Henrywood and Agarwal, Equation (12)

Percentage Accurate: 65.6% → 79.4%

Time: 24.2s

Alternatives: 18

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: 65.6% 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: 79.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{t_0}{\sqrt{-h}} \cdot \left(\frac{t_0}{\sqrt{-\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{+48}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{1}{\ell} \cdot \left(h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (- d))))
   (if (<= l -2e-310)
     (*
      (/ t_0 (sqrt (- h)))
      (*
       (/ t_0 (sqrt (- l)))
       (+ 1.0 (* (/ h l) (* (pow (* (/ M 2.0) (/ D d)) 2.0) -0.5)))))
     (if (<= l 1.25e+48)
       (*
        (* (/ (sqrt d) (sqrt h)) (sqrt (/ d l)))
        (- 1.0 (* 0.5 (* (/ 1.0 l) (* h (pow (* M (* D (/ 0.5 d))) 2.0))))))
       (*
        (* (sqrt (/ d h)) (/ (sqrt d) (sqrt l)))
        (- 1.0 (* 0.5 (pow (* (* (/ D d) (* M 0.5)) (sqrt (/ h l))) 2.0))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(-d);
	double tmp;
	if (l <= -2e-310) {
		tmp = (t_0 / sqrt(-h)) * ((t_0 / sqrt(-l)) * (1.0 + ((h / l) * (pow(((M / 2.0) * (D / d)), 2.0) * -0.5))));
	} else if (l <= 1.25e+48) {
		tmp = ((sqrt(d) / sqrt(h)) * sqrt((d / l))) * (1.0 - (0.5 * ((1.0 / l) * (h * pow((M * (D * (0.5 / d))), 2.0)))));
	} else {
		tmp = (sqrt((d / h)) * (sqrt(d) / sqrt(l))) * (1.0 - (0.5 * pow((((D / d) * (M * 0.5)) * sqrt((h / l))), 2.0)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(-d)
    if (l <= (-2d-310)) then
        tmp = (t_0 / sqrt(-h)) * ((t_0 / sqrt(-l)) * (1.0d0 + ((h / l) * ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (-0.5d0)))))
    else if (l <= 1.25d+48) then
        tmp = ((sqrt(d) / sqrt(h)) * sqrt((d / l))) * (1.0d0 - (0.5d0 * ((1.0d0 / l) * (h * ((m * (d_1 * (0.5d0 / d))) ** 2.0d0)))))
    else
        tmp = (sqrt((d / h)) * (sqrt(d) / sqrt(l))) * (1.0d0 - (0.5d0 * ((((d_1 / d) * (m * 0.5d0)) * sqrt((h / l))) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(-d);
	double tmp;
	if (l <= -2e-310) {
		tmp = (t_0 / Math.sqrt(-h)) * ((t_0 / Math.sqrt(-l)) * (1.0 + ((h / l) * (Math.pow(((M / 2.0) * (D / d)), 2.0) * -0.5))));
	} else if (l <= 1.25e+48) {
		tmp = ((Math.sqrt(d) / Math.sqrt(h)) * Math.sqrt((d / l))) * (1.0 - (0.5 * ((1.0 / l) * (h * Math.pow((M * (D * (0.5 / d))), 2.0)))));
	} else {
		tmp = (Math.sqrt((d / h)) * (Math.sqrt(d) / Math.sqrt(l))) * (1.0 - (0.5 * Math.pow((((D / d) * (M * 0.5)) * Math.sqrt((h / l))), 2.0)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(-d)
	tmp = 0
	if l <= -2e-310:
		tmp = (t_0 / math.sqrt(-h)) * ((t_0 / math.sqrt(-l)) * (1.0 + ((h / l) * (math.pow(((M / 2.0) * (D / d)), 2.0) * -0.5))))
	elif l <= 1.25e+48:
		tmp = ((math.sqrt(d) / math.sqrt(h)) * math.sqrt((d / l))) * (1.0 - (0.5 * ((1.0 / l) * (h * math.pow((M * (D * (0.5 / d))), 2.0)))))
	else:
		tmp = (math.sqrt((d / h)) * (math.sqrt(d) / math.sqrt(l))) * (1.0 - (0.5 * math.pow((((D / d) * (M * 0.5)) * math.sqrt((h / l))), 2.0)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(Float64(t_0 / sqrt(Float64(-l))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * -0.5)))));
	elseif (l <= 1.25e+48)
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(1.0 / l) * Float64(h * (Float64(M * Float64(D * Float64(0.5 / d))) ^ 2.0))))));
	else
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(sqrt(d) / sqrt(l))) * Float64(1.0 - Float64(0.5 * (Float64(Float64(Float64(D / d) * Float64(M * 0.5)) * sqrt(Float64(h / l))) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(-d);
	tmp = 0.0;
	if (l <= -2e-310)
		tmp = (t_0 / sqrt(-h)) * ((t_0 / sqrt(-l)) * (1.0 + ((h / l) * ((((M / 2.0) * (D / d)) ^ 2.0) * -0.5))));
	elseif (l <= 1.25e+48)
		tmp = ((sqrt(d) / sqrt(h)) * sqrt((d / l))) * (1.0 - (0.5 * ((1.0 / l) * (h * ((M * (D * (0.5 / d))) ^ 2.0)))));
	else
		tmp = (sqrt((d / h)) * (sqrt(d) / sqrt(l))) * (1.0 - (0.5 * ((((D / d) * (M * 0.5)) * sqrt((h / l))) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -2e-310], N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.25e+48], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(1.0 / l), $MachinePrecision] * N[(h * N[Power[N[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[Power[N[(N[(N[(D / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 71.0%

