Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.6% → 77.3%

Time: 15.1s

Alternatives: 5

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.6% 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: 77.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: d, h, l, 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 = sqrt((d / l))
    if (l <= (-5d-310)) then
        tmp = (sqrt(-d) / sqrt(-h)) * (t_0 * (1.0d0 + (h * (((m * ((d_1 / 2.0d0) / d)) ** 2.0d0) / (l / (-0.5d0))))))
    else
        tmp = (sqrt(d) / sqrt(h)) * (t_0 * (1.0d0 + ((h / l) * ((-0.5d0) * (((m / 2.0d0) * (d_1 / d)) ** 2.0d0)))))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 60.5%

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

   3. Simplified 59.8%

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

      1. expm1-log1p-u 34.9%

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

      2. expm1-udef 34.9%

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

      3. *-commutative 34.9%

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

      4. associate-*l* 34.9%

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

      5. div-inv 34.9%

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

      6. metadata-eval 34.9%

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

   5. Applied egg-rr 34.9%

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

      1. expm1-def 34.9%

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

      2. expm1-log1p 59.8%

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

      3. associate-*r* 59.8%

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

      4. *-commutative 59.8%

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

      5. associate-*r/ 61.5%

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

      6. associate-*l/ 62.8%

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

      7. *-commutative 62.8%

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

      8. *-commutative 62.8%

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

      9. associate-/l* 62.8%

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

   7. Simplified 62.8%

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

      1. frac-2neg 62.8%

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

      2. sqrt-div 80.9%

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

   9. Applied egg-rr 80.9%

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

   if -4.999999999999985e-310 < l

   1. Initial program 71.1%

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

   3. Simplified 71.0%

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

      1. sqrt-div 81.0%

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

      2. div-inv 81.0%

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

   5. Applied egg-rr 81.0%

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

      1. associate-*r/ 81.0%

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

      2. *-rgt-identity 81.0%

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

   7. Simplified 81.0%

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

4. Final simplification 81.0%

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

Alternative 2: 75.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 60.5%

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

   3. Simplified 59.8%

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

      1. expm1-log1p-u 34.9%

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

      2. expm1-udef 34.9%

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

      3. *-commutative 34.9%

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

      4. associate-*l* 34.9%

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

      5. div-inv 34.9%

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

      6. metadata-eval 34.9%

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

   5. Applied egg-rr 34.9%

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

      1. expm1-def 34.9%

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

      2. expm1-log1p 59.8%

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

      3. associate-*r* 59.8%

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

      4. *-commutative 59.8%

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

      5. associate-*r/ 61.5%

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

      6. associate-*l/ 62.8%

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

      7. *-commutative 62.8%

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

