	elif l <= -5e-310:
		tmp = t_0 * (t_1 + -1.0)
	elif l <= 1e-203:
		tmp = t_0 * (1.0 - (0.5 * ((h * math.pow((D_m * (M_m * (0.5 / d))), 2.0)) / l)))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 - t_1)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(d * (Float64(l * h) ^ -0.5))
	t_1 = Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0)))
	tmp = 0.0
	if (l <= -4.5e+213)
		tmp = Float64(Float64(1.0 - Float64(0.5 * (Float64(Float64(D_m * Float64(0.5 * Float64(M_m / d))) * sqrt(Float64(h / l))) ^ 2.0))) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	elseif (l <= -5e-310)
		tmp = Float64(t_0 * Float64(t_1 + -1.0));
	elseif (l <= 1e-203)
		tmp = Float64(t_0 * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(D_m * Float64(M_m * Float64(0.5 / d))) ^ 2.0)) / l))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - t_1));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = d * ((l * h) ^ -0.5);
	t_1 = 0.5 * ((h / l) * (((M_m / 2.0) * (D_m / d)) ^ 2.0));
	tmp = 0.0;
	if (l <= -4.5e+213)
		tmp = (1.0 - (0.5 * (((D_m * (0.5 * (M_m / d))) * sqrt((h / l))) ^ 2.0))) * sqrt(((d / l) * (d / h)));
	elseif (l <= -5e-310)
		tmp = t_0 * (t_1 + -1.0);
	elseif (l <= 1e-203)
		tmp = t_0 * (1.0 - (0.5 * ((h * ((D_m * (M_m * (0.5 / d))) ^ 2.0)) / l)));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - t_1);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4.5e+213], N[(N[(1.0 - N[(0.5 * N[Power[N[(N[(D$95$m * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(t$95$0 * N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1e-203], N[(t$95$0 * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.5000000000000002e213

   1. Initial program 54.2%

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

   3. Simplified 53.9%

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

      1. add-sqr-sqrt 1.8%

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

      2. pow2 1.8%

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

      3. sqrt-prod 1.8%

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

      4. sqrt-pow1 1.9%

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

      5. frac-times 2.6%

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

      6. associate-/r* 2.6%

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

      7. metadata-eval 2.6%

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

      8. pow1 2.6%

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

      9. div-inv 2.6%

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

      10. *-commutative 2.6%

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

      11. associate-*r* 2.6%

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

      12. div-inv 2.6%

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

      13. associate-*r/ 2.6%

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

      14. associate-/l/ 2.6%

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

      15. *-un-lft-identity 2.6%

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

      16. *-commutative 2.6%

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

      17. times-frac 2.6%

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

      18. metadata-eval 2.6%

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

   6. Applied egg-rr 67.3%

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

      1. div-inv 67.4%

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

   8. Applied egg-rr 67.4%

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

      1. pow1 67.4%

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

      2. sqrt-unprod 54.7%

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

      3. un-div-inv 54.7%

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

   10. Applied egg-rr 54.7%

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

       1. unpow1 54.7%

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

       2. *-commutative 54.7%

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

   12. Simplified 54.7%

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

   if -4.5000000000000002e213 < l < -4.999999999999985e-310

   1. Initial program 67.8%

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

   3. Simplified 67.0%

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

   5. Taylor expanded in d around 0 0.7%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 73.3%

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

      4. mul-1-neg 73.3%

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

      5. unpow-1 73.3%

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

      6. metadata-eval 73.3%

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

      7. pow-sqr 73.3%

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

      8. rem-sqrt-square 73.3%

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

      9. rem-square-sqrt 73.1%

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

      10. fabs-sqr 73.1%

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

      11. rem-square-sqrt 73.3%

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

   8. Simplified 73.3%

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

   if -4.999999999999985e-310 < l < 1e-203

   1. Initial program 69.3%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in d around 0 76.3%

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

   6. Taylor expanded in M around 0 76.3%

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

      1. *-commutative 76.3%

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

      2. associate-*l/ 76.3%

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

      3. associate-*l* 76.3%

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

      4. *-commutative 76.3%

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

      5. associate-*r/ 76.3%

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

      6. *-commutative 76.3%

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

      7. associate-/l* 76.3%

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

   8. Simplified 76.3%

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

      1. *-un-lft-identity 76.3%

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

      2. pow1/2 76.3%

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

      3. inv-pow 76.3%

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

      4. pow-pow 76.3%

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

      5. metadata-eval 76.3%

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

   10. Applied egg-rr 76.3%

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

       1. *-lft-identity 76.3%

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

   12. Simplified 76.3%

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

       1. associate-*r/ 93.1%

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

   14. Applied egg-rr 93.1%

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

   if 1e-203 < l

   1. Initial program 70.5%

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

   3. Simplified 70.4%

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

      1. *-commutative 70.4%

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

      2. sqrt-div 76.5%

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

      3. sqrt-div 82.0%

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

      4. frac-times 81.9%

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

      5. add-sqr-sqrt 82.0%

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

   6. Applied egg-rr 82.0%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+213}:\\ \;\;\;\;\left(1 - 0.5 \cdot {\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 10^{-203}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\\ t_1 := d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_1 \cdot \left(t\_0 + -1\right)\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{-203}:\\ \;\;\;\;t\_1 \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - t\_0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0))))
        (t_1 (* d (pow (* l h) -0.5))))
   (if (<= l -5e-310)
     (* t_1 (+ t_0 -1.0))
     (if (<= l 3.4e-203)
       (* t_1 (- 1.0 (* 0.5 (/ (* h (pow (* D_m (* M_m (/ 0.5 d))) 2.0)) l))))
       (* (/ d (* (sqrt h) (sqrt l))) (- 1.0 t_0))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 0.5 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0));
	double t_1 = d * pow((l * h), -0.5);
	double tmp;
	if (l <= -5e-310) {
		tmp = t_1 * (t_0 + -1.0);
	} else if (l <= 3.4e-203) {
		tmp = t_1 * (1.0 - (0.5 * ((h * pow((D_m * (M_m * (0.5 / d))), 2.0)) / l)));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - t_0);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_m / d)) ** 2.0d0))
    t_1 = d * ((l * h) ** (-0.5d0))
    if (l <= (-5d-310)) then
        tmp = t_1 * (t_0 + (-1.0d0))
    else if (l <= 3.4d-203) then
        tmp = t_1 * (1.0d0 - (0.5d0 * ((h * ((d_m * (m_m * (0.5d0 / d))) ** 2.0d0)) / l)))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 - t_0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D_m / d)), 2.0));
	double t_1 = d * Math.pow((l * h), -0.5);
	double tmp;
	if (l <= -5e-310) {
		tmp = t_1 * (t_0 + -1.0);
	} else if (l <= 3.4e-203) {
		tmp = t_1 * (1.0 - (0.5 * ((h * Math.pow((D_m * (M_m * (0.5 / d))), 2.0)) / l)));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 - t_0);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = 0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D_m / d)), 2.0))
	t_1 = d * math.pow((l * h), -0.5)
	tmp = 0
	if l <= -5e-310:
		tmp = t_1 * (t_0 + -1.0)
	elif l <= 3.4e-203:
		tmp = t_1 * (1.0 - (0.5 * ((h * math.pow((D_m * (M_m * (0.5 / d))), 2.0)) / l)))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 - t_0)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0)))
	t_1 = Float64(d * (Float64(l * h) ^ -0.5))
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(t_1 * Float64(t_0 + -1.0));
	elseif (l <= 3.4e-203)
		tmp = Float64(t_1 * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(D_m * Float64(M_m * Float64(0.5 / d))) ^ 2.0)) / l))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - t_0));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = 0.5 * ((h / l) * (((M_m / 2.0) * (D_m / d)) ^ 2.0));
	t_1 = d * ((l * h) ^ -0.5);
	tmp = 0.0;
	if (l <= -5e-310)
		tmp = t_1 * (t_0 + -1.0);
	elseif (l <= 3.4e-203)
		tmp = t_1 * (1.0 - (0.5 * ((h * ((D_m * (M_m * (0.5 / d))) ^ 2.0)) / l)));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - t_0);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5e-310], N[(t$95$1 * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.4e-203], N[(t$95$1 * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 66.2%