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

   3. Simplified 70.3%

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

      1. frac-2neg 70.3%

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

      2. sqrt-div 78.9%

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

   5. Applied egg-rr 78.9%

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

      1. frac-2neg 78.9%

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

      2. sqrt-div 85.3%

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

   7. Applied egg-rr 85.3%

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

   if -1.999999999999994e-310 < l < 1.24999999999999993e48

   1. Initial program 63.3%

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

   3. Simplified 61.9%

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

      1. associate-*r/ 70.1%

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

      2. clear-num 70.1%

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

      3. frac-times 71.5%

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

      4. *-commutative 71.5%

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

      5. frac-times 70.2%

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

      6. add-sqr-sqrt 50.2%

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

      7. add-sqr-sqrt 70.2%

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

      8. div-inv 70.2%

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

      9. metadata-eval 70.2%

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

   5. Applied egg-rr 70.2%

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

      1. associate-/r/ 70.2%

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

      2. *-commutative 70.2%

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

      3. associate-*l* 70.2%

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

   7. Simplified 70.2%

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

      1. *-commutative 70.2%

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

      2. associate-*r* 70.2%

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

      3. expm1-log1p-u 70.0%

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

      4. expm1-udef 65.5%

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

      5. associate-*r* 65.5%

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

      6. *-commutative 65.5%

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

      7. associate-*r/ 65.5%

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

   9. Applied egg-rr 65.5%

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

       1. expm1-def 70.0%

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

       2. expm1-log1p 70.2%

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

       3. associate-/l* 70.2%

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

       4. associate-/r/ 70.2%

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

   11. Simplified 70.2%

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

       1. sqrt-div 82.0%

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

       2. div-inv 82.0%

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

   13. Applied egg-rr 82.0%

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

       1. associate-*r/ 82.0%

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

       2. *-rgt-identity 82.0%

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

   15. Simplified 82.0%

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

   if 1.24999999999999993e48 < l

   1. Initial program 65.9%

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

   3. Simplified 65.9%

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

      1. add-sqr-sqrt 65.9%

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

      2. pow2 65.9%

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

      3. sqrt-prod 65.9%

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

      4. frac-times 65.8%

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

      5. *-commutative 65.8%

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

      6. frac-times 65.9%

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

      7. unpow2 65.9%

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

      8. sqrt-prod 46.8%

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

      9. add-sqr-sqrt 75.5%

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

      10. div-inv 75.5%

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

      11. metadata-eval 75.5%

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

   5. Applied egg-rr 75.5%

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

      1. sqrt-div 82.2%

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

      2. div-inv 82.2%

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

   7. Applied egg-rr 82.2%

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

      1. associate-*r/ 82.2%

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

      2. *-rgt-identity 82.2%

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

   9. Simplified 82.2%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{+48}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{1}{\ell} \cdot \left(h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \end{array} \]

Alternative 2: 69.2% accurate, 0.4× speedup

Math

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

FPCore

(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)))))
      INFINITY)
   (*
    (- 1.0 (* 0.5 (pow (* (* (/ D d) (* M 0.5)) (sqrt (/ h l))) 2.0)))
    (* (sqrt (/ d l)) (sqrt (/ d h))))
   (* d (sqrt (/ 1.0 (* l h))))))

C

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))))) <= ((double) INFINITY)) {
		tmp = (1.0 - (0.5 * pow((((D / d) * (M * 0.5)) * sqrt((h / l))), 2.0))) * (sqrt((d / l)) * sqrt((d / h)));
	} else {
		tmp = d * sqrt((1.0 / (l * h)));
	}
	return tmp;
}

Java

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))))) <= Double.POSITIVE_INFINITY) {
		tmp = (1.0 - (0.5 * Math.pow((((D / d) * (M * 0.5)) * Math.sqrt((h / l))), 2.0))) * (Math.sqrt((d / l)) * Math.sqrt((d / h)));
	} else {
		tmp = d * Math.sqrt((1.0 / (l * h)));
	}
	return tmp;
}

Python

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))))) <= math.inf:
		tmp = (1.0 - (0.5 * math.pow((((D / d) * (M * 0.5)) * math.sqrt((h / l))), 2.0))) * (math.sqrt((d / l)) * math.sqrt((d / h)))
	else:
		tmp = d * math.sqrt((1.0 / (l * h)))
	return tmp

