Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.8% → 78.7%

Time: 23.5s

Alternatives: 21

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 78.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: M should be positive before calling this function
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 (h <= (-5d-310)) then
        tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * (1.0d0 - (0.5d0 * ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (h / l)))))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 + ((h / l) * ((((d_1 * (0.5d0 * m)) / d) ** 2.0d0) * (-0.5d0))))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 63.3%

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

      1. associate-*l* 63.4%

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

      2. metadata-eval 63.4%

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

      3. unpow1/2 63.4%

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

      4. metadata-eval 63.4%

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

      5. unpow1/2 63.4%

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

      6. associate-*l* 63.4%

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

      7. metadata-eval 63.4%

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

      8. times-frac 61.9%

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

   3. Simplified 61.9%

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

      1. frac-2neg 61.9%

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

      2. sqrt-div 70.5%

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

   5. Applied egg-rr 70.5%

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

   if -4.999999999999985e-310 < h

   1. Initial program 68.6%

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

      1. metadata-eval 68.6%

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

      2. unpow1/2 68.6%

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

      3. metadata-eval 68.6%

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

      4. unpow1/2 68.6%

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

      5. *-commutative 68.6%

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

      6. associate-*l* 68.6%

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

      7. times-frac 68.6%

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

      8. metadata-eval 68.6%

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

   3. Simplified 68.6%

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

   5. Applied egg-rr 75.2%

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

      1. distribute-rgt1-in 78.4%

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

      2. +-commutative 78.4%

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

      3. *-commutative 78.4%

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

      4. *-commutative 78.4%

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

      5. associate-*r/ 78.4%

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

      6. *-commutative 78.4%

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

   7. Simplified 78.4%

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

4. Final simplification 74.3%

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

Alternative 2: 69.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq 4 \cdot 10^{+300}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - {\left(\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \sqrt{\frac{0.5}{\frac{\ell}{h}}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
       (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* M D) (* d 2.0)) 2.0)))))
      4e+300)
   (*
    (* (sqrt (/ d l)) (sqrt (/ d h)))
    (- 1.0 (pow (* (* (/ D d) (* 0.5 M)) (sqrt (/ 0.5 (/ l h)))) 2.0)))
   (* d (- (pow (* h l) -0.5)))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((M * D) / (d * 2.0)), 2.0))))) <= 4e+300) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - pow((((D / d) * (0.5 * M)) * sqrt((0.5 / (l / h)))), 2.0));
	} else {
		tmp = d * -pow((h * l), -0.5);
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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 / h) ** 0.5d0) * ((d / l) ** 0.5d0)) * (1.0d0 - ((h / l) * (0.5d0 * (((m * d_1) / (d * 2.0d0)) ** 2.0d0))))) <= 4d+300) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - ((((d_1 / d) * (0.5d0 * m)) * sqrt((0.5d0 / (l / h)))) ** 2.0d0))
    else
        tmp = d * -((h * l) ** (-0.5d0))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * Math.pow(((M * D) / (d * 2.0)), 2.0))))) <= 4e+300) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - Math.pow((((D / d) * (0.5 * M)) * Math.sqrt((0.5 / (l / h)))), 2.0));
	} else {
		tmp = d * -Math.pow((h * l), -0.5);
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if ((math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * math.pow(((M * D) / (d * 2.0)), 2.0))))) <= 4e+300:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - math.pow((((D / d) * (0.5 * M)) * math.sqrt((0.5 / (l / h)))), 2.0))
	else:
		tmp = d * -math.pow((h * l), -0.5)
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0))))) <= 4e+300)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - (Float64(Float64(Float64(D / d) * Float64(0.5 * M)) * sqrt(Float64(0.5 / Float64(l / h)))) ^ 2.0)));
	else
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((h / l) * (0.5 * (((M * D) / (d * 2.0)) ^ 2.0))))) <= 4e+300)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((D / d) * (0.5 * M)) * sqrt((0.5 / (l / h)))) ^ 2.0));
	else
		tmp = d * -((h * l) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+300], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Power[N[(N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.5 / N[(l / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 4.0000000000000002e300

   1. Initial program 84.9%

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

      1. metadata-eval 84.9%

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

      2. unpow1/2 84.9%

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

      3. metadata-eval 84.9%

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

      4. unpow1/2 85.0%

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

      5. *-commutative 85.0%

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

      6. associate-*l* 85.0%

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

      7. times-frac 83.9%

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

      8. metadata-eval 83.9%

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

   3. Simplified 83.9%

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

      1. associate-*r* 83.9%

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

      2. frac-times 85.0%

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

      3. *-commutative 85.0%

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

      4. metadata-eval 85.0%

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

      5. add-sqr-sqrt 85.0%

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

      6. pow2 85.0%

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

   5. Applied egg-rr 85.9%

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

   if 4.0000000000000002e300 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))

   1. Initial program 14.2%

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

      1. metadata-eval 14.2%

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

      2. unpow1/2 14.2%

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

      3. metadata-eval 14.2%

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

      4. unpow1/2 14.2%

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

      5. *-commutative 14.2%

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

      6. associate-*l* 14.2%

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

      7. times-frac 14.2%

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

      8. metadata-eval 14.2%

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

   3. Simplified 14.2%

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

   5. Applied egg-rr 14.2%

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

      1. associate-*l* 14.2%

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

      2. associate-*r/ 14.2%

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

      3. *-commutative 14.2%

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

   7. Simplified 14.2%

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

   8. Taylor expanded in d around -inf 34.6%

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

      1. mul-1-neg 34.6%

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

      2. distribute-rgt-neg-in 34.6%

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

      3. unpow-1 34.6%

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

      4. sqr-pow 34.6%

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

      5. rem-sqrt-square 34.6%

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

      6. metadata-eval 34.6%

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

      7. sqr-pow 34.4%

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

      8. fabs-sqr 34.4%

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

      9. sqr-pow 34.6%

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

      10. *-commutative 34.6%

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

   10. Simplified 34.6%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq 4 \cdot 10^{+300}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - {\left(\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \sqrt{\frac{0.5}{\frac{\ell}{h}}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \end{array} \]

Alternative 3: 69.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq 4 \cdot 10^{+300}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - {\left(\frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
       (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* M D) (* d 2.0)) 2.0)))))
      4e+300)
   (*
    (* (sqrt (/ d l)) (sqrt (/ d h)))
    (- 1.0 (* (pow (/ (* 0.5 (* M D)) d) 2.0) (* 0.5 (/ h l)))))
   (* d (- (pow (* h l) -0.5)))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((M * D) / (d * 2.0)), 2.0))))) <= 4e+300) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (pow(((0.5 * (M * D)) / d), 2.0) * (0.5 * (h / l))));
	} else {
		tmp = d * -pow((h * l), -0.5);
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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 / h) ** 0.5d0) * ((d / l) ** 0.5d0)) * (1.0d0 - ((h / l) * (0.5d0 * (((m * d_1) / (d * 2.0d0)) ** 2.0d0))))) <= 4d+300) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - ((((0.5d0 * (m * d_1)) / d) ** 2.0d0) * (0.5d0 * (h / l))))
    else
        tmp = d * -((h * l) ** (-0.5d0))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * Math.pow(((M * D) / (d * 2.0)), 2.0))))) <= 4e+300) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (Math.pow(((0.5 * (M * D)) / d), 2.0) * (0.5 * (h / l))));
	} else {
		tmp = d * -Math.pow((h * l), -0.5);
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if ((math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * math.pow(((M * D) / (d * 2.0)), 2.0))))) <= 4e+300:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (math.pow(((0.5 * (M * D)) / d), 2.0) * (0.5 * (h / l))))
	else:
		tmp = d * -math.pow((h * l), -0.5)
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0))))) <= 4e+300)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64((Float64(Float64(0.5 * Float64(M * D)) / d) ^ 2.0) * Float64(0.5 * Float64(h / l)))));
	else
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((h / l) * (0.5 * (((M * D) / (d * 2.0)) ^ 2.0))))) <= 4e+300)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((0.5 * (M * D)) / d) ^ 2.0) * (0.5 * (h / l))));
	else
		tmp = d * -((h * l) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+300], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[Power[N[(N[(0.5 * N[(M * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 4.0000000000000002e300

   1. Initial program 84.9%

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

      1. metadata-eval 84.9%

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

      2. unpow1/2 84.9%

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

      3. metadata-eval 84.9%

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

      4. unpow1/2 85.0%

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

      5. *-commutative 85.0%

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

      6. associate-*l* 85.0%

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

      7. times-frac 83.9%

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

      8. metadata-eval 83.9%

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

   3. Simplified 83.9%

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

      1. frac-times 85.0%

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

      2. associate-/r* 85.0%

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

      3. div-inv 85.0%

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

      4. metadata-eval 85.0%

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

   5. Applied egg-rr 85.0%

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

   if 4.0000000000000002e300 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))

   1. Initial program 14.2%

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

      1. metadata-eval 14.2%

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

      2. unpow1/2 14.2%

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

      3. metadata-eval 14.2%

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

      4. unpow1/2 14.2%

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

      5. *-commutative 14.2%

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

      6. associate-*l* 14.2%

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

      7. times-frac 14.2%

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

      8. metadata-eval 14.2%

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

   3. Simplified 14.2%

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

   5. Applied egg-rr 14.2%

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

      1. associate-*l* 14.2%

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

      2. associate-*r/ 14.2%

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

      3. *-commutative 14.2%

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

   7. Simplified 14.2%

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

   8. Taylor expanded in d around -inf 34.6%

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

      1. mul-1-neg 34.6%

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

      2. distribute-rgt-neg-in 34.6%

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

      3. unpow-1 34.6%

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

      4. sqr-pow 34.6%

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

      5. rem-sqrt-square 34.6%

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

      6. metadata-eval 34.6%

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

      7. sqr-pow 34.4%

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

      8. fabs-sqr 34.4%

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

      9. sqr-pow 34.6%

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

      10. *-commutative 34.6%

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

   10. Simplified 34.6%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq 4 \cdot 10^{+300}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - {\left(\frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \end{array} \]

Alternative 4: 75.5% accurate, 0.8× speedup

Math

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

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -2e-304)
   (*
    (sqrt (/ d h))
    (*
     (- 1.0 (* 0.5 (* (pow (* (/ M 2.0) (/ D d)) 2.0) (/ h l))))
     (/ (sqrt (- d)) (sqrt (- l)))))
   (*
    (/ d (* (sqrt l) (sqrt h)))
    (+ 1.0 (* (/ h l) (* (pow (/ (* D (* 0.5 M)) d) 2.0) -0.5))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2e-304) {
		tmp = sqrt((d / h)) * ((1.0 - (0.5 * (pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))) * (sqrt(-d) / sqrt(-l)));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + ((h / l) * (pow(((D * (0.5 * M)) / d), 2.0) * -0.5)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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 <= (-2d-304)) then
        tmp = sqrt((d / h)) * ((1.0d0 - (0.5d0 * ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (h / l)))) * (sqrt(-d) / sqrt(-l)))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 + ((h / l) * ((((d_1 * (0.5d0 * m)) / d) ** 2.0d0) * (-0.5d0))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2e-304) {
		tmp = Math.sqrt((d / h)) * ((1.0 - (0.5 * (Math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))) * (Math.sqrt(-d) / Math.sqrt(-l)));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 + ((h / l) * (Math.pow(((D * (0.5 * M)) / d), 2.0) * -0.5)));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -2e-304:
		tmp = math.sqrt((d / h)) * ((1.0 - (0.5 * (math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))) * (math.sqrt(-d) / math.sqrt(-l)))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 + ((h / l) * (math.pow(((D * (0.5 * M)) / d), 2.0) * -0.5)))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -2e-304)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(1.0 - Float64(0.5 * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * Float64(h / l)))) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l)))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(D * Float64(0.5 * M)) / d) ^ 2.0) * -0.5))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -2e-304)
		tmp = sqrt((d / h)) * ((1.0 - (0.5 * ((((M / 2.0) * (D / d)) ^ 2.0) * (h / l)))) * (sqrt(-d) / sqrt(-l)));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + ((h / l) * ((((D * (0.5 * M)) / d) ^ 2.0) * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -2e-304], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 - N[(0.5 * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.99999999999999994e-304