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

   3. Simplified 65.3%

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

   5. Taylor expanded in d around 0 0.9%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 67.0%

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

      4. mul-1-neg 67.0%

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

      5. unpow-1 67.0%

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

      6. metadata-eval 67.0%

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

      7. pow-sqr 67.0%

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

      8. rem-sqrt-square 67.0%

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

      9. rem-square-sqrt 66.8%

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

      10. fabs-sqr 66.8%

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

      11. rem-square-sqrt 67.0%

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

   8. Simplified 67.0%

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

   if -4.999999999999985e-310 < l < 3.3999999999999999e-203

   1. Initial program 69.3%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in d around 0 76.3%

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

   6. Taylor expanded in M around 0 76.3%

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

      1. *-commutative 76.3%

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

      2. associate-*l/ 76.3%

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

      3. associate-*l* 76.3%

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

      4. *-commutative 76.3%

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

      5. associate-*r/ 76.3%

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

      6. *-commutative 76.3%

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

      7. associate-/l* 76.3%

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

   8. Simplified 76.3%

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

      1. *-un-lft-identity 76.3%

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

      2. pow1/2 76.3%

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

      3. inv-pow 76.3%

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

      4. pow-pow 76.3%

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

      5. metadata-eval 76.3%

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

   10. Applied egg-rr 76.3%

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

       1. *-lft-identity 76.3%

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

   12. Simplified 76.3%

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

       1. associate-*r/ 93.1%

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

   14. Applied egg-rr 93.1%

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

   if 3.3999999999999999e-203 < l

   1. Initial program 70.5%

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

   3. Simplified 70.4%

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

      1. *-commutative 70.4%

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

      2. sqrt-div 76.5%

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

      3. sqrt-div 82.0%

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

      4. frac-times 81.9%

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

      5. add-sqr-sqrt 82.0%

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

   6. Applied egg-rr 82.0%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{-203}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.5 \cdot 10^{-202}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(D\_m \cdot \left(0.5 \cdot \frac{M\_m}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l 5.5e-202)
   (*
    (* (sqrt (/ d l)) (sqrt (/ d h)))
    (- 1.0 (* 0.5 (/ (* h (pow (* D_m (* 0.5 (/ M_m d))) 2.0)) l))))
   (*
    (/ d (* (sqrt h) (sqrt l)))
    (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 5.5e-202) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * ((h * pow((D_m * (0.5 * (M_m / d))), 2.0)) / l)));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (0.5 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= 5.5d-202) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (0.5d0 * ((h * ((d_m * (0.5d0 * (m_m / d))) ** 2.0d0)) / l)))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 - (0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_m / d)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 5.5e-202) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (0.5 * ((h * Math.pow((D_m * (0.5 * (M_m / d))), 2.0)) / l)));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 - (0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D_m / d)), 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= 5.5e-202:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (0.5 * ((h * math.pow((D_m * (0.5 * (M_m / d))), 2.0)) / l)))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 - (0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D_m / d)), 2.0))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= 5.5e-202)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(D_m * Float64(0.5 * Float64(M_m / d))) ^ 2.0)) / l))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= 5.5e-202)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * ((h * ((D_m * (0.5 * (M_m / d))) ^ 2.0)) / l)));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (0.5 * ((h / l) * (((M_m / 2.0) * (D_m / d)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, 5.5e-202], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(D$95$m * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.5e-202

   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 65.9%

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

      1. associate-*r/ 70.6%

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

   6. Applied egg-rr 70.7%

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

   if 5.5e-202 < l

   1. Initial program 70.8%

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

   3. Simplified 70.8%

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

      1. *-commutative 70.8%

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

      2. sqrt-div 77.0%

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

      3. sqrt-div 82.6%

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

      4. frac-times 82.5%

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

      5. add-sqr-sqrt 82.6%

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

   6. Applied egg-rr 82.6%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.5 \cdot 10^{-202}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.65 \cdot 10^{-203}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{-0.5 \cdot \left(h \cdot {\left(\frac{\frac{d}{M\_m}}{D\_m \cdot 0.5}\right)}^{-2}\right)}{\ell}\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l 1.65e-203)
   (*
    (sqrt (/ d l))
    (*
     (+ 1.0 (/ (* -0.5 (* h (pow (/ (/ d M_m) (* D_m 0.5)) -2.0))) l))
     (sqrt (/ d h))))
   (*
    (/ d (* (sqrt h) (sqrt l)))
    (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 1.65e-203) {
		tmp = sqrt((d / l)) * ((1.0 + ((-0.5 * (h * pow(((d / M_m) / (D_m * 0.5)), -2.0))) / l)) * sqrt((d / h)));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (0.5 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= 1.65d-203) then
        tmp = sqrt((d / l)) * ((1.0d0 + (((-0.5d0) * (h * (((d / m_m) / (d_m * 0.5d0)) ** (-2.0d0)))) / l)) * sqrt((d / h)))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 - (0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_m / d)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 1.65e-203) {
		tmp = Math.sqrt((d / l)) * ((1.0 + ((-0.5 * (h * Math.pow(((d / M_m) / (D_m * 0.5)), -2.0))) / l)) * Math.sqrt((d / h)));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 - (0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D_m / d)), 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= 1.65e-203:
		tmp = math.sqrt((d / l)) * ((1.0 + ((-0.5 * (h * math.pow(((d / M_m) / (D_m * 0.5)), -2.0))) / l)) * math.sqrt((d / h)))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 - (0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D_m / d)), 2.0))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= 1.65e-203)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(1.0 + Float64(Float64(-0.5 * Float64(h * (Float64(Float64(d / M_m) / Float64(D_m * 0.5)) ^ -2.0))) / l)) * sqrt(Float64(d / h))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= 1.65e-203)
		tmp = sqrt((d / l)) * ((1.0 + ((-0.5 * (h * (((d / M_m) / (D_m * 0.5)) ^ -2.0))) / l)) * sqrt((d / h)));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (0.5 * ((h / l) * (((M_m / 2.0) * (D_m / d)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, 1.65e-203], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 + N[(N[(-0.5 * N[(h * N[Power[N[(N[(d / M$95$m), $MachinePrecision] / N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.65000000000000012e-203