Julia

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))))) <= Inf)
		tmp = Float64(Float64(1.0 - Float64(0.5 * (Float64(Float64(Float64(D / d) * Float64(M * 0.5)) * sqrt(Float64(h / l))) ^ 2.0))) * Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))));
	else
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(l * h))));
	end
	return tmp
end

MATLAB

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))))) <= Inf)
		tmp = (1.0 - (0.5 * ((((D / d) * (M * 0.5)) * sqrt((h / l))) ^ 2.0))) * (sqrt((d / l)) * sqrt((d / h)));
	else
		tmp = d * sqrt((1.0 / (l * h)));
	end
	tmp_2 = tmp;
end

Wolfram

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], Infinity], N[(N[(1.0 - N[(0.5 * N[Power[N[(N[(N[(D / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $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)))) < +inf.0

   1. Initial program 86.1%

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

   3. Simplified 85.6%

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

      1. add-sqr-sqrt 85.6%

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

      2. pow2 85.6%

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

      3. sqrt-prod 85.6%

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

      4. frac-times 86.1%

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

      5. *-commutative 86.1%

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

      6. frac-times 85.2%

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

      7. unpow2 85.2%

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

      8. sqrt-prod 55.9%

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

      9. add-sqr-sqrt 88.9%

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

      10. div-inv 88.9%

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

      11. metadata-eval 88.9%

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

   5. Applied egg-rr 88.9%

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

   if +inf.0 < (*.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 0.0%

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

   3. Simplified 0.1%

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

   4. Taylor expanded in d around inf 24.1%

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

4. Final simplification 74.9%

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

Alternative 3: 76.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{D}{d} \cdot \left(M \cdot 0.5\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{-d}\\ t_3 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -2.95 \cdot 10^{+223}:\\ \;\;\;\;\frac{t_2}{\sqrt{-h}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot t_3\right)\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{t_2}{\sqrt{-\ell}} \cdot t_1\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {t_0}^{2}}{\ell}\right)\\ \mathbf{elif}\;h \leq 3.5 \cdot 10^{+40}:\\ \;\;\;\;\left(t_1 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(t_0 \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot t_3\right) \cdot \left(1 - 0.5 \cdot \left(\frac{1}{\ell} \cdot \left(h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ D d) (* M 0.5)))
        (t_1 (sqrt (/ d h)))
        (t_2 (sqrt (- d)))
        (t_3 (sqrt (/ d l))))
   (if (<= h -2.95e+223)
     (*
      (/ t_2 (sqrt (- h)))
      (* (+ 1.0 (* (/ h l) (* (pow (* (/ M 2.0) (/ D d)) 2.0) -0.5))) t_3))
     (if (<= h -2e-310)
       (*
        (* (/ t_2 (sqrt (- l))) t_1)
        (- 1.0 (* 0.5 (/ (* h (pow t_0 2.0)) l))))
       (if (<= h 3.5e+40)
         (*
          (* t_1 (/ (sqrt d) (sqrt l)))
          (- 1.0 (* 0.5 (pow (* t_0 (sqrt (/ h l))) 2.0))))
         (*
          (* (/ (sqrt d) (sqrt h)) t_3)
          (-
           1.0
           (* 0.5 (* (/ 1.0 l) (* h (pow (* M (* D (/ 0.5 d))) 2.0)))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (D / d) * (M * 0.5);
	double t_1 = sqrt((d / h));
	double t_2 = sqrt(-d);
	double t_3 = sqrt((d / l));
	double tmp;
	if (h <= -2.95e+223) {
		tmp = (t_2 / sqrt(-h)) * ((1.0 + ((h / l) * (pow(((M / 2.0) * (D / d)), 2.0) * -0.5))) * t_3);
	} else if (h <= -2e-310) {
		tmp = ((t_2 / sqrt(-l)) * t_1) * (1.0 - (0.5 * ((h * pow(t_0, 2.0)) / l)));