   1. Initial program 63.8%

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

      1. associate-*l* 63.8%

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

      2. metadata-eval 63.8%

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

      3. unpow1/2 63.8%

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

      4. metadata-eval 63.8%

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

      5. unpow1/2 63.9%

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

      6. associate-*l* 63.9%

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

      7. metadata-eval 63.9%

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

      8. times-frac 62.4%

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

   3. Simplified 62.4%

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

      1. frac-2neg 62.4%

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

      2. sqrt-div 68.0%

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

   5. Applied egg-rr 68.0%

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

   if -1.99999999999999994e-304 < l

   1. Initial program 68.1%

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

      1. metadata-eval 68.1%

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

      2. unpow1/2 68.1%

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

      3. metadata-eval 68.1%

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

      4. unpow1/2 68.1%

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

      5. *-commutative 68.1%

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

      6. associate-*l* 68.1%

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

      7. times-frac 68.1%

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

      8. metadata-eval 68.1%

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

   3. Simplified 68.1%

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

   5. Applied egg-rr 74.6%

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

      1. distribute-rgt1-in 77.8%

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

      2. +-commutative 77.8%

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

      3. *-commutative 77.8%

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

      4. *-commutative 77.8%

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

      5. associate-*r/ 77.7%

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

      6. *-commutative 77.7%

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

   7. Simplified 77.7%

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

4. Final simplification 72.7%

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

Alternative 5: 78.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: M should be positive before calling this function
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 (h <= (-5d-310)) then
        tmp = ((sqrt(-d) / sqrt(-h)) * sqrt((d / l))) * (1.0d0 - (((d_1 * (m / (d / 0.5d0))) ** 2.0d0) * (h * (0.5d0 / l))))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 + ((h / l) * ((((d_1 * (0.5d0 * m)) / d) ** 2.0d0) * (-0.5d0))))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 63.3%

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

      1. metadata-eval 63.3%

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

      2. unpow1/2 63.3%

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

      3. metadata-eval 63.3%

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

      4. unpow1/2 63.4%

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

      5. *-commutative 63.4%

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

      6. associate-*l* 63.4%

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

      7. times-frac 61.9%

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

      8. metadata-eval 61.9%

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

   3. Simplified 61.9%

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

      1. associate-*r* 61.9%

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

      2. frac-times 63.4%

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

      3. *-commutative 63.4%

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

      4. metadata-eval 63.4%

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

      5. expm1-log1p-u 63.2%

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

      6. expm1-udef 63.2%

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

   5. Applied egg-rr 61.9%

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

      1. +-commutative 61.9%

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

      2. associate--l+ 61.9%

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

      3. metadata-eval 61.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}} + \color{blue}{0}\right)\right) \]

      4. +-rgt-identity 61.9%

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

      5. associate-/r/ 61.9%

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

      6. associate-*r/ 63.4%

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

      7. associate-*l/ 63.4%

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

      8. *-commutative 63.4%

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

      9. associate-/l* 63.4%

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

      10. *-commutative 63.4%

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

   7. Simplified 63.4%

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

      1. frac-2neg 61.9%

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

      2. sqrt-div 70.5%

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

   9. Applied egg-rr 71.9%

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

   if -4.999999999999985e-310 < h

   1. Initial program 68.6%

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

      1. metadata-eval 68.6%

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

      2. unpow1/2 68.6%

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

      3. metadata-eval 68.6%

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

      4. unpow1/2 68.6%

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

      5. *-commutative 68.6%

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

      6. associate-*l* 68.6%

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

      7. times-frac 68.6%

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

      8. metadata-eval 68.6%

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

   3. Simplified 68.6%

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

   5. Applied egg-rr 75.2%

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

      1. distribute-rgt1-in 78.4%

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

      2. +-commutative 78.4%

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

      3. *-commutative 78.4%

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

      4. *-commutative 78.4%

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

      5. associate-*r/ 78.4%

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

      6. *-commutative 78.4%

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

   7. Simplified 78.4%

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

4. Final simplification 75.0%

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

Alternative 6: 68.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -8.2 \cdot 10^{-224}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot 0.25\right)\right)\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot \left(0.5 \cdot M\right)}{d}\right)}^{2} \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= h -8.2e-224)
   (*
    (/ 1.0 (sqrt (/ h d)))
    (*
     (sqrt (/ d l))
     (- 1.0 (* 0.5 (* (* (/ D d) (/ D d)) (* (/ (* M M) (/ l h)) 0.25))))))
   (if (<= h -5e-310)
     (* d (- (pow (* h l) -0.5)))
     (*
      (/ d (* (sqrt l) (sqrt h)))
      (+ 1.0 (* (/ h l) (* (pow (/ (* D (* 0.5 M)) d) 2.0) -0.5)))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -8.2e-224) {
		tmp = (1.0 / sqrt((h / d))) * (sqrt((d / l)) * (1.0 - (0.5 * (((D / d) * (D / d)) * (((M * M) / (l / h)) * 0.25)))));
	} else if (h <= -5e-310) {
		tmp = d * -pow((h * l), -0.5);
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + ((h / l) * (pow(((D * (0.5 * M)) / d), 2.0) * -0.5)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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 (h <= (-8.2d-224)) then
        tmp = (1.0d0 / sqrt((h / d))) * (sqrt((d / l)) * (1.0d0 - (0.5d0 * (((d_1 / d) * (d_1 / d)) * (((m * m) / (l / h)) * 0.25d0)))))
    else if (h <= (-5d-310)) then
        tmp = d * -((h * l) ** (-0.5d0))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 + ((h / l) * ((((d_1 * (0.5d0 * m)) / d) ** 2.0d0) * (-0.5d0))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -8.2e-224) {
		tmp = (1.0 / Math.sqrt((h / d))) * (Math.sqrt((d / l)) * (1.0 - (0.5 * (((D / d) * (D / d)) * (((M * M) / (l / h)) * 0.25)))));
	} else if (h <= -5e-310) {
		tmp = d * -Math.pow((h * l), -0.5);
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 + ((h / l) * (Math.pow(((D * (0.5 * M)) / d), 2.0) * -0.5)));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if h <= -8.2e-224:
		tmp = (1.0 / math.sqrt((h / d))) * (math.sqrt((d / l)) * (1.0 - (0.5 * (((D / d) * (D / d)) * (((M * M) / (l / h)) * 0.25)))))
	elif h <= -5e-310:
		tmp = d * -math.pow((h * l), -0.5)
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 + ((h / l) * (math.pow(((D * (0.5 * M)) / d), 2.0) * -0.5)))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -8.2e-224)
		tmp = Float64(Float64(1.0 / sqrt(Float64(h / d))) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(0.5 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(Float64(M * M) / Float64(l / h)) * 0.25))))));
	elseif (h <= -5e-310)
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(D * Float64(0.5 * M)) / d) ^ 2.0) * -0.5))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= -8.2e-224)
		tmp = (1.0 / sqrt((h / d))) * (sqrt((d / l)) * (1.0 - (0.5 * (((D / d) * (D / d)) * (((M * M) / (l / h)) * 0.25)))));
	elseif (h <= -5e-310)
		tmp = d * -((h * l) ^ -0.5);
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + ((h / l) * ((((D * (0.5 * M)) / d) ^ 2.0) * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[h, -8.2e-224], N[(N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-310], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -8.19999999999999972e-224

   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. 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. associate-*l* 64.8%

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

      7. metadata-eval 64.8%

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

      8. times-frac 63.2%

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

   3. Simplified 63.2%

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

   4. Taylor expanded in M around 0 44.1%

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

      1. associate-*r/ 44.1%

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

      2. *-commutative 44.1%

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

      3. associate-*r/ 44.1%

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

      4. *-commutative 44.1%

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

      5. times-frac 44.1%

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

      6. unpow2 44.1%

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

      7. unpow2 44.1%

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

      8. times-frac 57.2%

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

      9. associate-*l* 57.2%

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

      10. associate-/l* 56.3%

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

      11. unpow2 56.3%

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

   6. Simplified 56.3%

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

      1. clear-num 56.2%

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

      2. sqrt-div 56.2%

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

      3. metadata-eval 56.2%

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

   8. Applied egg-rr 56.2%

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

   if -8.19999999999999972e-224 < h < -4.999999999999985e-310

   1. Initial program 47.2%

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

      1. metadata-eval 47.2%

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

      2. unpow1/2 47.2%

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

      3. metadata-eval 47.2%

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

      4. unpow1/2 47.2%

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

      5. *-commutative 47.2%

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

      6. associate-*l* 47.2%

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

      7. times-frac 47.2%

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

      8. metadata-eval 47.2%

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

   3. Simplified 47.2%

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

   5. Applied egg-rr 30.0%

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

      1. associate-*l* 30.0%

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

      2. associate-*r/ 30.0%

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

      3. *-commutative 30.0%

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

   7. Simplified 30.0%

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

   8. Taylor expanded in d around -inf 82.2%

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

      1. mul-1-neg 82.2%

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

      2. distribute-rgt-neg-in 82.2%

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

      3. unpow-1 82.2%

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

      4. sqr-pow 82.2%

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

      5. rem-sqrt-square 82.2%

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

      6. metadata-eval 82.2%

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

      7. sqr-pow 82.0%

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

      8. fabs-sqr 82.0%

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

      9. sqr-pow 82.2%

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

      10. *-commutative 82.2%

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

   10. Simplified 82.2%

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

   if -4.999999999999985e-310 < h

   1. Initial program 68.6%

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

      1. metadata-eval 68.6%

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

      2. unpow1/2 68.6%

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

      3. metadata-eval 68.6%

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

      4. unpow1/2 68.6%

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

      5. *-commutative 68.6%

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

      6. associate-*l* 68.6%

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

      7. times-frac 68.6%

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

      8. metadata-eval 68.6%

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

   3. Simplified 68.6%

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

   5. Applied egg-rr 75.2%

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

      1. distribute-rgt1-in 78.4%

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

      2. +-commutative 78.4%

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

      3. *-commutative 78.4%

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

      4. *-commutative 78.4%

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

      5. associate-*r/ 78.4%

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

      6. *-commutative 78.4%

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

   7. Simplified 78.4%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -8.2 \cdot 10^{-224}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot 0.25\right)\right)\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot \left(0.5 \cdot M\right)}{d}\right)}^{2} \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 7: 73.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7.5 \cdot 10^{-293}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot \left(0.5 \cdot M\right)}{d}\right)}^{2} \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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-293)
   (*
    (*
     (sqrt (/ d l))
     (- 1.0 (* 0.5 (* (pow (* (/ M 2.0) (/ D d)) 2.0) (/ h l)))))
    (sqrt (/ d h)))
   (*
    (/ d (* (sqrt l) (sqrt h)))
    (+ 1.0 (* (/ h l) (* (pow (/ (* D (* 0.5 M)) d) 2.0) -0.5))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 7.5e-293) {
		tmp = (sqrt((d / l)) * (1.0 - (0.5 * (pow(((M / 2.0) * (D / d)), 2.0) * (h / l))))) * sqrt((d / h));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + ((h / l) * (pow(((D * (0.5 * M)) / d), 2.0) * -0.5)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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-293) then
        tmp = (sqrt((d / l)) * (1.0d0 - (0.5d0 * ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (h / l))))) * sqrt((d / h))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 + ((h / l) * ((((d_1 * (0.5d0 * m)) / d) ** 2.0d0) * (-0.5d0))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 7.5e-293) {
		tmp = (Math.sqrt((d / l)) * (1.0 - (0.5 * (Math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l))))) * Math.sqrt((d / h));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 + ((h / l) * (Math.pow(((D * (0.5 * M)) / d), 2.0) * -0.5)));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= 7.5e-293:
		tmp = (math.sqrt((d / l)) * (1.0 - (0.5 * (math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l))))) * math.sqrt((d / h))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 + ((h / l) * (math.pow(((D * (0.5 * M)) / d), 2.0) * -0.5)))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 7.5e-293)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(0.5 * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * Float64(h / l))))) * sqrt(Float64(d / h)));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(D * Float64(0.5 * M)) / d) ^ 2.0) * -0.5))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 7.5e-293)
		tmp = (sqrt((d / l)) * (1.0 - (0.5 * ((((M / 2.0) * (D / d)) ^ 2.0) * (h / l))))) * sqrt((d / h));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + ((h / l) * ((((D * (0.5 * M)) / d) ^ 2.0) * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
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-293], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.50000000000000038e-293