   1. Initial program 66.8%

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

   3. Simplified 66.1%

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

      1. associate-*r/ 66.8%

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

      2. div-inv 66.8%

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

      3. associate-*r* 66.8%

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

      4. *-commutative 66.8%

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

      5. div-inv 66.8%

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

      6. associate-/r* 66.8%

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

      7. frac-times 66.1%

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

      8. associate-*r/ 66.8%

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

      9. div-inv 66.8%

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

      10. metadata-eval 66.8%

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

   6. Applied egg-rr 66.8%

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

      1. metadata-eval 66.8%

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

      2. div-inv 66.8%

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

      3. *-commutative 66.8%

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

      4. associate-*r/ 66.1%

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

      5. expm1-log1p-u 33.8%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)}\right)\right) \]

      6. log1p-define 33.8%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)}\right)\right)\right) \]

      7. expm1-undefine 33.8%

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

   8. Applied egg-rr 66.1%

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

   10. Simplified 70.9%

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

   if 1.65000000000000012e-203 < l

   1. Initial program 70.5%

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

   3. Simplified 70.4%

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

      1. *-commutative 70.4%

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

      2. sqrt-div 76.5%

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

      3. sqrt-div 82.0%

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

      4. frac-times 81.9%

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

      5. add-sqr-sqrt 82.0%

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

   6. Applied egg-rr 82.0%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.65 \cdot 10^{-203}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{-0.5 \cdot \left(h \cdot {\left(\frac{\frac{d}{M}}{D \cdot 0.5}\right)}^{-2}\right)}{\ell}\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -5.4 \cdot 10^{-218}:\\ \;\;\;\;\sqrt{\frac{d \cdot \frac{d}{\ell}}{h}} \cdot \left(1 + -0.5 \cdot \frac{h \cdot {\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-300}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -5.4e-218)
   (*
    (sqrt (/ (* d (/ d l)) h))
    (+ 1.0 (* -0.5 (/ (* h (pow (* M_m (* 0.5 (/ D_m d))) 2.0)) l))))
   (if (<= d 1.05e-300)
     (* d (pow (+ -1.0 (fma h l 1.0)) -0.5))
     (*
      (* d (pow (* l h) -0.5))
      (- 1.0 (* 0.5 (/ (* h (pow (* D_m (* M_m (/ 0.5 d))) 2.0)) l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -5.4e-218) {
		tmp = sqrt(((d * (d / l)) / h)) * (1.0 + (-0.5 * ((h * pow((M_m * (0.5 * (D_m / d))), 2.0)) / l)));
	} else if (d <= 1.05e-300) {
		tmp = d * pow((-1.0 + fma(h, l, 1.0)), -0.5);
	} else {
		tmp = (d * pow((l * h), -0.5)) * (1.0 - (0.5 * ((h * pow((D_m * (M_m * (0.5 / d))), 2.0)) / l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -5.4e-218)
		tmp = Float64(sqrt(Float64(Float64(d * Float64(d / l)) / h)) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h * (Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0)) / l))));
	elseif (d <= 1.05e-300)
		tmp = Float64(d * (Float64(-1.0 + fma(h, l, 1.0)) ^ -0.5));
	else
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(D_m * Float64(M_m * Float64(0.5 / d))) ^ 2.0)) / l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -5.4e-218], N[(N[Sqrt[N[(N[(d * N[(d / l), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.05e-300], N[(d * N[Power[N[(-1.0 + N[(h * l + 1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.3999999999999999e-218

   1. Initial program 74.5%

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

   3. Simplified 74.5%

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

      1. add-sqr-sqrt 0.9%

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

      2. pow2 0.9%

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

      3. sqrt-prod 0.9%

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

      4. sqrt-pow1 1.0%

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

      5. frac-times 1.1%

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

      6. associate-/r* 1.1%

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

      7. metadata-eval 1.1%

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

      8. pow1 1.1%

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

      9. div-inv 1.1%

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

      10. *-commutative 1.1%

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

      11. associate-*r* 1.1%

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

      12. div-inv 1.1%

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

      13. associate-*r/ 1.2%

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

      14. associate-/l/ 1.2%

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

      15. *-un-lft-identity 1.2%

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

      16. *-commutative 1.2%

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

      17. times-frac 1.2%

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

      18. metadata-eval 1.2%

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

   6. Applied egg-rr 77.4%

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

      1. pow1 77.4%

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

   8. Applied egg-rr 62.8%

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

      1. unpow1 62.8%

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

      2. associate-*l/ 60.9%

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

      3. *-commutative 60.9%

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

      4. associate-*l/ 63.1%

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

      5. *-commutative 63.1%

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

      6. associate-*r* 63.1%

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

   10. Simplified 63.1%

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

   if -5.3999999999999999e-218 < d < 1.05000000000000002e-300

   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 23.1%

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

      1. add-sqr-sqrt 0.5%

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

      2. pow2 0.5%

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

      3. sqrt-prod 0.5%

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

      4. sqrt-pow1 0.5%

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

      5. frac-times 0.9%

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

      6. associate-/r* 0.9%

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

      7. metadata-eval 0.9%

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

      8. pow1 0.9%

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

      9. div-inv 0.9%

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

      10. *-commutative 0.9%

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

      11. associate-*r* 0.9%

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

      12. div-inv 0.9%

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

      13. associate-*r/ 0.9%

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

      14. associate-/l/ 0.9%

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

      15. *-un-lft-identity 0.9%

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

      16. *-commutative 0.9%

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

      17. times-frac 0.9%

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

      18. metadata-eval 0.9%

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

   6. Applied egg-rr 35.8%

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

   7. Taylor expanded in d around inf 11.3%

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

      1. unpow-1 11.3%

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

      2. metadata-eval 11.3%

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

      3. pow-sqr 11.3%

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

      4. rem-sqrt-square 11.3%

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

      5. rem-square-sqrt 11.3%

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

      6. fabs-sqr 11.3%

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

      7. rem-square-sqrt 11.3%

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

   9. Simplified 11.3%

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

       1. expm1-log1p-u 11.3%

          \[\leadsto d \cdot {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(h \cdot \ell\right)\right)\right)}}^{-0.5} \]

       2. expm1-undefine 28.6%

          \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1\right)}}^{-0.5} \]

   11. Applied egg-rr 28.6%

       \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1\right)}}^{-0.5} \]
   12. Step-by-step derivation

       1. sub-neg 28.6%

          \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \left(-1\right)\right)}}^{-0.5} \]

       2. metadata-eval 28.6%

          \[\leadsto d \cdot {\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \color{blue}{-1}\right)}^{-0.5} \]

       3. +-commutative 28.6%

          \[\leadsto d \cdot {\color{blue}{\left(-1 + e^{\mathsf{log1p}\left(h \cdot \ell\right)}\right)}}^{-0.5} \]

       4. log1p-undefine 28.6%

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

       5. rem-exp-log 28.6%

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

       6. +-commutative 28.6%

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

       7. fma-define 28.6%

          \[\leadsto d \cdot {\left(-1 + \color{blue}{\mathsf{fma}\left(h, \ell, 1\right)}\right)}^{-0.5} \]