	} else if (h <= 3.5e+40) {
		tmp = (t_1 * (sqrt(d) / sqrt(l))) * (1.0 - (0.5 * pow((t_0 * sqrt((h / l))), 2.0)));
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * t_3) * (1.0 - (0.5 * ((1.0 / l) * (h * pow((M * (D * (0.5 / d))), 2.0)))));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (d_1 / d) * (m * 0.5d0)
    t_1 = sqrt((d / h))
    t_2 = sqrt(-d)
    t_3 = sqrt((d / l))
    if (h <= (-2.95d+223)) then
        tmp = (t_2 / sqrt(-h)) * ((1.0d0 + ((h / l) * ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (-0.5d0)))) * t_3)
    else if (h <= (-2d-310)) then
        tmp = ((t_2 / sqrt(-l)) * t_1) * (1.0d0 - (0.5d0 * ((h * (t_0 ** 2.0d0)) / l)))
    else if (h <= 3.5d+40) then
        tmp = (t_1 * (sqrt(d) / sqrt(l))) * (1.0d0 - (0.5d0 * ((t_0 * sqrt((h / l))) ** 2.0d0)))
    else
        tmp = ((sqrt(d) / sqrt(h)) * t_3) * (1.0d0 - (0.5d0 * ((1.0d0 / l) * (h * ((m * (d_1 * (0.5d0 / d))) ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (D / d) * (M * 0.5);
	double t_1 = Math.sqrt((d / h));
	double t_2 = Math.sqrt(-d);
	double t_3 = Math.sqrt((d / l));
	double tmp;
	if (h <= -2.95e+223) {
		tmp = (t_2 / Math.sqrt(-h)) * ((1.0 + ((h / l) * (Math.pow(((M / 2.0) * (D / d)), 2.0) * -0.5))) * t_3);
	} else if (h <= -2e-310) {
		tmp = ((t_2 / Math.sqrt(-l)) * t_1) * (1.0 - (0.5 * ((h * Math.pow(t_0, 2.0)) / l)));
	} else if (h <= 3.5e+40) {
		tmp = (t_1 * (Math.sqrt(d) / Math.sqrt(l))) * (1.0 - (0.5 * Math.pow((t_0 * Math.sqrt((h / l))), 2.0)));
	} else {
		tmp = ((Math.sqrt(d) / Math.sqrt(h)) * t_3) * (1.0 - (0.5 * ((1.0 / l) * (h * Math.pow((M * (D * (0.5 / d))), 2.0)))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (D / d) * (M * 0.5)
	t_1 = math.sqrt((d / h))
	t_2 = math.sqrt(-d)
	t_3 = math.sqrt((d / l))
	tmp = 0
	if h <= -2.95e+223:
		tmp = (t_2 / math.sqrt(-h)) * ((1.0 + ((h / l) * (math.pow(((M / 2.0) * (D / d)), 2.0) * -0.5))) * t_3)
	elif h <= -2e-310:
		tmp = ((t_2 / math.sqrt(-l)) * t_1) * (1.0 - (0.5 * ((h * math.pow(t_0, 2.0)) / l)))
	elif h <= 3.5e+40:
		tmp = (t_1 * (math.sqrt(d) / math.sqrt(l))) * (1.0 - (0.5 * math.pow((t_0 * math.sqrt((h / l))), 2.0)))
	else:
		tmp = ((math.sqrt(d) / math.sqrt(h)) * t_3) * (1.0 - (0.5 * ((1.0 / l) * (h * math.pow((M * (D * (0.5 / d))), 2.0)))))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(D / d) * Float64(M * 0.5))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(-d))
	t_3 = sqrt(Float64(d / l))
	tmp = 0.0
	if (h <= -2.95e+223)
		tmp = Float64(Float64(t_2 / sqrt(Float64(-h))) * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * -0.5))) * t_3));
	elseif (h <= -2e-310)
		tmp = Float64(Float64(Float64(t_2 / sqrt(Float64(-l))) * t_1) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (t_0 ^ 2.0)) / l))));
	elseif (h <= 3.5e+40)
		tmp = Float64(Float64(t_1 * Float64(sqrt(d) / sqrt(l))) * Float64(1.0 - Float64(0.5 * (Float64(t_0 * sqrt(Float64(h / l))) ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * t_3) * Float64(1.0 - Float64(0.5 * Float64(Float64(1.0 / l) * Float64(h * (Float64(M * Float64(D * Float64(0.5 / d))) ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (D / d) * (M * 0.5);
	t_1 = sqrt((d / h));
	t_2 = sqrt(-d);
	t_3 = sqrt((d / l));
	tmp = 0.0;
	if (h <= -2.95e+223)
		tmp = (t_2 / sqrt(-h)) * ((1.0 + ((h / l) * ((((M / 2.0) * (D / d)) ^ 2.0) * -0.5))) * t_3);
	elseif (h <= -2e-310)
		tmp = ((t_2 / sqrt(-l)) * t_1) * (1.0 - (0.5 * ((h * (t_0 ^ 2.0)) / l)));
	elseif (h <= 3.5e+40)
		tmp = (t_1 * (sqrt(d) / sqrt(l))) * (1.0 - (0.5 * ((t_0 * sqrt((h / l))) ^ 2.0)));
	else
		tmp = ((sqrt(d) / sqrt(h)) * t_3) * (1.0 - (0.5 * ((1.0 / l) * (h * ((M * (D * (0.5 / d))) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(D / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -2.95e+223], N[(N[(t$95$2 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -2e-310], N[(N[(N[(t$95$2 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 3.5e+40], N[(N[(t$95$1 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[Power[N[(t$95$0 * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(1.0 / l), $MachinePrecision] * N[(h * N[Power[N[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -2.95000000000000011e223

