Henrywood and Agarwal, Equation (12)

Percentage Accurate: 67.4% → 75.5%

Time: 25.2s

Alternatives: 19

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.4% 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: 75.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right) + -1\right)\\ \mathbf{if}\;d \leq -3.9 \cdot 10^{+54}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -3.1 \cdot 10^{-167}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot {\left({\left(\frac{-1}{\ell}\right)}^{0.25} \cdot {\left(\frac{-1}{d}\right)}^{-0.25}\right)}^{2}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* d (sqrt (/ 1.0 (* h l))))
          (+ (* (pow (* M (* 0.5 (/ D d))) 2.0) (* 0.5 (/ h l))) -1.0))))
   (if (<= d -3.9e+54)
     t_0
     (if (<= d -3.1e-167)
       (*
        (*
         (sqrt (/ d h))
         (pow (* (pow (/ -1.0 l) 0.25) (pow (/ -1.0 d) -0.25)) 2.0))
        (- 1.0 (/ (* h (* 0.5 (pow (/ (* D (* M 0.5)) d) 2.0))) l)))
       (if (<= d -5e-310)
         t_0
         (*
          (/ (sqrt d) (sqrt h))
          (*
           (sqrt (/ d l))
           (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D d) (/ M 2.0)) 2.0)))))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (d * sqrt((1.0 / (h * l)))) * ((pow((M * (0.5 * (D / d))), 2.0) * (0.5 * (h / l))) + -1.0);
	double tmp;
	if (d <= -3.9e+54) {
		tmp = t_0;
	} else if (d <= -3.1e-167) {
		tmp = (sqrt((d / h)) * pow((pow((-1.0 / l), 0.25) * pow((-1.0 / d), -0.25)), 2.0)) * (1.0 - ((h * (0.5 * pow(((D * (M * 0.5)) / d), 2.0))) / l));
	} else if (d <= -5e-310) {
		tmp = t_0;
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * pow(((D / d) * (M / 2.0)), 2.0)))));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (d * sqrt((1.0d0 / (h * l)))) * ((((m * (0.5d0 * (d_1 / d))) ** 2.0d0) * (0.5d0 * (h / l))) + (-1.0d0))
    if (d <= (-3.9d+54)) then
        tmp = t_0
    else if (d <= (-3.1d-167)) then
        tmp = (sqrt((d / h)) * (((((-1.0d0) / l) ** 0.25d0) * (((-1.0d0) / d) ** (-0.25d0))) ** 2.0d0)) * (1.0d0 - ((h * (0.5d0 * (((d_1 * (m * 0.5d0)) / d) ** 2.0d0))) / l))
    else if (d <= (-5d-310)) then
        tmp = t_0
    else
        tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * (1.0d0 - (0.5d0 * ((h / l) * (((d_1 / d) * (m / 2.0d0)) ** 2.0d0)))))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (d * Math.sqrt((1.0 / (h * l)))) * ((Math.pow((M * (0.5 * (D / d))), 2.0) * (0.5 * (h / l))) + -1.0);
	double tmp;
	if (d <= -3.9e+54) {
		tmp = t_0;
	} else if (d <= -3.1e-167) {
		tmp = (Math.sqrt((d / h)) * Math.pow((Math.pow((-1.0 / l), 0.25) * Math.pow((-1.0 / d), -0.25)), 2.0)) * (1.0 - ((h * (0.5 * Math.pow(((D * (M * 0.5)) / d), 2.0))) / l));
	} else if (d <= -5e-310) {
		tmp = t_0;
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * (Math.sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * Math.pow(((D / d) * (M / 2.0)), 2.0)))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = (d * math.sqrt((1.0 / (h * l)))) * ((math.pow((M * (0.5 * (D / d))), 2.0) * (0.5 * (h / l))) + -1.0)
	tmp = 0
	if d <= -3.9e+54:
		tmp = t_0
	elif d <= -3.1e-167:
		tmp = (math.sqrt((d / h)) * math.pow((math.pow((-1.0 / l), 0.25) * math.pow((-1.0 / d), -0.25)), 2.0)) * (1.0 - ((h * (0.5 * math.pow(((D * (M * 0.5)) / d), 2.0))) / l))
	elif d <= -5e-310:
		tmp = t_0
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * (math.sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * math.pow(((D / d) * (M / 2.0)), 2.0)))))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(Float64((Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0) * Float64(0.5 * Float64(h / l))) + -1.0))
	tmp = 0.0
	if (d <= -3.9e+54)
		tmp = t_0;
	elseif (d <= -3.1e-167)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * (Float64((Float64(-1.0 / l) ^ 0.25) * (Float64(-1.0 / d) ^ -0.25)) ^ 2.0)) * Float64(1.0 - Float64(Float64(h * Float64(0.5 * (Float64(Float64(D * Float64(M * 0.5)) / d) ^ 2.0))) / l)));
	elseif (d <= -5e-310)
		tmp = t_0;
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0))))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (d * sqrt((1.0 / (h * l)))) * ((((M * (0.5 * (D / d))) ^ 2.0) * (0.5 * (h / l))) + -1.0);
	tmp = 0.0;
	if (d <= -3.9e+54)
		tmp = t_0;
	elseif (d <= -3.1e-167)
		tmp = (sqrt((d / h)) * ((((-1.0 / l) ^ 0.25) * ((-1.0 / d) ^ -0.25)) ^ 2.0)) * (1.0 - ((h * (0.5 * (((D * (M * 0.5)) / d) ^ 2.0))) / l));
	elseif (d <= -5e-310)
		tmp = t_0;
	else
		tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * (((D / d) * (M / 2.0)) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.9e+54], t$95$0, If[LessEqual[d, -3.1e-167], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Power[N[(N[Power[N[(-1.0 / l), $MachinePrecision], 0.25], $MachinePrecision] * N[Power[N[(-1.0 / d), $MachinePrecision], -0.25], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h * N[(0.5 * N[Power[N[(N[(D * N[(M * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], t$95$0, N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.9000000000000003e54 or -3.1e-167 < d < -4.999999999999985e-310

   1. Initial program 57.0%

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

      1. metadata-eval 57.0%

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

      2. unpow1/2 57.0%

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

      3. metadata-eval 57.0%

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

      4. unpow1/2 57.0%

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

      5. *-commutative 57.0%

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

      6. associate-*l* 57.0%

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

      7. times-frac 57.6%

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

      8. metadata-eval 57.6%

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

   3. Simplified 57.6%

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

      1. expm1-log1p-u 29.1%

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

      2. expm1-udef 27.0%

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

   5. Applied egg-rr 24.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.5\right)\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 25.3%

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

      2. expm1-log1p 48.8%

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

      3. sub-neg 48.8%

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

      4. *-commutative 48.8%

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

      5. sub-neg 48.8%

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

      6. associate-*l* 48.8%

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

      7. *-commutative 48.8%

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

   7. Simplified 48.8%

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

   8. Taylor expanded in d around -inf 78.2%

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

      1. mul-1-neg 78.2%

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

      2. *-commutative 78.2%

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

      3. distribute-rgt-neg-in 78.2%

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

      4. *-commutative 78.2%

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

   10. Simplified 78.2%

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

   if -3.9000000000000003e54 < d < -3.1e-167

   1. Initial program 78.6%

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

      1. metadata-eval 78.6%

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

      2. unpow1/2 78.6%

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

      3. metadata-eval 78.6%

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

      4. unpow1/2 78.6%

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

      5. *-commutative 78.6%

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

      6. associate-*l* 78.6%

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

      7. times-frac 78.7%

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

      8. metadata-eval 78.7%

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

   3. Simplified 78.7%

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

      1. associate-*r* 78.7%

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

      2. frac-times 78.6%

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

      3. *-commutative 78.6%

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

      4. metadata-eval 78.6%

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

      5. associate-*r/ 82.6%

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

      6. metadata-eval 82.6%

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

      7. *-commutative 82.6%

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

      8. frac-times 82.6%

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

      9. div-inv 82.6%

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

      10. metadata-eval 82.6%

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

   5. Applied egg-rr 82.6%

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

      1. metadata-eval 82.6%

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

      2. div-inv 82.6%

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

      3. associate-*r/ 82.6%

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

      4. div-inv 82.6%

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

      5. metadata-eval 82.6%

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

   7. Applied egg-rr 82.6%

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

      1. pow1/2 82.6%

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

      2. sqr-pow 82.5%

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

      3. pow2 82.5%

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

      4. metadata-eval 82.5%

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

   9. Applied egg-rr 82.5%

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

   10. Taylor expanded in d around -inf 86.5%

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

       1. distribute-lft-in 86.5%

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

       2. exp-sum 86.8%

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

       3. *-commutative 86.8%

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

       4. exp-to-pow 87.7%

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

       5. *-commutative 87.7%

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

       6. *-commutative 87.7%

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

       7. associate-*l* 87.7%

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

       8. metadata-eval 87.7%

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

       9. metadata-eval 87.7%

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

       10. metadata-eval 87.7%

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

       11. exp-to-pow 89.6%

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

       12. metadata-eval 89.6%

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

       13. metadata-eval 89.6%

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

   12. Simplified 89.6%

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

   if -4.999999999999985e-310 < d

   1. Initial program 58.1%

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

      1. associate-*l* 58.1%

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

      2. metadata-eval 58.1%

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

      3. unpow1/2 58.1%

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

      4. metadata-eval 58.1%

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

      5. unpow1/2 58.1%

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

      6. associate-*l* 58.1%

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

      7. metadata-eval 58.1%

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

      8. times-frac 57.3%

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

   3. Simplified 57.3%

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

      1. sqrt-div 68.1%

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

   5. Applied egg-rr 68.1%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.9 \cdot 10^{+54}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right) + -1\right)\\ \mathbf{elif}\;d \leq -3.1 \cdot 10^{-167}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot {\left({\left(\frac{-1}{\ell}\right)}^{0.25} \cdot {\left(\frac{-1}{d}\right)}^{-0.25}\right)}^{2}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \end{array} \]

Alternative 2: 73.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (l <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((((m * (0.5d0 * (d_1 / d))) ** 2.0d0) * (0.5d0 * (h / l))) + (-1.0d0))
    else
        tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * (1.0d0 - (0.5d0 * ((h / l) * (((d_1 / d) * (m / 2.0d0)) ** 2.0d0)))))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 64.8%

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

      1. metadata-eval 64.8%

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

      2. unpow1/2 64.8%

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

      3. metadata-eval 64.8%

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

      4. unpow1/2 64.8%

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

      5. *-commutative 64.8%

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

      6. associate-*l* 64.8%

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

      7. times-frac 65.2%

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

      8. metadata-eval 65.2%

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

   3. Simplified 65.2%

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

      1. expm1-log1p-u 28.6%

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

      2. expm1-udef 21.4%

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

   5. Applied egg-rr 19.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.5\right)\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 24.8%

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

      2. expm1-log1p 53.9%

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

      3. sub-neg 53.9%

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

      4. *-commutative 53.9%

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

      5. sub-neg 53.9%

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

      6. associate-*l* 53.9%

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

      7. *-commutative 53.9%

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

   7. Simplified 53.9%

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

   8. Taylor expanded in d around -inf 75.6%

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

      1. mul-1-neg 75.6%

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

      2. *-commutative 75.6%

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

      3. distribute-rgt-neg-in 75.6%

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

      4. *-commutative 75.6%

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

   10. Simplified 75.6%

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

   if -4.999999999999985e-310 < l

   1. Initial program 58.1%

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

      1. associate-*l* 58.1%

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

      2. metadata-eval 58.1%

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

      3. unpow1/2 58.1%

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

      4. metadata-eval 58.1%

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

      5. unpow1/2 58.1%

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

      6. associate-*l* 58.1%

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

      7. metadata-eval 58.1%

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

      8. times-frac 57.3%

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

   3. Simplified 57.3%

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

      1. sqrt-div 68.1%

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

   5. Applied egg-rr 68.1%

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

4. Final simplification 72.1%

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

Alternative 3: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.1 \cdot 10^{-290}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.1e-290)
   (*
    (* d (sqrt (/ 1.0 (* h l))))
    (+ (* (pow (* M (* 0.5 (/ D d))) 2.0) (* 0.5 (/ h l))) -1.0))
   (if (<= l 1.15e+95)
     (*
      (sqrt (/ d h))
      (*
       (sqrt (/ d l))
       (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D d) (/ M 2.0)) 2.0))))))
     (* d (* (pow h -0.5) (pow l -0.5))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.1e-290) {
		tmp = (d * sqrt((1.0 / (h * l)))) * ((pow((M * (0.5 * (D / d))), 2.0) * (0.5 * (h / l))) + -1.0);
	} else if (l <= 1.15e+95) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * pow(((D / d) * (M / 2.0)), 2.0)))));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (l <= (-1.1d-290)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((((m * (0.5d0 * (d_1 / d))) ** 2.0d0) * (0.5d0 * (h / l))) + (-1.0d0))
    else if (l <= 1.15d+95) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 - (0.5d0 * ((h / l) * (((d_1 / d) * (m / 2.0d0)) ** 2.0d0)))))
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.1e-290) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * ((Math.pow((M * (0.5 * (D / d))), 2.0) * (0.5 * (h / l))) + -1.0);
	} else if (l <= 1.15e+95) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * Math.pow(((D / d) * (M / 2.0)), 2.0)))));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -1.1e-290:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * ((math.pow((M * (0.5 * (D / d))), 2.0) * (0.5 * (h / l))) + -1.0)
	elif l <= 1.15e+95:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * math.pow(((D / d) * (M / 2.0)), 2.0)))))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.1e-290)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(Float64((Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0) * Float64(0.5 * Float64(h / l))) + -1.0));
	elseif (l <= 1.15e+95)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0))))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1.1e-290)
		tmp = (d * sqrt((1.0 / (h * l)))) * ((((M * (0.5 * (D / d))) ^ 2.0) * (0.5 * (h / l))) + -1.0);
	elseif (l <= 1.15e+95)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * (((D / d) * (M / 2.0)) ^ 2.0)))));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.1e-290], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.15e+95], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.1e-290

   1. Initial program 64.3%

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

      1. metadata-eval 64.3%

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

      2. unpow1/2 64.3%

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

      3. metadata-eval 64.3%

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

      4. unpow1/2 64.3%

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

      5. *-commutative 64.3%

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

      6. associate-*l* 64.3%

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

      7. times-frac 64.7%

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

      8. metadata-eval 64.7%

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

   3. Simplified 64.7%

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

      1. expm1-log1p-u 29.0%

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

      2. expm1-udef 21.7%

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

   5. Applied egg-rr 19.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.5\right)\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 25.1%