      8. *-commutative 62.8%

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

      9. associate-/l* 62.8%

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

   7. Simplified 62.8%

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

      1. frac-2neg 62.8%

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

      2. sqrt-div 68.2%

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

   9. Applied egg-rr 68.2%

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

   if -4.999999999999985e-310 < l

   1. Initial program 71.1%

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

   3. Simplified 71.0%

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

      1. sqrt-div 81.0%

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

      2. div-inv 81.0%

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

   5. Applied egg-rr 81.0%

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

      1. associate-*r/ 81.0%

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

      2. *-rgt-identity 81.0%

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

   7. Simplified 81.0%

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

4. Final simplification 74.0%

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

Alternative 3: 71.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \left(D \cdot \frac{M}{d}\right) \cdot 0.5\\ t_1 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;h \leq 7.2 \cdot 10^{-182}:\\ \;\;\;\;t_1 \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(\frac{t_0}{\ell} \cdot \frac{t_0}{-2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\left(1 + h \cdot \frac{{\left(M \cdot \frac{\frac{D}{2}}{d}\right)}^{2}}{\frac{\ell}{-0.5}}\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* D (/ M d)) 0.5)) (t_1 (sqrt (/ d h))))
   (if (<= h 7.2e-182)
     (* t_1 (* (sqrt (/ d l)) (+ 1.0 (* h (* (/ t_0 l) (/ t_0 -2.0))))))
     (*
      t_1
      (*
       (+ 1.0 (* h (/ (pow (* M (/ (/ D 2.0) d)) 2.0) (/ l -0.5))))
       (/ (sqrt d) (sqrt l)))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (D * (M / d)) * 0.5;
	double t_1 = sqrt((d / h));
	double tmp;
	if (h <= 7.2e-182) {
		tmp = t_1 * (sqrt((d / l)) * (1.0 + (h * ((t_0 / l) * (t_0 / -2.0)))));
	} else {
		tmp = t_1 * ((1.0 + (h * (pow((M * ((D / 2.0) / d)), 2.0) / (l / -0.5)))) * (sqrt(d) / sqrt(l)));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, 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) :: t_1
    real(8) :: tmp
    t_0 = (d_1 * (m / d)) * 0.5d0
    t_1 = sqrt((d / h))
    if (h <= 7.2d-182) then
        tmp = t_1 * (sqrt((d / l)) * (1.0d0 + (h * ((t_0 / l) * (t_0 / (-2.0d0))))))
    else
        tmp = t_1 * ((1.0d0 + (h * (((m * ((d_1 / 2.0d0) / d)) ** 2.0d0) / (l / (-0.5d0))))) * (sqrt(d) / sqrt(l)))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (D * (M / d)) * 0.5;
	double t_1 = Math.sqrt((d / h));
	double tmp;
	if (h <= 7.2e-182) {
		tmp = t_1 * (Math.sqrt((d / l)) * (1.0 + (h * ((t_0 / l) * (t_0 / -2.0)))));
	} else {
		tmp = t_1 * ((1.0 + (h * (Math.pow((M * ((D / 2.0) / d)), 2.0) / (l / -0.5)))) * (Math.sqrt(d) / Math.sqrt(l)));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	t_0 = (D * (M / d)) * 0.5
	t_1 = math.sqrt((d / h))
	tmp = 0
	if h <= 7.2e-182:
		tmp = t_1 * (math.sqrt((d / l)) * (1.0 + (h * ((t_0 / l) * (t_0 / -2.0)))))
	else:
		tmp = t_1 * ((1.0 + (h * (math.pow((M * ((D / 2.0) / d)), 2.0) / (l / -0.5)))) * (math.sqrt(d) / math.sqrt(l)))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(D * Float64(M / d)) * 0.5)
	t_1 = sqrt(Float64(d / h))
	tmp = 0.0
	if (h <= 7.2e-182)
		tmp = Float64(t_1 * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(h * Float64(Float64(t_0 / l) * Float64(t_0 / -2.0))))));
	else
		tmp = Float64(t_1 * Float64(Float64(1.0 + Float64(h * Float64((Float64(M * Float64(Float64(D / 2.0) / d)) ^ 2.0) / Float64(l / -0.5)))) * Float64(sqrt(d) / sqrt(l))));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (D * (M / d)) * 0.5;
	t_1 = sqrt((d / h));
	tmp = 0.0;
	if (h <= 7.2e-182)
		tmp = t_1 * (sqrt((d / l)) * (1.0 + (h * ((t_0 / l) * (t_0 / -2.0)))));
	else
		tmp = t_1 * ((1.0 + (h * (((M * ((D / 2.0) / d)) ^ 2.0) / (l / -0.5)))) * (sqrt(d) / sqrt(l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, 7.2e-182], N[(t$95$1 * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h * N[(N[(t$95$0 / l), $MachinePrecision] * N[(t$95$0 / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(1.0 + N[(h * N[(N[Power[N[(M * N[(N[(D / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l / -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if h < 7.19999999999999954e-182