   1. Initial program 28.9%

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

   3. Simplified 28.9%

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

      1. frac-2neg 28.9%

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

      2. sqrt-div 71.8%

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

   5. Applied egg-rr 71.8%

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

   if -2.95000000000000011e223 < h < -1.999999999999994e-310

   1. Initial program 79.5%

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

   3. Simplified 79.5%

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

      1. associate-*r/ 81.6%

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

      2. frac-times 81.6%

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

      3. *-commutative 81.6%

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

      4. frac-times 80.7%

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

      5. add-sqr-sqrt 50.6%

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

      6. add-sqr-sqrt 80.7%

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

      7. div-inv 80.7%

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

      8. metadata-eval 80.7%

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

   5. Applied egg-rr 80.7%

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

      1. frac-2neg 80.3%

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

      2. sqrt-div 87.1%

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

   7. Applied egg-rr 88.5%

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

   if -1.999999999999994e-310 < h < 3.4999999999999999e40

   1. Initial program 70.9%

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

   3. Simplified 71.0%

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

      1. add-sqr-sqrt 71.0%

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

      2. pow2 71.0%

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

      3. sqrt-prod 71.0%

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

      4. frac-times 70.9%

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

      5. *-commutative 70.9%

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

      6. frac-times 69.7%

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

      7. unpow2 69.7%

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

      8. sqrt-prod 51.5%

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

      9. add-sqr-sqrt 77.3%

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

      10. div-inv 77.3%

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

      11. metadata-eval 77.3%

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

   5. Applied egg-rr 77.3%

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

      1. sqrt-div 83.7%

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

      2. div-inv 83.6%

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

   7. Applied egg-rr 83.6%

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

      1. associate-*r/ 83.7%

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

      2. *-rgt-identity 83.7%

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

   9. Simplified 83.7%

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

   if 3.4999999999999999e40 < h

   1. Initial program 56.9%

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

   3. Simplified 55.1%

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

      1. associate-*r/ 65.1%

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

      2. clear-num 65.1%

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

      3. frac-times 66.8%

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

      4. *-commutative 66.8%

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

      5. frac-times 66.7%

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

      6. add-sqr-sqrt 45.5%

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

      7. add-sqr-sqrt 66.7%

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

      8. div-inv 66.7%

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

      9. metadata-eval 66.7%

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

   5. Applied egg-rr 66.7%

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

      1. associate-/r/ 66.7%

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

      2. *-commutative 66.7%

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

      3. associate-*l* 66.7%

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

   7. Simplified 66.7%

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

      1. *-commutative 66.7%

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

      2. associate-*r* 66.7%

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

      3. expm1-log1p-u 66.0%

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

      4. expm1-udef 60.4%

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

      5. associate-*r* 60.4%

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

      6. *-commutative 60.4%

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

      7. associate-*r/ 60.4%

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

   9. Applied egg-rr 60.4%

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

       1. expm1-def 66.0%

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

       2. expm1-log1p 66.7%

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

       3. associate-/l* 66.7%

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

       4. associate-/r/ 66.7%

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

   11. Simplified 66.7%

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

       1. sqrt-div 78.3%

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

       2. div-inv 78.3%

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

   13. Applied egg-rr 78.3%

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

       1. associate-*r/ 78.3%

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

       2. *-rgt-identity 78.3%

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

   15. Simplified 78.3%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.95 \cdot 10^{+223}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;h \leq 3.5 \cdot 10^{+40}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{1}{\ell} \cdot \left(h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \end{array} \]