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

      2. expm1-log1p 53.2%

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

      3. sub-neg 53.2%

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

      4. *-commutative 53.2%

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

      5. sub-neg 53.2%

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

      6. associate-*l* 53.2%

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

      7. *-commutative 53.2%

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

   7. Simplified 53.2%

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

   8. Taylor expanded in d around -inf 75.9%

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

      1. mul-1-neg 75.9%

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

      2. *-commutative 75.9%

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

      3. distribute-rgt-neg-in 75.9%

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

      4. *-commutative 75.9%

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

   10. Simplified 75.9%

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

   if -1.1e-290 < l < 1.14999999999999999e95

   1. Initial program 77.7%

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

      1. associate-*l* 77.7%

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

      2. metadata-eval 77.7%

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

      3. unpow1/2 77.7%

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

      4. metadata-eval 77.7%

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

      5. unpow1/2 77.7%

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

      6. associate-*l* 77.7%

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

      7. metadata-eval 77.7%

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

      8. times-frac 76.4%

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

   3. Simplified 76.4%

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

   if 1.14999999999999999e95 < l

   1. Initial program 31.3%

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

      1. metadata-eval 31.3%

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

      2. unpow1/2 31.3%

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

      3. metadata-eval 31.3%

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

      4. unpow1/2 31.3%

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

      5. *-commutative 31.3%

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

      6. associate-*l* 31.3%

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

      7. times-frac 31.3%

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

      8. metadata-eval 31.3%

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

   3. Simplified 31.3%

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

   4. Taylor expanded in d around inf 44.1%

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

      1. *-un-lft-identity 44.1%

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

      2. *-commutative 44.1%

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

   6. Applied egg-rr 44.1%

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

      1. *-lft-identity 44.1%

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

      2. unpow-1 44.1%

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

      3. sqr-pow 44.1%

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

      4. rem-sqrt-square 44.1%

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

      5. sqr-pow 43.9%

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

      6. fabs-sqr 43.9%

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

      7. sqr-pow 44.1%

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

      8. metadata-eval 44.1%

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

   8. Simplified 44.1%

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

      1. unpow-prod-down 47.9%

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

   10. Applied egg-rr 47.9%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.1 \cdot 10^{-290}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 4: 69.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7.5 \cdot 10^{-291}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -7.5e-291)
   (*
    (* d (sqrt (/ 1.0 (* h l))))
    (+ (* (pow (* M (* 0.5 (/ D d))) 2.0) (* 0.5 (/ h l))) -1.0))
   (if (<= l 1.6e+95)
     (*
      (sqrt (/ d h))
      (*
       (sqrt (/ d l))
       (- 1.0 (* 0.5 (* (/ h l) (pow (/ (/ (* M D) 2.0) d) 2.0))))))
     (* d (* (pow h -0.5) (pow l -0.5))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -7.5e-291) {
		tmp = (d * sqrt((1.0 / (h * l)))) * ((pow((M * (0.5 * (D / d))), 2.0) * (0.5 * (h / l))) + -1.0);
	} else if (l <= 1.6e+95) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * pow((((M * D) / 2.0) / d), 2.0)))));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (l <= (-7.5d-291)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((((m * (0.5d0 * (d_1 / d))) ** 2.0d0) * (0.5d0 * (h / l))) + (-1.0d0))
    else if (l <= 1.6d+95) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 - (0.5d0 * ((h / l) * ((((m * d_1) / 2.0d0) / d) ** 2.0d0)))))
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -7.5e-291) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * ((Math.pow((M * (0.5 * (D / d))), 2.0) * (0.5 * (h / l))) + -1.0);
	} else if (l <= 1.6e+95) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * Math.pow((((M * D) / 2.0) / d), 2.0)))));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -7.5e-291:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * ((math.pow((M * (0.5 * (D / d))), 2.0) * (0.5 * (h / l))) + -1.0)
	elif l <= 1.6e+95:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * math.pow((((M * D) / 2.0) / d), 2.0)))))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -7.5e-291)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(Float64((Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0) * Float64(0.5 * Float64(h / l))) + -1.0));
	elseif (l <= 1.6e+95)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(Float64(M * D) / 2.0) / d) ^ 2.0))))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -7.5e-291)
		tmp = (d * sqrt((1.0 / (h * l)))) * ((((M * (0.5 * (D / d))) ^ 2.0) * (0.5 * (h / l))) + -1.0);
	elseif (l <= 1.6e+95)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * ((((M * D) / 2.0) / d) ^ 2.0)))));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -7.5e-291], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.6e+95], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(N[(M * D), $MachinePrecision] / 2.0), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -7.49999999999999981e-291

   1. Initial program 64.3%

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

      1. metadata-eval 64.3%

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

      2. unpow1/2 64.3%

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

      3. metadata-eval 64.3%

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

      4. unpow1/2 64.3%

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

      5. *-commutative 64.3%

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

      6. associate-*l* 64.3%

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

      7. times-frac 64.7%

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

      8. metadata-eval 64.7%

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

   3. Simplified 64.7%

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

      1. expm1-log1p-u 29.0%

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

      2. expm1-udef 21.7%

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

   5. Applied egg-rr 19.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.5\right)\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 25.1%

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

      2. expm1-log1p 53.2%

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

      3. sub-neg 53.2%

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

      4. *-commutative 53.2%

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

      5. sub-neg 53.2%

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

      6. associate-*l* 53.2%

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

      7. *-commutative 53.2%

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

   7. Simplified 53.2%

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

   8. Taylor expanded in d around -inf 75.9%

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

      1. mul-1-neg 75.9%

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

      2. *-commutative 75.9%

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

      3. distribute-rgt-neg-in 75.9%

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

      4. *-commutative 75.9%

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

   10. Simplified 75.9%

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

   if -7.49999999999999981e-291 < l < 1.6e95

   1. Initial program 77.7%

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

      1. associate-*l* 77.7%

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

      2. metadata-eval 77.7%

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

      3. unpow1/2 77.7%

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

      4. metadata-eval 77.7%

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

      5. unpow1/2 77.7%

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

      6. associate-*l* 77.7%

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

      7. metadata-eval 77.7%

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

      8. times-frac 76.4%

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

   3. Simplified 76.4%

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

      1. frac-times 77.7%

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

      2. associate-/r* 77.7%

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

   5. Applied egg-rr 77.7%

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

   if 1.6e95 < l

   1. Initial program 31.3%

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

      1. metadata-eval 31.3%

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

      2. unpow1/2 31.3%

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

      3. metadata-eval 31.3%

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

      4. unpow1/2 31.3%

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

      5. *-commutative 31.3%

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

      6. associate-*l* 31.3%

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

      7. times-frac 31.3%

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

      8. metadata-eval 31.3%

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

   3. Simplified 31.3%

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

   4. Taylor expanded in d around inf 44.1%

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

      1. *-un-lft-identity 44.1%

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

      2. *-commutative 44.1%

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

   6. Applied egg-rr 44.1%

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

      1. *-lft-identity 44.1%

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

      2. unpow-1 44.1%

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

      3. sqr-pow 44.1%

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

      4. rem-sqrt-square 44.1%

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

      5. sqr-pow 43.9%

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

      6. fabs-sqr 43.9%

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

      7. sqr-pow 44.1%

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

      8. metadata-eval 44.1%

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

   8. Simplified 44.1%

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

      1. unpow-prod-down 47.9%

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

   10. Applied egg-rr 47.9%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.5 \cdot 10^{-291}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 5: 70.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{-290}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+95}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.4e-290)
   (*
    (* d (sqrt (/ 1.0 (* h l))))
    (+ (* (pow (* M (* 0.5 (/ D d))) 2.0) (* 0.5 (/ h l))) -1.0))
   (if (<= l 1.6e+95)
     (*
      (* (sqrt (/ d h)) (sqrt (/ d l)))
      (- 1.0 (/ (* h (* 0.5 (pow (* (/ D d) (* M 0.5)) 2.0))) l)))
     (* d (* (pow h -0.5) (pow l -0.5))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.4e-290) {
		tmp = (d * sqrt((1.0 / (h * l)))) * ((pow((M * (0.5 * (D / d))), 2.0) * (0.5 * (h / l))) + -1.0);
	} else if (l <= 1.6e+95) {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - ((h * (0.5 * pow(((D / d) * (M * 0.5)), 2.0))) / l));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (l <= (-1.4d-290)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((((m * (0.5d0 * (d_1 / d))) ** 2.0d0) * (0.5d0 * (h / l))) + (-1.0d0))
    else if (l <= 1.6d+95) then
        tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0d0 - ((h * (0.5d0 * (((d_1 / d) * (m * 0.5d0)) ** 2.0d0))) / l))
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.4e-290) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * ((Math.pow((M * (0.5 * (D / d))), 2.0) * (0.5 * (h / l))) + -1.0);
	} else if (l <= 1.6e+95) {
		tmp = (Math.sqrt((d / h)) * Math.sqrt((d / l))) * (1.0 - ((h * (0.5 * Math.pow(((D / d) * (M * 0.5)), 2.0))) / l));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -1.4e-290:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * ((math.pow((M * (0.5 * (D / d))), 2.0) * (0.5 * (h / l))) + -1.0)
	elif l <= 1.6e+95:
		tmp = (math.sqrt((d / h)) * math.sqrt((d / l))) * (1.0 - ((h * (0.5 * math.pow(((D / d) * (M * 0.5)), 2.0))) / l))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.4e-290)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(Float64((Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0) * Float64(0.5 * Float64(h / l))) + -1.0));
	elseif (l <= 1.6e+95)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(Float64(h * Float64(0.5 * (Float64(Float64(D / d) * Float64(M * 0.5)) ^ 2.0))) / l)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1.4e-290)
		tmp = (d * sqrt((1.0 / (h * l)))) * ((((M * (0.5 * (D / d))) ^ 2.0) * (0.5 * (h / l))) + -1.0);
	elseif (l <= 1.6e+95)
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - ((h * (0.5 * (((D / d) * (M * 0.5)) ^ 2.0))) / l));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.4e-290], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.6e+95], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h * N[(0.5 * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.39999999999999998e-290

   1. Initial program 64.3%

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

      1. metadata-eval 64.3%

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

      2. unpow1/2 64.3%

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

      3. metadata-eval 64.3%

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

      4. unpow1/2 64.3%

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

      5. *-commutative 64.3%

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

      6. associate-*l* 64.3%

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

      7. times-frac 64.7%

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

      8. metadata-eval 64.7%

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

   3. Simplified 64.7%

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

      1. expm1-log1p-u 29.0%

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

      2. expm1-udef 21.7%

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

   5. Applied egg-rr 19.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.5\right)\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 25.1%

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

      2. expm1-log1p 53.2%

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

      3. sub-neg 53.2%

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

      4. *-commutative 53.2%

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

      5. sub-neg 53.2%

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

      6. associate-*l* 53.2%

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

      7. *-commutative 53.2%

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

   7. Simplified 53.2%

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

   8. Taylor expanded in d around -inf 75.9%

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

      1. mul-1-neg 75.9%

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

      2. *-commutative 75.9%

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

      3. distribute-rgt-neg-in 75.9%

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

      4. *-commutative 75.9%

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

   10. Simplified 75.9%

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

   if -1.39999999999999998e-290 < l < 1.6e95

   1. Initial program 77.7%

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

      1. metadata-eval 77.7%

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

      2. unpow1/2 77.7%

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

      3. metadata-eval 77.7%

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

      4. unpow1/2 77.7%

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

      5. *-commutative 77.7%

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

      6. associate-*l* 77.7%

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

      7. times-frac 76.4%

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

      8. metadata-eval 76.4%

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

   3. Simplified 76.4%

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

      1. associate-*r* 76.4%

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

      2. frac-times 77.7%

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

      3. *-commutative 77.7%

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

      4. metadata-eval 77.7%

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

      5. associate-*r/ 82.1%

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

      6. metadata-eval 82.1%

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

      7. *-commutative 82.1%

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

      8. frac-times 80.8%

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

      9. div-inv 80.8%

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

      10. metadata-eval 80.8%

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

   5. Applied egg-rr 80.8%

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

   if 1.6e95 < l

   1. Initial program 31.3%

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

      1. metadata-eval 31.3%

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

      2. unpow1/2 31.3%

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

      3. metadata-eval 31.3%

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

      4. unpow1/2 31.3%

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

      5. *-commutative 31.3%

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

      6. associate-*l* 31.3%

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

      7. times-frac 31.3%

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

      8. metadata-eval 31.3%

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

   3. Simplified 31.3%

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

   4. Taylor expanded in d around inf 44.1%

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

      1. *-un-lft-identity 44.1%

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

      2. *-commutative 44.1%

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

   6. Applied egg-rr 44.1%

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

      1. *-lft-identity 44.1%

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

      2. unpow-1 44.1%

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

      3. sqr-pow 44.1%

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

      4. rem-sqrt-square 44.1%

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

      5. sqr-pow 43.9%

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

      6. fabs-sqr 43.9%

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

      7. sqr-pow 44.1%

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

      8. metadata-eval 44.1%

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

   8. Simplified 44.1%

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

      1. unpow-prod-down 47.9%

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

   10. Applied egg-rr 47.9%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{-290}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+95}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 6: 70.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -9.5 \cdot 10^{-291}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+95}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2}\right)}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -9.5e-291)
   (*
    (* d (sqrt (/ 1.0 (* h l))))
    (+ (* (pow (* M (* 0.5 (/ D d))) 2.0) (* 0.5 (/ h l))) -1.0))
   (if (<= l 1.6e+95)
     (*
      (- 1.0 (/ (* h (* 0.5 (pow (/ (* D (* M 0.5)) d) 2.0))) l))
      (* (sqrt (/ d h)) (sqrt (/ d l))))
     (* d (* (pow h -0.5) (pow l -0.5))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -9.5e-291) {
		tmp = (d * sqrt((1.0 / (h * l)))) * ((pow((M * (0.5 * (D / d))), 2.0) * (0.5 * (h / l))) + -1.0);
	} else if (l <= 1.6e+95) {
		tmp = (1.0 - ((h * (0.5 * pow(((D * (M * 0.5)) / d), 2.0))) / l)) * (sqrt((d / h)) * sqrt((d / l)));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (l <= (-9.5d-291)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((((m * (0.5d0 * (d_1 / d))) ** 2.0d0) * (0.5d0 * (h / l))) + (-1.0d0))
    else if (l <= 1.6d+95) then
        tmp = (1.0d0 - ((h * (0.5d0 * (((d_1 * (m * 0.5d0)) / d) ** 2.0d0))) / l)) * (sqrt((d / h)) * sqrt((d / l)))
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -9.5e-291) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * ((Math.pow((M * (0.5 * (D / d))), 2.0) * (0.5 * (h / l))) + -1.0);
	} else if (l <= 1.6e+95) {
		tmp = (1.0 - ((h * (0.5 * Math.pow(((D * (M * 0.5)) / d), 2.0))) / l)) * (Math.sqrt((d / h)) * Math.sqrt((d / l)));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -9.5e-291:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * ((math.pow((M * (0.5 * (D / d))), 2.0) * (0.5 * (h / l))) + -1.0)
	elif l <= 1.6e+95:
		tmp = (1.0 - ((h * (0.5 * math.pow(((D * (M * 0.5)) / d), 2.0))) / l)) * (math.sqrt((d / h)) * math.sqrt((d / l)))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -9.5e-291)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(Float64((Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0) * Float64(0.5 * Float64(h / l))) + -1.0));
	elseif (l <= 1.6e+95)
		tmp = Float64(Float64(1.0 - Float64(Float64(h * Float64(0.5 * (Float64(Float64(D * Float64(M * 0.5)) / d) ^ 2.0))) / l)) * Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -9.5e-291)
		tmp = (d * sqrt((1.0 / (h * l)))) * ((((M * (0.5 * (D / d))) ^ 2.0) * (0.5 * (h / l))) + -1.0);
	elseif (l <= 1.6e+95)
		tmp = (1.0 - ((h * (0.5 * (((D * (M * 0.5)) / d) ^ 2.0))) / l)) * (sqrt((d / h)) * sqrt((d / l)));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -9.5e-291], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.6e+95], N[(N[(1.0 - N[(N[(h * N[(0.5 * N[Power[N[(N[(D * N[(M * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.49999999999999938e-291

   1. Initial program 64.3%

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

      1. metadata-eval 64.3%

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

      2. unpow1/2 64.3%

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

      3. metadata-eval 64.3%

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

      4. unpow1/2 64.3%

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

      5. *-commutative 64.3%

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

      6. associate-*l* 64.3%

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

      7. times-frac 64.7%

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

      8. metadata-eval 64.7%

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

   3. Simplified 64.7%

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

      1. expm1-log1p-u 29.0%

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

      2. expm1-udef 21.7%

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

   5. Applied egg-rr 19.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.5\right)\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 25.1%