   1. Initial program 60.2%

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

   3. Simplified 59.6%

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

      1. expm1-log1p-u 34.5%

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

      2. expm1-udef 34.5%

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

      3. *-commutative 34.5%

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

      4. associate-*l* 34.5%

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

      5. div-inv 34.5%

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

      6. metadata-eval 34.5%

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

   5. Applied egg-rr 34.5%

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

      1. expm1-def 34.5%

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

      2. expm1-log1p 59.6%

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

      3. associate-*r* 59.6%

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

      4. *-commutative 59.6%

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

      5. associate-*r/ 61.1%

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

      6. associate-*l/ 62.2%

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

      7. *-commutative 62.2%

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

      8. *-commutative 62.2%

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

      9. associate-/l* 62.2%

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

   7. Simplified 62.2%

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

      1. unpow2 62.2%

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

      2. div-inv 62.2%

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

      3. metadata-eval 62.2%

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

      4. metadata-eval 62.2%

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

      5. times-frac 65.3%

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

      6. associate-*r/ 65.2%

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

      7. associate-*l/ 64.1%

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

      8. div-inv 64.1%

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

      9. metadata-eval 64.1%

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

      10. associate-*r* 64.1%

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

      11. *-commutative 64.1%

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

      12. associate-*r/ 65.3%

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

      13. associate-*l/ 65.4%

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

      14. div-inv 65.4%

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

      15. metadata-eval 65.4%

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

      16. associate-*r* 65.4%

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

      17. *-commutative 65.4%

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

      18. metadata-eval 65.4%

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

   9. Applied egg-rr 65.4%

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

   if 7.19999999999999954e-182 < h

   1. Initial program 74.2%

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

   3. Simplified 74.2%

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

      1. expm1-log1p-u 24.9%

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

      2. expm1-udef 24.9%

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

      3. *-commutative 24.9%

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

      4. associate-*l* 24.9%

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

      5. div-inv 24.9%

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

      6. metadata-eval 24.9%

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

   5. Applied egg-rr 24.9%

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

      1. expm1-def 24.9%

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

      2. expm1-log1p 74.2%

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

      3. associate-*r* 74.2%

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

      4. *-commutative 74.2%

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

      5. associate-*r/ 72.2%

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

      6. associate-*l/ 75.3%

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

      7. *-commutative 75.3%

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

      8. *-commutative 75.3%

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

      9. associate-/l* 75.3%

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

   7. Simplified 75.3%

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

      1. sqrt-div 81.7%

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

      2. div-inv 81.7%

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

   9. Applied egg-rr 81.7%

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

       1. associate-*r/ 81.7%

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

       2. *-rgt-identity 81.7%

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

   11. Simplified 81.7%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 7.2 \cdot 10^{-182}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(\frac{\left(D \cdot \frac{M}{d}\right) \cdot 0.5}{\ell} \cdot \frac{\left(D \cdot \frac{M}{d}\right) \cdot 0.5}{-2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 + h \cdot \frac{{\left(M \cdot \frac{\frac{D}{2}}{d}\right)}^{2}}{\frac{\ell}{-0.5}}\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \]