   13. Simplified 28.6%

       \[\leadsto d \cdot {\color{blue}{\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}}^{-0.5} \]

   if 1.05000000000000002e-300 < d

   1. Initial program 70.7%

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

   3. Simplified 70.7%

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

   5. Taylor expanded in d around 0 73.0%

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

   6. Taylor expanded in M around 0 73.0%

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

      1. *-commutative 73.0%

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

      2. associate-*l/ 70.1%

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

      3. associate-*l* 70.1%

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

      4. *-commutative 70.1%

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

      5. associate-*r/ 70.1%

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

      6. *-commutative 70.1%

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

      7. associate-/l* 70.1%

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

   8. Simplified 70.1%

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

      1. *-un-lft-identity 70.1%

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

      2. pow1/2 70.1%

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

      3. inv-pow 70.1%

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

      4. pow-pow 70.1%

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

      5. metadata-eval 70.1%

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

   10. Applied egg-rr 70.1%

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

       1. *-lft-identity 70.1%

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

   12. Simplified 70.1%

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

       1. associate-*r/ 75.2%

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

   14. Applied egg-rr 75.2%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.4 \cdot 10^{-218}:\\ \;\;\;\;\sqrt{\frac{d \cdot \frac{d}{\ell}}{h}} \cdot \left(1 + -0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-300}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{if}\;\ell \leq -5.1 \cdot 10^{-179}:\\ \;\;\;\;d \cdot \left(-t\_0\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(d \cdot t\_0\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (pow (* l h) -0.5)))
   (if (<= l -5.1e-179)
     (* d (- t_0))
     (if (<= l -5e-310)
       (* d (pow (+ -1.0 (fma h l 1.0)) -0.5))
       (*
        (* d t_0)
        (- 1.0 (* 0.5 (/ (* h (pow (* D_m (* M_m (/ 0.5 d))) 2.0)) l))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = pow((l * h), -0.5);
	double tmp;
	if (l <= -5.1e-179) {
		tmp = d * -t_0;
	} else if (l <= -5e-310) {
		tmp = d * pow((-1.0 + fma(h, l, 1.0)), -0.5);
	} else {
		tmp = (d * t_0) * (1.0 - (0.5 * ((h * pow((D_m * (M_m * (0.5 / d))), 2.0)) / l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(l * h) ^ -0.5
	tmp = 0.0
	if (l <= -5.1e-179)
		tmp = Float64(d * Float64(-t_0));
	elseif (l <= -5e-310)
		tmp = Float64(d * (Float64(-1.0 + fma(h, l, 1.0)) ^ -0.5));
	else
		tmp = Float64(Float64(d * t_0) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(D_m * Float64(M_m * Float64(0.5 / d))) ^ 2.0)) / l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[l, -5.1e-179], N[(d * (-t$95$0)), $MachinePrecision], If[LessEqual[l, -5e-310], N[(d * N[Power[N[(-1.0 + N[(h * l + 1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(N[(d * t$95$0), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.10000000000000028e-179

   1. Initial program 61.7%

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

   3. Simplified 60.5%

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

      1. add-sqr-sqrt 1.1%

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

      2. pow2 1.1%

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

      3. sqrt-prod 1.1%

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

      4. sqrt-pow1 1.1%

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

      5. frac-times 1.4%

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

      6. associate-/r* 1.4%

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

      7. metadata-eval 1.4%

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

      8. pow1 1.4%

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

      9. div-inv 1.4%

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

      10. *-commutative 1.4%

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

      11. associate-*r* 1.4%

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

      12. div-inv 1.4%

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

      13. associate-*r/ 1.4%

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

      14. associate-/l/ 1.4%

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

      15. *-un-lft-identity 1.4%

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

      16. *-commutative 1.4%

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

      17. times-frac 1.4%

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

      18. metadata-eval 1.4%

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

   6. Applied egg-rr 65.9%

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

   7. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow-1 0.0%

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

      3. metadata-eval 0.0%

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

      4. pow-sqr 0.0%

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

      5. rem-sqrt-square 0.0%

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

      6. rem-square-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. rem-square-sqrt 0.0%

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

      9. *-commutative 0.0%

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

      10. unpow2 0.0%

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

      11. rem-square-sqrt 46.0%

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

      12. neg-mul-1 46.0%

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

   9. Simplified 46.0%

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

   if -5.10000000000000028e-179 < l < -4.999999999999985e-310

   1. Initial program 78.7%

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

   3. Simplified 78.7%

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

      1. add-sqr-sqrt 0.1%

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

      2. pow2 0.1%

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

      3. sqrt-prod 0.1%

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

      4. sqrt-pow1 0.2%

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

      5. frac-times 0.1%

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

      6. associate-/r* 0.1%

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

      7. metadata-eval 0.1%

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

      8. pow1 0.1%

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

      9. div-inv 0.1%

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

      10. *-commutative 0.1%

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

      11. associate-*r* 0.1%

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

      12. div-inv 0.1%

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

      13. associate-*r/ 0.2%

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

      14. associate-/l/ 0.2%

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

      15. *-un-lft-identity 0.2%

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

      16. *-commutative 0.2%

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

      17. times-frac 0.2%

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

      18. metadata-eval 0.2%

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

   6. Applied egg-rr 81.8%

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

   7. Taylor expanded in d around inf 32.8%

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

      1. unpow-1 32.8%

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

      2. metadata-eval 32.8%

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

      3. pow-sqr 32.8%

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

      4. rem-sqrt-square 32.8%

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

      5. rem-square-sqrt 32.8%

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

      6. fabs-sqr 32.8%

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

      7. rem-square-sqrt 32.8%

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

   9. Simplified 32.8%

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

       1. expm1-log1p-u 32.8%

          \[\leadsto d \cdot {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(h \cdot \ell\right)\right)\right)}}^{-0.5} \]

       2. expm1-undefine 50.7%

          \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1\right)}}^{-0.5} \]

   11. Applied egg-rr 50.7%

       \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1\right)}}^{-0.5} \]
   12. Step-by-step derivation

       1. sub-neg 50.7%

          \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \left(-1\right)\right)}}^{-0.5} \]

       2. metadata-eval 50.7%

          \[\leadsto d \cdot {\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \color{blue}{-1}\right)}^{-0.5} \]

       3. +-commutative 50.7%

          \[\leadsto d \cdot {\color{blue}{\left(-1 + e^{\mathsf{log1p}\left(h \cdot \ell\right)}\right)}}^{-0.5} \]

       4. log1p-undefine 50.7%

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

       5. rem-exp-log 50.7%

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

       6. +-commutative 50.7%

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

       7. fma-define 50.7%

          \[\leadsto d \cdot {\left(-1 + \color{blue}{\mathsf{fma}\left(h, \ell, 1\right)}\right)}^{-0.5} \]