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

      2. expm1-log1p 53.2%

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

      3. sub-neg 53.2%

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

      4. *-commutative 53.2%

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

      5. sub-neg 53.2%

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

      6. associate-*l* 53.2%

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

      7. *-commutative 53.2%

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

   7. Simplified 53.2%

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

   8. Taylor expanded in d around -inf 75.9%

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

      1. mul-1-neg 75.9%

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

      2. *-commutative 75.9%

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

      3. distribute-rgt-neg-in 75.9%

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

      4. *-commutative 75.9%

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

   10. Simplified 75.9%

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

   if -9.49999999999999938e-291 < l < 1.6e95

   1. Initial program 77.7%

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

      1. metadata-eval 77.7%

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

      2. unpow1/2 77.7%

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

      3. metadata-eval 77.7%

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

      4. unpow1/2 77.7%

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

      5. *-commutative 77.7%

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

      6. associate-*l* 77.7%

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

      7. times-frac 76.4%

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

      8. metadata-eval 76.4%

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

   3. Simplified 76.4%

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

      1. associate-*r* 76.4%

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

      2. frac-times 77.7%

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

      3. *-commutative 77.7%

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

      4. metadata-eval 77.7%

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

      5. associate-*r/ 82.1%

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

      6. metadata-eval 82.1%

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

      7. *-commutative 82.1%

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

      8. frac-times 80.8%

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

      9. div-inv 80.8%

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

      10. metadata-eval 80.8%

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

   5. Applied egg-rr 80.8%

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

      1. metadata-eval 80.8%

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

      2. div-inv 80.8%

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

      3. associate-*r/ 82.1%

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

      4. div-inv 82.1%

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

      5. metadata-eval 82.1%

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

   7. Applied egg-rr 82.1%

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

   if 1.6e95 < l

   1. Initial program 31.3%

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

      1. metadata-eval 31.3%

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

      2. unpow1/2 31.3%

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

      3. metadata-eval 31.3%

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

      4. unpow1/2 31.3%

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

      5. *-commutative 31.3%

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

      6. associate-*l* 31.3%

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

      7. times-frac 31.3%

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

      8. metadata-eval 31.3%

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

   3. Simplified 31.3%

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

   4. Taylor expanded in d around inf 44.1%

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

      1. *-un-lft-identity 44.1%

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

      2. *-commutative 44.1%

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

   6. Applied egg-rr 44.1%

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

      1. *-lft-identity 44.1%

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

      2. unpow-1 44.1%

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

      3. sqr-pow 44.1%

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

      4. rem-sqrt-square 44.1%

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

      5. sqr-pow 43.9%

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

      6. fabs-sqr 43.9%

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

      7. sqr-pow 44.1%

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

      8. metadata-eval 44.1%

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

   8. Simplified 44.1%

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

      1. unpow-prod-down 47.9%

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

   10. Applied egg-rr 47.9%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9.5 \cdot 10^{-291}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+95}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2}\right)}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 7: 66.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\\ \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(t_0 \cdot \left(0.5 \cdot \frac{h}{\ell}\right) + -1\right)\\ \mathbf{elif}\;d \leq 10^{-157}:\\ \;\;\;\;-0.125 \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-53}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \frac{M}{\frac{\ell}{M}}\right)\right)\right)\\ \mathbf{elif}\;d \leq 6.7 \cdot 10^{+184}:\\ \;\;\;\;\sqrt{\frac{d}{\frac{h \cdot \ell}{d}}} \cdot \left(1 - \frac{0.5}{\frac{\frac{\ell}{h}}{t_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (* M (* 0.5 (/ D d))) 2.0)))
   (if (<= d -5e-310)
     (* (* d (sqrt (/ 1.0 (* h l)))) (+ (* t_0 (* 0.5 (/ h l))) -1.0))
     (if (<= d 1e-157)
       (* -0.125 (/ (* (* M D) (* M D)) (/ d (sqrt (/ h (pow l 3.0))))))
       (if (<= d 1.9e-53)
         (*
          (* (sqrt (/ d h)) (sqrt (/ d l)))
          (- 1.0 (* 0.125 (* (* (/ D d) (/ D d)) (* h (/ M (/ l M)))))))
         (if (<= d 6.7e+184)
           (* (sqrt (/ d (/ (* h l) d))) (- 1.0 (/ 0.5 (/ (/ l h) t_0))))
           (* d (* (pow h -0.5) (pow l -0.5)))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = pow((M * (0.5 * (D / d))), 2.0);
	double tmp;
	if (d <= -5e-310) {
		tmp = (d * sqrt((1.0 / (h * l)))) * ((t_0 * (0.5 * (h / l))) + -1.0);
	} else if (d <= 1e-157) {
		tmp = -0.125 * (((M * D) * (M * D)) / (d / sqrt((h / pow(l, 3.0)))));
	} else if (d <= 1.9e-53) {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - (0.125 * (((D / d) * (D / d)) * (h * (M / (l / M))))));
	} else if (d <= 6.7e+184) {
		tmp = sqrt((d / ((h * l) / d))) * (1.0 - (0.5 / ((l / h) / t_0)));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (m * (0.5d0 * (d_1 / d))) ** 2.0d0
    if (d <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((t_0 * (0.5d0 * (h / l))) + (-1.0d0))
    else if (d <= 1d-157) then
        tmp = (-0.125d0) * (((m * d_1) * (m * d_1)) / (d / sqrt((h / (l ** 3.0d0)))))
    else if (d <= 1.9d-53) then
        tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0d0 - (0.125d0 * (((d_1 / d) * (d_1 / d)) * (h * (m / (l / m))))))
    else if (d <= 6.7d+184) then
        tmp = sqrt((d / ((h * l) / d))) * (1.0d0 - (0.5d0 / ((l / h) / t_0)))
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.pow((M * (0.5 * (D / d))), 2.0);
	double tmp;
	if (d <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * ((t_0 * (0.5 * (h / l))) + -1.0);
	} else if (d <= 1e-157) {
		tmp = -0.125 * (((M * D) * (M * D)) / (d / Math.sqrt((h / Math.pow(l, 3.0)))));
	} else if (d <= 1.9e-53) {
		tmp = (Math.sqrt((d / h)) * Math.sqrt((d / l))) * (1.0 - (0.125 * (((D / d) * (D / d)) * (h * (M / (l / M))))));
	} else if (d <= 6.7e+184) {
		tmp = Math.sqrt((d / ((h * l) / d))) * (1.0 - (0.5 / ((l / h) / t_0)));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.pow((M * (0.5 * (D / d))), 2.0)
	tmp = 0
	if d <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * ((t_0 * (0.5 * (h / l))) + -1.0)
	elif d <= 1e-157:
		tmp = -0.125 * (((M * D) * (M * D)) / (d / math.sqrt((h / math.pow(l, 3.0)))))
	elif d <= 1.9e-53:
		tmp = (math.sqrt((d / h)) * math.sqrt((d / l))) * (1.0 - (0.125 * (((D / d) * (D / d)) * (h * (M / (l / M))))))
	elif d <= 6.7e+184:
		tmp = math.sqrt((d / ((h * l) / d))) * (1.0 - (0.5 / ((l / h) / t_0)))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0
	tmp = 0.0
	if (d <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(Float64(t_0 * Float64(0.5 * Float64(h / l))) + -1.0));
	elseif (d <= 1e-157)
		tmp = Float64(-0.125 * Float64(Float64(Float64(M * D) * Float64(M * D)) / Float64(d / sqrt(Float64(h / (l ^ 3.0))))));
	elseif (d <= 1.9e-53)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(0.125 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(h * Float64(M / Float64(l / M)))))));
	elseif (d <= 6.7e+184)
		tmp = Float64(sqrt(Float64(d / Float64(Float64(h * l) / d))) * Float64(1.0 - Float64(0.5 / Float64(Float64(l / h) / t_0))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (M * (0.5 * (D / d))) ^ 2.0;
	tmp = 0.0;
	if (d <= -5e-310)
		tmp = (d * sqrt((1.0 / (h * l)))) * ((t_0 * (0.5 * (h / l))) + -1.0);
	elseif (d <= 1e-157)
		tmp = -0.125 * (((M * D) * (M * D)) / (d / sqrt((h / (l ^ 3.0)))));
	elseif (d <= 1.9e-53)
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - (0.125 * (((D / d) * (D / d)) * (h * (M / (l / M))))));
	elseif (d <= 6.7e+184)
		tmp = sqrt((d / ((h * l) / d))) * (1.0 - (0.5 / ((l / h) / t_0)));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[d, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1e-157], N[(-0.125 * N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(d / N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.9e-53], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6.7e+184], N[(N[Sqrt[N[(d / N[(N[(h * l), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 / N[(N[(l / h), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -4.999999999999985e-310

   1. Initial program 64.8%

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

      1. metadata-eval 64.8%

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

      2. unpow1/2 64.8%

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

      3. metadata-eval 64.8%

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

      4. unpow1/2 64.8%

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

      5. *-commutative 64.8%

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

      6. associate-*l* 64.8%

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

      7. times-frac 65.2%

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

      8. metadata-eval 65.2%

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

   3. Simplified 65.2%

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

      1. expm1-log1p-u 28.6%

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

      2. expm1-udef 21.4%

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

   5. Applied egg-rr 19.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.5\right)\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 24.8%

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

      2. expm1-log1p 53.9%

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

      3. sub-neg 53.9%

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

      4. *-commutative 53.9%

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

      5. sub-neg 53.9%

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

      6. associate-*l* 53.9%

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

      7. *-commutative 53.9%

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

   7. Simplified 53.9%

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

   8. Taylor expanded in d around -inf 75.6%

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

      1. mul-1-neg 75.6%

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

      2. *-commutative 75.6%

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

      3. distribute-rgt-neg-in 75.6%

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

      4. *-commutative 75.6%

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

   10. Simplified 75.6%

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

   if -4.999999999999985e-310 < d < 9.99999999999999943e-158

   1. Initial program 32.0%

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

      1. metadata-eval 32.0%

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

      2. unpow1/2 32.0%

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

      3. metadata-eval 32.0%

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

      4. unpow1/2 32.0%

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

      5. *-commutative 32.0%

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

      6. associate-*l* 32.0%

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

      7. times-frac 32.0%

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

      8. metadata-eval 32.0%

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

   3. Simplified 32.0%

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

      1. associate-*r* 32.0%

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

      2. frac-times 32.0%

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

      3. *-commutative 32.0%

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

      4. metadata-eval 32.0%

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

      5. associate-*r/ 32.0%

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

      6. metadata-eval 32.0%

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

      7. *-commutative 32.0%

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

      8. frac-times 32.0%

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

      9. div-inv 32.0%

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

      10. metadata-eval 32.0%

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

   5. Applied egg-rr 32.0%

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

   6. Taylor expanded in d around 0 32.1%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 34.8%

         \[\leadsto -0.125 \cdot \color{blue}{\frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}} \]

      2. associate-/l* 34.8%

         \[\leadsto -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}} \]

      3. unpow2 34.8%

         \[\leadsto -0.125 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}} \]

      4. unpow2 34.8%

         \[\leadsto -0.125 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}} \]

      5. unswap-sqr 47.0%

         \[\leadsto -0.125 \cdot \frac{\color{blue}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}} \]

   8. Simplified 47.0%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}} \]

   if 9.99999999999999943e-158 < d < 1.8999999999999999e-53

   1. Initial program 59.2%

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

      1. metadata-eval 59.2%

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

      2. unpow1/2 59.2%

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

      3. metadata-eval 59.2%

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

      4. unpow1/2 59.2%

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

      5. *-commutative 59.2%

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

      6. associate-*l* 59.2%

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

      7. times-frac 55.4%

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

      8. metadata-eval 55.4%

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

   3. Simplified 55.4%

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

      1. associate-*r* 55.4%

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

      2. frac-times 59.2%

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

      3. *-commutative 59.2%

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

      4. metadata-eval 59.2%

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

      5. associate-*r/ 63.6%

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

      6. metadata-eval 63.6%

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

      7. *-commutative 63.6%

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

      8. frac-times 59.7%

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

      9. div-inv 59.7%

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

      10. metadata-eval 59.7%

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

   5. Applied egg-rr 59.7%

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

      1. metadata-eval 59.7%

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

      2. div-inv 59.7%

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

      3. associate-*r/ 63.6%

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

      4. div-inv 63.6%

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

      5. metadata-eval 63.6%

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

   7. Applied egg-rr 63.6%

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

   8. Taylor expanded in M around 0 46.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   9. Step-by-step derivation

      1. *-commutative 46.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right) \]

      2. times-frac 47.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \]

      3. unpow2 47.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \]

      4. unpow2 47.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \]

      5. times-frac 47.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right) \]

      6. associate-*l/ 46.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(\frac{{M}^{2}}{\ell} \cdot h\right)}\right)\right) \]

      7. *-commutative 46.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(h \cdot \frac{{M}^{2}}{\ell}\right)}\right)\right) \]

      8. unpow2 46.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \frac{\color{blue}{M \cdot M}}{\ell}\right)\right)\right) \]

      9. associate-/l* 51.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \color{blue}{\frac{M}{\frac{\ell}{M}}}\right)\right)\right) \]

   10. Simplified 51.7%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \frac{M}{\frac{\ell}{M}}\right)\right)}\right) \]

   if 1.8999999999999999e-53 < d < 6.7e184

   1. Initial program 77.9%

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

      1. metadata-eval 77.9%

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

      2. unpow1/2 77.9%

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

      3. metadata-eval 77.9%

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

      4. unpow1/2 77.9%

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

      5. *-commutative 77.9%

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

      6. associate-*l* 77.9%

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

      7. times-frac 77.9%

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

      8. metadata-eval 77.9%

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

   3. Simplified 77.9%

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

      1. associate-*r* 77.9%

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

      2. frac-times 77.9%

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

      3. *-commutative 77.9%

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

      4. metadata-eval 77.9%

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

      5. associate-*r/ 73.2%

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

      6. metadata-eval 73.2%

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

      7. *-commutative 73.2%

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

      8. frac-times 73.2%

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

      9. div-inv 73.2%

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

      10. metadata-eval 73.2%

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

   5. Applied egg-rr 73.2%

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

      1. metadata-eval 73.2%

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

      2. div-inv 73.2%

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

      3. associate-*r/ 73.2%

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

      4. div-inv 73.2%

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

      5. metadata-eval 73.2%

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

   7. Applied egg-rr 73.2%

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

      1. pow1 73.2%

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

      2. sqrt-unprod 65.8%

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

      3. *-commutative 65.8%

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

      4. frac-times 73.0%

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

      5. *-commutative 73.0%

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

      6. associate-/l* 75.5%

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

      7. *-commutative 75.5%

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

      8. associate-*r/ 75.5%

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

      9. associate-*r* 75.5%

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

   9. Applied egg-rr 75.5%

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

       1. sub-neg 75.5%

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

       2. sub-neg 75.5%

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

       3. associate-/l* 73.0%

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

       4. associate-*r/ 75.4%

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

       5. unpow1 75.4%

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

       6. associate-/l* 75.5%

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

   11. Simplified 75.5%

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

   if 6.7e184 < d

   1. Initial program 61.2%

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

      1. metadata-eval 61.2%

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

      2. unpow1/2 61.2%

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

      3. metadata-eval 61.2%

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

      4. unpow1/2 61.2%

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

      5. *-commutative 61.2%

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

      6. associate-*l* 61.2%

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

      7. times-frac 61.2%

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

      8. metadata-eval 61.2%

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

   3. Simplified 61.2%

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

   4. Taylor expanded in d around inf 75.2%

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

      1. *-un-lft-identity 75.2%

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

      2. *-commutative 75.2%

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

   6. Applied egg-rr 75.2%

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

      1. *-lft-identity 75.2%

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

      2. unpow-1 75.2%

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

      3. sqr-pow 75.3%

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

      4. rem-sqrt-square 75.3%

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

      5. sqr-pow 75.1%

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

      6. fabs-sqr 75.1%

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

      7. sqr-pow 75.3%

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

      8. metadata-eval 75.3%

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

   8. Simplified 75.3%

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

      1. unpow-prod-down 84.9%

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

   10. Applied egg-rr 84.9%

       \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 5 regimes into one program.