Alternative 4: 72.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \left(D \cdot \frac{M}{d}\right) \cdot 0.5\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq 3.1 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t_1 \cdot \left(1 + h \cdot \left(\frac{t_0}{\ell} \cdot \frac{t_0}{-2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t_1 \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* D (/ M d)) 0.5)) (t_1 (sqrt (/ d l))))
   (if (<= l 3.1e-305)
     (* (sqrt (/ d h)) (* t_1 (+ 1.0 (* h (* (/ t_0 l) (/ t_0 -2.0))))))
     (*
      (/ (sqrt d) (sqrt h))
      (* t_1 (+ 1.0 (* (/ h l) (* -0.5 (pow (* (/ M 2.0) (/ D d)) 2.0)))))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (D * (M / d)) * 0.5;
	double t_1 = sqrt((d / l));
	double tmp;
	if (l <= 3.1e-305) {
		tmp = sqrt((d / h)) * (t_1 * (1.0 + (h * ((t_0 / l) * (t_0 / -2.0)))));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (t_1 * (1.0 + ((h / l) * (-0.5 * pow(((M / 2.0) * (D / d)), 2.0)))));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, 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) :: t_1
    real(8) :: tmp
    t_0 = (d_1 * (m / d)) * 0.5d0
    t_1 = sqrt((d / l))
    if (l <= 3.1d-305) then
        tmp = sqrt((d / h)) * (t_1 * (1.0d0 + (h * ((t_0 / l) * (t_0 / (-2.0d0))))))
    else
        tmp = (sqrt(d) / sqrt(h)) * (t_1 * (1.0d0 + ((h / l) * ((-0.5d0) * (((m / 2.0d0) * (d_1 / d)) ** 2.0d0)))))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (D * (M / d)) * 0.5;
	double t_1 = Math.sqrt((d / l));
	double tmp;
	if (l <= 3.1e-305) {
		tmp = Math.sqrt((d / h)) * (t_1 * (1.0 + (h * ((t_0 / l) * (t_0 / -2.0)))));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * (t_1 * (1.0 + ((h / l) * (-0.5 * Math.pow(((M / 2.0) * (D / d)), 2.0)))));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	t_0 = (D * (M / d)) * 0.5
	t_1 = math.sqrt((d / l))
	tmp = 0
	if l <= 3.1e-305:
		tmp = math.sqrt((d / h)) * (t_1 * (1.0 + (h * ((t_0 / l) * (t_0 / -2.0)))))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * (t_1 * (1.0 + ((h / l) * (-0.5 * math.pow(((M / 2.0) * (D / d)), 2.0)))))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(D * Float64(M / d)) * 0.5)
	t_1 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= 3.1e-305)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(t_1 * Float64(1.0 + Float64(h * Float64(Float64(t_0 / l) * Float64(t_0 / -2.0))))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(t_1 * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0))))));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (D * (M / d)) * 0.5;
	t_1 = sqrt((d / l));
	tmp = 0.0;
	if (l <= 3.1e-305)
		tmp = sqrt((d / h)) * (t_1 * (1.0 + (h * ((t_0 / l) * (t_0 / -2.0)))));
	else
		tmp = (sqrt(d) / sqrt(h)) * (t_1 * (1.0 + ((h / l) * (-0.5 * (((M / 2.0) * (D / d)) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, 3.1e-305], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[(1.0 + N[(h * N[(N[(t$95$0 / l), $MachinePrecision] * N[(t$95$0 / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if l < 3.0999999999999998e-305