   13. Simplified 50.7%

       \[\leadsto d \cdot {\color{blue}{\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}}^{-0.5} \]

   if -4.999999999999985e-310 < l

   1. Initial program 70.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in d around 0 72.4%

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

   6. Taylor expanded in M around 0 72.4%

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

      1. *-commutative 72.4%

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

      2. associate-*l/ 69.5%

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

      3. associate-*l* 69.5%

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

      4. *-commutative 69.5%

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

      5. associate-*r/ 69.5%

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

      6. *-commutative 69.5%

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

      7. associate-/l* 69.6%

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

   8. Simplified 69.6%

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

      1. *-un-lft-identity 69.6%

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

      2. pow1/2 69.6%

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

      3. inv-pow 69.6%

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

      4. pow-pow 69.5%

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

      5. metadata-eval 69.5%

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

   10. Applied egg-rr 69.5%

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

       1. *-lft-identity 69.5%

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

   12. Simplified 69.5%

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

       1. associate-*r/ 74.6%

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

   14. Applied egg-rr 74.6%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.1 \cdot 10^{-179}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{if}\;\ell \leq -5.8 \cdot 10^{-175}:\\ \;\;\;\;d \cdot \left(-t\_0\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(t\_0 \cdot \left(1 + -0.5 \cdot \frac{h \cdot {\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (pow (* l h) -0.5)))
   (if (<= l -5.8e-175)
     (* d (- t_0))
     (if (<= l -5e-310)
       (* d (pow (+ -1.0 (fma h l 1.0)) -0.5))
       (*
        d
        (*
         t_0
         (+ 1.0 (* -0.5 (/ (* h (pow (* M_m (* 0.5 (/ D_m d))) 2.0)) l)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = pow((l * h), -0.5);
	double tmp;
	if (l <= -5.8e-175) {
		tmp = d * -t_0;
	} else if (l <= -5e-310) {
		tmp = d * pow((-1.0 + fma(h, l, 1.0)), -0.5);
	} else {
		tmp = d * (t_0 * (1.0 + (-0.5 * ((h * pow((M_m * (0.5 * (D_m / d))), 2.0)) / l))));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(l * h) ^ -0.5
	tmp = 0.0
	if (l <= -5.8e-175)
		tmp = Float64(d * Float64(-t_0));
	elseif (l <= -5e-310)
		tmp = Float64(d * (Float64(-1.0 + fma(h, l, 1.0)) ^ -0.5));
	else
		tmp = Float64(d * Float64(t_0 * Float64(1.0 + Float64(-0.5 * Float64(Float64(h * (Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0)) / l)))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[l, -5.8e-175], N[(d * (-t$95$0)), $MachinePrecision], If[LessEqual[l, -5e-310], N[(d * N[Power[N[(-1.0 + N[(h * l + 1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(d * N[(t$95$0 * N[(1.0 + N[(-0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.79999999999999998e-175

   1. Initial program 61.7%

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

   3. Simplified 60.5%

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

      1. add-sqr-sqrt 1.1%

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

      2. pow2 1.1%

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

      3. sqrt-prod 1.1%

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

      4. sqrt-pow1 1.1%

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

      5. frac-times 1.4%

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

      6. associate-/r* 1.4%

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

      7. metadata-eval 1.4%

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

      8. pow1 1.4%

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

      9. div-inv 1.4%

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

      10. *-commutative 1.4%

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

      11. associate-*r* 1.4%

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

      12. div-inv 1.4%

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

      13. associate-*r/ 1.4%

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

      14. associate-/l/ 1.4%

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

      15. *-un-lft-identity 1.4%

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

      16. *-commutative 1.4%

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

      17. times-frac 1.4%

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

      18. metadata-eval 1.4%

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

   6. Applied egg-rr 65.9%

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

   7. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow-1 0.0%

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

      3. metadata-eval 0.0%

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

      4. pow-sqr 0.0%

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

      5. rem-sqrt-square 0.0%

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

      6. rem-square-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. rem-square-sqrt 0.0%

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

      9. *-commutative 0.0%

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

      10. unpow2 0.0%

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

      11. rem-square-sqrt 46.0%

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

      12. neg-mul-1 46.0%

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

   9. Simplified 46.0%

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

   if -5.79999999999999998e-175 < l < -4.999999999999985e-310

   1. Initial program 78.7%

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

   3. Simplified 78.7%

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

      1. add-sqr-sqrt 0.1%

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

      2. pow2 0.1%

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

      3. sqrt-prod 0.1%

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

      4. sqrt-pow1 0.2%

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

      5. frac-times 0.1%

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

      6. associate-/r* 0.1%

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

      7. metadata-eval 0.1%

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

      8. pow1 0.1%

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

      9. div-inv 0.1%

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

      10. *-commutative 0.1%

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

      11. associate-*r* 0.1%

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

      12. div-inv 0.1%

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

      13. associate-*r/ 0.2%

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

      14. associate-/l/ 0.2%

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

      15. *-un-lft-identity 0.2%

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

      16. *-commutative 0.2%

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

      17. times-frac 0.2%

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

      18. metadata-eval 0.2%

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

   6. Applied egg-rr 81.8%

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

   7. Taylor expanded in d around inf 32.8%

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

      1. unpow-1 32.8%

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

      2. metadata-eval 32.8%

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

      3. pow-sqr 32.8%

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

      4. rem-sqrt-square 32.8%

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

      5. rem-square-sqrt 32.8%

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

      6. fabs-sqr 32.8%

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

      7. rem-square-sqrt 32.8%

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

   9. Simplified 32.8%

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

       1. expm1-log1p-u 32.8%

          \[\leadsto d \cdot {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(h \cdot \ell\right)\right)\right)}}^{-0.5} \]

       2. expm1-undefine 50.7%

          \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1\right)}}^{-0.5} \]

   11. Applied egg-rr 50.7%

       \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1\right)}}^{-0.5} \]
   12. Step-by-step derivation

       1. sub-neg 50.7%

          \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \left(-1\right)\right)}}^{-0.5} \]

       2. metadata-eval 50.7%

          \[\leadsto d \cdot {\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \color{blue}{-1}\right)}^{-0.5} \]

       3. +-commutative 50.7%

          \[\leadsto d \cdot {\color{blue}{\left(-1 + e^{\mathsf{log1p}\left(h \cdot \ell\right)}\right)}}^{-0.5} \]

       4. log1p-undefine 50.7%

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

       5. rem-exp-log 50.7%

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

       6. +-commutative 50.7%

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

       7. fma-define 50.7%

          \[\leadsto d \cdot {\left(-1 + \color{blue}{\mathsf{fma}\left(h, \ell, 1\right)}\right)}^{-0.5} \]

   13. Simplified 50.7%

       \[\leadsto d \cdot {\color{blue}{\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}}^{-0.5} \]

   if -4.999999999999985e-310 < l

   1. Initial program 70.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in d around 0 72.4%

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

      1. pow1 72.4%

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

   7. Applied egg-rr 72.4%

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

      1. unpow1 72.4%

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

      2. associate-*l* 73.2%

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

      3. *-commutative 73.2%

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

      4. associate-*l/ 77.6%

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

      5. *-commutative 77.6%

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

      6. associate-*r* 77.6%

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

   9. Simplified 77.6%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.8 \cdot 10^{-175}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\left(\ell \cdot h\right)}^{-0.5} \cdot \left(1 + -0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 72.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 66.2%