   1. Initial program 64.4%

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

      1. associate-*l* 64.4%

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

      2. metadata-eval 64.4%

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

      3. unpow1/2 64.4%

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

      4. metadata-eval 64.4%

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

      5. unpow1/2 64.4%

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

      6. associate-*l* 64.4%

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

      7. metadata-eval 64.4%

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

      8. times-frac 63.0%

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

   3. Simplified 63.0%

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

   if 7.50000000000000038e-293 < l

   1. Initial program 67.6%

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

      1. metadata-eval 67.6%

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

      2. unpow1/2 67.6%

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

      3. metadata-eval 67.6%

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

      4. unpow1/2 67.6%

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

      5. *-commutative 67.6%

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

      6. associate-*l* 67.6%

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

      7. times-frac 67.6%

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

      8. metadata-eval 67.6%

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

   3. Simplified 67.6%

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

   5. Applied egg-rr 74.4%

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

      1. distribute-rgt1-in 77.7%

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

      2. +-commutative 77.7%

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

      3. *-commutative 77.7%

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

      4. *-commutative 77.7%

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

      5. associate-*r/ 77.7%

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

      6. *-commutative 77.7%

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

   7. Simplified 77.7%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.5 \cdot 10^{-293}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot \left(0.5 \cdot M\right)}{d}\right)}^{2} \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 8: 44.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 4.2 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{\frac{d}{\frac{\ell}{\frac{d}{h}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 - 0.5 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot 0.25\right)\right)\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= M 4.2e-237)
   (sqrt (/ d (/ l (/ d h))))
   (*
    (sqrt (/ d h))
    (*
     (- 1.0 (* 0.5 (* (* (/ D d) (/ D d)) (* (/ (* M M) (/ l h)) 0.25))))
     (/ 1.0 (sqrt (/ l d)))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 4.2e-237) {
		tmp = sqrt((d / (l / (d / h))));
	} else {
		tmp = sqrt((d / h)) * ((1.0 - (0.5 * (((D / d) * (D / d)) * (((M * M) / (l / h)) * 0.25)))) * (1.0 / sqrt((l / d))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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 <= 4.2d-237) then
        tmp = sqrt((d / (l / (d / h))))
    else
        tmp = sqrt((d / h)) * ((1.0d0 - (0.5d0 * (((d_1 / d) * (d_1 / d)) * (((m * m) / (l / h)) * 0.25d0)))) * (1.0d0 / sqrt((l / d))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 4.2e-237) {
		tmp = Math.sqrt((d / (l / (d / h))));
	} else {
		tmp = Math.sqrt((d / h)) * ((1.0 - (0.5 * (((D / d) * (D / d)) * (((M * M) / (l / h)) * 0.25)))) * (1.0 / Math.sqrt((l / d))));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if M <= 4.2e-237:
		tmp = math.sqrt((d / (l / (d / h))))
	else:
		tmp = math.sqrt((d / h)) * ((1.0 - (0.5 * (((D / d) * (D / d)) * (((M * M) / (l / h)) * 0.25)))) * (1.0 / math.sqrt((l / d))))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (M <= 4.2e-237)
		tmp = sqrt(Float64(d / Float64(l / Float64(d / h))));
	else
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(Float64(M * M) / Float64(l / h)) * 0.25)))) * Float64(1.0 / sqrt(Float64(l / d)))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (M <= 4.2e-237)
		tmp = sqrt((d / (l / (d / h))));
	else
		tmp = sqrt((d / h)) * ((1.0 - (0.5 * (((D / d) * (D / d)) * (((M * M) / (l / h)) * 0.25)))) * (1.0 / sqrt((l / d))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[M, 4.2e-237], N[Sqrt[N[(d / N[(l / N[(d / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 - N[(0.5 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.2000000000000002e-237

   1. Initial program 66.8%

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

      1. metadata-eval 66.8%

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

      2. unpow1/2 66.8%

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

      3. metadata-eval 66.8%

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

      4. unpow1/2 66.8%

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

      5. *-commutative 66.8%

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

      6. associate-*l* 66.8%

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

      7. times-frac 65.5%

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

      8. metadata-eval 65.5%

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

   3. Simplified 65.5%

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

   5. Applied egg-rr 32.5%

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

      1. associate-*l/ 29.5%

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

      2. associate-/l* 33.1%

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

      3. +-commutative 33.1%

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

      4. associate-*r* 33.1%

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

      5. *-commutative 33.1%

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

      6. associate-*l* 33.1%

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

      7. fma-def 33.1%

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

      8. associate-*r/ 33.1%

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

      9. associate-*l/ 33.1%

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

      10. *-commutative 33.1%

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

      11. associate-/l* 33.1%

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

   7. Simplified 33.1%

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

   8. Taylor expanded in D around 0 35.9%

      \[\leadsto \sqrt{\frac{d}{\frac{\ell}{\frac{d}{h}}} \cdot {\color{blue}{1}}^{2}} \]

   if 4.2000000000000002e-237 < 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* 63.9%

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

      2. metadata-eval 63.9%

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

      3. unpow1/2 63.9%

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

      4. metadata-eval 63.9%

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

      5. unpow1/2 63.9%

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

      6. associate-*l* 63.9%

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

      7. metadata-eval 63.9%

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

      8. times-frac 63.9%

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

   3. Simplified 63.9%

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

   4. Taylor expanded in M around 0 45.2%

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

      1. associate-*r/ 45.2%

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

      2. *-commutative 45.2%

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

      3. associate-*r/ 45.2%

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

      4. *-commutative 45.2%

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

      5. times-frac 46.1%

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

      6. unpow2 46.1%

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

      7. unpow2 46.1%

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

      8. times-frac 54.3%

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

      9. associate-*l* 54.3%

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

      10. associate-/l* 53.4%

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

      11. unpow2 53.4%

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

   6. Simplified 53.4%

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

      1. clear-num 53.4%

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

      2. sqrt-div 53.5%

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

      3. metadata-eval 53.5%

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

   8. Applied egg-rr 53.5%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot 0.25\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4.2 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{\frac{d}{\frac{\ell}{\frac{d}{h}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 - 0.5 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot 0.25\right)\right)\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)\\ \end{array} \]

Alternative 9: 45.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 4.5 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{\frac{d}{\frac{\ell}{\frac{d}{h}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot 0.25\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= M 4.5e-237)
   (sqrt (/ d (/ l (/ d h))))
   (*
    (/ 1.0 (sqrt (/ h d)))
    (*
     (sqrt (/ d l))
     (- 1.0 (* 0.5 (* (* (/ D d) (/ D d)) (* (/ (* M M) (/ l h)) 0.25))))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 4.5e-237) {
		tmp = sqrt((d / (l / (d / h))));
	} else {
		tmp = (1.0 / sqrt((h / d))) * (sqrt((d / l)) * (1.0 - (0.5 * (((D / d) * (D / d)) * (((M * M) / (l / h)) * 0.25)))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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 <= 4.5d-237) then
        tmp = sqrt((d / (l / (d / h))))
    else
        tmp = (1.0d0 / sqrt((h / d))) * (sqrt((d / l)) * (1.0d0 - (0.5d0 * (((d_1 / d) * (d_1 / d)) * (((m * m) / (l / h)) * 0.25d0)))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 4.5e-237) {
		tmp = Math.sqrt((d / (l / (d / h))));
	} else {
		tmp = (1.0 / Math.sqrt((h / d))) * (Math.sqrt((d / l)) * (1.0 - (0.5 * (((D / d) * (D / d)) * (((M * M) / (l / h)) * 0.25)))));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if M <= 4.5e-237:
		tmp = math.sqrt((d / (l / (d / h))))
	else:
		tmp = (1.0 / math.sqrt((h / d))) * (math.sqrt((d / l)) * (1.0 - (0.5 * (((D / d) * (D / d)) * (((M * M) / (l / h)) * 0.25)))))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (M <= 4.5e-237)
		tmp = sqrt(Float64(d / Float64(l / Float64(d / h))));
	else
		tmp = Float64(Float64(1.0 / sqrt(Float64(h / d))) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(0.5 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(Float64(M * M) / Float64(l / h)) * 0.25))))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (M <= 4.5e-237)
		tmp = sqrt((d / (l / (d / h))));
	else
		tmp = (1.0 / sqrt((h / d))) * (sqrt((d / l)) * (1.0 - (0.5 * (((D / d) * (D / d)) * (((M * M) / (l / h)) * 0.25)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[M, 4.5e-237], N[Sqrt[N[(d / N[(l / N[(d / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.50000000000000009e-237

   1. Initial program 66.8%

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

      1. metadata-eval 66.8%

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

      2. unpow1/2 66.8%

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

      3. metadata-eval 66.8%

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

      4. unpow1/2 66.8%

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

      5. *-commutative 66.8%

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

      6. associate-*l* 66.8%

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

      7. times-frac 65.5%

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

      8. metadata-eval 65.5%

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

   3. Simplified 65.5%

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

   5. Applied egg-rr 32.5%

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

      1. associate-*l/ 29.5%

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

      2. associate-/l* 33.1%

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

      3. +-commutative 33.1%

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

      4. associate-*r* 33.1%

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

      5. *-commutative 33.1%

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

      6. associate-*l* 33.1%

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

      7. fma-def 33.1%

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

      8. associate-*r/ 33.1%

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

      9. associate-*l/ 33.1%

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

      10. *-commutative 33.1%

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

      11. associate-/l* 33.1%

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

   7. Simplified 33.1%

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

   8. Taylor expanded in D around 0 35.9%

      \[\leadsto \sqrt{\frac{d}{\frac{\ell}{\frac{d}{h}}} \cdot {\color{blue}{1}}^{2}} \]

   if 4.50000000000000009e-237 < 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* 63.9%

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

      2. metadata-eval 63.9%

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

      3. unpow1/2 63.9%

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

      4. metadata-eval 63.9%

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

      5. unpow1/2 63.9%

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

      6. associate-*l* 63.9%

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

      7. metadata-eval 63.9%

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

      8. times-frac 63.9%

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

   3. Simplified 63.9%

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

   4. Taylor expanded in M around 0 45.2%

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

      1. associate-*r/ 45.2%

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

      2. *-commutative 45.2%

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

      3. associate-*r/ 45.2%

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

      4. *-commutative 45.2%

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

      5. times-frac 46.1%

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

      6. unpow2 46.1%

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

      7. unpow2 46.1%

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

      8. times-frac 54.3%

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

      9. associate-*l* 54.3%

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

      10. associate-/l* 53.4%

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

      11. unpow2 53.4%

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

   6. Simplified 53.4%

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

      1. clear-num 53.4%

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

      2. sqrt-div 53.4%

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

      3. metadata-eval 53.4%

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

   8. Applied egg-rr 53.4%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot 0.25\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4.5 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{\frac{d}{\frac{\ell}{\frac{d}{h}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot 0.25\right)\right)\right)\right)\\ \end{array} \]

Alternative 10: 45.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 5 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{\frac{d}{\frac{\ell}{\frac{d}{h}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot 0.25\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= M 5e-237)
   (sqrt (/ d (/ l (/ d h))))
   (*
    (sqrt (/ d h))
    (*
     (sqrt (/ d l))
     (- 1.0 (* 0.5 (* (* (/ D d) (/ D d)) (* (/ (* M M) (/ l h)) 0.25))))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 5e-237) {
		tmp = sqrt((d / (l / (d / h))));
	} else {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * (((D / d) * (D / d)) * (((M * M) / (l / h)) * 0.25)))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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 <= 5d-237) then
        tmp = sqrt((d / (l / (d / h))))
    else
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 - (0.5d0 * (((d_1 / d) * (d_1 / d)) * (((m * m) / (l / h)) * 0.25d0)))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 5e-237) {
		tmp = Math.sqrt((d / (l / (d / h))));
	} else {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - (0.5 * (((D / d) * (D / d)) * (((M * M) / (l / h)) * 0.25)))));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if M <= 5e-237:
		tmp = math.sqrt((d / (l / (d / h))))
	else:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - (0.5 * (((D / d) * (D / d)) * (((M * M) / (l / h)) * 0.25)))))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (M <= 5e-237)
		tmp = sqrt(Float64(d / Float64(l / Float64(d / h))));
	else
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(0.5 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(Float64(M * M) / Float64(l / h)) * 0.25))))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (M <= 5e-237)
		tmp = sqrt((d / (l / (d / h))));
	else
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * (((D / d) * (D / d)) * (((M * M) / (l / h)) * 0.25)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[M, 5e-237], N[Sqrt[N[(d / N[(l / N[(d / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 5.0000000000000002e-237

   1. Initial program 66.8%

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

      1. metadata-eval 66.8%

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

      2. unpow1/2 66.8%

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

      3. metadata-eval 66.8%

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

      4. unpow1/2 66.8%

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

      5. *-commutative 66.8%

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

      6. associate-*l* 66.8%

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

      7. times-frac 65.5%

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

      8. metadata-eval 65.5%

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

   3. Simplified 65.5%

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

   5. Applied egg-rr 32.5%

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

      1. associate-*l/ 29.5%

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

      2. associate-/l* 33.1%

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

      3. +-commutative 33.1%

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

      4. associate-*r* 33.1%

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

      5. *-commutative 33.1%

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

      6. associate-*l* 33.1%

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

      7. fma-def 33.1%

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

      8. associate-*r/ 33.1%

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

      9. associate-*l/ 33.1%

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

      10. *-commutative 33.1%

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

      11. associate-/l* 33.1%

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

   7. Simplified 33.1%

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

   8. Taylor expanded in D around 0 35.9%

      \[\leadsto \sqrt{\frac{d}{\frac{\ell}{\frac{d}{h}}} \cdot {\color{blue}{1}}^{2}} \]

   if 5.0000000000000002e-237 < 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* 63.9%

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

      2. metadata-eval 63.9%

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

      3. unpow1/2 63.9%

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

      4. metadata-eval 63.9%

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

      5. unpow1/2 63.9%

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

      6. associate-*l* 63.9%

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

      7. metadata-eval 63.9%

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

      8. times-frac 63.9%

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

   3. Simplified 63.9%

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

   4. Taylor expanded in M around 0 45.2%

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

      1. associate-*r/ 45.2%

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

      2. *-commutative 45.2%

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

      3. associate-*r/ 45.2%

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

      4. *-commutative 45.2%

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

      5. times-frac 46.1%

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

      6. unpow2 46.1%

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

      7. unpow2 46.1%

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

      8. times-frac 54.3%

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

      9. associate-*l* 54.3%

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

      10. associate-/l* 53.4%

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

      11. unpow2 53.4%

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

   6. Simplified 53.4%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot 0.25\right)\right)}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 5 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{\frac{d}{\frac{\ell}{\frac{d}{h}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot 0.25\right)\right)\right)\right)\\ \end{array} \]