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right) + -1\right)\\ \mathbf{elif}\;d \leq 10^{-157}:\\ \;\;\;\;-0.125 \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-53}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \frac{M}{\frac{\ell}{M}}\right)\right)\right)\\ \mathbf{elif}\;d \leq 6.7 \cdot 10^{+184}:\\ \;\;\;\;\sqrt{\frac{d}{\frac{h \cdot \ell}{d}}} \cdot \left(1 - \frac{0.5}{\frac{\frac{\ell}{h}}{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 8: 65.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right) + -1\right)\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{-157}:\\ \;\;\;\;-0.125 \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{+181}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{\frac{d}{0.5}}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d -5e-310)
   (*
    (* d (sqrt (/ 1.0 (* h l))))
    (+ (* (pow (* M (* 0.5 (/ D d))) 2.0) (* 0.5 (/ h l))) -1.0))
   (if (<= d 2.8e-157)
     (* -0.125 (/ (* (* M D) (* M D)) (/ d (sqrt (/ h (pow l 3.0))))))
     (if (<= d 4.2e+181)
       (*
        (sqrt (* d (/ d (* h l))))
        (+ 1.0 (* -0.5 (* (/ h l) (pow (* D (/ M (/ d 0.5))) 2.0)))))
       (* d (* (pow h -0.5) (pow l -0.5)))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -5e-310) {
		tmp = (d * sqrt((1.0 / (h * l)))) * ((pow((M * (0.5 * (D / d))), 2.0) * (0.5 * (h / l))) + -1.0);
	} else if (d <= 2.8e-157) {
		tmp = -0.125 * (((M * D) * (M * D)) / (d / sqrt((h / pow(l, 3.0)))));
	} else if (d <= 4.2e+181) {
		tmp = sqrt((d * (d / (h * l)))) * (1.0 + (-0.5 * ((h / l) * pow((D * (M / (d / 0.5))), 2.0))));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (d <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((((m * (0.5d0 * (d_1 / d))) ** 2.0d0) * (0.5d0 * (h / l))) + (-1.0d0))
    else if (d <= 2.8d-157) then
        tmp = (-0.125d0) * (((m * d_1) * (m * d_1)) / (d / sqrt((h / (l ** 3.0d0)))))
    else if (d <= 4.2d+181) then
        tmp = sqrt((d * (d / (h * l)))) * (1.0d0 + ((-0.5d0) * ((h / l) * ((d_1 * (m / (d / 0.5d0))) ** 2.0d0))))
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * ((Math.pow((M * (0.5 * (D / d))), 2.0) * (0.5 * (h / l))) + -1.0);
	} else if (d <= 2.8e-157) {
		tmp = -0.125 * (((M * D) * (M * D)) / (d / Math.sqrt((h / Math.pow(l, 3.0)))));
	} else if (d <= 4.2e+181) {
		tmp = Math.sqrt((d * (d / (h * l)))) * (1.0 + (-0.5 * ((h / l) * Math.pow((D * (M / (d / 0.5))), 2.0))));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if d <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * ((math.pow((M * (0.5 * (D / d))), 2.0) * (0.5 * (h / l))) + -1.0)
	elif d <= 2.8e-157:
		tmp = -0.125 * (((M * D) * (M * D)) / (d / math.sqrt((h / math.pow(l, 3.0)))))
	elif d <= 4.2e+181:
		tmp = math.sqrt((d * (d / (h * l)))) * (1.0 + (-0.5 * ((h / l) * math.pow((D * (M / (d / 0.5))), 2.0))))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(Float64((Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0) * Float64(0.5 * Float64(h / l))) + -1.0));
	elseif (d <= 2.8e-157)
		tmp = Float64(-0.125 * Float64(Float64(Float64(M * D) * Float64(M * D)) / Float64(d / sqrt(Float64(h / (l ^ 3.0))))));
	elseif (d <= 4.2e+181)
		tmp = Float64(sqrt(Float64(d * Float64(d / Float64(h * l)))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(D * Float64(M / Float64(d / 0.5))) ^ 2.0)))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -5e-310)
		tmp = (d * sqrt((1.0 / (h * l)))) * ((((M * (0.5 * (D / d))) ^ 2.0) * (0.5 * (h / l))) + -1.0);
	elseif (d <= 2.8e-157)
		tmp = -0.125 * (((M * D) * (M * D)) / (d / sqrt((h / (l ^ 3.0)))));
	elseif (d <= 4.2e+181)
		tmp = sqrt((d * (d / (h * l)))) * (1.0 + (-0.5 * ((h / l) * ((D * (M / (d / 0.5))) ^ 2.0))));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.8e-157], N[(-0.125 * N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(d / N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.2e+181], N[(N[Sqrt[N[(d * N[(d / N[(h * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M / N[(d / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -4.999999999999985e-310

   1. Initial program 64.8%

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

      1. metadata-eval 64.8%

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

      2. unpow1/2 64.8%

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

      3. metadata-eval 64.8%

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

      4. unpow1/2 64.8%

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

      5. *-commutative 64.8%

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

      6. associate-*l* 64.8%

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

      7. times-frac 65.2%

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

      8. metadata-eval 65.2%

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

   3. Simplified 65.2%

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

      1. expm1-log1p-u 28.6%

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

      2. expm1-udef 21.4%

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

   5. Applied egg-rr 19.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.5\right)\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 24.8%

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

      2. expm1-log1p 53.9%

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

      3. sub-neg 53.9%

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

      4. *-commutative 53.9%

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

      5. sub-neg 53.9%

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

      6. associate-*l* 53.9%

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

      7. *-commutative 53.9%

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

   7. Simplified 53.9%

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

   8. Taylor expanded in d around -inf 75.6%

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

      1. mul-1-neg 75.6%

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

      2. *-commutative 75.6%

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

      3. distribute-rgt-neg-in 75.6%

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

      4. *-commutative 75.6%

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

   10. Simplified 75.6%

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

   if -4.999999999999985e-310 < d < 2.8000000000000001e-157

   1. Initial program 32.0%

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

      1. metadata-eval 32.0%

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

      2. unpow1/2 32.0%

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

      3. metadata-eval 32.0%

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

      4. unpow1/2 32.0%

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

      5. *-commutative 32.0%

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

      6. associate-*l* 32.0%

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

      7. times-frac 32.0%

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

      8. metadata-eval 32.0%

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

   3. Simplified 32.0%

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

      1. associate-*r* 32.0%

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

      2. frac-times 32.0%

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

      3. *-commutative 32.0%

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

      4. metadata-eval 32.0%

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

      5. associate-*r/ 32.0%

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

      6. metadata-eval 32.0%

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

      7. *-commutative 32.0%

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

      8. frac-times 32.0%

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

      9. div-inv 32.0%

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

      10. metadata-eval 32.0%

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

   5. Applied egg-rr 32.0%

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

   6. Taylor expanded in d around 0 32.1%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 34.8%

         \[\leadsto -0.125 \cdot \color{blue}{\frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}} \]

      2. associate-/l* 34.8%

         \[\leadsto -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}} \]

      3. unpow2 34.8%

         \[\leadsto -0.125 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}} \]

      4. unpow2 34.8%

         \[\leadsto -0.125 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}} \]

      5. unswap-sqr 47.0%

         \[\leadsto -0.125 \cdot \frac{\color{blue}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}} \]

   8. Simplified 47.0%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}} \]

   if 2.8000000000000001e-157 < d < 4.19999999999999995e181

   1. Initial program 70.9%

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

      1. metadata-eval 70.9%

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

      2. unpow1/2 70.9%

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

      3. metadata-eval 70.9%

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

      4. unpow1/2 70.9%

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

      5. *-commutative 70.9%

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

      6. associate-*l* 70.9%

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

      7. times-frac 69.5%

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

      8. metadata-eval 69.5%

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

   3. Simplified 69.5%

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

      1. associate-*r* 69.5%

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

      2. frac-times 70.9%

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

      3. *-commutative 70.9%

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

      4. metadata-eval 70.9%

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

      5. associate-*r/ 69.6%

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

      6. metadata-eval 69.6%

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

      7. *-commutative 69.6%

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

      8. frac-times 68.1%

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

      9. div-inv 68.1%

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

      10. metadata-eval 68.1%

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

   5. Applied egg-rr 68.1%

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

      1. metadata-eval 68.1%

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

      2. div-inv 68.1%

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

      3. associate-*r/ 69.6%

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

      4. div-inv 69.6%

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

      5. metadata-eval 69.6%

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

   7. Applied egg-rr 69.6%

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

      1. pow1 69.6%

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

      2. sqrt-unprod 59.2%

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

      3. *-commutative 59.2%

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

      4. frac-times 61.1%

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

      5. *-commutative 61.1%

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

      6. associate-/l* 61.0%

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

      7. *-commutative 61.0%

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

      8. associate-*r/ 60.4%

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

      9. associate-*r* 60.4%

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

   9. Applied egg-rr 60.4%

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

       1. associate-/l* 60.4%

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

       2. associate-*r/ 60.3%

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

       3. unpow1 60.3%

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

       4. associate-/l* 60.7%

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

       5. associate-/r/ 60.6%

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

       6. associate-*l* 60.6%

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

       7. cancel-sign-sub-inv 60.6%

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

       8. metadata-eval 60.6%

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

   11. Simplified 61.2%

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

   if 4.19999999999999995e181 < d

   1. Initial program 61.2%

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

      1. metadata-eval 61.2%

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

      2. unpow1/2 61.2%

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

      3. metadata-eval 61.2%

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

      4. unpow1/2 61.2%

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

      5. *-commutative 61.2%

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

      6. associate-*l* 61.2%

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

      7. times-frac 61.2%

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

      8. metadata-eval 61.2%

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

   3. Simplified 61.2%

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

   4. Taylor expanded in d around inf 75.2%

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

      1. *-un-lft-identity 75.2%

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

      2. *-commutative 75.2%

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

   6. Applied egg-rr 75.2%

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

      1. *-lft-identity 75.2%

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

      2. unpow-1 75.2%

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

      3. sqr-pow 75.3%

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

      4. rem-sqrt-square 75.3%

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

      5. sqr-pow 75.1%

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

      6. fabs-sqr 75.1%

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

      7. sqr-pow 75.3%

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

      8. metadata-eval 75.3%

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

   8. Simplified 75.3%

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

      1. unpow-prod-down 84.9%

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

   10. Applied egg-rr 84.9%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right) + -1\right)\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{-157}:\\ \;\;\;\;-0.125 \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{+181}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{\frac{d}{0.5}}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 9: 57.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.02 \cdot 10^{-25}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(\frac{D}{2 \cdot \frac{d}{M}}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+101}:\\ \;\;\;\;-0.125 \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{+162}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l 1.02e-25)
   (*
    (sqrt (* (/ d h) (/ d l)))
    (- 1.0 (* (pow (/ D (* 2.0 (/ d M))) 2.0) (/ 0.5 (/ l h)))))
   (if (<= l 8e+101)
     (* -0.125 (/ (* (* M D) (* M D)) (/ d (sqrt (/ h (pow l 3.0))))))
     (if (<= l 1.25e+162)
       (* d (* (pow h -0.5) (pow l -0.5)))
       (* d (/ 1.0 (* (sqrt h) (sqrt l))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 1.02e-25) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 - (pow((D / (2.0 * (d / M))), 2.0) * (0.5 / (l / h))));
	} else if (l <= 8e+101) {
		tmp = -0.125 * (((M * D) * (M * D)) / (d / sqrt((h / pow(l, 3.0)))));
	} else if (l <= 1.25e+162) {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	} else {
		tmp = d * (1.0 / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (l <= 1.02d-25) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 - (((d_1 / (2.0d0 * (d / m))) ** 2.0d0) * (0.5d0 / (l / h))))
    else if (l <= 8d+101) then
        tmp = (-0.125d0) * (((m * d_1) * (m * d_1)) / (d / sqrt((h / (l ** 3.0d0)))))
    else if (l <= 1.25d+162) then
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    else
        tmp = d * (1.0d0 / (sqrt(h) * sqrt(l)))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 1.02e-25) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 - (Math.pow((D / (2.0 * (d / M))), 2.0) * (0.5 / (l / h))));
	} else if (l <= 8e+101) {
		tmp = -0.125 * (((M * D) * (M * D)) / (d / Math.sqrt((h / Math.pow(l, 3.0)))));
	} else if (l <= 1.25e+162) {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	} else {
		tmp = d * (1.0 / (Math.sqrt(h) * Math.sqrt(l)));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= 1.02e-25:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 - (math.pow((D / (2.0 * (d / M))), 2.0) * (0.5 / (l / h))))
	elif l <= 8e+101:
		tmp = -0.125 * (((M * D) * (M * D)) / (d / math.sqrt((h / math.pow(l, 3.0)))))
	elif l <= 1.25e+162:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	else:
		tmp = d * (1.0 / (math.sqrt(h) * math.sqrt(l)))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 1.02e-25)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 - Float64((Float64(D / Float64(2.0 * Float64(d / M))) ^ 2.0) * Float64(0.5 / Float64(l / h)))));
	elseif (l <= 8e+101)
		tmp = Float64(-0.125 * Float64(Float64(Float64(M * D) * Float64(M * D)) / Float64(d / sqrt(Float64(h / (l ^ 3.0))))));
	elseif (l <= 1.25e+162)
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	else
		tmp = Float64(d * Float64(1.0 / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 1.02e-25)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 - (((D / (2.0 * (d / M))) ^ 2.0) * (0.5 / (l / h))));
	elseif (l <= 8e+101)
		tmp = -0.125 * (((M * D) * (M * D)) / (d / sqrt((h / (l ^ 3.0)))));
	elseif (l <= 1.25e+162)
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	else
		tmp = d * (1.0 / (sqrt(h) * sqrt(l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, 1.02e-25], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[Power[N[(D / N[(2.0 * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8e+101], N[(-0.125 * N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(d / N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.25e+162], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(1.0 / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < 1.01999999999999998e-25

   1. Initial program 69.3%

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

      1. metadata-eval 69.3%

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

      2. unpow1/2 69.3%

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

      3. metadata-eval 69.3%

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

      4. unpow1/2 69.3%

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

      5. *-commutative 69.3%

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

      6. associate-*l* 69.3%

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

      7. times-frac 69.1%

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

      8. metadata-eval 69.1%

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

   3. Simplified 69.1%

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

      1. frac-times 69.3%

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

      2. associate-/l* 69.2%

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

      3. *-commutative 69.2%

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

   5. Applied egg-rr 69.2%

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

      1. associate-/l* 69.3%

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

      2. associate-/r* 69.3%

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

      3. *-commutative 69.3%

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

      4. associate-/l* 69.3%

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

      5. associate-/l/ 69.3%

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

   7. Simplified 69.3%

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

      1. pow1 69.3%

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

      2. sqrt-unprod 59.6%

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

      3. associate-*r/ 59.6%

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

   9. Applied egg-rr 59.6%

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

       1. unpow1 59.6%

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

       2. associate-/l* 59.6%

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

   11. Simplified 59.6%

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

   if 1.01999999999999998e-25 < l < 7.9999999999999998e101

   1. Initial program 60.7%

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

      1. metadata-eval 60.7%

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

      2. unpow1/2 60.7%

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

      3. metadata-eval 60.7%

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

      4. unpow1/2 60.7%

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

      5. *-commutative 60.7%

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

      6. associate-*l* 60.7%

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

      7. times-frac 60.7%

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

      8. metadata-eval 60.7%

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

   3. Simplified 60.7%

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

      1. associate-*r* 60.7%

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

      2. frac-times 60.7%

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

      3. *-commutative 60.7%

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

      4. metadata-eval 60.7%

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

      5. associate-*r/ 61.1%

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

      6. metadata-eval 61.1%

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

      7. *-commutative 61.1%

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

      8. frac-times 61.1%

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

      9. div-inv 61.1%

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

      10. metadata-eval 61.1%

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

   5. Applied egg-rr 61.1%

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

   6. Taylor expanded in d around 0 48.5%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 53.8%

         \[\leadsto -0.125 \cdot \color{blue}{\frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}} \]