Alternative 4: 76.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\ell}\\ t_1 := \sqrt{-d}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -4.6 \cdot 10^{+223}:\\ \;\;\;\;t_0 \cdot \left(\frac{t_1}{\sqrt{-h}} \cdot t_2\right)\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{t_1}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot t_2\right) \cdot \left(1 - 0.5 \cdot \left(\frac{1}{\ell} \cdot \left(h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (- 1.0 (* 0.5 (/ (* h (pow (* (/ D d) (* M 0.5)) 2.0)) l))))
        (t_1 (sqrt (- d)))
        (t_2 (sqrt (/ d l))))
   (if (<= h -4.6e+223)
     (* t_0 (* (/ t_1 (sqrt (- h))) t_2))
     (if (<= h -2e-310)
       (* (* (/ t_1 (sqrt (- l))) (sqrt (/ d h))) t_0)
       (*
        (* (/ (sqrt d) (sqrt h)) t_2)
        (-
         1.0
         (* 0.5 (* (/ 1.0 l) (* h (pow (* M (* D (/ 0.5 d))) 2.0))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 - (0.5 * ((h * pow(((D / d) * (M * 0.5)), 2.0)) / l));
	double t_1 = sqrt(-d);
	double t_2 = sqrt((d / l));
	double tmp;
	if (h <= -4.6e+223) {
		tmp = t_0 * ((t_1 / sqrt(-h)) * t_2);
	} else if (h <= -2e-310) {
		tmp = ((t_1 / sqrt(-l)) * sqrt((d / h))) * t_0;
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * t_2) * (1.0 - (0.5 * ((1.0 / l) * (h * pow((M * (D * (0.5 / d))), 2.0)))));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 - (0.5d0 * ((h * (((d_1 / d) * (m * 0.5d0)) ** 2.0d0)) / l))
    t_1 = sqrt(-d)
    t_2 = sqrt((d / l))
    if (h <= (-4.6d+223)) then
        tmp = t_0 * ((t_1 / sqrt(-h)) * t_2)
    else if (h <= (-2d-310)) then
        tmp = ((t_1 / sqrt(-l)) * sqrt((d / h))) * t_0
    else
        tmp = ((sqrt(d) / sqrt(h)) * t_2) * (1.0d0 - (0.5d0 * ((1.0d0 / l) * (h * ((m * (d_1 * (0.5d0 / d))) ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 - (0.5 * ((h * Math.pow(((D / d) * (M * 0.5)), 2.0)) / l));
	double t_1 = Math.sqrt(-d);
	double t_2 = Math.sqrt((d / l));
	double tmp;
	if (h <= -4.6e+223) {
		tmp = t_0 * ((t_1 / Math.sqrt(-h)) * t_2);
	} else if (h <= -2e-310) {
		tmp = ((t_1 / Math.sqrt(-l)) * Math.sqrt((d / h))) * t_0;
	} else {
		tmp = ((Math.sqrt(d) / Math.sqrt(h)) * t_2) * (1.0 - (0.5 * ((1.0 / l) * (h * Math.pow((M * (D * (0.5 / d))), 2.0)))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = 1.0 - (0.5 * ((h * math.pow(((D / d) * (M * 0.5)), 2.0)) / l))
	t_1 = math.sqrt(-d)
	t_2 = math.sqrt((d / l))
	tmp = 0
	if h <= -4.6e+223:
		tmp = t_0 * ((t_1 / math.sqrt(-h)) * t_2)
	elif h <= -2e-310:
		tmp = ((t_1 / math.sqrt(-l)) * math.sqrt((d / h))) * t_0
	else:
		tmp = ((math.sqrt(d) / math.sqrt(h)) * t_2) * (1.0 - (0.5 * ((1.0 / l) * (h * math.pow((M * (D * (0.5 / d))), 2.0)))))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(D / d) * Float64(M * 0.5)) ^ 2.0)) / l)))
	t_1 = sqrt(Float64(-d))
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (h <= -4.6e+223)
		tmp = Float64(t_0 * Float64(Float64(t_1 / sqrt(Float64(-h))) * t_2));
	elseif (h <= -2e-310)
		tmp = Float64(Float64(Float64(t_1 / sqrt(Float64(-l))) * sqrt(Float64(d / h))) * t_0);
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * t_2) * Float64(1.0 - Float64(0.5 * Float64(Float64(1.0 / l) * Float64(h * (Float64(M * Float64(D * Float64(0.5 / d))) ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = 1.0 - (0.5 * ((h * (((D / d) * (M * 0.5)) ^ 2.0)) / l));
	t_1 = sqrt(-d);
	t_2 = sqrt((d / l));
	tmp = 0.0;
	if (h <= -4.6e+223)
		tmp = t_0 * ((t_1 / sqrt(-h)) * t_2);
	elseif (h <= -2e-310)
		tmp = ((t_1 / sqrt(-l)) * sqrt((d / h))) * t_0;
	else
		tmp = ((sqrt(d) / sqrt(h)) * t_2) * (1.0 - (0.5 * ((1.0 / l) * (h * ((M * (D * (0.5 / d))) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -4.6e+223], N[(t$95$0 * N[(N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -2e-310], N[(N[(N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(1.0 / l), $MachinePrecision] * N[(h * N[Power[N[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -4.60000000000000009e223