Alternative 11: 55.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq 2.4 \cdot 10^{+144}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot M}{d \cdot \frac{d}{h}}\right) \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - {\left(D \cdot \frac{M}{\frac{d}{0.5}}\right)}^{2} \cdot \frac{h \cdot 0.5}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= D 2.4e+144)
   (*
    (* (sqrt (/ d l)) (sqrt (/ d h)))
    (- 1.0 (* (* (/ D (/ l D)) (/ (* M M) (* d (/ d h)))) 0.125)))
   (*
    (sqrt (/ (/ d h) (/ l d)))
    (- 1.0 (* (pow (* D (/ M (/ d 0.5))) 2.0) (/ (* h 0.5) l))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (D <= 2.4e+144) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((D / (l / D)) * ((M * M) / (d * (d / h)))) * 0.125));
	} else {
		tmp = sqrt(((d / h) / (l / d))) * (1.0 - (pow((D * (M / (d / 0.5))), 2.0) * ((h * 0.5) / l)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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 <= 2.4d+144) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (((d_1 / (l / d_1)) * ((m * m) / (d * (d / h)))) * 0.125d0))
    else
        tmp = sqrt(((d / h) / (l / d))) * (1.0d0 - (((d_1 * (m / (d / 0.5d0))) ** 2.0d0) * ((h * 0.5d0) / l)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (D <= 2.4e+144) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (((D / (l / D)) * ((M * M) / (d * (d / h)))) * 0.125));
	} else {
		tmp = Math.sqrt(((d / h) / (l / d))) * (1.0 - (Math.pow((D * (M / (d / 0.5))), 2.0) * ((h * 0.5) / l)));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if D <= 2.4e+144:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (((D / (l / D)) * ((M * M) / (d * (d / h)))) * 0.125))
	else:
		tmp = math.sqrt(((d / h) / (l / d))) * (1.0 - (math.pow((D * (M / (d / 0.5))), 2.0) * ((h * 0.5) / l)))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (D <= 2.4e+144)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(D / Float64(l / D)) * Float64(Float64(M * M) / Float64(d * Float64(d / h)))) * 0.125)));
	else
		tmp = Float64(sqrt(Float64(Float64(d / h) / Float64(l / d))) * Float64(1.0 - Float64((Float64(D * Float64(M / Float64(d / 0.5))) ^ 2.0) * Float64(Float64(h * 0.5) / l))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (D <= 2.4e+144)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((D / (l / D)) * ((M * M) / (d * (d / h)))) * 0.125));
	else
		tmp = sqrt(((d / h) / (l / d))) * (1.0 - (((D * (M / (d / 0.5))) ^ 2.0) * ((h * 0.5) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[D, 2.4e+144], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / N[(d * N[(d / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] / N[(l / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[Power[N[(D * N[(M / N[(d / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h * 0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 2.4000000000000001e144

   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. metadata-eval 66.1%

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

      2. unpow1/2 66.1%

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

      3. metadata-eval 66.1%

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

      4. unpow1/2 66.1%

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

      5. *-commutative 66.1%

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

      6. associate-*l* 66.1%

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

      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. Taylor expanded in M around 0 43.8%

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

      1. *-commutative 43.8%

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

      2. times-frac 47.7%

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

      3. unpow2 47.7%

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

      4. associate-/l* 47.7%

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

      5. associate-/l* 46.8%

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

      6. unpow2 46.8%

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

      7. unpow2 46.8%

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

      8. associate-*r/ 50.3%

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

   6. Simplified 50.3%

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

   if 2.4000000000000001e144 < D

   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. metadata-eval 64.7%

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

      2. unpow1/2 64.7%

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

      3. metadata-eval 64.7%

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

      4. unpow1/2 64.7%

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

      5. *-commutative 64.7%

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

      6. associate-*l* 64.7%

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

      7. times-frac 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. associate-*r* 64.7%

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

      2. frac-times 64.7%

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

      3. *-commutative 64.7%

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

      4. metadata-eval 64.7%

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

      5. add-sqr-sqrt 64.7%

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

      6. pow2 64.7%

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

   5. Applied egg-rr 66.9%

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

      1. pow1 66.9%

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

      2. sqrt-unprod 66.9%

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

      3. *-commutative 66.9%

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

      4. div-inv 66.9%

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

      5. clear-num 66.9%

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

      6. associate-*l* 66.9%

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

   7. Applied egg-rr 66.9%

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

      1. unpow1 66.9%

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

      2. associate-*r/ 57.5%

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

      3. associate-/l* 66.9%

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

      4. unpow2 66.9%

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

      5. swap-sqr 64.6%

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

      6. rem-square-sqrt 64.7%

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

      7. associate-*r/ 64.7%

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

      8. unpow2 64.7%

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

      9. associate-*r* 64.7%

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

      10. associate-*r/ 64.6%

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

      11. associate-*l/ 62.3%

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

      12. associate-*l/ 62.3%

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

      13. *-commutative 62.3%

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

      14. associate-*l/ 62.3%

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

      15. associate-/l* 62.3%

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

   9. Simplified 62.3%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 2.4 \cdot 10^{+144}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot M}{d \cdot \frac{d}{h}}\right) \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - {\left(D \cdot \frac{M}{\frac{d}{0.5}}\right)}^{2} \cdot \frac{h \cdot 0.5}{\ell}\right)\\ \end{array} \]

Alternative 12: 46.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \frac{1}{h \cdot \ell}\\ t_1 := d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{if}\;\ell \leq -2.6 \cdot 10^{-216}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -3.1 \cdot 10^{-304}:\\ \;\;\;\;d \cdot \sqrt[3]{t_0 \cdot \sqrt{t_0}}\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{-293}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{-127} \lor \neg \left(\ell \leq 1.55 \cdot 10^{-21}\right) \land \ell \leq 1.5 \cdot 10^{+51}:\\ \;\;\;\;-0.125 \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \left(D \cdot D\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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 (/ 1.0 (* h l))) (t_1 (* d (- (pow (* h l) -0.5)))))
   (if (<= l -2.6e-216)
     t_1
     (if (<= l -3.1e-304)
       (* d (cbrt (* t_0 (sqrt t_0))))
       (if (<= l 7.5e-293)
         t_1
         (if (or (<= l 5.8e-127) (and (not (<= l 1.55e-21)) (<= l 1.5e+51)))
           (* -0.125 (/ (* (sqrt (/ h (pow l 3.0))) (* (* M M) (* D D))) d))
           (* d (* (pow h -0.5) (pow l -0.5)))))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 / (h * l);
	double t_1 = d * -pow((h * l), -0.5);
	double tmp;
	if (l <= -2.6e-216) {
		tmp = t_1;
	} else if (l <= -3.1e-304) {
		tmp = d * cbrt((t_0 * sqrt(t_0)));
	} else if (l <= 7.5e-293) {
		tmp = t_1;
	} else if ((l <= 5.8e-127) || (!(l <= 1.55e-21) && (l <= 1.5e+51))) {
		tmp = -0.125 * ((sqrt((h / pow(l, 3.0))) * ((M * M) * (D * D))) / d);
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 / (h * l);
	double t_1 = d * -Math.pow((h * l), -0.5);
	double tmp;
	if (l <= -2.6e-216) {
		tmp = t_1;
	} else if (l <= -3.1e-304) {
		tmp = d * Math.cbrt((t_0 * Math.sqrt(t_0)));
	} else if (l <= 7.5e-293) {
		tmp = t_1;
	} else if ((l <= 5.8e-127) || (!(l <= 1.55e-21) && (l <= 1.5e+51))) {
		tmp = -0.125 * ((Math.sqrt((h / Math.pow(l, 3.0))) * ((M * M) * (D * D))) / d);
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(1.0 / Float64(h * l))
	t_1 = Float64(d * Float64(-(Float64(h * l) ^ -0.5)))
	tmp = 0.0
	if (l <= -2.6e-216)
		tmp = t_1;
	elseif (l <= -3.1e-304)
		tmp = Float64(d * cbrt(Float64(t_0 * sqrt(t_0))));
	elseif (l <= 7.5e-293)
		tmp = t_1;
	elseif ((l <= 5.8e-127) || (!(l <= 1.55e-21) && (l <= 1.5e+51)))
		tmp = Float64(-0.125 * Float64(Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(Float64(M * M) * Float64(D * D))) / d));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
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[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[l, -2.6e-216], t$95$1, If[LessEqual[l, -3.1e-304], N[(d * N[Power[N[(t$95$0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.5e-293], t$95$1, If[Or[LessEqual[l, 5.8e-127], And[N[Not[LessEqual[l, 1.55e-21]], $MachinePrecision], LessEqual[l, 1.5e+51]]], N[(-0.125 * N[(N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $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 l < -2.5999999999999999e-216 or -3.09999999999999985e-304 < l < 7.50000000000000038e-293

   1. Initial program 64.1%

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

      1. metadata-eval 64.1%

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

      2. unpow1/2 64.1%

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

      3. metadata-eval 64.1%

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

      4. unpow1/2 64.1%

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

      5. *-commutative 64.1%

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

      6. associate-*l* 64.1%

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

      7. times-frac 63.3%

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

      8. metadata-eval 63.3%

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

   3. Simplified 63.3%

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

   5. Applied egg-rr 35.4%

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

      1. associate-*l* 35.4%

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

      2. associate-*r/ 35.4%

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

      3. *-commutative 35.4%

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

   7. Simplified 35.4%

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

   8. Taylor expanded in d around -inf 51.6%

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

      1. mul-1-neg 51.6%

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

      2. distribute-rgt-neg-in 51.6%

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

      3. unpow-1 51.6%

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

      4. sqr-pow 51.7%

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

      5. rem-sqrt-square 51.7%

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

      6. metadata-eval 51.7%

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

      7. sqr-pow 51.4%

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

      8. fabs-sqr 51.4%

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

      9. sqr-pow 51.7%

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

      10. *-commutative 51.7%

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

   10. Simplified 51.7%

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

   if -2.5999999999999999e-216 < l < -3.09999999999999985e-304

   1. Initial program 67.3%

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

      1. metadata-eval 67.3%

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

      2. unpow1/2 67.3%

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

      3. metadata-eval 67.3%

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

      4. unpow1/2 67.3%

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

      5. *-commutative 67.3%

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

      6. associate-*l* 67.3%

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

      7. times-frac 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. Taylor expanded in d around inf 35.6%

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

      1. add-cbrt-cube 42.0%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}}} \cdot d \]

      2. add-sqr-sqrt 42.0%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{\ell \cdot h}} \cdot \sqrt{\frac{1}{\ell \cdot h}}} \cdot d \]

   6. Applied egg-rr 42.0%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{\ell \cdot h} \cdot \sqrt{\frac{1}{\ell \cdot h}}}} \cdot d \]

   if 7.50000000000000038e-293 < l < 5.8000000000000001e-127 or 1.5499999999999999e-21 < l < 1.5e51

   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. Taylor expanded in d around 0 58.7%