   1. Initial program 60.0%

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

   3. Simplified 59.3%

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

      1. expm1-log1p-u 34.6%

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

      2. expm1-udef 34.6%

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

      3. *-commutative 34.6%

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

      4. associate-*l* 34.6%

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

      5. div-inv 34.6%

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

      6. metadata-eval 34.6%

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

   5. Applied egg-rr 34.6%

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

      1. expm1-def 34.6%

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

      2. expm1-log1p 59.3%

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

      3. associate-*r* 59.3%

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

      4. *-commutative 59.3%

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

      5. associate-*r/ 61.8%

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

      6. associate-*l/ 63.1%

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

      7. *-commutative 63.1%

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

      8. *-commutative 63.1%

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

      9. associate-/l* 63.1%

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

   7. Simplified 63.1%

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

      1. unpow2 63.1%

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

      2. div-inv 63.1%

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

      3. metadata-eval 63.1%

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

      4. metadata-eval 63.1%

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

      5. times-frac 65.2%

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

      6. associate-*r/ 65.2%

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

      7. associate-*l/ 63.9%

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

      8. div-inv 63.9%

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

      9. metadata-eval 63.9%

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

      10. associate-*r* 63.9%

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

      11. *-commutative 63.9%

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

      12. associate-*r/ 65.3%

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

      13. associate-*l/ 65.3%

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

      14. div-inv 65.3%

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

      15. metadata-eval 65.3%

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

      16. associate-*r* 65.3%

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

      17. *-commutative 65.3%

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

      18. metadata-eval 65.3%

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

   9. Applied egg-rr 65.3%

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

   if 3.0999999999999998e-305 < l

   1. Initial program 71.7%

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

   3. Simplified 71.7%

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

      1. sqrt-div 81.8%

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

      2. div-inv 81.7%

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

   5. Applied egg-rr 81.7%

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

      1. associate-*r/ 81.8%

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

      2. *-rgt-identity 81.8%

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

   7. Simplified 81.8%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.1 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(\frac{\left(D \cdot \frac{M}{d}\right) \cdot 0.5}{\ell} \cdot \frac{\left(D \cdot \frac{M}{d}\right) \cdot 0.5}{-2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)\\ \end{array} \]

Alternative 5: 69.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \left(D \cdot \frac{M}{d}\right) \cdot 0.5\\ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(\frac{t_0}{\ell} \cdot \frac{t_0}{-2}\right)\right)\right) \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* D (/ M d)) 0.5)))
   (*
    (sqrt (/ d h))
    (* (sqrt (/ d l)) (+ 1.0 (* h (* (/ t_0 l) (/ t_0 -2.0))))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (D * (M / d)) * 0.5;
	return sqrt((d / h)) * (sqrt((d / l)) * (1.0 + (h * ((t_0 / l) * (t_0 / -2.0)))));
}

Fortran

NOTE: d, h, l, 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
    t_0 = (d_1 * (m / d)) * 0.5d0
    code = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 + (h * ((t_0 / l) * (t_0 / (-2.0d0))))))
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (D * (M / d)) * 0.5;
	return Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 + (h * ((t_0 / l) * (t_0 / -2.0)))));
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	t_0 = (D * (M / d)) * 0.5
	return math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 + (h * ((t_0 / l) * (t_0 / -2.0)))))

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(D * Float64(M / d)) * 0.5)
	return Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(h * Float64(Float64(t_0 / l) * Float64(t_0 / -2.0))))))
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp = code(d, h, l, M, D)
	t_0 = (D * (M / d)) * 0.5;
	tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + (h * ((t_0 / l) * (t_0 / -2.0)))));
end

Wolfram

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

Derivation

1. Initial program 65.2%

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

3. Simplified 64.8%

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

   1. expm1-log1p-u 31.0%

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

   2. expm1-udef 31.0%

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

   3. *-commutative 31.0%

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

   4. associate-*l* 31.0%

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

   5. div-inv 31.0%

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

   6. metadata-eval 31.0%

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

5. Applied egg-rr 31.0%

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

   1. expm1-def 31.0%

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

   2. expm1-log1p 64.8%

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

   3. associate-*r* 64.8%

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

   4. *-commutative 64.8%

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

   5. associate-*r/ 65.1%

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

   6. associate-*l/ 66.9%

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

   7. *-commutative 66.9%

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

   8. *-commutative 66.9%

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

   9. associate-/l* 66.9%

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

7. Simplified 66.9%

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

   1. unpow2 66.9%

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

   2. div-inv 66.9%

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

   3. metadata-eval 66.9%

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

   4. metadata-eval 66.9%

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

   5. times-frac 69.3%

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

   6. associate-*r/ 69.2%

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

   7. associate-*l/ 68.5%

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

   8. div-inv 68.5%

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

   9. metadata-eval 68.5%

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

   10. associate-*r* 68.5%

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

   11. *-commutative 68.5%

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

   12. associate-*r/ 69.3%

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

   13. associate-*l/ 69.3%

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

   14. div-inv 69.3%

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

   15. metadata-eval 69.3%

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

   16. associate-*r* 69.3%

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

   17. *-commutative 69.3%

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

   18. metadata-eval 69.3%

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

9. Applied egg-rr 69.3%

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

10. Final simplification 69.3%

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

Reproduce

herbie shell --seed 2023321 
(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)))))