      2. associate-/l* 53.8%

         \[\leadsto -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}} \]

      3. unpow2 53.8%

         \[\leadsto -0.125 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}} \]

      4. unpow2 53.8%

         \[\leadsto -0.125 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}} \]

      5. unswap-sqr 70.9%

         \[\leadsto -0.125 \cdot \frac{\color{blue}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}} \]

   8. Simplified 70.9%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}} \]

   if 7.9999999999999998e101 < l < 1.2499999999999999e162

   1. Initial program 27.6%

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

      1. metadata-eval 27.6%

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

      2. unpow1/2 27.6%

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

      3. metadata-eval 27.6%

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

      4. unpow1/2 27.6%

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

      5. *-commutative 27.6%

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

      6. associate-*l* 27.6%

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

      7. times-frac 27.6%

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

      8. metadata-eval 27.6%

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

   3. Simplified 27.6%

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

   4. Taylor expanded in d around inf 50.8%

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

      1. *-un-lft-identity 50.8%

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

      2. *-commutative 50.8%

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

   6. Applied egg-rr 50.8%

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

      1. *-lft-identity 50.8%

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

      2. unpow-1 50.8%

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

      3. sqr-pow 50.8%

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

      4. rem-sqrt-square 50.8%

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

      5. sqr-pow 50.7%

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

      6. fabs-sqr 50.7%

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

      7. sqr-pow 50.8%

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

      8. metadata-eval 50.8%

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

   8. Simplified 50.8%

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

      1. unpow-prod-down 51.0%

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

   10. Applied egg-rr 51.0%

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

   if 1.2499999999999999e162 < l

   1. Initial program 33.7%

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

      1. metadata-eval 33.7%

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

      2. unpow1/2 33.7%

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

      3. metadata-eval 33.7%

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

      4. unpow1/2 33.7%

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

      5. *-commutative 33.7%

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

      6. associate-*l* 33.7%

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

      7. times-frac 33.7%

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

      8. metadata-eval 33.7%

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

   3. Simplified 33.7%

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

   4. Taylor expanded in d around inf 42.6%

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

      1. sqrt-div 42.5%

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

      2. metadata-eval 42.5%

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

      3. *-commutative 42.5%

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

   6. Applied egg-rr 42.5%

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

      1. sqrt-prod 48.1%

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

   8. Applied egg-rr 48.1%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \cdot d \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.02 \cdot 10^{-25}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(\frac{D}{2 \cdot \frac{d}{M}}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right)\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+101}:\\ \;\;\;\;-0.125 \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{+162}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 10: 57.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 9.5 \cdot 10^{-26}:\\ \;\;\;\;\left(1 - {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+95}:\\ \;\;\;\;-0.125 \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l 9.5e-26)
   (*
    (- 1.0 (* (pow (* M (* 0.5 (/ D d))) 2.0) (* 0.5 (/ h l))))
    (sqrt (* (/ d h) (/ d l))))
   (if (<= l 1.75e+95)
     (* -0.125 (/ (* (* M D) (* M D)) (/ d (sqrt (/ h (pow l 3.0))))))
     (* d (* (pow h -0.5) (pow l -0.5))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 9.5e-26) {
		tmp = (1.0 - (pow((M * (0.5 * (D / d))), 2.0) * (0.5 * (h / l)))) * sqrt(((d / h) * (d / l)));
	} else if (l <= 1.75e+95) {
		tmp = -0.125 * (((M * D) * (M * D)) / (d / sqrt((h / pow(l, 3.0)))));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (l <= 9.5d-26) then
        tmp = (1.0d0 - (((m * (0.5d0 * (d_1 / d))) ** 2.0d0) * (0.5d0 * (h / l)))) * sqrt(((d / h) * (d / l)))
    else if (l <= 1.75d+95) then
        tmp = (-0.125d0) * (((m * d_1) * (m * d_1)) / (d / sqrt((h / (l ** 3.0d0)))))
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 9.5e-26) {
		tmp = (1.0 - (Math.pow((M * (0.5 * (D / d))), 2.0) * (0.5 * (h / l)))) * Math.sqrt(((d / h) * (d / l)));
	} else if (l <= 1.75e+95) {
		tmp = -0.125 * (((M * D) * (M * D)) / (d / Math.sqrt((h / Math.pow(l, 3.0)))));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= 9.5e-26:
		tmp = (1.0 - (math.pow((M * (0.5 * (D / d))), 2.0) * (0.5 * (h / l)))) * math.sqrt(((d / h) * (d / l)))
	elif l <= 1.75e+95:
		tmp = -0.125 * (((M * D) * (M * D)) / (d / math.sqrt((h / math.pow(l, 3.0)))))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 9.5e-26)
		tmp = Float64(Float64(1.0 - Float64((Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0) * Float64(0.5 * Float64(h / l)))) * sqrt(Float64(Float64(d / h) * Float64(d / l))));
	elseif (l <= 1.75e+95)
		tmp = Float64(-0.125 * Float64(Float64(Float64(M * D) * Float64(M * D)) / Float64(d / sqrt(Float64(h / (l ^ 3.0))))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 9.5e-26)
		tmp = (1.0 - (((M * (0.5 * (D / d))) ^ 2.0) * (0.5 * (h / l)))) * sqrt(((d / h) * (d / l)));
	elseif (l <= 1.75e+95)
		tmp = -0.125 * (((M * D) * (M * D)) / (d / sqrt((h / (l ^ 3.0)))));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, 9.5e-26], N[(N[(1.0 - N[(N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.75e+95], N[(-0.125 * N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(d / N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 9.4999999999999995e-26

   1. Initial program 69.3%

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

      1. metadata-eval 69.3%

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

      2. unpow1/2 69.3%

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

      3. metadata-eval 69.3%

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

      4. unpow1/2 69.3%

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

      5. *-commutative 69.3%

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

      6. associate-*l* 69.3%

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

      7. times-frac 69.1%

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

      8. metadata-eval 69.1%

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

   3. Simplified 69.1%

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

      1. expm1-log1p-u 29.2%

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

      2. expm1-udef 22.0%

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

   5. Applied egg-rr 19.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.5\right)\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 24.7%

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

      2. expm1-log1p 59.1%

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

      3. sub-neg 59.1%

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

      4. *-commutative 59.1%

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

      5. sub-neg 59.1%

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

      6. associate-*l* 59.1%

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

      7. *-commutative 59.1%

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

   7. Simplified 59.1%

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

   if 9.4999999999999995e-26 < l < 1.75e95

   1. Initial program 60.7%

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

      1. metadata-eval 60.7%

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

      2. unpow1/2 60.7%

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

      3. metadata-eval 60.7%

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

      4. unpow1/2 60.7%

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

      5. *-commutative 60.7%

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

      6. associate-*l* 60.7%

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

      7. times-frac 60.7%

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

      8. metadata-eval 60.7%

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

   3. Simplified 60.7%

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

      1. associate-*r* 60.7%

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

      2. frac-times 60.7%

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

      3. *-commutative 60.7%

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

      4. metadata-eval 60.7%

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

      5. associate-*r/ 61.1%

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

      6. metadata-eval 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. div-inv 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. metadata-eval 61.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 61.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

   6. Taylor expanded in d around 0 48.5%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 53.8%

         \[\leadsto -0.125 \cdot \color{blue}{\frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}} \]

      2. associate-/l* 53.8%

         \[\leadsto -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}} \]

      3. unpow2 53.8%

         \[\leadsto -0.125 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}} \]

      4. unpow2 53.8%

         \[\leadsto -0.125 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}} \]

      5. unswap-sqr 70.9%

         \[\leadsto -0.125 \cdot \frac{\color{blue}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}} \]

   8. Simplified 70.9%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}} \]

   if 1.75e95 < l

   1. Initial program 31.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 31.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 31.9%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 31.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 31.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 31.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 31.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 31.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 31.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 31.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 45.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 45.0%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 45.0%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   6. Applied egg-rr 45.0%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   7. Step-by-step derivation

      1. *-lft-identity 45.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 45.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 45.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 45.0%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. sqr-pow 44.8%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      6. fabs-sqr 44.8%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      7. sqr-pow 45.0%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      8. metadata-eval 45.0%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   8. Simplified 45.0%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   9. Step-by-step derivation

      1. unpow-prod-down 48.8%

         \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   10. Applied egg-rr 48.8%

       \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9.5 \cdot 10^{-26}:\\ \;\;\;\;\left(1 - {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+95}:\\ \;\;\;\;-0.125 \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 11: 57.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.25 \cdot 10^{-26}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - {\left(\frac{\frac{M}{\frac{d}{D}}}{2}\right)}^{2} \cdot \left(h \cdot \frac{0.5}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{+98}:\\ \;\;\;\;-0.125 \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l 1.25e-26)
   (*
    (sqrt (/ (/ d h) (/ l d)))
    (- 1.0 (* (pow (/ (/ M (/ d D)) 2.0) 2.0) (* h (/ 0.5 l)))))
   (if (<= l 5.5e+98)
     (* -0.125 (/ (* (* M D) (* M D)) (/ d (sqrt (/ h (pow l 3.0))))))
     (* d (* (pow h -0.5) (pow l -0.5))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 1.25e-26) {
		tmp = sqrt(((d / h) / (l / d))) * (1.0 - (pow(((M / (d / D)) / 2.0), 2.0) * (h * (0.5 / l))));
	} else if (l <= 5.5e+98) {
		tmp = -0.125 * (((M * D) * (M * D)) / (d / sqrt((h / pow(l, 3.0)))));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (l <= 1.25d-26) then
        tmp = sqrt(((d / h) / (l / d))) * (1.0d0 - ((((m / (d / d_1)) / 2.0d0) ** 2.0d0) * (h * (0.5d0 / l))))
    else if (l <= 5.5d+98) then
        tmp = (-0.125d0) * (((m * d_1) * (m * d_1)) / (d / sqrt((h / (l ** 3.0d0)))))
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 1.25e-26) {
		tmp = Math.sqrt(((d / h) / (l / d))) * (1.0 - (Math.pow(((M / (d / D)) / 2.0), 2.0) * (h * (0.5 / l))));
	} else if (l <= 5.5e+98) {
		tmp = -0.125 * (((M * D) * (M * D)) / (d / Math.sqrt((h / Math.pow(l, 3.0)))));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= 1.25e-26:
		tmp = math.sqrt(((d / h) / (l / d))) * (1.0 - (math.pow(((M / (d / D)) / 2.0), 2.0) * (h * (0.5 / l))))
	elif l <= 5.5e+98:
		tmp = -0.125 * (((M * D) * (M * D)) / (d / math.sqrt((h / math.pow(l, 3.0)))))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 1.25e-26)
		tmp = Float64(sqrt(Float64(Float64(d / h) / Float64(l / d))) * Float64(1.0 - Float64((Float64(Float64(M / Float64(d / D)) / 2.0) ^ 2.0) * Float64(h * Float64(0.5 / l)))));
	elseif (l <= 5.5e+98)
		tmp = Float64(-0.125 * Float64(Float64(Float64(M * D) * Float64(M * D)) / Float64(d / sqrt(Float64(h / (l ^ 3.0))))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 1.25e-26)
		tmp = sqrt(((d / h) / (l / d))) * (1.0 - ((((M / (d / D)) / 2.0) ^ 2.0) * (h * (0.5 / l))));
	elseif (l <= 5.5e+98)
		tmp = -0.125 * (((M * D) * (M * D)) / (d / sqrt((h / (l ^ 3.0)))));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, 1.25e-26], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] / N[(l / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[Power[N[(N[(M / N[(d / D), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * N[(0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.5e+98], N[(-0.125 * N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(d / N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.25000000000000005e-26

   1. Initial program 69.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 69.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 69.3%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 69.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 69.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 69.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 69.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 69.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 69.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 69.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. frac-times 69.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      2. associate-/l* 69.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      3. *-commutative 69.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{\frac{\color{blue}{d \cdot 2}}{D}}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   5. Applied egg-rr 69.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. associate-/l* 69.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M \cdot D}{d \cdot 2}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      2. associate-/r* 69.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{\frac{M \cdot D}{d}}{2}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      3. *-commutative 69.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{\frac{\color{blue}{D \cdot M}}{d}}{2}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      4. associate-/l* 69.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{\color{blue}{\frac{D}{\frac{d}{M}}}}{2}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      5. associate-/l/ 69.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{D}{2 \cdot \frac{d}{M}}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   7. Simplified 69.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{D}{2 \cdot \frac{d}{M}}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
   8. Step-by-step derivation

      1. pow1 69.3%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{D}{2 \cdot \frac{d}{M}}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)}^{1}} \]

      2. sqrt-unprod 59.6%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - {\left(\frac{D}{2 \cdot \frac{d}{M}}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)}^{1} \]

      3. associate-*r/ 59.6%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(\frac{D}{2 \cdot \frac{d}{M}}\right)}^{2} \cdot \color{blue}{\frac{0.5 \cdot h}{\ell}}\right)\right)}^{1} \]

   9. Applied egg-rr 59.6%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(\frac{D}{2 \cdot \frac{d}{M}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 59.6%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(\frac{D}{2 \cdot \frac{d}{M}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

       2. associate-*r/ 50.7%

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{d}{h} \cdot d}{\ell}}} \cdot \left(1 - {\left(\frac{D}{2 \cdot \frac{d}{M}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

       3. associate-/l* 59.6%

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{d}{h}}{\frac{\ell}{d}}}} \cdot \left(1 - {\left(\frac{D}{2 \cdot \frac{d}{M}}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) \]

       4. associate-/l* 59.6%

          \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - {\left(\frac{D}{2 \cdot \frac{d}{M}}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

       5. associate-/l/ 59.6%

          \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - {\color{blue}{\left(\frac{\frac{D}{\frac{d}{M}}}{2}\right)}}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

       6. associate-/l* 59.1%

          \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - {\left(\frac{\color{blue}{\frac{D \cdot M}{d}}}{2}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

       7. *-commutative 59.1%

          \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - {\left(\frac{\frac{\color{blue}{M \cdot D}}{d}}{2}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

       8. associate-/l* 59.1%

          \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - {\left(\frac{\color{blue}{\frac{M}{\frac{d}{D}}}}{2}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}}\right) \]

       9. associate-/r/ 59.1%

          \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - {\left(\frac{\frac{M}{\frac{d}{D}}}{2}\right)}^{2} \cdot \color{blue}{\left(\frac{0.5}{\ell} \cdot h\right)}\right) \]

   11. Simplified 59.1%

       \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - {\left(\frac{\frac{M}{\frac{d}{D}}}{2}\right)}^{2} \cdot \left(\frac{0.5}{\ell} \cdot h\right)\right)} \]

   if 1.25000000000000005e-26 < l < 5.49999999999999946e98

   1. Initial program 60.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 60.7%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 60.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 60.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 60.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 60.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 60.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 60.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 60.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 60.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 60.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 60.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 60.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 60.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r/ 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      6. metadata-eval 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}\right) \]

      7. *-commutative 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot 0.5\right)} \cdot h}{\ell}\right) \]

      8. frac-times 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. div-inv 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      10. metadata-eval 61.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied egg-rr 61.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

   6. Taylor expanded in d around 0 48.5%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 53.8%

         \[\leadsto -0.125 \cdot \color{blue}{\frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}} \]

      2. associate-/l* 53.8%

         \[\leadsto -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}} \]

      3. unpow2 53.8%

         \[\leadsto -0.125 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}} \]

      4. unpow2 53.8%

         \[\leadsto -0.125 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}} \]