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

      1. associate-*l/ 58.7%

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

      2. unpow2 58.7%

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

      3. unpow2 58.7%

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

   6. Simplified 58.7%

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

   if 5.8000000000000001e-127 < l < 1.5499999999999999e-21 or 1.5e51 < l

   1. Initial program 63.1%

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

      1. metadata-eval 63.1%

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

      2. unpow1/2 63.1%

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

      3. metadata-eval 63.1%

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

      4. unpow1/2 63.1%

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

      5. *-commutative 63.1%

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

      6. associate-*l* 63.1%

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

      7. times-frac 63.1%

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

      8. metadata-eval 63.1%

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

   3. Simplified 63.1%

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

   4. Taylor expanded in d around inf 41.0%

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

      1. *-un-lft-identity 41.0%

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

   6. Applied egg-rr 41.0%

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

      1. *-lft-identity 41.0%

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

      2. unpow-1 41.0%

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

      3. sqr-pow 41.1%

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

      4. rem-sqrt-square 41.1%

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

      5. metadata-eval 41.1%

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

      6. sqr-pow 40.9%

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

      7. fabs-sqr 40.9%

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

      8. sqr-pow 41.1%

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

      9. *-commutative 41.1%

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

   8. Simplified 41.1%

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

      1. unpow-prod-down 54.5%

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

   10. Applied egg-rr 54.5%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.6 \cdot 10^{-216}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq -3.1 \cdot 10^{-304}:\\ \;\;\;\;d \cdot \sqrt[3]{\frac{1}{h \cdot \ell} \cdot \sqrt{\frac{1}{h \cdot \ell}}}\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{-293}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{-127} \lor \neg \left(\ell \leq 1.55 \cdot 10^{-21}\right) \land \ell \leq 1.5 \cdot 10^{+51}:\\ \;\;\;\;-0.125 \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \left(D \cdot D\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 13: 48.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \frac{1}{h \cdot \ell}\\ \mathbf{if}\;\ell \leq -2.4 \cdot 10^{-214}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt[3]{t_0 \cdot \sqrt{t_0}}\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{-126} \lor \neg \left(\ell \leq 3.1 \cdot 10^{-72}\right) \land \ell \leq 1.4 \cdot 10^{+54}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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 (/ 1.0 (* h l))))
   (if (<= l -2.4e-214)
     (* d (- (pow (* h l) -0.5)))
     (if (<= l -2e-310)
       (* d (cbrt (* t_0 (sqrt t_0))))
       (if (or (<= l 2.1e-126) (and (not (<= l 3.1e-72)) (<= l 1.4e+54)))
         (* -0.125 (/ (* (* (* M D) (* M D)) (sqrt (/ h (pow l 3.0)))) d))
         (* d (* (pow h -0.5) (pow l -0.5))))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 / (h * l);
	double tmp;
	if (l <= -2.4e-214) {
		tmp = d * -pow((h * l), -0.5);
	} else if (l <= -2e-310) {
		tmp = d * cbrt((t_0 * sqrt(t_0)));
	} else if ((l <= 2.1e-126) || (!(l <= 3.1e-72) && (l <= 1.4e+54))) {
		tmp = -0.125 * ((((M * D) * (M * D)) * sqrt((h / pow(l, 3.0)))) / d);
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 / (h * l);
	double tmp;
	if (l <= -2.4e-214) {
		tmp = d * -Math.pow((h * l), -0.5);
	} else if (l <= -2e-310) {
		tmp = d * Math.cbrt((t_0 * Math.sqrt(t_0)));
	} else if ((l <= 2.1e-126) || (!(l <= 3.1e-72) && (l <= 1.4e+54))) {
		tmp = -0.125 * ((((M * D) * (M * D)) * Math.sqrt((h / Math.pow(l, 3.0)))) / d);
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(1.0 / Float64(h * l))
	tmp = 0.0
	if (l <= -2.4e-214)
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	elseif (l <= -2e-310)
		tmp = Float64(d * cbrt(Float64(t_0 * sqrt(t_0))));
	elseif ((l <= 2.1e-126) || (!(l <= 3.1e-72) && (l <= 1.4e+54)))
		tmp = Float64(-0.125 * Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * sqrt(Float64(h / (l ^ 3.0)))) / d));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
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[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.4e-214], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, -2e-310], N[(d * N[Power[N[(t$95$0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, 2.1e-126], And[N[Not[LessEqual[l, 3.1e-72]], $MachinePrecision], LessEqual[l, 1.4e+54]]], N[(-0.125 * N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / d), $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 l < -2.4000000000000002e-214

   1. Initial program 63.4%

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

      1. metadata-eval 63.4%

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

      2. unpow1/2 63.4%

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

      3. metadata-eval 63.4%

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

      4. unpow1/2 63.4%

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

      5. *-commutative 63.4%

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

      6. associate-*l* 63.4%

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

      7. times-frac 62.5%

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

      8. metadata-eval 62.5%

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

   3. Simplified 62.5%

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

   5. Applied egg-rr 36.0%

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

      1. associate-*l* 36.0%

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

      2. associate-*r/ 36.1%

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

      3. *-commutative 36.1%

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

   7. Simplified 36.1%

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

   8. Taylor expanded in d around -inf 50.4%

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

      1. mul-1-neg 50.4%

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

      2. distribute-rgt-neg-in 50.4%

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

      3. unpow-1 50.4%

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

      4. sqr-pow 50.5%

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

      5. rem-sqrt-square 50.5%

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

      6. metadata-eval 50.5%

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

      7. sqr-pow 50.2%

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

      8. fabs-sqr 50.2%

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

      9. sqr-pow 50.5%

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

      10. *-commutative 50.5%

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

   10. Simplified 50.5%

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

   if -2.4000000000000002e-214 < l < -1.999999999999994e-310

   1. Initial program 63.1%

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

      1. metadata-eval 63.1%

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

      2. unpow1/2 63.1%

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

      3. metadata-eval 63.1%

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

      4. unpow1/2 63.1%

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

      5. *-commutative 63.1%

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

      6. associate-*l* 63.1%

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

      7. times-frac 56.9%

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

      8. metadata-eval 56.9%

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

   3. Simplified 56.9%

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

   4. Taylor expanded in d around inf 33.5%

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

      1. add-cbrt-cube 39.4%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}}} \cdot d \]

      2. add-sqr-sqrt 39.4%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{\ell \cdot h}} \cdot \sqrt{\frac{1}{\ell \cdot h}}} \cdot d \]

   6. Applied egg-rr 39.4%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{\ell \cdot h} \cdot \sqrt{\frac{1}{\ell \cdot h}}}} \cdot d \]

   if -1.999999999999994e-310 < l < 2.0999999999999999e-126 or 3.0999999999999998e-72 < l < 1.40000000000000008e54

   1. Initial program 78.9%

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

      1. metadata-eval 78.9%

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

      2. unpow1/2 78.9%

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

      3. metadata-eval 78.9%

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

      4. unpow1/2 78.9%

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

      5. *-commutative 78.9%

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

      6. associate-*l* 78.9%

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

      7. times-frac 78.9%

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

      8. metadata-eval 78.9%

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

   3. Simplified 78.9%

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

      1. associate-*r* 78.9%

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

      2. frac-times 78.9%

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

      3. *-commutative 78.9%

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

      4. metadata-eval 78.9%

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

      5. expm1-log1p-u 78.2%

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

      6. expm1-udef 78.2%

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

   5. Applied egg-rr 78.9%

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

      1. +-commutative 78.9%

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

      2. associate--l+ 78.9%

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

      3. metadata-eval 78.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}} + \color{blue}{0}\right)\right) \]

      4. +-rgt-identity 78.9%

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

      5. associate-/r/ 79.0%

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

      6. associate-*r/ 78.9%

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

      7. associate-*l/ 77.2%

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

      8. *-commutative 77.2%

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

      9. associate-/l* 77.2%

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

      10. *-commutative 77.2%

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

   7. Simplified 77.2%

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

   8. Taylor expanded in d around 0 52.3%

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

      1. associate-*l/ 52.3%

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

      2. unpow2 52.3%

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

      3. unpow2 52.3%

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

      4. unswap-sqr 66.5%

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

   10. Simplified 66.5%

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

   if 2.0999999999999999e-126 < l < 3.0999999999999998e-72 or 1.40000000000000008e54 < l

   1. Initial program 60.0%

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

      1. metadata-eval 60.0%

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

      2. unpow1/2 60.0%

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

      3. metadata-eval 60.0%

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

      4. unpow1/2 60.0%

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

      5. *-commutative 60.0%

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

      6. associate-*l* 60.0%

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

      7. times-frac 60.0%

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

      8. metadata-eval 60.0%

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

   3. Simplified 60.0%

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

   4. Taylor expanded in d around inf 41.6%

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

      1. *-un-lft-identity 41.6%

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

   6. Applied egg-rr 41.6%

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

      1. *-lft-identity 41.6%

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

      2. unpow-1 41.6%

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

      3. sqr-pow 41.7%

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

      4. rem-sqrt-square 41.7%

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

      5. metadata-eval 41.7%

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

      6. sqr-pow 41.5%

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

      7. fabs-sqr 41.5%

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

      8. sqr-pow 41.7%

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

      9. *-commutative 41.7%

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

   8. Simplified 41.7%

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

      1. unpow-prod-down 57.0%

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

   10. Applied egg-rr 57.0%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{-214}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt[3]{\frac{1}{h \cdot \ell} \cdot \sqrt{\frac{1}{h \cdot \ell}}}\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{-126} \lor \neg \left(\ell \leq 3.1 \cdot 10^{-72}\right) \land \ell \leq 1.4 \cdot 10^{+54}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 14: 49.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \frac{1}{h \cdot \ell}\\ \mathbf{if}\;\ell \leq -1.3 \cdot 10^{-204}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\sqrt[3]{t_0 \cdot \left(t_0 \cdot t_0\right)}}\\ \mathbf{elif}\;\ell \leq 9.8 \cdot 10^{-126} \lor \neg \left(\ell \leq 4.2 \cdot 10^{-72}\right) \land \ell \leq 1.5 \cdot 10^{+51}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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 (/ 1.0 (* h l))))
   (if (<= l -1.3e-204)
     (* d (- (pow (* h l) -0.5)))
     (if (<= l -2e-310)
       (* d (sqrt (cbrt (* t_0 (* t_0 t_0)))))
       (if (or (<= l 9.8e-126) (and (not (<= l 4.2e-72)) (<= l 1.5e+51)))
         (* -0.125 (/ (* (* (* M D) (* M D)) (sqrt (/ h (pow l 3.0)))) d))
         (* d (* (pow h -0.5) (pow l -0.5))))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 / (h * l);
	double tmp;
	if (l <= -1.3e-204) {
		tmp = d * -pow((h * l), -0.5);
	} else if (l <= -2e-310) {
		tmp = d * sqrt(cbrt((t_0 * (t_0 * t_0))));
	} else if ((l <= 9.8e-126) || (!(l <= 4.2e-72) && (l <= 1.5e+51))) {
		tmp = -0.125 * ((((M * D) * (M * D)) * sqrt((h / pow(l, 3.0)))) / d);
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 / (h * l);
	double tmp;
	if (l <= -1.3e-204) {
		tmp = d * -Math.pow((h * l), -0.5);
	} else if (l <= -2e-310) {
		tmp = d * Math.sqrt(Math.cbrt((t_0 * (t_0 * t_0))));
	} else if ((l <= 9.8e-126) || (!(l <= 4.2e-72) && (l <= 1.5e+51))) {
		tmp = -0.125 * ((((M * D) * (M * D)) * Math.sqrt((h / Math.pow(l, 3.0)))) / d);
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(1.0 / Float64(h * l))
	tmp = 0.0
	if (l <= -1.3e-204)
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	elseif (l <= -2e-310)
		tmp = Float64(d * sqrt(cbrt(Float64(t_0 * Float64(t_0 * t_0)))));
	elseif ((l <= 9.8e-126) || (!(l <= 4.2e-72) && (l <= 1.5e+51)))
		tmp = Float64(-0.125 * Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * sqrt(Float64(h / (l ^ 3.0)))) / d));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
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[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.3e-204], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, -2e-310], N[(d * N[Sqrt[N[Power[N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, 9.8e-126], And[N[Not[LessEqual[l, 4.2e-72]], $MachinePrecision], LessEqual[l, 1.5e+51]]], N[(-0.125 * N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / d), $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 l < -1.29999999999999991e-204

   1. Initial program 63.4%

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

      1. metadata-eval 63.4%

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

      2. unpow1/2 63.4%

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

      3. metadata-eval 63.4%

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

      4. unpow1/2 63.4%

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

      5. *-commutative 63.4%

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

      6. associate-*l* 63.4%

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

      7. times-frac 62.5%

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

      8. metadata-eval 62.5%

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

   3. Simplified 62.5%

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

   5. Applied egg-rr 36.0%

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

      1. associate-*l* 36.0%

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

      2. associate-*r/ 36.1%

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

      3. *-commutative 36.1%

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

   7. Simplified 36.1%

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

   8. Taylor expanded in d around -inf 50.4%

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

      1. mul-1-neg 50.4%

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

      2. distribute-rgt-neg-in 50.4%

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

      3. unpow-1 50.4%

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

      4. sqr-pow 50.5%

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

      5. rem-sqrt-square 50.5%

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

      6. metadata-eval 50.5%

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

      7. sqr-pow 50.2%

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

      8. fabs-sqr 50.2%

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

      9. sqr-pow 50.5%

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

      10. *-commutative 50.5%

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

   10. Simplified 50.5%

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

   if -1.29999999999999991e-204 < l < -1.999999999999994e-310

   1. Initial program 63.1%

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

      1. metadata-eval 63.1%

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

      2. unpow1/2 63.1%

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

      3. metadata-eval 63.1%

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

      4. unpow1/2 63.1%

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

      5. *-commutative 63.1%

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

      6. associate-*l* 63.1%

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

      7. times-frac 56.9%

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

      8. metadata-eval 56.9%

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

   3. Simplified 56.9%

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

   4. Taylor expanded in d around inf 33.5%

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

      1. add-cbrt-cube 51.2%

         \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\frac{1}{\ell \cdot h} \cdot \frac{1}{\ell \cdot h}\right) \cdot \frac{1}{\ell \cdot h}}}} \cdot d \]