   1. Initial program 30.3%

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

   3. Simplified 30.3%

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

      1. associate-*r/ 30.4%

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

      2. frac-times 30.4%

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

      3. *-commutative 30.4%

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

      4. frac-times 30.4%

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

      5. add-sqr-sqrt 15.5%

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

      6. add-sqr-sqrt 30.4%

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

      7. div-inv 30.4%

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

      8. metadata-eval 30.4%

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

   5. Applied egg-rr 30.4%

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

      1. frac-2neg 30.3%

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

      2. sqrt-div 70.4%

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

   7. Applied egg-rr 60.3%

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

   if -4.60000000000000009e223 < h < -1.999999999999994e-310

   1. Initial program 78.8%

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

   3. Simplified 78.8%

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

      1. associate-*r/ 80.8%

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

      2. frac-times 80.8%

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

      3. *-commutative 80.8%

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

      4. frac-times 79.9%

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

      5. add-sqr-sqrt 50.1%

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

      6. add-sqr-sqrt 79.9%

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

      7. div-inv 79.9%

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

      8. metadata-eval 79.9%

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

   5. Applied egg-rr 79.9%

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

      1. frac-2neg 80.5%

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

      2. sqrt-div 87.3%

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

   7. Applied egg-rr 87.7%

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

   if -1.999999999999994e-310 < h

   1. Initial program 64.4%

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

   3. Simplified 63.6%

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

      1. associate-*r/ 68.2%

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

      2. clear-num 68.2%

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

      3. frac-times 69.0%

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

      4. *-commutative 69.0%

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

      5. frac-times 68.3%

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

      6. add-sqr-sqrt 46.1%

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

      7. add-sqr-sqrt 68.3%

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

      8. div-inv 68.3%

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

      9. metadata-eval 68.3%

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

   5. Applied egg-rr 68.3%

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

      1. associate-/r/ 68.3%

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

      2. *-commutative 68.3%

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

      3. associate-*l* 68.3%

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

   7. Simplified 68.3%

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

      1. *-commutative 68.3%

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

      2. associate-*r* 68.3%

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

      3. expm1-log1p-u 68.0%

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

      4. expm1-udef 65.4%

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

      5. associate-*r* 65.4%

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

      6. *-commutative 65.4%

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

      7. associate-*r/ 65.4%

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

   9. Applied egg-rr 65.4%

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

       1. expm1-def 68.0%

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

       2. expm1-log1p 68.3%

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

       3. associate-/l* 68.3%

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

       4. associate-/r/ 68.3%

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

   11. Simplified 68.3%

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

       1. sqrt-div 75.8%

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

       2. div-inv 75.8%

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

   13. Applied egg-rr 75.8%

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

       1. associate-*r/ 75.8%

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

       2. *-rgt-identity 75.8%

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

   15. Simplified 75.8%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -4.6 \cdot 10^{+223}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\ell}\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{1}{\ell} \cdot \left(h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \end{array} \]

Alternative 5: 76.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -3.8 \cdot 10^{+222}:\\ \;\;\;\;\frac{t_0}{\sqrt{-h}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot t_1\right)\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{t_0}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot t_1\right) \cdot \left(1 - 0.5 \cdot \left(\frac{1}{\ell} \cdot \left(h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (- d))) (t_1 (sqrt (/ d l))))
   (if (<= h -3.8e+222)
     (*
      (/ t_0 (sqrt (- h)))
      (* (+ 1.0 (* (/ h l) (* (pow (* (/ M 2.0) (/ D d)) 2.0) -0.5))) t_1))
     (if (<= h -2e-310)
       (*
        (* (/ t_0 (sqrt (- l))) (sqrt (/ d h)))
        (- 1.0 (* 0.5 (/ (* h (pow (* (/ D d) (* M 0.5)) 2.0)) l))))
       (*
        (* (/ (sqrt d) (sqrt h)) t_1)
        (-
         1.0
         (* 0.5 (* (/ 1.0 l) (* h (pow (* M (* D (/ 0.5 d))) 2.0))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(-d);
	double t_1 = sqrt((d / l));
	double tmp;
	if (h <= -3.8e+222) {
		tmp = (t_0 / sqrt(-h)) * ((1.0 + ((h / l) * (pow(((M / 2.0) * (D / d)), 2.0) * -0.5))) * t_1);
	} else if (h <= -2e-310) {
		tmp = ((t_0 / sqrt(-l)) * sqrt((d / h))) * (1.0 - (0.5 * ((h * pow(((D / d) * (M * 0.5)), 2.0)) / l)));
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * t_1) * (1.0 - (0.5 * ((1.0 / l) * (h * pow((M * (D * (0.5 / d))), 2.0)))));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(-d)
    t_1 = sqrt((d / l))
    if (h <= (-3.8d+222)) then
        tmp = (t_0 / sqrt(-h)) * ((1.0d0 + ((h / l) * ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (-0.5d0)))) * t_1)
    else if (h <= (-2d-310)) then
        tmp = ((t_0 / sqrt(-l)) * sqrt((d / h))) * (1.0d0 - (0.5d0 * ((h * (((d_1 / d) * (m * 0.5d0)) ** 2.0d0)) / l)))
    else
        tmp = ((sqrt(d) / sqrt(h)) * t_1) * (1.0d0 - (0.5d0 * ((1.0d0 / l) * (h * ((m * (d_1 * (0.5d0 / d))) ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(-d);
	double t_1 = Math.sqrt((d / l));
	double tmp;
	if (h <= -3.8e+222) {
		tmp = (t_0 / Math.sqrt(-h)) * ((1.0 + ((h / l) * (Math.pow(((M / 2.0) * (D / d)), 2.0) * -0.5))) * t_1);
	} else if (h <= -2e-310) {
		tmp = ((t_0 / Math.sqrt(-l)) * Math.sqrt((d / h))) * (1.0 - (0.5 * ((h * Math.pow(((D / d) * (M * 0.5)), 2.0)) / l)));
	} else {
		tmp = ((Math.sqrt(d) / Math.sqrt(h)) * t_1) * (1.0 - (0.5 * ((1.0 / l) * (h * Math.pow((M * (D * (0.5 / d))), 2.0)))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(-d)
	t_1 = math.sqrt((d / l))
	tmp = 0
	if h <= -3.8e+222:
		tmp = (t_0 / math.sqrt(-h)) * ((1.0 + ((h / l) * (math.pow(((M / 2.0) * (D / d)), 2.0) * -0.5))) * t_1)
	elif h <= -2e-310:
		tmp = ((t_0 / math.sqrt(-l)) * math.sqrt((d / h))) * (1.0 - (0.5 * ((h * math.pow(((D / d) * (M * 0.5)), 2.0)) / l)))
	else:
		tmp = ((math.sqrt(d) / math.sqrt(h)) * t_1) * (1.0 - (0.5 * ((1.0 / l) * (h * math.pow((M * (D * (0.5 / d))), 2.0)))))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(-d))
	t_1 = sqrt(Float64(d / l))
	tmp = 0.0
	if (h <= -3.8e+222)
		tmp = Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * -0.5))) * t_1));
	elseif (h <= -2e-310)
		tmp = Float64(Float64(Float64(t_0 / sqrt(Float64(-l))) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(D / d) * Float64(M * 0.5)) ^ 2.0)) / l))));
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * t_1) * Float64(1.0 - Float64(0.5 * Float64(Float64(1.0 / l) * Float64(h * (Float64(M * Float64(D * Float64(0.5 / d))) ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(-d);
	t_1 = sqrt((d / l));
	tmp = 0.0;
	if (h <= -3.8e+222)
		tmp = (t_0 / sqrt(-h)) * ((1.0 + ((h / l) * ((((M / 2.0) * (D / d)) ^ 2.0) * -0.5))) * t_1);
	elseif (h <= -2e-310)
		tmp = ((t_0 / sqrt(-l)) * sqrt((d / h))) * (1.0 - (0.5 * ((h * (((D / d) * (M * 0.5)) ^ 2.0)) / l)));
	else
		tmp = ((sqrt(d) / sqrt(h)) * t_1) * (1.0 - (0.5 * ((1.0 / l) * (h * ((M * (D * (0.5 / d))) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -3.8e+222], N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -2e-310], N[(N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(1.0 / l), $MachinePrecision] * N[(h * N[Power[N[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -3.80000000000000018e222