      5. unswap-sqr 70.9%

         \[\leadsto -0.125 \cdot \frac{\color{blue}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}} \]

   8. Simplified 70.9%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}} \]

   if 5.49999999999999946e98 < l

   1. Initial program 31.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 31.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 31.9%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 31.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 31.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 31.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 31.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 31.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 31.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 31.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 45.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 45.0%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 45.0%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   6. Applied egg-rr 45.0%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   7. Step-by-step derivation

      1. *-lft-identity 45.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 45.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 45.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 45.0%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. sqr-pow 44.8%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      6. fabs-sqr 44.8%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      7. sqr-pow 45.0%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      8. metadata-eval 45.0%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   8. Simplified 45.0%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   9. Step-by-step derivation

      1. unpow-prod-down 48.8%

         \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   10. Applied egg-rr 48.8%

       \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.25 \cdot 10^{-26}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - {\left(\frac{\frac{M}{\frac{d}{D}}}{2}\right)}^{2} \cdot \left(h \cdot \frac{0.5}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{+98}:\\ \;\;\;\;-0.125 \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\frac{d}{\sqrt{\frac{h}{{\ell}^{3}}}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 12: 48.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq -3.1 \cdot 10^{+153} \lor \neg \left(M \leq 8.5 \cdot 10^{-95}\right):\\ \;\;\;\;\sqrt{\frac{d}{\frac{h \cdot \ell}{d}}} \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(-0.125 \cdot \left(h \cdot \frac{M \cdot M}{\ell}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (or (<= M -3.1e+153) (not (<= M 8.5e-95)))
   (*
    (sqrt (/ d (/ (* h l) d)))
    (* (/ D d) (* (/ D d) (* -0.125 (* h (/ (* M M) l))))))
   (* (sqrt (/ d h)) (sqrt (/ d l)))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if ((M <= -3.1e+153) || !(M <= 8.5e-95)) {
		tmp = sqrt((d / ((h * l) / d))) * ((D / d) * ((D / d) * (-0.125 * (h * ((M * M) / l)))));
	} else {
		tmp = sqrt((d / h)) * sqrt((d / l));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if ((m <= (-3.1d+153)) .or. (.not. (m <= 8.5d-95))) then
        tmp = sqrt((d / ((h * l) / d))) * ((d_1 / d) * ((d_1 / d) * ((-0.125d0) * (h * ((m * m) / l)))))
    else
        tmp = sqrt((d / h)) * sqrt((d / l))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if ((M <= -3.1e+153) || !(M <= 8.5e-95)) {
		tmp = Math.sqrt((d / ((h * l) / d))) * ((D / d) * ((D / d) * (-0.125 * (h * ((M * M) / l)))));
	} else {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if (M <= -3.1e+153) or not (M <= 8.5e-95):
		tmp = math.sqrt((d / ((h * l) / d))) * ((D / d) * ((D / d) * (-0.125 * (h * ((M * M) / l)))))
	else:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if ((M <= -3.1e+153) || !(M <= 8.5e-95))
		tmp = Float64(sqrt(Float64(d / Float64(Float64(h * l) / d))) * Float64(Float64(D / d) * Float64(Float64(D / d) * Float64(-0.125 * Float64(h * Float64(Float64(M * M) / l))))));
	else
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if ((M <= -3.1e+153) || ~((M <= 8.5e-95)))
		tmp = sqrt((d / ((h * l) / d))) * ((D / d) * ((D / d) * (-0.125 * (h * ((M * M) / l)))));
	else
		tmp = sqrt((d / h)) * sqrt((d / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[Or[LessEqual[M, -3.1e+153], N[Not[LessEqual[M, 8.5e-95]], $MachinePrecision]], N[(N[Sqrt[N[(d / N[(N[(h * l), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(-0.125 * N[(h * N[(N[(M * M), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -3.1e153 or 8.4999999999999995e-95 < M

   1. Initial program 64.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 64.8%

         \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      2. metadata-eval 64.8%

         \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. unpow1/2 64.8%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval 64.8%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. unpow1/2 64.8%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. sub-neg 64.8%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      7. +-commutative 64.8%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]

      8. *-commutative 64.8%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) + 1\right)\right) \]

      9. distribute-rgt-neg-in 64.8%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{h}{\ell} \cdot \left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} + 1\right)\right) \]

      10. fma-def 64.8%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, -\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}, 1\right)}\right) \]

   3. Simplified 63.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2} \cdot -0.5, 1\right)\right)} \]

   4. Taylor expanded in h around inf 39.5%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}\right)}\right) \]
   5. Step-by-step derivation

      1. *-commutative 39.5%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}} \cdot -0.125\right)}\right) \]

      2. *-commutative 39.5%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\color{blue}{{d}^{2} \cdot \ell}} \cdot -0.125\right)\right) \]

      3. times-frac 38.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)} \cdot -0.125\right)\right) \]

      4. *-commutative 38.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right) \cdot -0.125\right)\right) \]

      5. unpow2 38.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot -0.125\right)\right) \]

      6. unpow2 38.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot -0.125\right)\right) \]

      7. unpow2 38.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell}\right) \cdot -0.125\right)\right) \]

   6. Simplified 38.6%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)}\right) \]
   7. Step-by-step derivation

      1. sqrt-div 20.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right) \]

   8. Applied egg-rr 20.1%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right) \]
   9. Step-by-step derivation

      1. pow1 20.1%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)\right)}^{1}} \]

      2. sqrt-div 38.6%

         \[\leadsto {\left(\sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)\right)}^{1} \]

      3. associate-*r* 38.6%

         \[\leadsto {\color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)}}^{1} \]

      4. sqrt-unprod 29.2%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)}^{1} \]

      5. *-commutative 29.2%

         \[\leadsto {\left(\sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)}^{1} \]

      6. frac-times 20.9%

         \[\leadsto {\left(\sqrt{\color{blue}{\frac{d \cdot d}{\ell \cdot h}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)}^{1} \]

      7. *-commutative 20.9%

         \[\leadsto {\left(\sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)}^{1} \]

      8. associate-*l* 20.9%

         \[\leadsto {\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \color{blue}{\left(\frac{D \cdot D}{d \cdot d} \cdot \left(\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot -0.125\right)\right)}\right)}^{1} \]

      9. times-frac 26.6%

         \[\leadsto {\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot -0.125\right)\right)\right)}^{1} \]

      10. associate-/l* 25.5%

          \[\leadsto {\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\color{blue}{\frac{M \cdot M}{\frac{\ell}{h}}} \cdot -0.125\right)\right)\right)}^{1} \]

   10. Applied egg-rr 25.5%

       \[\leadsto \color{blue}{{\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot -0.125\right)\right)\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 25.5%

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot -0.125\right)\right)} \]

       2. associate-/l* 34.8%

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\frac{h \cdot \ell}{d}}}} \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot -0.125\right)\right) \]

       3. associate-*l* 41.8%

          \[\leadsto \sqrt{\frac{d}{\frac{h \cdot \ell}{d}}} \cdot \color{blue}{\left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot -0.125\right)\right)\right)} \]

       4. *-commutative 41.8%

          \[\leadsto \sqrt{\frac{d}{\frac{h \cdot \ell}{d}}} \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \color{blue}{\left(-0.125 \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\right)\right) \]

       5. unpow2 41.8%

          \[\leadsto \sqrt{\frac{d}{\frac{h \cdot \ell}{d}}} \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(-0.125 \cdot \frac{\color{blue}{{M}^{2}}}{\frac{\ell}{h}}\right)\right)\right) \]

       6. associate-/r/ 42.9%

          \[\leadsto \sqrt{\frac{d}{\frac{h \cdot \ell}{d}}} \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(-0.125 \cdot \color{blue}{\left(\frac{{M}^{2}}{\ell} \cdot h\right)}\right)\right)\right) \]

       7. unpow2 42.9%

          \[\leadsto \sqrt{\frac{d}{\frac{h \cdot \ell}{d}}} \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(-0.125 \cdot \left(\frac{\color{blue}{M \cdot M}}{\ell} \cdot h\right)\right)\right)\right) \]

   12. Simplified 42.9%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\frac{h \cdot \ell}{d}}} \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(-0.125 \cdot \left(\frac{M \cdot M}{\ell} \cdot h\right)\right)\right)\right)} \]

   if -3.1e153 < M < 8.4999999999999995e-95

   1. Initial program 59.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 59.5%

         \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      2. metadata-eval 59.5%

         \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. unpow1/2 59.5%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval 59.5%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. unpow1/2 59.5%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. sub-neg 59.5%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      7. +-commutative 59.5%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]

      8. *-commutative 59.5%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) + 1\right)\right) \]

      9. distribute-rgt-neg-in 59.5%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{h}{\ell} \cdot \left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} + 1\right)\right) \]

      10. fma-def 59.5%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, -\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}, 1\right)}\right) \]

   3. Simplified 60.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2} \cdot -0.5, 1\right)\right)} \]

   4. Taylor expanded in h around 0 42.1%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -3.1 \cdot 10^{+153} \lor \neg \left(M \leq 8.5 \cdot 10^{-95}\right):\\ \;\;\;\;\sqrt{\frac{d}{\frac{h \cdot \ell}{d}}} \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(-0.125 \cdot \left(h \cdot \frac{M \cdot M}{\ell}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \]

Alternative 13: 37.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.2 \cdot 10^{-260}:\\ \;\;\;\;\sqrt{\frac{d}{\frac{h \cdot \ell}{d}}} \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(-0.125 \cdot \left(h \cdot \frac{M \cdot M}{\ell}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l 3.2e-260)
   (*
    (sqrt (/ d (/ (* h l) d)))
    (* (/ D d) (* (/ D d) (* -0.125 (* h (/ (* M M) l))))))
   (* d (* (pow h -0.5) (pow l -0.5)))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 3.2e-260) {
		tmp = sqrt((d / ((h * l) / d))) * ((D / d) * ((D / d) * (-0.125 * (h * ((M * M) / l)))));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (l <= 3.2d-260) then
        tmp = sqrt((d / ((h * l) / d))) * ((d_1 / d) * ((d_1 / d) * ((-0.125d0) * (h * ((m * m) / l)))))
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 3.2e-260) {
		tmp = Math.sqrt((d / ((h * l) / d))) * ((D / d) * ((D / d) * (-0.125 * (h * ((M * M) / l)))));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= 3.2e-260:
		tmp = math.sqrt((d / ((h * l) / d))) * ((D / d) * ((D / d) * (-0.125 * (h * ((M * M) / l)))))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 3.2e-260)
		tmp = Float64(sqrt(Float64(d / Float64(Float64(h * l) / d))) * Float64(Float64(D / d) * Float64(Float64(D / d) * Float64(-0.125 * Float64(h * Float64(Float64(M * M) / l))))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 3.2e-260)
		tmp = sqrt((d / ((h * l) / d))) * ((D / d) * ((D / d) * (-0.125 * (h * ((M * M) / l)))));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, 3.2e-260], N[(N[Sqrt[N[(d / N[(N[(h * l), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(-0.125 * N[(h * N[(N[(M * M), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.19999999999999995e-260

   1. Initial program 66.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 66.1%

         \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      2. metadata-eval 66.1%

         \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. unpow1/2 66.1%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval 66.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. unpow1/2 66.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. sub-neg 66.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      7. +-commutative 66.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]

      8. *-commutative 66.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) + 1\right)\right) \]

      9. distribute-rgt-neg-in 66.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{h}{\ell} \cdot \left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} + 1\right)\right) \]

      10. fma-def 66.1%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, -\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}, 1\right)}\right) \]

   3. Simplified 66.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2} \cdot -0.5, 1\right)\right)} \]

   4. Taylor expanded in h around inf 27.6%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}\right)}\right) \]
   5. Step-by-step derivation

      1. *-commutative 27.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}} \cdot -0.125\right)}\right) \]

      2. *-commutative 27.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\color{blue}{{d}^{2} \cdot \ell}} \cdot -0.125\right)\right) \]

      3. times-frac 27.0%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)} \cdot -0.125\right)\right) \]

      4. *-commutative 27.0%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right) \cdot -0.125\right)\right) \]

      5. unpow2 27.0%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot -0.125\right)\right) \]

      6. unpow2 27.0%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot -0.125\right)\right) \]

      7. unpow2 27.0%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell}\right) \cdot -0.125\right)\right) \]

   6. Simplified 27.0%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)}\right) \]
   7. Step-by-step derivation

      1. sqrt-div 5.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right) \]

   8. Applied egg-rr 5.4%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right) \]
   9. Step-by-step derivation

      1. pow1 5.4%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)\right)}^{1}} \]

      2. sqrt-div 27.0%

         \[\leadsto {\left(\sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)\right)}^{1} \]

      3. associate-*r* 27.0%

         \[\leadsto {\color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)}}^{1} \]

      4. sqrt-unprod 20.9%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)}^{1} \]

      5. *-commutative 20.9%

         \[\leadsto {\left(\sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)}^{1} \]

      6. frac-times 16.8%

         \[\leadsto {\left(\sqrt{\color{blue}{\frac{d \cdot d}{\ell \cdot h}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)}^{1} \]

      7. *-commutative 16.8%

         \[\leadsto {\left(\sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)}^{1} \]

      8. associate-*l* 16.8%

         \[\leadsto {\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \color{blue}{\left(\frac{D \cdot D}{d \cdot d} \cdot \left(\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot -0.125\right)\right)}\right)}^{1} \]

      9. times-frac 21.2%

         \[\leadsto {\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot -0.125\right)\right)\right)}^{1} \]

      10. associate-/l* 21.5%

          \[\leadsto {\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\color{blue}{\frac{M \cdot M}{\frac{\ell}{h}}} \cdot -0.125\right)\right)\right)}^{1} \]

   10. Applied egg-rr 21.5%

       \[\leadsto \color{blue}{{\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot -0.125\right)\right)\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 21.5%

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot -0.125\right)\right)} \]

       2. associate-/l* 27.0%

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\frac{h \cdot \ell}{d}}}} \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot -0.125\right)\right) \]

       3. associate-*l* 32.1%

          \[\leadsto \sqrt{\frac{d}{\frac{h \cdot \ell}{d}}} \cdot \color{blue}{\left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot -0.125\right)\right)\right)} \]

       4. *-commutative 32.1%

          \[\leadsto \sqrt{\frac{d}{\frac{h \cdot \ell}{d}}} \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \color{blue}{\left(-0.125 \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\right)\right) \]

       5. unpow2 32.1%

          \[\leadsto \sqrt{\frac{d}{\frac{h \cdot \ell}{d}}} \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(-0.125 \cdot \frac{\color{blue}{{M}^{2}}}{\frac{\ell}{h}}\right)\right)\right) \]

       6. associate-/r/ 32.9%

          \[\leadsto \sqrt{\frac{d}{\frac{h \cdot \ell}{d}}} \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(-0.125 \cdot \color{blue}{\left(\frac{{M}^{2}}{\ell} \cdot h\right)}\right)\right)\right) \]

       7. unpow2 32.9%

          \[\leadsto \sqrt{\frac{d}{\frac{h \cdot \ell}{d}}} \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(-0.125 \cdot \left(\frac{\color{blue}{M \cdot M}}{\ell} \cdot h\right)\right)\right)\right) \]

   12. Simplified 32.9%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\frac{h \cdot \ell}{d}}} \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(-0.125 \cdot \left(\frac{M \cdot M}{\ell} \cdot h\right)\right)\right)\right)} \]

   if 3.19999999999999995e-260 < l

   1. Initial program 55.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 55.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 55.6%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 55.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 55.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 55.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 55.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 54.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 54.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 54.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 38.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 38.4%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 38.4%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   6. Applied egg-rr 38.4%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   7. Step-by-step derivation