   6. Applied egg-rr 51.2%

      \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\frac{1}{\ell \cdot h} \cdot \frac{1}{\ell \cdot h}\right) \cdot \frac{1}{\ell \cdot h}}}} \cdot d \]

   if -1.999999999999994e-310 < l < 9.8000000000000002e-126 or 4.2e-72 < l < 1.5e51

   1. Initial program 78.9%

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

      1. metadata-eval 78.9%

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

      2. unpow1/2 78.9%

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

      3. metadata-eval 78.9%

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

      4. unpow1/2 78.9%

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

      5. *-commutative 78.9%

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

      6. associate-*l* 78.9%

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

      7. times-frac 78.9%

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

      8. metadata-eval 78.9%

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

   3. Simplified 78.9%

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

      1. associate-*r* 78.9%

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

      2. frac-times 78.9%

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

      3. *-commutative 78.9%

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

      4. metadata-eval 78.9%

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

      5. expm1-log1p-u 78.2%

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

      6. expm1-udef 78.2%

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

   5. Applied egg-rr 78.9%

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

      1. +-commutative 78.9%

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

      2. associate--l+ 78.9%

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

      3. metadata-eval 78.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{0.5}{\frac{\ell}{h}} + \color{blue}{0}\right)\right) \]

      4. +-rgt-identity 78.9%

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

      5. associate-/r/ 79.0%

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

      6. associate-*r/ 78.9%

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

      7. associate-*l/ 77.2%

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

      8. *-commutative 77.2%

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

      9. associate-/l* 77.2%

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

      10. *-commutative 77.2%

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

   7. Simplified 77.2%

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

   8. Taylor expanded in d around 0 52.3%

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

      1. associate-*l/ 52.3%

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

      2. unpow2 52.3%

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

      3. unpow2 52.3%

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

      4. unswap-sqr 66.5%

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

   10. Simplified 66.5%

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

   if 9.8000000000000002e-126 < l < 4.2e-72 or 1.5e51 < l

   1. Initial program 60.0%

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

      1. metadata-eval 60.0%

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

      2. unpow1/2 60.0%

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

      3. metadata-eval 60.0%

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

      4. unpow1/2 60.0%

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

      5. *-commutative 60.0%

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

      6. associate-*l* 60.0%

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

      7. times-frac 60.0%

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

      8. metadata-eval 60.0%

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

   3. Simplified 60.0%

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

   4. Taylor expanded in d around inf 41.6%

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

      1. *-un-lft-identity 41.6%

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

   6. Applied egg-rr 41.6%

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

      1. *-lft-identity 41.6%

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

      2. unpow-1 41.6%

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

      3. sqr-pow 41.7%

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

      4. rem-sqrt-square 41.7%

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

      5. metadata-eval 41.7%

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

      6. sqr-pow 41.5%

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

      7. fabs-sqr 41.5%

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

      8. sqr-pow 41.7%

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

      9. *-commutative 41.7%

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

   8. Simplified 41.7%

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

      1. unpow-prod-down 57.0%

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

   10. Applied egg-rr 57.0%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.3 \cdot 10^{-204}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\sqrt[3]{\frac{1}{h \cdot \ell} \cdot \left(\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}\right)}}\\ \mathbf{elif}\;\ell \leq 9.8 \cdot 10^{-126} \lor \neg \left(\ell \leq 4.2 \cdot 10^{-72}\right) \land \ell \leq 1.5 \cdot 10^{+51}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 15: 56.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.1 \cdot 10^{+135}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - {\left(D \cdot \frac{M}{\frac{d}{0.5}}\right)}^{2} \cdot \frac{h \cdot 0.5}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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+135)
   (* d (- (pow (* h l) -0.5)))
   (*
    (sqrt (/ (/ d h) (/ l d)))
    (- 1.0 (* (pow (* D (/ M (/ d 0.5))) 2.0) (/ (* h 0.5) l))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.1e+135) {
		tmp = d * -pow((h * l), -0.5);
	} else {
		tmp = sqrt(((d / h) / (l / d))) * (1.0 - (pow((D * (M / (d / 0.5))), 2.0) * ((h * 0.5) / l)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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+135)) then
        tmp = d * -((h * l) ** (-0.5d0))
    else
        tmp = sqrt(((d / h) / (l / d))) * (1.0d0 - (((d_1 * (m / (d / 0.5d0))) ** 2.0d0) * ((h * 0.5d0) / l)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.1e+135) {
		tmp = d * -Math.pow((h * l), -0.5);
	} else {
		tmp = Math.sqrt(((d / h) / (l / d))) * (1.0 - (Math.pow((D * (M / (d / 0.5))), 2.0) * ((h * 0.5) / l)));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -1.1e+135:
		tmp = d * -math.pow((h * l), -0.5)
	else:
		tmp = math.sqrt(((d / h) / (l / d))) * (1.0 - (math.pow((D * (M / (d / 0.5))), 2.0) * ((h * 0.5) / l)))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.1e+135)
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	else
		tmp = Float64(sqrt(Float64(Float64(d / h) / Float64(l / d))) * Float64(1.0 - Float64((Float64(D * Float64(M / Float64(d / 0.5))) ^ 2.0) * Float64(Float64(h * 0.5) / l))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1.1e+135)
		tmp = d * -((h * l) ^ -0.5);
	else
		tmp = sqrt(((d / h) / (l / d))) * (1.0 - (((D * (M / (d / 0.5))) ^ 2.0) * ((h * 0.5) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
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+135], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] / N[(l / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[Power[N[(D * N[(M / N[(d / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h * 0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.1e135

   1. Initial program 45.6%

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

      1. metadata-eval 45.6%

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

      2. unpow1/2 45.6%

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

      3. metadata-eval 45.6%

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

      4. unpow1/2 45.6%

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

      5. *-commutative 45.6%

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

      6. associate-*l* 45.6%

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

      7. times-frac 45.6%

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

      8. metadata-eval 45.6%

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

   3. Simplified 45.6%

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

   5. Applied egg-rr 30.9%

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

      1. associate-*l* 30.9%

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

      2. associate-*r/ 30.9%

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

      3. *-commutative 30.9%

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

   7. Simplified 30.9%

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

   8. Taylor expanded in d around -inf 50.3%

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

      1. mul-1-neg 50.3%

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

      2. distribute-rgt-neg-in 50.3%

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

      3. unpow-1 50.3%

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

      4. sqr-pow 50.3%

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

      5. rem-sqrt-square 50.3%

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

      6. metadata-eval 50.3%

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

      7. sqr-pow 50.0%

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

      8. fabs-sqr 50.0%

         \[\leadsto d \cdot \left(-\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)}}\right) \]

      9. sqr-pow 50.3%

         \[\leadsto d \cdot \left(-\color{blue}{{\left(\ell \cdot h\right)}^{-0.5}}\right) \]

      10. *-commutative 50.3%

          \[\leadsto d \cdot \left(-{\color{blue}{\left(h \cdot \ell\right)}}^{-0.5}\right) \]

   10. Simplified 50.3%

       \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if -1.1e135 < l

   1. Initial program 69.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 69.7%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 69.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 69.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 69.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 69.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 69.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 68.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 68.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 68.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 68.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}}\right) \]

      2. frac-times 69.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative 69.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval 69.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. add-sqr-sqrt 69.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}} \cdot \sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}}\right) \]

      6. pow2 69.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

   5. Applied egg-rr 70.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{0.5}{\frac{\ell}{h}}}\right)}^{2}}\right) \]
   6. Step-by-step derivation

      1. pow1 70.2%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{0.5}{\frac{\ell}{h}}}\right)}^{2}\right)\right)}^{1}} \]

      2. sqrt-unprod 62.2%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - {\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{0.5}{\frac{\ell}{h}}}\right)}^{2}\right)\right)}^{1} \]

      3. *-commutative 62.2%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\color{blue}{\left(\sqrt{\frac{0.5}{\frac{\ell}{h}}} \cdot \left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)\right)}}^{2}\right)\right)}^{1} \]

      4. div-inv 62.2%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(\sqrt{\color{blue}{0.5 \cdot \frac{1}{\frac{\ell}{h}}}} \cdot \left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)}^{1} \]

      5. clear-num 62.2%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(\sqrt{0.5 \cdot \color{blue}{\frac{h}{\ell}}} \cdot \left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)}^{1} \]

      6. associate-*l* 62.2%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(\sqrt{0.5 \cdot \frac{h}{\ell}} \cdot \color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}\right)}^{2}\right)\right)}^{1} \]

   7. Applied egg-rr 62.2%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(\sqrt{0.5 \cdot \frac{h}{\ell}} \cdot \left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)\right)}^{2}\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 62.2%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(\sqrt{0.5 \cdot \frac{h}{\ell}} \cdot \left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)\right)}^{2}\right)} \]

      2. associate-*r/ 56.5%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{d}{h} \cdot d}{\ell}}} \cdot \left(1 - {\left(\sqrt{0.5 \cdot \frac{h}{\ell}} \cdot \left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)\right)}^{2}\right) \]

      3. associate-/l* 62.2%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{d}{h}}{\frac{\ell}{d}}}} \cdot \left(1 - {\left(\sqrt{0.5 \cdot \frac{h}{\ell}} \cdot \left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)\right)}^{2}\right) \]

      4. unpow2 62.2%

         \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - \color{blue}{\left(\sqrt{0.5 \cdot \frac{h}{\ell}} \cdot \left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)\right) \cdot \left(\sqrt{0.5 \cdot \frac{h}{\ell}} \cdot \left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)\right)}\right) \]

      5. swap-sqr 60.9%

         \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - \color{blue}{\left(\sqrt{0.5 \cdot \frac{h}{\ell}} \cdot \sqrt{0.5 \cdot \frac{h}{\ell}}\right) \cdot \left(\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)\right)}\right) \]

      6. rem-square-sqrt 60.9%

         \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - \color{blue}{\left(0.5 \cdot \frac{h}{\ell}\right)} \cdot \left(\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)\right)\right) \]

      7. associate-*r/ 60.9%

         \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - \color{blue}{\frac{0.5 \cdot h}{\ell}} \cdot \left(\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)\right)\right) \]

      8. unpow2 60.9%

         \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - \frac{0.5 \cdot h}{\ell} \cdot \color{blue}{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}\right) \]

      9. associate-*r* 60.9%

         \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - \frac{0.5 \cdot h}{\ell} \cdot {\color{blue}{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}}^{2}\right) \]

      10. associate-*r/ 61.4%

          \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - \frac{0.5 \cdot h}{\ell} \cdot {\color{blue}{\left(\frac{\left(M \cdot 0.5\right) \cdot D}{d}\right)}}^{2}\right) \]

      11. associate-*l/ 60.9%

          \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - \frac{0.5 \cdot h}{\ell} \cdot {\color{blue}{\left(\frac{M \cdot 0.5}{d} \cdot D\right)}}^{2}\right) \]

      12. associate-*l/ 60.9%

          \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - \frac{0.5 \cdot h}{\ell} \cdot {\left(\color{blue}{\left(\frac{M}{d} \cdot 0.5\right)} \cdot D\right)}^{2}\right) \]

      13. *-commutative 60.9%

          \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - \frac{0.5 \cdot h}{\ell} \cdot {\color{blue}{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}}^{2}\right) \]

      14. associate-*l/ 60.9%

          \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - \frac{0.5 \cdot h}{\ell} \cdot {\left(D \cdot \color{blue}{\frac{M \cdot 0.5}{d}}\right)}^{2}\right) \]

      15. associate-/l* 60.9%

          \[\leadsto \sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - \frac{0.5 \cdot h}{\ell} \cdot {\left(D \cdot \color{blue}{\frac{M}{\frac{d}{0.5}}}\right)}^{2}\right) \]

   9. Simplified 60.9%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - \frac{0.5 \cdot h}{\ell} \cdot {\left(D \cdot \frac{M}{\frac{d}{0.5}}\right)}^{2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.1 \cdot 10^{+135}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}} \cdot \left(1 - {\left(D \cdot \frac{M}{\frac{d}{0.5}}\right)}^{2} \cdot \frac{h \cdot 0.5}{\ell}\right)\\ \end{array} \]

Alternative 16: 45.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \frac{1}{h \cdot \ell}\\ t_1 := d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{if}\;\ell \leq -1.4 \cdot 10^{-208}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -3.6 \cdot 10^{-302}:\\ \;\;\;\;d \cdot \sqrt[3]{t_0 \cdot \sqrt{t_0}}\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{-224}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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 (/ 1.0 (* h l))) (t_1 (* d (- (pow (* h l) -0.5)))))
   (if (<= l -1.4e-208)
     t_1
     (if (<= l -3.6e-302)
       (* d (cbrt (* t_0 (sqrt t_0))))
       (if (<= l 4.8e-224) t_1 (* d (* (pow h -0.5) (pow l -0.5))))))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 / (h * l);
	double t_1 = d * -pow((h * l), -0.5);
	double tmp;
	if (l <= -1.4e-208) {
		tmp = t_1;
	} else if (l <= -3.6e-302) {
		tmp = d * cbrt((t_0 * sqrt(t_0)));
	} else if (l <= 4.8e-224) {
		tmp = t_1;
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 / (h * l);
	double t_1 = d * -Math.pow((h * l), -0.5);
	double tmp;
	if (l <= -1.4e-208) {
		tmp = t_1;
	} else if (l <= -3.6e-302) {
		tmp = d * Math.cbrt((t_0 * Math.sqrt(t_0)));
	} else if (l <= 4.8e-224) {
		tmp = t_1;
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(1.0 / Float64(h * l))
	t_1 = Float64(d * Float64(-(Float64(h * l) ^ -0.5)))
	tmp = 0.0
	if (l <= -1.4e-208)
		tmp = t_1;
	elseif (l <= -3.6e-302)
		tmp = Float64(d * cbrt(Float64(t_0 * sqrt(t_0))));
	elseif (l <= 4.8e-224)
		tmp = t_1;
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
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[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[l, -1.4e-208], t$95$1, If[LessEqual[l, -3.6e-302], N[(d * N[Power[N[(t$95$0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.8e-224], t$95$1, 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 < -1.40000000000000001e-208 or -3.6000000000000001e-302 < l < 4.80000000000000028e-224