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

   3. Simplified 65.3%

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

   5. Taylor expanded in d around 0 0.9%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 67.0%

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

      4. mul-1-neg 67.0%

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

      5. unpow-1 67.0%

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

      6. metadata-eval 67.0%

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

      7. pow-sqr 67.0%

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

      8. rem-sqrt-square 67.0%

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

      9. rem-square-sqrt 66.8%

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

      10. fabs-sqr 66.8%

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

      11. rem-square-sqrt 67.0%

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

   8. Simplified 67.0%

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

   if -4.999999999999985e-310 < l

   1. Initial program 70.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in d around 0 72.4%

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

      1. associate-*r/ 77.5%

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

      2. clear-num 77.5%

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

   7. Applied egg-rr 74.6%

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

      1. associate-/r/ 74.6%

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

      2. *-commutative 74.6%

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

      3. associate-*r/ 74.6%

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

      4. *-commutative 74.6%

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

      5. associate-/l* 74.7%

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

   9. Simplified 74.7%

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

4. Final simplification 71.1%

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

Alternative 14: 65.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 66.2%

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

   3. Simplified 65.3%

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

      1. add-sqr-sqrt 0.9%

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

      2. pow2 0.9%

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

      3. sqrt-prod 0.9%

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

      4. sqrt-pow1 0.9%

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

      5. frac-times 1.1%

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

      6. associate-/r* 1.1%

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

      7. metadata-eval 1.1%

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

      8. pow1 1.1%

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

      9. div-inv 1.1%

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

      10. *-commutative 1.1%

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

      11. associate-*r* 1.1%

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

      12. div-inv 1.1%

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

      13. associate-*r/ 1.1%

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

      14. associate-/l/ 1.1%

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

      15. *-un-lft-identity 1.1%

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

      16. *-commutative 1.1%

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

      17. times-frac 1.1%

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

      18. metadata-eval 1.1%

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

   6. Applied egg-rr 70.1%

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

      1. div-inv 70.1%

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

   8. Applied egg-rr 70.1%

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

      1. pow1 70.1%

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

   10. Applied egg-rr 53.2%

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

       1. unpow1 53.2%

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

       2. +-commutative 53.2%

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

       3. *-commutative 53.2%

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

       4. +-commutative 53.2%

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

       5. associate-*r* 53.2%

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

   12. Simplified 53.2%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(\left(D \cdot 0.5\right) \cdot \frac{M}{d}\right)}^{2}\right)} \]

   if -4.999999999999985e-310 < l

   1. Initial program 70.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 70.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around 0 72.4%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Taylor expanded in M around 0 72.4%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(0.5 \cdot \frac{D \cdot M}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. *-commutative 72.4%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(0.5 \cdot \frac{\color{blue}{M \cdot D}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. associate-*l/ 69.5%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(0.5 \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. associate-*l* 69.5%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\left(0.5 \cdot \frac{M}{d}\right) \cdot D\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. *-commutative 69.5%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. associate-*r/ 69.5%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \color{blue}{\frac{0.5 \cdot M}{d}}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. *-commutative 69.5%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\color{blue}{M \cdot 0.5}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      7. associate-/l* 69.6%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \color{blue}{\left(M \cdot \frac{0.5}{d}\right)}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. Simplified 69.6%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   9. Step-by-step derivation

      1. *-un-lft-identity 69.6%

         \[\leadsto \left(d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. pow1/2 69.6%

         \[\leadsto \left(d \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right)\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. inv-pow 69.6%

         \[\leadsto \left(d \cdot \left(1 \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right)\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. pow-pow 69.5%

         \[\leadsto \left(d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right)\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. metadata-eval 69.5%

         \[\leadsto \left(d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right)\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   10. Applied egg-rr 69.5%

       \[\leadsto \left(d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)}\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   11. Step-by-step derivation

       1. *-lft-identity 69.5%

          \[\leadsto \left(d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   12. Simplified 69.5%

       \[\leadsto \left(d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   13. Step-by-step derivation

       1. associate-*r/ 74.6%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot h}{\ell}}\right) \]

   14. Applied egg-rr 74.6%

       \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot h}{\ell}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 72.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* d (pow (* l h) -0.5))))
   (if (<= l -5e-310)
     (* t_0 (+ (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D_m d)) 2.0))) -1.0))
     (* t_0 (- 1.0 (* 0.5 (/ (* h (pow (* D_m (* M_m (/ 0.5 d))) 2.0)) l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = d * pow((l * h), -0.5);
	double tmp;
	if (l <= -5e-310) {
		tmp = t_0 * ((0.5 * ((h / l) * pow(((M_m / 2.0) * (D_m / d)), 2.0))) + -1.0);
	} else {
		tmp = t_0 * (1.0 - (0.5 * ((h * pow((D_m * (M_m * (0.5 / d))), 2.0)) / l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d * ((l * h) ** (-0.5d0))
    if (l <= (-5d-310)) then
        tmp = t_0 * ((0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_m / d)) ** 2.0d0))) + (-1.0d0))
    else
        tmp = t_0 * (1.0d0 - (0.5d0 * ((h * ((d_m * (m_m * (0.5d0 / d))) ** 2.0d0)) / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = d * Math.pow((l * h), -0.5);
	double tmp;
	if (l <= -5e-310) {
		tmp = t_0 * ((0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D_m / d)), 2.0))) + -1.0);
	} else {
		tmp = t_0 * (1.0 - (0.5 * ((h * Math.pow((D_m * (M_m * (0.5 / d))), 2.0)) / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = d * math.pow((l * h), -0.5)
	tmp = 0
	if l <= -5e-310:
		tmp = t_0 * ((0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D_m / d)), 2.0))) + -1.0)
	else:
		tmp = t_0 * (1.0 - (0.5 * ((h * math.pow((D_m * (M_m * (0.5 / d))), 2.0)) / l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(d * (Float64(l * h) ^ -0.5))
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(t_0 * Float64(Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0))) + -1.0));
	else
		tmp = Float64(t_0 * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(D_m * Float64(M_m * Float64(0.5 / d))) ^ 2.0)) / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = d * ((l * h) ^ -0.5);
	tmp = 0.0;
	if (l <= -5e-310)
		tmp = t_0 * ((0.5 * ((h / l) * (((M_m / 2.0) * (D_m / d)) ^ 2.0))) + -1.0);
	else
		tmp = t_0 * (1.0 - (0.5 * ((h * ((D_m * (M_m * (0.5 / d))) ^ 2.0)) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5e-310], N[(t$95$0 * N[(N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 65.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around 0 0.9%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Taylor expanded in h around -inf 0.0%

      \[\leadsto \left(d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \left(d \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. unpow2 0.0%

         \[\leadsto \left(d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. rem-square-sqrt 67.0%

         \[\leadsto \left(d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. mul-1-neg 67.0%

         \[\leadsto \left(d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. unpow-1 67.0%

         \[\leadsto \left(d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right)\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. metadata-eval 67.0%

         \[\leadsto \left(d \cdot \left(-\sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right)\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      7. pow-sqr 67.0%

         \[\leadsto \left(d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right)\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. rem-sqrt-square 67.0%

         \[\leadsto \left(d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right)\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      9. rem-square-sqrt 66.8%

         \[\leadsto \left(d \cdot \left(-\left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right)\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      10. fabs-sqr 66.8%

          \[\leadsto \left(d \cdot \left(-\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right)\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      11. rem-square-sqrt 67.0%

          \[\leadsto \left(d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right)\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. Simplified 67.0%

      \[\leadsto \left(d \cdot \color{blue}{\left(-{\left(h \cdot \ell\right)}^{-0.5}\right)}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   if -4.999999999999985e-310 < l