      1. *-lft-identity 38.4%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 38.4%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 38.4%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 38.4%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. sqr-pow 38.3%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      6. fabs-sqr 38.3%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      7. sqr-pow 38.4%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      8. metadata-eval 38.4%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   8. Simplified 38.4%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   9. Step-by-step derivation

      1. unpow-prod-down 44.6%

         \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   10. Applied egg-rr 44.6%

       \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.2 \cdot 10^{-260}:\\ \;\;\;\;\sqrt{\frac{d}{\frac{h \cdot \ell}{d}}} \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(-0.125 \cdot \left(h \cdot \frac{M \cdot M}{\ell}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 14: 34.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq -5.5 \cdot 10^{-14} \lor \neg \left(M \leq 1.9 \cdot 10^{-96}\right):\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \frac{M}{\frac{\ell}{M}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (or (<= M -5.5e-14) (not (<= M 1.9e-96)))
   (*
    (sqrt (* (/ d h) (/ d l)))
    (* -0.125 (* (* (/ D d) (/ D d)) (* h (/ M (/ l M))))))
   (* d (sqrt (/ 1.0 (* h l))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if ((M <= -5.5e-14) || !(M <= 1.9e-96)) {
		tmp = sqrt(((d / h) * (d / l))) * (-0.125 * (((D / d) * (D / d)) * (h * (M / (l / M)))));
	} else {
		tmp = d * sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if ((m <= (-5.5d-14)) .or. (.not. (m <= 1.9d-96))) then
        tmp = sqrt(((d / h) * (d / l))) * ((-0.125d0) * (((d_1 / d) * (d_1 / d)) * (h * (m / (l / m)))))
    else
        tmp = d * sqrt((1.0d0 / (h * l)))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if ((M <= -5.5e-14) || !(M <= 1.9e-96)) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (-0.125 * (((D / d) * (D / d)) * (h * (M / (l / M)))));
	} else {
		tmp = d * Math.sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if (M <= -5.5e-14) or not (M <= 1.9e-96):
		tmp = math.sqrt(((d / h) * (d / l))) * (-0.125 * (((D / d) * (D / d)) * (h * (M / (l / M)))))
	else:
		tmp = d * math.sqrt((1.0 / (h * l)))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if ((M <= -5.5e-14) || !(M <= 1.9e-96))
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(h * Float64(M / Float64(l / M))))));
	else
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(h * l))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if ((M <= -5.5e-14) || ~((M <= 1.9e-96)))
		tmp = sqrt(((d / h) * (d / l))) * (-0.125 * (((D / d) * (D / d)) * (h * (M / (l / M)))));
	else
		tmp = d * sqrt((1.0 / (h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[Or[LessEqual[M, -5.5e-14], N[Not[LessEqual[M, 1.9e-96]], $MachinePrecision]], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -5.49999999999999991e-14 or 1.9e-96 < M

   1. Initial program 62.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 62.6%

         \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      2. metadata-eval 62.6%

         \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. unpow1/2 62.6%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval 62.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. unpow1/2 62.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. sub-neg 62.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      7. +-commutative 62.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]

      8. *-commutative 62.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) + 1\right)\right) \]

      9. distribute-rgt-neg-in 62.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{h}{\ell} \cdot \left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} + 1\right)\right) \]

      10. fma-def 62.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, -\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}, 1\right)}\right) \]

   3. Simplified 62.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2} \cdot -0.5, 1\right)\right)} \]

   4. Taylor expanded in h around inf 37.0%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}\right)}\right) \]
   5. Step-by-step derivation

      1. *-commutative 37.0%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}} \cdot -0.125\right)}\right) \]

      2. *-commutative 37.0%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\color{blue}{{d}^{2} \cdot \ell}} \cdot -0.125\right)\right) \]

      3. times-frac 35.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)} \cdot -0.125\right)\right) \]

      4. *-commutative 35.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right) \cdot -0.125\right)\right) \]

      5. unpow2 35.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot -0.125\right)\right) \]

      6. unpow2 35.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot -0.125\right)\right) \]

      7. unpow2 35.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell}\right) \cdot -0.125\right)\right) \]

   6. Simplified 35.7%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)}\right) \]
   7. Step-by-step derivation

      1. sqrt-div 18.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right) \]

   8. Applied egg-rr 18.4%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right) \]
   9. Step-by-step derivation

      1. pow1 18.4%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)\right)}^{1}} \]

      2. sqrt-div 35.7%

         \[\leadsto {\left(\sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)\right)}^{1} \]

      3. associate-*r* 35.7%

         \[\leadsto {\color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)}}^{1} \]

      4. sqrt-unprod 26.6%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)}^{1} \]

      5. *-commutative 26.6%

         \[\leadsto {\left(\sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)}^{1} \]

      6. frac-times 20.5%

         \[\leadsto {\left(\sqrt{\color{blue}{\frac{d \cdot d}{\ell \cdot h}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)}^{1} \]

      7. *-commutative 20.5%

         \[\leadsto {\left(\sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)}^{1} \]

      8. associate-*l* 20.5%

         \[\leadsto {\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \color{blue}{\left(\frac{D \cdot D}{d \cdot d} \cdot \left(\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot -0.125\right)\right)}\right)}^{1} \]

      9. times-frac 26.2%

         \[\leadsto {\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot -0.125\right)\right)\right)}^{1} \]

      10. associate-/l* 25.4%

          \[\leadsto {\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\color{blue}{\frac{M \cdot M}{\frac{\ell}{h}}} \cdot -0.125\right)\right)\right)}^{1} \]

   10. Applied egg-rr 25.4%

       \[\leadsto \color{blue}{{\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot -0.125\right)\right)\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 25.4%

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot -0.125\right)\right)} \]

       2. times-frac 31.6%

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot -0.125\right)\right) \]

       3. *-commutative 31.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(\left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot -0.125\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)} \]

       4. *-commutative 31.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(\color{blue}{\left(-0.125 \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)} \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \]

       5. associate-*l* 31.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(-0.125 \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\right)} \]

       6. *-commutative 31.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(-0.125 \cdot \color{blue}{\left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\right) \]

       7. unpow2 31.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{\color{blue}{{M}^{2}}}{\frac{\ell}{h}}\right)\right) \]

       8. associate-/r/ 32.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(\frac{{M}^{2}}{\ell} \cdot h\right)}\right)\right) \]

       9. *-commutative 32.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(h \cdot \frac{{M}^{2}}{\ell}\right)}\right)\right) \]

       10. unpow2 32.4%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \frac{\color{blue}{M \cdot M}}{\ell}\right)\right)\right) \]

       11. associate-/l* 33.2%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \color{blue}{\frac{M}{\frac{\ell}{M}}}\right)\right)\right) \]

   12. Simplified 33.2%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \frac{M}{\frac{\ell}{M}}\right)\right)\right)} \]

   if -5.49999999999999991e-14 < M < 1.9e-96

   1. Initial program 60.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 60.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 60.6%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 60.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 60.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 60.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 60.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 59.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 59.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 59.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 30.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -5.5 \cdot 10^{-14} \lor \neg \left(M \leq 1.9 \cdot 10^{-96}\right):\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \frac{M}{\frac{\ell}{M}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]

Alternative 15: 36.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq -6 \cdot 10^{-14} \lor \neg \left(M \leq 9.2 \cdot 10^{-98}\right):\\ \;\;\;\;\sqrt{\frac{d}{\frac{h \cdot \ell}{d}}} \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(-0.125 \cdot \left(h \cdot \frac{M \cdot M}{\ell}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (or (<= M -6e-14) (not (<= M 9.2e-98)))
   (*
    (sqrt (/ d (/ (* h l) d)))
    (* (/ D d) (* (/ D d) (* -0.125 (* h (/ (* M M) l))))))
   (* d (sqrt (/ 1.0 (* h l))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if ((M <= -6e-14) || !(M <= 9.2e-98)) {
		tmp = sqrt((d / ((h * l) / d))) * ((D / d) * ((D / d) * (-0.125 * (h * ((M * M) / l)))));
	} else {
		tmp = d * sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if ((m <= (-6d-14)) .or. (.not. (m <= 9.2d-98))) then
        tmp = sqrt((d / ((h * l) / d))) * ((d_1 / d) * ((d_1 / d) * ((-0.125d0) * (h * ((m * m) / l)))))
    else
        tmp = d * sqrt((1.0d0 / (h * l)))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if ((M <= -6e-14) || !(M <= 9.2e-98)) {
		tmp = Math.sqrt((d / ((h * l) / d))) * ((D / d) * ((D / d) * (-0.125 * (h * ((M * M) / l)))));
	} else {
		tmp = d * Math.sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if (M <= -6e-14) or not (M <= 9.2e-98):
		tmp = math.sqrt((d / ((h * l) / d))) * ((D / d) * ((D / d) * (-0.125 * (h * ((M * M) / l)))))
	else:
		tmp = d * math.sqrt((1.0 / (h * l)))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if ((M <= -6e-14) || !(M <= 9.2e-98))
		tmp = Float64(sqrt(Float64(d / Float64(Float64(h * l) / d))) * Float64(Float64(D / d) * Float64(Float64(D / d) * Float64(-0.125 * Float64(h * Float64(Float64(M * M) / l))))));
	else
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(h * l))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if ((M <= -6e-14) || ~((M <= 9.2e-98)))
		tmp = sqrt((d / ((h * l) / d))) * ((D / d) * ((D / d) * (-0.125 * (h * ((M * M) / l)))));
	else
		tmp = d * sqrt((1.0 / (h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[Or[LessEqual[M, -6e-14], N[Not[LessEqual[M, 9.2e-98]], $MachinePrecision]], N[(N[Sqrt[N[(d / N[(N[(h * l), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(-0.125 * N[(h * N[(N[(M * M), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -5.9999999999999997e-14 or 9.20000000000000002e-98 < M

   1. Initial program 62.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 62.6%

         \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      2. metadata-eval 62.6%

         \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. unpow1/2 62.6%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval 62.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. unpow1/2 62.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. sub-neg 62.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      7. +-commutative 62.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]

      8. *-commutative 62.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) + 1\right)\right) \]

      9. distribute-rgt-neg-in 62.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{h}{\ell} \cdot \left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} + 1\right)\right) \]

      10. fma-def 62.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, -\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}, 1\right)}\right) \]

   3. Simplified 62.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2} \cdot -0.5, 1\right)\right)} \]

   4. Taylor expanded in h around inf 37.0%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}\right)}\right) \]
   5. Step-by-step derivation

      1. *-commutative 37.0%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}} \cdot -0.125\right)}\right) \]

      2. *-commutative 37.0%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\color{blue}{{d}^{2} \cdot \ell}} \cdot -0.125\right)\right) \]

      3. times-frac 35.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)} \cdot -0.125\right)\right) \]

      4. *-commutative 35.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right) \cdot -0.125\right)\right) \]

      5. unpow2 35.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot -0.125\right)\right) \]

      6. unpow2 35.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot -0.125\right)\right) \]

      7. unpow2 35.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell}\right) \cdot -0.125\right)\right) \]

   6. Simplified 35.7%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)}\right) \]
   7. Step-by-step derivation

      1. sqrt-div 18.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right) \]

   8. Applied egg-rr 18.4%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right) \]
   9. Step-by-step derivation

      1. pow1 18.4%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)\right)}^{1}} \]

      2. sqrt-div 35.7%

         \[\leadsto {\left(\sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)\right)}^{1} \]

      3. associate-*r* 35.7%

         \[\leadsto {\color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)}}^{1} \]

      4. sqrt-unprod 26.6%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)}^{1} \]

      5. *-commutative 26.6%

         \[\leadsto {\left(\sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)}^{1} \]

      6. frac-times 20.5%

         \[\leadsto {\left(\sqrt{\color{blue}{\frac{d \cdot d}{\ell \cdot h}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)}^{1} \]

      7. *-commutative 20.5%

         \[\leadsto {\left(\sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)}^{1} \]

      8. associate-*l* 20.5%

         \[\leadsto {\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \color{blue}{\left(\frac{D \cdot D}{d \cdot d} \cdot \left(\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot -0.125\right)\right)}\right)}^{1} \]

      9. times-frac 26.2%

         \[\leadsto {\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot -0.125\right)\right)\right)}^{1} \]

      10. associate-/l* 25.4%

          \[\leadsto {\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\color{blue}{\frac{M \cdot M}{\frac{\ell}{h}}} \cdot -0.125\right)\right)\right)}^{1} \]

   10. Applied egg-rr 25.4%

       \[\leadsto \color{blue}{{\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot -0.125\right)\right)\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 25.4%

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot -0.125\right)\right)} \]

       2. associate-/l* 33.0%

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\frac{h \cdot \ell}{d}}}} \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot -0.125\right)\right) \]

       3. associate-*l* 38.9%

          \[\leadsto \sqrt{\frac{d}{\frac{h \cdot \ell}{d}}} \cdot \color{blue}{\left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot -0.125\right)\right)\right)} \]

       4. *-commutative 38.9%

          \[\leadsto \sqrt{\frac{d}{\frac{h \cdot \ell}{d}}} \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \color{blue}{\left(-0.125 \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\right)\right) \]

       5. unpow2 38.9%

          \[\leadsto \sqrt{\frac{d}{\frac{h \cdot \ell}{d}}} \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(-0.125 \cdot \frac{\color{blue}{{M}^{2}}}{\frac{\ell}{h}}\right)\right)\right) \]

       6. associate-/r/ 39.7%

          \[\leadsto \sqrt{\frac{d}{\frac{h \cdot \ell}{d}}} \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(-0.125 \cdot \color{blue}{\left(\frac{{M}^{2}}{\ell} \cdot h\right)}\right)\right)\right) \]

       7. unpow2 39.7%

          \[\leadsto \sqrt{\frac{d}{\frac{h \cdot \ell}{d}}} \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(-0.125 \cdot \left(\frac{\color{blue}{M \cdot M}}{\ell} \cdot h\right)\right)\right)\right) \]

   12. Simplified 39.7%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\frac{h \cdot \ell}{d}}} \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(-0.125 \cdot \left(\frac{M \cdot M}{\ell} \cdot h\right)\right)\right)\right)} \]

   if -5.9999999999999997e-14 < M < 9.20000000000000002e-98

   1. Initial program 60.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 60.6%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 60.6%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 60.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 60.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 60.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 60.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 59.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 59.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 59.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 30.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -6 \cdot 10^{-14} \lor \neg \left(M \leq 9.2 \cdot 10^{-98}\right):\\ \;\;\;\;\sqrt{\frac{d}{\frac{h \cdot \ell}{d}}} \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(-0.125 \cdot \left(h \cdot \frac{M \cdot M}{\ell}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]