   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. metadata-eval 64.7%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 64.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 64.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 64.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 64.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 64.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 64.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 64.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 64.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   5. Applied egg-rr 34.0%

      \[\leadsto \color{blue}{\sqrt{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right) \cdot {\left(1 + \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)}^{2}}} \]
   6. Step-by-step derivation

      1. associate-*l* 34.0%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \left(\frac{d}{h} \cdot {\left(1 + \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)}^{2}\right)}} \]

      2. associate-*r/ 34.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \left(\frac{d}{h} \cdot {\left(1 + \frac{h}{\ell} \cdot \left({\color{blue}{\left(\frac{\left(M \cdot 0.5\right) \cdot D}{d}\right)}}^{2} \cdot -0.5\right)\right)}^{2}\right)} \]

      3. *-commutative 34.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \left(\frac{d}{h} \cdot {\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{\color{blue}{D \cdot \left(M \cdot 0.5\right)}}{d}\right)}^{2} \cdot -0.5\right)\right)}^{2}\right)} \]

   7. Simplified 34.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \left(\frac{d}{h} \cdot {\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2} \cdot -0.5\right)\right)}^{2}\right)}} \]

   8. Taylor expanded in d around -inf 50.1%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg 50.1%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. distribute-rgt-neg-in 50.1%

         \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)} \]

      3. unpow-1 50.1%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}}\right) \]

      4. sqr-pow 50.2%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}}}\right) \]

      5. rem-sqrt-square 50.2%

         \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}\right|}\right) \]

      6. metadata-eval 50.2%

         \[\leadsto d \cdot \left(-\left|{\left(\ell \cdot h\right)}^{\color{blue}{-0.5}}\right|\right) \]

      7. sqr-pow 49.9%

         \[\leadsto d \cdot \left(-\left|\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)}}\right|\right) \]

      8. fabs-sqr 49.9%

         \[\leadsto d \cdot \left(-\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)}}\right) \]

      9. sqr-pow 50.2%

         \[\leadsto d \cdot \left(-\color{blue}{{\left(\ell \cdot h\right)}^{-0.5}}\right) \]

      10. *-commutative 50.2%

          \[\leadsto d \cdot \left(-{\color{blue}{\left(h \cdot \ell\right)}}^{-0.5}\right) \]

   10. Simplified 50.2%

       \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if -1.40000000000000001e-208 < l < -3.6000000000000001e-302

   1. Initial program 67.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 67.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 67.3%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 67.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 67.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 67.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 67.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 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. Taylor expanded in d around inf 35.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. add-cbrt-cube 42.0%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}}} \cdot d \]

      2. add-sqr-sqrt 42.0%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{\ell \cdot h}} \cdot \sqrt{\frac{1}{\ell \cdot h}}} \cdot d \]

   6. Applied egg-rr 42.0%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{\ell \cdot h} \cdot \sqrt{\frac{1}{\ell \cdot h}}}} \cdot d \]

   if 4.80000000000000028e-224 < l

   1. Initial program 67.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 67.0%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 67.0%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 67.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 67.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 67.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 67.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 67.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 67.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 67.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 31.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 31.9%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

   6. Applied egg-rr 31.9%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]
   7. Step-by-step derivation

      1. *-lft-identity 31.9%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \cdot d \]

      2. unpow-1 31.9%

         \[\leadsto \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \cdot d \]

      3. sqr-pow 32.0%

         \[\leadsto \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 32.0%

         \[\leadsto \color{blue}{\left|{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. metadata-eval 32.0%

         \[\leadsto \left|{\left(\ell \cdot h\right)}^{\color{blue}{-0.5}}\right| \cdot d \]

      6. sqr-pow 31.9%

         \[\leadsto \left|\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)}}\right| \cdot d \]

      7. fabs-sqr 31.9%

         \[\leadsto \color{blue}{\left({\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \cdot d \]

      8. sqr-pow 32.0%

         \[\leadsto \color{blue}{{\left(\ell \cdot h\right)}^{-0.5}} \cdot d \]

      9. *-commutative 32.0%

         \[\leadsto {\color{blue}{\left(h \cdot \ell\right)}}^{-0.5} \cdot d \]

   8. Simplified 32.0%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   9. Step-by-step derivation

      1. unpow-prod-down 44.5%

         \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   10. Applied egg-rr 44.5%

       \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{-208}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq -3.6 \cdot 10^{-302}:\\ \;\;\;\;d \cdot \sqrt[3]{\frac{1}{h \cdot \ell} \cdot \sqrt{\frac{1}{h \cdot \ell}}}\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{-224}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 17: 45.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 2.15 \cdot 10^{-259}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d 2.15e-259)
   (* d (- (pow (* h l) -0.5)))
   (* d (* (pow h -0.5) (pow l -0.5)))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 2.15e-259) {
		tmp = d * -pow((h * l), -0.5);
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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 <= 2.15d-259) then
        tmp = d * -((h * l) ** (-0.5d0))
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 2.15e-259) {
		tmp = d * -Math.pow((h * l), -0.5);
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if d <= 2.15e-259:
		tmp = d * -math.pow((h * l), -0.5)
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= 2.15e-259)
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= 2.15e-259)
		tmp = d * -((h * l) ^ -0.5);
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, 2.15e-259], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $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 d < 2.15e-259

   1. Initial program 60.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 60.5%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 60.5%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 60.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 60.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 60.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 60.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 59.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 59.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 59.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   5. Applied egg-rr 30.9%

      \[\leadsto \color{blue}{\sqrt{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right) \cdot {\left(1 + \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)}^{2}}} \]
   6. Step-by-step derivation

      1. associate-*l* 30.9%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \left(\frac{d}{h} \cdot {\left(1 + \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)}^{2}\right)}} \]

      2. associate-*r/ 31.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \left(\frac{d}{h} \cdot {\left(1 + \frac{h}{\ell} \cdot \left({\color{blue}{\left(\frac{\left(M \cdot 0.5\right) \cdot D}{d}\right)}}^{2} \cdot -0.5\right)\right)}^{2}\right)} \]

      3. *-commutative 31.0%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \left(\frac{d}{h} \cdot {\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{\color{blue}{D \cdot \left(M \cdot 0.5\right)}}{d}\right)}^{2} \cdot -0.5\right)\right)}^{2}\right)} \]

   7. Simplified 31.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \left(\frac{d}{h} \cdot {\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2} \cdot -0.5\right)\right)}^{2}\right)}} \]

   8. Taylor expanded in d around -inf 43.6%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg 43.6%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. distribute-rgt-neg-in 43.6%

         \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)} \]

      3. unpow-1 43.6%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}}\right) \]

      4. sqr-pow 43.7%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}}}\right) \]

      5. rem-sqrt-square 43.7%

         \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}\right|}\right) \]

      6. metadata-eval 43.7%

         \[\leadsto d \cdot \left(-\left|{\left(\ell \cdot h\right)}^{\color{blue}{-0.5}}\right|\right) \]

      7. sqr-pow 43.4%

         \[\leadsto d \cdot \left(-\left|\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)}}\right|\right) \]

      8. fabs-sqr 43.4%

         \[\leadsto d \cdot \left(-\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)}}\right) \]

      9. sqr-pow 43.7%

         \[\leadsto d \cdot \left(-\color{blue}{{\left(\ell \cdot h\right)}^{-0.5}}\right) \]

      10. *-commutative 43.7%

          \[\leadsto d \cdot \left(-{\color{blue}{\left(h \cdot \ell\right)}}^{-0.5}\right) \]

   10. Simplified 43.7%

       \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if 2.15e-259 < d

   1. Initial program 72.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 72.7%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 72.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 72.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 72.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 72.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 72.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 72.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 72.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 72.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 32.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 32.1%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

   6. Applied egg-rr 32.1%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]
   7. Step-by-step derivation

      1. *-lft-identity 32.1%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \cdot d \]

      2. unpow-1 32.1%

         \[\leadsto \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \cdot d \]

      3. sqr-pow 32.2%

         \[\leadsto \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 32.2%

         \[\leadsto \color{blue}{\left|{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. metadata-eval 32.2%

         \[\leadsto \left|{\left(\ell \cdot h\right)}^{\color{blue}{-0.5}}\right| \cdot d \]

      6. sqr-pow 32.1%

         \[\leadsto \left|\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)}}\right| \cdot d \]

      7. fabs-sqr 32.1%

         \[\leadsto \color{blue}{\left({\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \cdot d \]

      8. sqr-pow 32.2%

         \[\leadsto \color{blue}{{\left(\ell \cdot h\right)}^{-0.5}} \cdot d \]

      9. *-commutative 32.2%

         \[\leadsto {\color{blue}{\left(h \cdot \ell\right)}}^{-0.5} \cdot d \]

   8. Simplified 32.2%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   9. Step-by-step derivation

      1. unpow-prod-down 44.5%

         \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   10. Applied egg-rr 44.5%

       \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2.15 \cdot 10^{-259}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 18: 40.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{if}\;\ell \leq -1.6 \cdot 10^{-212} \lor \neg \left(\ell \leq -1.2 \cdot 10^{-303}\right) \land \ell \leq 4.4 \cdot 10^{-127}:\\ \;\;\;\;d \cdot \left(-t_0\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot t_0\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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 (* h l) -0.5)))
   (if (or (<= l -1.6e-212) (and (not (<= l -1.2e-303)) (<= l 4.4e-127)))
     (* d (- t_0))
     (* d t_0))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = pow((h * l), -0.5);
	double tmp;
	if ((l <= -1.6e-212) || (!(l <= -1.2e-303) && (l <= 4.4e-127))) {
		tmp = d * -t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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 = (h * l) ** (-0.5d0)
    if ((l <= (-1.6d-212)) .or. (.not. (l <= (-1.2d-303))) .and. (l <= 4.4d-127)) then
        tmp = d * -t_0
    else
        tmp = d * t_0
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.pow((h * l), -0.5);
	double tmp;
	if ((l <= -1.6e-212) || (!(l <= -1.2e-303) && (l <= 4.4e-127))) {
		tmp = d * -t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.pow((h * l), -0.5)
	tmp = 0
	if (l <= -1.6e-212) or (not (l <= -1.2e-303) and (l <= 4.4e-127)):
		tmp = d * -t_0
	else:
		tmp = d * t_0
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(h * l) ^ -0.5
	tmp = 0.0
	if ((l <= -1.6e-212) || (!(l <= -1.2e-303) && (l <= 4.4e-127)))
		tmp = Float64(d * Float64(-t_0));
	else
		tmp = Float64(d * t_0);
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (h * l) ^ -0.5;
	tmp = 0.0;
	if ((l <= -1.6e-212) || (~((l <= -1.2e-303)) && (l <= 4.4e-127)))
		tmp = d * -t_0;
	else
		tmp = d * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
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[(h * l), $MachinePrecision], -0.5], $MachinePrecision]}, If[Or[LessEqual[l, -1.6e-212], And[N[Not[LessEqual[l, -1.2e-303]], $MachinePrecision], LessEqual[l, 4.4e-127]]], N[(d * (-t$95$0)), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -1.5999999999999999e-212 or -1.2e-303 < l < 4.4000000000000003e-127

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 66.7%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 66.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 66.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 66.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   5. Applied egg-rr 31.5%

      \[\leadsto \color{blue}{\sqrt{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right) \cdot {\left(1 + \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)}^{2}}} \]
   6. Step-by-step derivation

      1. associate-*l* 31.5%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \left(\frac{d}{h} \cdot {\left(1 + \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)}^{2}\right)}} \]

      2. associate-*r/ 31.5%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \left(\frac{d}{h} \cdot {\left(1 + \frac{h}{\ell} \cdot \left({\color{blue}{\left(\frac{\left(M \cdot 0.5\right) \cdot D}{d}\right)}}^{2} \cdot -0.5\right)\right)}^{2}\right)} \]

      3. *-commutative 31.5%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \left(\frac{d}{h} \cdot {\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{\color{blue}{D \cdot \left(M \cdot 0.5\right)}}{d}\right)}^{2} \cdot -0.5\right)\right)}^{2}\right)} \]