   1. Initial program 70.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 70.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around 0 72.4%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Taylor expanded in M around 0 72.4%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(0.5 \cdot \frac{D \cdot M}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. *-commutative 72.4%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(0.5 \cdot \frac{\color{blue}{M \cdot D}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. associate-*l/ 69.5%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(0.5 \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. associate-*l* 69.5%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\left(0.5 \cdot \frac{M}{d}\right) \cdot D\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. *-commutative 69.5%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. associate-*r/ 69.5%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \color{blue}{\frac{0.5 \cdot M}{d}}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. *-commutative 69.5%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\color{blue}{M \cdot 0.5}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      7. associate-/l* 69.6%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \color{blue}{\left(M \cdot \frac{0.5}{d}\right)}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. Simplified 69.6%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   9. Step-by-step derivation

      1. *-un-lft-identity 69.6%

         \[\leadsto \left(d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. pow1/2 69.6%

         \[\leadsto \left(d \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right)\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. inv-pow 69.6%

         \[\leadsto \left(d \cdot \left(1 \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right)\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. pow-pow 69.5%

         \[\leadsto \left(d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right)\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. metadata-eval 69.5%

         \[\leadsto \left(d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right)\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   10. Applied egg-rr 69.5%

       \[\leadsto \left(d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)}\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   11. Step-by-step derivation

       1. *-lft-identity 69.5%

          \[\leadsto \left(d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   12. Simplified 69.5%

       \[\leadsto \left(d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   13. Step-by-step derivation

       1. associate-*r/ 74.6%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot h}{\ell}}\right) \]

   14. Applied egg-rr 74.6%

       \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot h}{\ell}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 46.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 8.2 \cdot 10^{-265}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l 8.2e-265)
   (* d (- (pow (* l h) -0.5)))
   (* d (* (pow l -0.5) (pow h -0.5)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 8.2e-265) {
		tmp = d * -pow((l * h), -0.5);
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= 8.2d-265) then
        tmp = d * -((l * h) ** (-0.5d0))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 8.2e-265) {
		tmp = d * -Math.pow((l * h), -0.5);
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= 8.2e-265:
		tmp = d * -math.pow((l * h), -0.5)
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= 8.2e-265)
		tmp = Float64(d * Float64(-(Float64(l * h) ^ -0.5)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= 8.2e-265)
		tmp = d * -((l * h) ^ -0.5);
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, 8.2e-265], N[(d * (-N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 8.2e-265

   1. Initial program 66.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 7.7%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 7.7%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 7.7%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 7.8%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. frac-times 7.9%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-/r* 7.9%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. metadata-eval 7.9%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. pow1 7.9%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. div-inv 7.9%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. *-commutative 7.9%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. associate-*r* 7.9%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot \frac{1}{2}\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. div-inv 7.9%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \color{blue}{\frac{M}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. associate-*r/ 8.0%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{\frac{M}{2}}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      14. associate-/l/ 8.0%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      15. *-un-lft-identity 8.0%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{\color{blue}{1 \cdot M}}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      16. *-commutative 8.0%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      17. times-frac 8.0%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      18. metadata-eval 8.0%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 70.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. unpow-1 0.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      3. metadata-eval 0.0%

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      4. pow-sqr 0.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      5. rem-sqrt-square 0.0%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      6. rem-square-sqrt 0.0%

         \[\leadsto \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      7. fabs-sqr 0.0%

         \[\leadsto \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      8. rem-square-sqrt 0.0%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      9. *-commutative 0.0%

         \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      10. unpow2 0.0%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      11. rem-square-sqrt 42.3%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      12. neg-mul-1 42.3%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left(-d\right)} \]

   9. Simplified 42.3%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if 8.2e-265 < l

   1. Initial program 70.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 70.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 71.0%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 71.0%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 70.9%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 73.1%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. frac-times 74.7%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-/r* 74.7%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. metadata-eval 74.7%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. pow1 74.7%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. div-inv 74.7%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. *-commutative 74.7%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. associate-*r* 74.7%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot \frac{1}{2}\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. div-inv 74.7%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \color{blue}{\frac{M}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. associate-*r/ 70.8%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{\frac{M}{2}}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      14. associate-/l/ 70.8%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      15. *-un-lft-identity 70.8%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{\color{blue}{1 \cdot M}}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      16. *-commutative 70.8%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      17. times-frac 70.8%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      18. metadata-eval 70.8%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 70.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 42.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. unpow-1 42.1%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. metadata-eval 42.1%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      3. pow-sqr 42.1%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      4. rem-sqrt-square 42.1%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      5. rem-square-sqrt 41.9%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

      6. fabs-sqr 41.9%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      7. rem-square-sqrt 42.1%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 42.1%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. *-commutative 42.1%

          \[\leadsto d \cdot {\color{blue}{\left(\ell \cdot h\right)}}^{-0.5} \]

       2. unpow-prod-down 47.3%

          \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]

   11. Applied egg-rr 47.3%

       \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8.2 \cdot 10^{-265}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 43.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.22 \cdot 10^{-263}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l 1.22e-263)
   (* d (- (pow (* l h) -0.5)))
   (* d (sqrt (/ (/ 1.0 h) l)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 1.22e-263) {
		tmp = d * -pow((l * h), -0.5);
	} else {
		tmp = d * sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= 1.22d-263) then
        tmp = d * -((l * h) ** (-0.5d0))
    else
        tmp = d * sqrt(((1.0d0 / h) / l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 1.22e-263) {
		tmp = d * -Math.pow((l * h), -0.5);
	} else {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= 1.22e-263:
		tmp = d * -math.pow((l * h), -0.5)
	else:
		tmp = d * math.sqrt(((1.0 / h) / l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= 1.22e-263)
		tmp = Float64(d * Float64(-(Float64(l * h) ^ -0.5)));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= 1.22e-263)
		tmp = d * -((l * h) ^ -0.5);
	else
		tmp = d * sqrt(((1.0 / h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, 1.22e-263], N[(d * (-N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.22000000000000005e-263

   1. Initial program 66.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 7.7%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 7.7%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 7.7%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 7.8%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. frac-times 7.9%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-/r* 7.9%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. metadata-eval 7.9%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. pow1 7.9%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. div-inv 7.9%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. *-commutative 7.9%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. associate-*r* 7.9%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot \frac{1}{2}\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. div-inv 7.9%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \color{blue}{\frac{M}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. associate-*r/ 8.0%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{\frac{M}{2}}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      14. associate-/l/ 8.0%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      15. *-un-lft-identity 8.0%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{\color{blue}{1 \cdot M}}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      16. *-commutative 8.0%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      17. times-frac 8.0%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      18. metadata-eval 8.0%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 70.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. unpow-1 0.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      3. metadata-eval 0.0%

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      4. pow-sqr 0.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      5. rem-sqrt-square 0.0%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      6. rem-square-sqrt 0.0%

         \[\leadsto \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      7. fabs-sqr 0.0%

         \[\leadsto \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      8. rem-square-sqrt 0.0%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      9. *-commutative 0.0%

         \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      10. unpow2 0.0%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      11. rem-square-sqrt 42.3%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      12. neg-mul-1 42.3%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left(-d\right)} \]

   9. Simplified 42.3%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if 1.22000000000000005e-263 < l

   1. Initial program 70.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 70.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 71.0%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 71.0%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 70.9%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. sqrt-pow1 73.1%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. frac-times 74.7%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-/r* 74.7%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. metadata-eval 74.7%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. pow1 74.7%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. div-inv 74.7%