Alternative 16: 35.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq -2.8 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \left(-0.125 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)\right)\right)\\ \mathbf{elif}\;D \leq 1.05 \cdot 10^{+51}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \frac{M}{\frac{\ell}{M}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= D -2.8e-161)
   (*
    (sqrt (* d (/ d (* h l))))
    (* -0.125 (* (/ D (/ l D)) (* (/ (* M M) d) (/ h d)))))
   (if (<= D 1.05e+51)
     (* d (pow (* h l) -0.5))
     (*
      (sqrt (* (/ d h) (/ d l)))
      (* -0.125 (* (* (/ D d) (/ D d)) (* h (/ M (/ l M)))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (D <= -2.8e-161) {
		tmp = sqrt((d * (d / (h * l)))) * (-0.125 * ((D / (l / D)) * (((M * M) / d) * (h / d))));
	} else if (D <= 1.05e+51) {
		tmp = d * pow((h * l), -0.5);
	} else {
		tmp = sqrt(((d / h) * (d / l))) * (-0.125 * (((D / d) * (D / d)) * (h * (M / (l / M)))));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (d_1 <= (-2.8d-161)) then
        tmp = sqrt((d * (d / (h * l)))) * ((-0.125d0) * ((d_1 / (l / d_1)) * (((m * m) / d) * (h / d))))
    else if (d_1 <= 1.05d+51) then
        tmp = d * ((h * l) ** (-0.5d0))
    else
        tmp = sqrt(((d / h) * (d / l))) * ((-0.125d0) * (((d_1 / d) * (d_1 / d)) * (h * (m / (l / m)))))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (D <= -2.8e-161) {
		tmp = Math.sqrt((d * (d / (h * l)))) * (-0.125 * ((D / (l / D)) * (((M * M) / d) * (h / d))));
	} else if (D <= 1.05e+51) {
		tmp = d * Math.pow((h * l), -0.5);
	} else {
		tmp = Math.sqrt(((d / h) * (d / l))) * (-0.125 * (((D / d) * (D / d)) * (h * (M / (l / M)))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if D <= -2.8e-161:
		tmp = math.sqrt((d * (d / (h * l)))) * (-0.125 * ((D / (l / D)) * (((M * M) / d) * (h / d))))
	elif D <= 1.05e+51:
		tmp = d * math.pow((h * l), -0.5)
	else:
		tmp = math.sqrt(((d / h) * (d / l))) * (-0.125 * (((D / d) * (D / d)) * (h * (M / (l / M)))))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (D <= -2.8e-161)
		tmp = Float64(sqrt(Float64(d * Float64(d / Float64(h * l)))) * Float64(-0.125 * Float64(Float64(D / Float64(l / D)) * Float64(Float64(Float64(M * M) / d) * Float64(h / d)))));
	elseif (D <= 1.05e+51)
		tmp = Float64(d * (Float64(h * l) ^ -0.5));
	else
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(h * Float64(M / Float64(l / M))))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (D <= -2.8e-161)
		tmp = sqrt((d * (d / (h * l)))) * (-0.125 * ((D / (l / D)) * (((M * M) / d) * (h / d))));
	elseif (D <= 1.05e+51)
		tmp = d * ((h * l) ^ -0.5);
	else
		tmp = sqrt(((d / h) * (d / l))) * (-0.125 * (((D / d) * (D / d)) * (h * (M / (l / M)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[D, -2.8e-161], N[(N[Sqrt[N[(d * N[(d / N[(h * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 1.05e+51], N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if D < -2.79999999999999992e-161

   1. Initial program 59.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 59.9%

         \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      2. metadata-eval 59.9%

         \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. unpow1/2 59.9%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval 59.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. unpow1/2 59.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. sub-neg 59.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      7. +-commutative 59.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]

      8. *-commutative 59.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) + 1\right)\right) \]

      9. distribute-rgt-neg-in 59.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{h}{\ell} \cdot \left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} + 1\right)\right) \]

      10. fma-def 59.9%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, -\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}, 1\right)}\right) \]

   3. Simplified 60.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2} \cdot -0.5, 1\right)\right)} \]

   4. Taylor expanded in h around inf 29.9%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}\right)}\right) \]
   5. Step-by-step derivation

      1. *-commutative 29.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}} \cdot -0.125\right)}\right) \]

      2. *-commutative 29.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\color{blue}{{d}^{2} \cdot \ell}} \cdot -0.125\right)\right) \]

      3. times-frac 29.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)} \cdot -0.125\right)\right) \]

      4. *-commutative 29.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right) \cdot -0.125\right)\right) \]

      5. unpow2 29.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot -0.125\right)\right) \]

      6. unpow2 29.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot -0.125\right)\right) \]

      7. unpow2 29.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell}\right) \cdot -0.125\right)\right) \]

   6. Simplified 29.1%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)}\right) \]
   7. Step-by-step derivation

      1. sqrt-div 15.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right) \]

   8. Applied egg-rr 15.7%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right) \]
   9. Step-by-step derivation

      1. pow1 15.7%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)\right)}^{1}} \]

      2. sqrt-div 29.1%

         \[\leadsto {\left(\sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)\right)}^{1} \]

      3. associate-*r* 29.1%

         \[\leadsto {\color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)}}^{1} \]

      4. sqrt-unprod 23.2%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)}^{1} \]

      5. *-commutative 23.2%

         \[\leadsto {\left(\sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)}^{1} \]

      6. frac-times 21.2%

         \[\leadsto {\left(\sqrt{\color{blue}{\frac{d \cdot d}{\ell \cdot h}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)}^{1} \]

      7. *-commutative 21.2%

         \[\leadsto {\left(\sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)}^{1} \]

      8. associate-*l* 21.2%

         \[\leadsto {\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \color{blue}{\left(\frac{D \cdot D}{d \cdot d} \cdot \left(\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot -0.125\right)\right)}\right)}^{1} \]

      9. times-frac 26.1%

         \[\leadsto {\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot -0.125\right)\right)\right)}^{1} \]

      10. associate-/l* 26.4%

          \[\leadsto {\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\color{blue}{\frac{M \cdot M}{\frac{\ell}{h}}} \cdot -0.125\right)\right)\right)}^{1} \]

   10. Applied egg-rr 26.4%

       \[\leadsto \color{blue}{{\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot -0.125\right)\right)\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 26.4%

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot -0.125\right)\right)} \]

       2. associate-/l* 29.4%

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\frac{h \cdot \ell}{d}}}} \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot -0.125\right)\right) \]

       3. associate-/r/ 29.4%

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h \cdot \ell} \cdot d}} \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot -0.125\right)\right) \]

       4. *-commutative 29.4%

          \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \color{blue}{\left(\left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot -0.125\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)} \]

       5. *-commutative 29.4%

          \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \left(\color{blue}{\left(-0.125 \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)} \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \]

       6. associate-*l* 29.4%

          \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \color{blue}{\left(-0.125 \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\right)} \]

       7. unpow2 29.4%

          \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \left(-0.125 \cdot \left(\frac{\color{blue}{{M}^{2}}}{\frac{\ell}{h}} \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\right) \]

       8. associate-/l* 29.1%

          \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \left(-0.125 \cdot \left(\color{blue}{\frac{{M}^{2} \cdot h}{\ell}} \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\right) \]

       9. *-commutative 29.1%

          \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \left(-0.125 \cdot \left(\frac{\color{blue}{h \cdot {M}^{2}}}{\ell} \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\right) \]

       10. times-frac 24.2%

           \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \left(-0.125 \cdot \left(\frac{h \cdot {M}^{2}}{\ell} \cdot \color{blue}{\frac{D \cdot D}{d \cdot d}}\right)\right) \]

       11. unpow2 24.2%

           \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \left(-0.125 \cdot \left(\frac{h \cdot {M}^{2}}{\ell} \cdot \frac{\color{blue}{{D}^{2}}}{d \cdot d}\right)\right) \]

       12. unpow2 24.2%

           \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \left(-0.125 \cdot \left(\frac{h \cdot {M}^{2}}{\ell} \cdot \frac{{D}^{2}}{\color{blue}{{d}^{2}}}\right)\right) \]

   12. Simplified 28.4%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \left(-0.125 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)\right)\right)} \]

   if -2.79999999999999992e-161 < D < 1.0500000000000001e51

   1. Initial program 59.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 59.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 59.4%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 59.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 59.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 59.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 59.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 59.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 59.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 59.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 26.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 26.8%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 26.8%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   6. Applied egg-rr 26.8%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   7. Step-by-step derivation

      1. *-lft-identity 26.8%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 26.8%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 26.9%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 26.9%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. sqr-pow 26.8%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      6. fabs-sqr 26.8%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      7. sqr-pow 26.9%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      8. metadata-eval 26.9%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   8. Simplified 26.9%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

   if 1.0500000000000001e51 < D

   1. Initial program 68.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 68.7%

         \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      2. metadata-eval 68.7%

         \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. unpow1/2 68.7%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval 68.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. unpow1/2 68.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. sub-neg 68.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      7. +-commutative 68.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]

      8. *-commutative 68.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) + 1\right)\right) \]

      9. distribute-rgt-neg-in 68.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{h}{\ell} \cdot \left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} + 1\right)\right) \]

      10. fma-def 68.7%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, -\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}, 1\right)}\right) \]

   3. Simplified 70.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2} \cdot -0.5, 1\right)\right)} \]

   4. Taylor expanded in h around inf 37.4%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}\right)}\right) \]
   5. Step-by-step derivation

      1. *-commutative 37.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}} \cdot -0.125\right)}\right) \]

      2. *-commutative 37.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\color{blue}{{d}^{2} \cdot \ell}} \cdot -0.125\right)\right) \]

      3. times-frac 37.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)} \cdot -0.125\right)\right) \]

      4. *-commutative 37.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right) \cdot -0.125\right)\right) \]

      5. unpow2 37.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot -0.125\right)\right) \]

      6. unpow2 37.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot -0.125\right)\right) \]

      7. unpow2 37.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell}\right) \cdot -0.125\right)\right) \]

   6. Simplified 37.4%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)}\right) \]
   7. Step-by-step derivation

      1. sqrt-div 17.8%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right) \]

   8. Applied egg-rr 17.8%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right) \]
   9. Step-by-step derivation

      1. pow1 17.8%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)\right)}^{1}} \]

      2. sqrt-div 37.4%

         \[\leadsto {\left(\sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)\right)}^{1} \]

      3. associate-*r* 37.4%

         \[\leadsto {\color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)}}^{1} \]

      4. sqrt-unprod 25.1%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)}^{1} \]

      5. *-commutative 25.1%

         \[\leadsto {\left(\sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)}^{1} \]

      6. frac-times 14.5%

         \[\leadsto {\left(\sqrt{\color{blue}{\frac{d \cdot d}{\ell \cdot h}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)}^{1} \]

      7. *-commutative 14.5%

         \[\leadsto {\left(\sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(\left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot -0.125\right)\right)}^{1} \]

      8. associate-*l* 14.5%

         \[\leadsto {\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \color{blue}{\left(\frac{D \cdot D}{d \cdot d} \cdot \left(\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot -0.125\right)\right)}\right)}^{1} \]

      9. times-frac 19.9%

         \[\leadsto {\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(\frac{\left(M \cdot M\right) \cdot h}{\ell} \cdot -0.125\right)\right)\right)}^{1} \]

      10. associate-/l* 19.9%

          \[\leadsto {\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\color{blue}{\frac{M \cdot M}{\frac{\ell}{h}}} \cdot -0.125\right)\right)\right)}^{1} \]

   10. Applied egg-rr 19.9%

       \[\leadsto \color{blue}{{\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot -0.125\right)\right)\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 19.9%

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}} \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot -0.125\right)\right)} \]

       2. times-frac 30.4%

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot -0.125\right)\right) \]

       3. *-commutative 30.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(\left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot -0.125\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)} \]

       4. *-commutative 30.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(\color{blue}{\left(-0.125 \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)} \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \]

       5. associate-*l* 30.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(-0.125 \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\right)} \]

       6. *-commutative 30.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(-0.125 \cdot \color{blue}{\left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)}\right) \]

       7. unpow2 30.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{\color{blue}{{M}^{2}}}{\frac{\ell}{h}}\right)\right) \]

       8. associate-/r/ 32.2%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(\frac{{M}^{2}}{\ell} \cdot h\right)}\right)\right) \]

       9. *-commutative 32.2%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(h \cdot \frac{{M}^{2}}{\ell}\right)}\right)\right) \]

       10. unpow2 32.2%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \frac{\color{blue}{M \cdot M}}{\ell}\right)\right)\right) \]

       11. associate-/l* 35.7%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \color{blue}{\frac{M}{\frac{\ell}{M}}}\right)\right)\right) \]

   12. Simplified 35.7%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \frac{M}{\frac{\ell}{M}}\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 29.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq -2.8 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}} \cdot \left(-0.125 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)\right)\right)\\ \mathbf{elif}\;D \leq 1.05 \cdot 10^{+51}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \frac{M}{\frac{\ell}{M}}\right)\right)\right)\\ \end{array} \]

Alternative 17: 25.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ d \cdot \sqrt{\frac{1}{h \cdot \ell}} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D) :precision binary64 (* d (sqrt (/ 1.0 (* h l)))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	return d * sqrt((1.0 / (h * l)));
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
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 * sqrt((1.0d0 / (h * l)))
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	return d * Math.sqrt((1.0 / (h * l)));
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	return d * math.sqrt((1.0 / (h * l)))

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	return Float64(d * sqrt(Float64(1.0 / Float64(h * l))))
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp = code(d, h, l, M, D)
	tmp = d * sqrt((1.0 / (h * l)));
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.7%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Step-by-step derivation

   1. metadata-eval 61.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. unpow1/2 61.7%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. metadata-eval 61.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. unpow1/2 61.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. *-commutative 61.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

   6. associate-*l* 61.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

   7. times-frac 61.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. metadata-eval 61.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

3. Simplified 61.6%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

4. Taylor expanded in d around inf 22.3%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

5. Final simplification 22.3%

   \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

Alternative 18: 25.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ d \cdot {\left(h \cdot \ell\right)}^{-0.5} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D) :precision binary64 (* d (pow (* h l) -0.5)))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	return d * pow((h * l), -0.5);
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
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 * l) ** (-0.5d0))
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	return d * Math.pow((h * l), -0.5);
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	return d * math.pow((h * l), -0.5)

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	return Float64(d * (Float64(h * l) ^ -0.5))
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp = code(d, h, l, M, D)
	tmp = d * ((h * l) ^ -0.5);
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.7%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Step-by-step derivation

   1. metadata-eval 61.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. unpow1/2 61.7%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. metadata-eval 61.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. unpow1/2 61.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. *-commutative 61.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

   6. associate-*l* 61.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

   7. times-frac 61.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. metadata-eval 61.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

3. Simplified 61.6%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

4. Taylor expanded in d around inf 22.3%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
5. Step-by-step derivation

   1. *-un-lft-identity 22.3%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

   2. *-commutative 22.3%

      \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

6. Applied egg-rr 22.3%

   \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
7. Step-by-step derivation

   1. *-lft-identity 22.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

   2. unpow-1 22.3%

      \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

   3. sqr-pow 22.3%

      \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

   4. rem-sqrt-square 22.0%

      \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

   5. sqr-pow 21.9%

      \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

   6. fabs-sqr 21.9%

      \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

   7. sqr-pow 22.0%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

   8. metadata-eval 22.0%

      \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

8. Simplified 22.0%

   \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

9. Final simplification 22.0%

   \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{-0.5} \]

Alternative 19: 25.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \frac{d}{\sqrt{h \cdot \ell}} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D) :precision binary64 (/ d (sqrt (* h l))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	return d / sqrt((h * l));
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
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 / sqrt((h * l))
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	return d / Math.sqrt((h * l));
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	return d / math.sqrt((h * l))

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	return Float64(d / sqrt(Float64(h * l)))
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp = code(d, h, l, M, D)
	tmp = d / sqrt((h * l));
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.7%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Step-by-step derivation

   1. metadata-eval 61.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. unpow1/2 61.7%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. metadata-eval 61.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. unpow1/2 61.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. *-commutative 61.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

   6. associate-*l* 61.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

   7. times-frac 61.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. metadata-eval 61.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

3. Simplified 61.6%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

4. Taylor expanded in d around inf 22.3%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
5. Step-by-step derivation

   1. sqrt-div 22.0%

      \[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \cdot d \]

   2. metadata-eval 22.0%

      \[\leadsto \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}} \cdot d \]

   3. *-commutative 22.0%

      \[\leadsto \frac{1}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot d \]

6. Applied egg-rr 22.0%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{h \cdot \ell}}} \cdot d \]
7. Step-by-step derivation

   1. associate-*l/ 21.9%

      \[\leadsto \color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}} \]

   2. *-un-lft-identity 21.9%

      \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   3. *-commutative 21.9%

      \[\leadsto \frac{d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

8. Applied egg-rr 21.9%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

9. Final simplification 21.9%

   \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

Reproduce

herbie shell --seed 2023173 
(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)))))