   7. Simplified 31.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \left(\frac{d}{h} \cdot {\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2} \cdot -0.5\right)\right)}^{2}\right)}} \]

   8. Taylor expanded in d around -inf 46.6%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg 46.6%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

      2. distribute-rgt-neg-in 46.6%

         \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)} \]

      3. unpow-1 46.6%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}}\right) \]

      4. sqr-pow 46.6%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}}}\right) \]

      5. rem-sqrt-square 46.6%

         \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}\right|}\right) \]

      6. metadata-eval 46.6%

         \[\leadsto d \cdot \left(-\left|{\left(\ell \cdot h\right)}^{\color{blue}{-0.5}}\right|\right) \]

      7. sqr-pow 46.4%

         \[\leadsto d \cdot \left(-\left|\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)}}\right|\right) \]

      8. fabs-sqr 46.4%

         \[\leadsto d \cdot \left(-\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)}}\right) \]

      9. sqr-pow 46.6%

         \[\leadsto d \cdot \left(-\color{blue}{{\left(\ell \cdot h\right)}^{-0.5}}\right) \]

      10. *-commutative 46.6%

          \[\leadsto d \cdot \left(-{\color{blue}{\left(h \cdot \ell\right)}}^{-0.5}\right) \]

   10. Simplified 46.6%

       \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if -1.5999999999999999e-212 < l < -1.2e-303 or 4.4000000000000003e-127 < l

   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. metadata-eval 64.7%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 64.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 64.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 64.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 64.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 64.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 63.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 36.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 36.3%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

   6. Applied egg-rr 36.3%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]
   7. Step-by-step derivation

      1. *-lft-identity 36.3%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \cdot d \]

      2. unpow-1 36.3%

         \[\leadsto \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \cdot d \]

      3. sqr-pow 36.3%

         \[\leadsto \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 36.3%

         \[\leadsto \color{blue}{\left|{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. metadata-eval 36.3%

         \[\leadsto \left|{\left(\ell \cdot h\right)}^{\color{blue}{-0.5}}\right| \cdot d \]

      6. sqr-pow 36.2%

         \[\leadsto \left|\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)}}\right| \cdot d \]

      7. fabs-sqr 36.2%

         \[\leadsto \color{blue}{\left({\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \cdot d \]

      8. sqr-pow 36.3%

         \[\leadsto \color{blue}{{\left(\ell \cdot h\right)}^{-0.5}} \cdot d \]

      9. *-commutative 36.3%

         \[\leadsto {\color{blue}{\left(h \cdot \ell\right)}}^{-0.5} \cdot d \]

   8. Simplified 36.3%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.6 \cdot 10^{-212} \lor \neg \left(\ell \leq -1.2 \cdot 10^{-303}\right) \land \ell \leq 4.4 \cdot 10^{-127}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 19: 37.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.1 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -4.1e-237) (sqrt (* (/ d l) (/ d h))) (* d (pow (* h l) -0.5))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -4.1e-237) {
		tmp = sqrt(((d / l) * (d / h)));
	} else {
		tmp = d * pow((h * l), -0.5);
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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 <= (-4.1d-237)) then
        tmp = sqrt(((d / l) * (d / h)))
    else
        tmp = d * ((h * l) ** (-0.5d0))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -4.1e-237) {
		tmp = Math.sqrt(((d / l) * (d / h)));
	} else {
		tmp = d * Math.pow((h * l), -0.5);
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -4.1e-237:
		tmp = math.sqrt(((d / l) * (d / h)))
	else:
		tmp = d * math.pow((h * l), -0.5)
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -4.1e-237)
		tmp = sqrt(Float64(Float64(d / l) * Float64(d / h)));
	else
		tmp = Float64(d * (Float64(h * l) ^ -0.5));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -4.1e-237)
		tmp = sqrt(((d / l) * (d / h)));
	else
		tmp = d * ((h * l) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -4.1e-237], N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.1000000000000001e-237

   1. Initial program 63.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 63.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 63.3%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 63.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 63.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 63.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 63.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 61.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 61.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 61.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   5. Applied egg-rr 36.2%

      \[\leadsto \color{blue}{\sqrt{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right) \cdot {\left(1 + \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)}^{2}}} \]
   6. Step-by-step derivation

      1. associate-*l* 36.2%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \left(\frac{d}{h} \cdot {\left(1 + \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)}^{2}\right)}} \]

      2. associate-*r/ 36.2%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \left(\frac{d}{h} \cdot {\left(1 + \frac{h}{\ell} \cdot \left({\color{blue}{\left(\frac{\left(M \cdot 0.5\right) \cdot D}{d}\right)}}^{2} \cdot -0.5\right)\right)}^{2}\right)} \]

      3. *-commutative 36.2%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \left(\frac{d}{h} \cdot {\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{\color{blue}{D \cdot \left(M \cdot 0.5\right)}}{d}\right)}^{2} \cdot -0.5\right)\right)}^{2}\right)} \]

   7. Simplified 36.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \left(\frac{d}{h} \cdot {\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2} \cdot -0.5\right)\right)}^{2}\right)}} \]

   8. Taylor expanded in d around inf 38.6%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \color{blue}{\frac{d}{h}}} \]

   if -4.1000000000000001e-237 < l

   1. Initial program 68.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 68.2%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 68.2%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 68.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 68.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 68.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 68.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 68.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 68.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 68.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 31.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 31.1%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

   6. Applied egg-rr 31.1%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]
   7. Step-by-step derivation

      1. *-lft-identity 31.1%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \cdot d \]

      2. unpow-1 31.1%

         \[\leadsto \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \cdot d \]

      3. sqr-pow 31.2%

         \[\leadsto \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 31.2%

         \[\leadsto \color{blue}{\left|{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. metadata-eval 31.2%

         \[\leadsto \left|{\left(\ell \cdot h\right)}^{\color{blue}{-0.5}}\right| \cdot d \]

      6. sqr-pow 31.1%

         \[\leadsto \left|\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)}}\right| \cdot d \]

      7. fabs-sqr 31.1%

         \[\leadsto \color{blue}{\left({\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \cdot d \]

      8. sqr-pow 31.2%

         \[\leadsto \color{blue}{{\left(\ell \cdot h\right)}^{-0.5}} \cdot d \]

      9. *-commutative 31.2%

         \[\leadsto {\color{blue}{\left(h \cdot \ell\right)}}^{-0.5} \cdot d \]

   8. Simplified 31.2%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.1 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 20: 37.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.35 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -2.35e-237) (sqrt (/ (/ d h) (/ l d))) (* d (pow (* h l) -0.5))))

C

M = abs(M);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2.35e-237) {
		tmp = sqrt(((d / h) / (l / d)));
	} else {
		tmp = d * pow((h * l), -0.5);
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
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 <= (-2.35d-237)) then
        tmp = sqrt(((d / h) / (l / d)))
    else
        tmp = d * ((h * l) ** (-0.5d0))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2.35e-237) {
		tmp = Math.sqrt(((d / h) / (l / d)));
	} else {
		tmp = d * Math.pow((h * l), -0.5);
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -2.35e-237:
		tmp = math.sqrt(((d / h) / (l / d)))
	else:
		tmp = d * math.pow((h * l), -0.5)
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -2.35e-237)
		tmp = sqrt(Float64(Float64(d / h) / Float64(l / d)));
	else
		tmp = Float64(d * (Float64(h * l) ^ -0.5));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -2.35e-237)
		tmp = sqrt(((d / h) / (l / d)));
	else
		tmp = d * ((h * l) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -2.35e-237], N[Sqrt[N[(N[(d / h), $MachinePrecision] / N[(l / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.3499999999999999e-237

   1. Initial program 63.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 63.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 63.3%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 63.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 63.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 63.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 63.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 61.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 61.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 61.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   5. Applied egg-rr 36.2%

      \[\leadsto \color{blue}{\sqrt{\left(\frac{d}{\ell} \cdot \frac{d}{h}\right) \cdot {\left(1 + \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)}^{2}}} \]
   6. Step-by-step derivation

      1. associate-*l* 36.2%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \left(\frac{d}{h} \cdot {\left(1 + \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)}^{2}\right)}} \]

      2. associate-*r/ 36.2%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \left(\frac{d}{h} \cdot {\left(1 + \frac{h}{\ell} \cdot \left({\color{blue}{\left(\frac{\left(M \cdot 0.5\right) \cdot D}{d}\right)}}^{2} \cdot -0.5\right)\right)}^{2}\right)} \]

      3. *-commutative 36.2%

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \left(\frac{d}{h} \cdot {\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{\color{blue}{D \cdot \left(M \cdot 0.5\right)}}{d}\right)}^{2} \cdot -0.5\right)\right)}^{2}\right)} \]

   7. Simplified 36.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \left(\frac{d}{h} \cdot {\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2} \cdot -0.5\right)\right)}^{2}\right)}} \]

   8. Taylor expanded in d around inf 29.0%

      \[\leadsto \sqrt{\color{blue}{\frac{{d}^{2}}{\ell \cdot h}}} \]
   9. Step-by-step derivation

      1. unpow2 29.0%

         \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{\ell \cdot h}} \]

      2. *-commutative 29.0%

         \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \]

      3. times-frac 38.6%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

      4. associate-*r/ 35.9%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{d}{h} \cdot d}{\ell}}} \]

      5. associate-/l* 38.6%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{d}{h}}{\frac{\ell}{d}}}} \]

   10. Simplified 38.6%

       \[\leadsto \sqrt{\color{blue}{\frac{\frac{d}{h}}{\frac{\ell}{d}}}} \]

   if -2.3499999999999999e-237 < l

   1. Initial program 68.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 68.2%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 68.2%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 68.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 68.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 68.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 68.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 68.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 68.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 68.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 31.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 31.1%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

   6. Applied egg-rr 31.1%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]
   7. Step-by-step derivation

      1. *-lft-identity 31.1%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \cdot d \]

      2. unpow-1 31.1%

         \[\leadsto \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \cdot d \]

      3. sqr-pow 31.2%

         \[\leadsto \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 31.2%

         \[\leadsto \color{blue}{\left|{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. metadata-eval 31.2%

         \[\leadsto \left|{\left(\ell \cdot h\right)}^{\color{blue}{-0.5}}\right| \cdot d \]

      6. sqr-pow 31.1%

         \[\leadsto \left|\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)}}\right| \cdot d \]

      7. fabs-sqr 31.1%

         \[\leadsto \color{blue}{\left({\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \cdot d \]

      8. sqr-pow 31.2%

         \[\leadsto \color{blue}{{\left(\ell \cdot h\right)}^{-0.5}} \cdot d \]

      9. *-commutative 31.2%

         \[\leadsto {\color{blue}{\left(h \cdot \ell\right)}}^{-0.5} \cdot d \]

   8. Simplified 31.2%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.35 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 21: 26.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ d \cdot {\left(h \cdot \ell\right)}^{-0.5} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
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

M = abs(M);
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 should be positive before calling this function
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

M = Math.abs(M);
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 = abs(M)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	return d * math.pow((h * l), -0.5)

Julia

M = abs(M)
M, D = sort([M, D])
function code(d, h, l, M, D)
	return Float64(d * (Float64(h * l) ^ -0.5))
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp = code(d, h, l, M, D)
	tmp = d * ((h * l) ^ -0.5);
end

Wolfram

NOTE: M should be positive before calling this function
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 65.9%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Step-by-step derivation

   1. metadata-eval 65.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. unpow1/2 65.9%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. metadata-eval 65.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. unpow1/2 65.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. *-commutative 65.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

   6. associate-*l* 65.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

   7. times-frac 65.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. metadata-eval 65.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

3. Simplified 65.1%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

4. Taylor expanded in d around inf 19.2%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
5. Step-by-step derivation

   1. *-un-lft-identity 19.2%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

6. Applied egg-rr 19.2%

   \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]
7. Step-by-step derivation

   1. *-lft-identity 19.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \cdot d \]

   2. unpow-1 19.2%

      \[\leadsto \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \cdot d \]

   3. sqr-pow 19.2%

      \[\leadsto \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

   4. rem-sqrt-square 19.2%

      \[\leadsto \color{blue}{\left|{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

   5. metadata-eval 19.2%

      \[\leadsto \left|{\left(\ell \cdot h\right)}^{\color{blue}{-0.5}}\right| \cdot d \]

   6. sqr-pow 19.2%

      \[\leadsto \left|\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)}}\right| \cdot d \]

   7. fabs-sqr 19.2%

      \[\leadsto \color{blue}{\left({\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \cdot d \]

   8. sqr-pow 19.2%

      \[\leadsto \color{blue}{{\left(\ell \cdot h\right)}^{-0.5}} \cdot d \]

   9. *-commutative 19.2%

      \[\leadsto {\color{blue}{\left(h \cdot \ell\right)}}^{-0.5} \cdot d \]

8. Simplified 19.2%

   \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

9. Final simplification 19.2%

   \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{-0.5} \]

Reproduce

herbie shell --seed 2023238 
(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)))))