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. *-commutative 74.7%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. associate-*r* 74.7%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot \frac{1}{2}\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. div-inv 74.7%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \color{blue}{\frac{M}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      13. associate-*r/ 70.8%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{\frac{M}{2}}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      14. associate-/l/ 70.8%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      15. *-un-lft-identity 70.8%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{\color{blue}{1 \cdot M}}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      16. *-commutative 70.8%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      17. times-frac 70.8%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      18. metadata-eval 70.8%

          \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 70.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   7. Taylor expanded in d around inf 42.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. unpow-1 42.1%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. metadata-eval 42.1%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      3. pow-sqr 42.1%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      4. rem-sqrt-square 42.1%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      5. rem-square-sqrt 41.9%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

      6. fabs-sqr 41.9%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      7. rem-square-sqrt 42.1%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 42.1%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. metadata-eval 42.1%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{\left(-1 \cdot 0.5\right)}} \]

       2. pow-pow 42.1%

          \[\leadsto d \cdot \color{blue}{{\left({\left(h \cdot \ell\right)}^{-1}\right)}^{0.5}} \]

       3. inv-pow 42.1%

          \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{0.5} \]

       4. pow1/2 42.1%

          \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

       5. associate-/r* 42.1%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   11. Applied egg-rr 42.1%

       \[\leadsto d \cdot \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.22 \cdot 10^{-263}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 26.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (* d (sqrt (/ (/ 1.0 h) l))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return d * sqrt(((1.0 / h) / l));
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = d * sqrt(((1.0d0 / h) / l))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return d * Math.sqrt(((1.0 / h) / l));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return d * math.sqrt(((1.0 / h) / l))

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(d * sqrt(Float64(Float64(1.0 / h) / l)))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = d * sqrt(((1.0 / h) / l));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.3%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 67.9%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 38.6%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

   2. pow2 38.6%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

   3. sqrt-prod 38.6%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

   4. sqrt-pow1 39.7%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   5. frac-times 40.5%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. associate-/r* 40.5%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   7. metadata-eval 40.5%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   8. pow1 40.5%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   9. div-inv 40.5%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   10. *-commutative 40.5%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   11. associate-*r* 40.5%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot \frac{1}{2}\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   12. div-inv 40.5%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \color{blue}{\frac{M}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   13. associate-*r/ 38.6%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{\frac{M}{2}}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   14. associate-/l/ 38.6%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   15. *-un-lft-identity 38.6%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{\color{blue}{1 \cdot M}}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   16. *-commutative 38.6%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   17. times-frac 38.6%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   18. metadata-eval 38.6%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

6. Applied egg-rr 70.4%

   \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

7. Taylor expanded in d around inf 26.7%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
8. Step-by-step derivation

   1. unpow-1 26.7%

      \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

   2. metadata-eval 26.7%

      \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

   3. pow-sqr 26.7%

      \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

   4. rem-sqrt-square 26.7%

      \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

   5. rem-square-sqrt 26.6%

      \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

   6. fabs-sqr 26.6%

      \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

   7. rem-square-sqrt 26.7%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

9. Simplified 26.7%

   \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
10. Step-by-step derivation

    1. metadata-eval 26.7%

       \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{\left(-1 \cdot 0.5\right)}} \]

    2. pow-pow 26.7%

       \[\leadsto d \cdot \color{blue}{{\left({\left(h \cdot \ell\right)}^{-1}\right)}^{0.5}} \]

    3. inv-pow 26.7%

       \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{0.5} \]

    4. pow1/2 26.7%

       \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

    5. associate-/r* 26.7%

       \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

11. Applied egg-rr 26.7%

    \[\leadsto d \cdot \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}}} \]
12. Add Preprocessing

Alternative 19: 26.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ d \cdot \sqrt{\frac{1}{\ell \cdot h}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (* d (sqrt (/ 1.0 (* l h)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return d * sqrt((1.0 / (l * h)));
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = d * sqrt((1.0d0 / (l * h)))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return d * Math.sqrt((1.0 / (l * h)));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return d * math.sqrt((1.0 / (l * h)))

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(d * sqrt(Float64(1.0 / Float64(l * h))))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = d * sqrt((1.0 / (l * h)));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.3%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 67.9%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 38.6%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

   2. pow2 38.6%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

   3. sqrt-prod 38.6%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

   4. sqrt-pow1 39.7%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   5. frac-times 40.5%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. associate-/r* 40.5%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   7. metadata-eval 40.5%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   8. pow1 40.5%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   9. div-inv 40.5%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   10. *-commutative 40.5%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   11. associate-*r* 40.5%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot \frac{1}{2}\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   12. div-inv 40.5%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \color{blue}{\frac{M}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   13. associate-*r/ 38.6%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{\frac{M}{2}}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   14. associate-/l/ 38.6%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   15. *-un-lft-identity 38.6%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{\color{blue}{1 \cdot M}}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   16. *-commutative 38.6%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   17. times-frac 38.6%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   18. metadata-eval 38.6%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

6. Applied egg-rr 70.4%

   \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

7. Taylor expanded in d around inf 26.7%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

8. Final simplification 26.7%

   \[\leadsto d \cdot \sqrt{\frac{1}{\ell \cdot h}} \]
9. Add Preprocessing

Alternative 20: 26.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ d \cdot {\left(\ell \cdot h\right)}^{-0.5} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (* d (pow (* l h) -0.5)))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return d * pow((l * h), -0.5);
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = d * ((l * h) ** (-0.5d0))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return d * Math.pow((l * h), -0.5);
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return d * math.pow((l * h), -0.5)

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(d * (Float64(l * h) ^ -0.5))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = d * ((l * h) ^ -0.5);
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.3%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 67.9%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 38.6%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

   2. pow2 38.6%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

   3. sqrt-prod 38.6%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

   4. sqrt-pow1 39.7%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   5. frac-times 40.5%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. associate-/r* 40.5%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   7. metadata-eval 40.5%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   8. pow1 40.5%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   9. div-inv 40.5%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot \frac{1}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   10. *-commutative 40.5%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   11. associate-*r* 40.5%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(M \cdot \frac{1}{2}\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   12. div-inv 40.5%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \color{blue}{\frac{M}{2}}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   13. associate-*r/ 38.6%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{\frac{M}{2}}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   14. associate-/l/ 38.6%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\frac{M}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   15. *-un-lft-identity 38.6%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{\color{blue}{1 \cdot M}}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   16. *-commutative 38.6%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \frac{1 \cdot M}{\color{blue}{2 \cdot d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   17. times-frac 38.6%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   18. metadata-eval 38.6%

       \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

6. Applied egg-rr 70.4%

   \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

7. Taylor expanded in d around inf 26.7%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
8. Step-by-step derivation

   1. unpow-1 26.7%

      \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

   2. metadata-eval 26.7%

      \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

   3. pow-sqr 26.7%

      \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

   4. rem-sqrt-square 26.7%

      \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

   5. rem-square-sqrt 26.6%

      \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

   6. fabs-sqr 26.6%

      \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

   7. rem-square-sqrt 26.7%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

9. Simplified 26.7%

   \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

10. Final simplification 26.7%

    \[\leadsto d \cdot {\left(\ell \cdot h\right)}^{-0.5} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024178 
(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)))))