   1. Initial program 28.9%

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

   3. Simplified 28.9%

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

      1. frac-2neg 28.9%

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

      2. sqrt-div 71.8%

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

   5. Applied egg-rr 71.8%

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

   if -3.80000000000000018e222 < h < -1.999999999999994e-310

   1. Initial program 79.5%

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

   3. Simplified 79.5%

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

      1. associate-*r/ 81.6%

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

      2. frac-times 81.6%

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

      3. *-commutative 81.6%

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

      4. frac-times 80.7%

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

      5. add-sqr-sqrt 50.6%

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

      6. add-sqr-sqrt 80.7%

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

      7. div-inv 80.7%

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

      8. metadata-eval 80.7%

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

   5. Applied egg-rr 80.7%

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

      1. frac-2neg 80.3%

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

      2. sqrt-div 87.1%

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

   7. Applied egg-rr 88.5%

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

   if -1.999999999999994e-310 < h

   1. Initial program 64.4%

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

   3. Simplified 63.6%

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

      1. associate-*r/ 68.2%

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

      2. clear-num 68.2%

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

      3. frac-times 69.0%

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

      4. *-commutative 69.0%

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

      5. frac-times 68.3%

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

      6. add-sqr-sqrt 46.1%

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

      7. add-sqr-sqrt 68.3%

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

      8. div-inv 68.3%

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

      9. metadata-eval 68.3%

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

   5. Applied egg-rr 68.3%

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

      1. associate-/r/ 68.3%

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

      2. *-commutative 68.3%

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

      3. associate-*l* 68.3%

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

   7. Simplified 68.3%

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

      1. *-commutative 68.3%

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

      2. associate-*r* 68.3%

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

      3. expm1-log1p-u 68.0%

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

      4. expm1-udef 65.4%

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

      5. associate-*r* 65.4%

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

      6. *-commutative 65.4%

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

      7. associate-*r/ 65.4%

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

   9. Applied egg-rr 65.4%

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

       1. expm1-def 68.0%

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

       2. expm1-log1p 68.3%

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

       3. associate-/l* 68.3%

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

       4. associate-/r/ 68.3%

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

   11. Simplified 68.3%

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

       1. sqrt-div 75.8%

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

       2. div-inv 75.8%

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

   13. Applied egg-rr 75.8%

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

       1. associate-*r/ 75.8%

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

       2. *-rgt-identity 75.8%

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

   15. Simplified 75.8%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -3.8 \cdot 10^{+222}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{1}{\ell} \cdot \left(h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\right)\\ \end{array} \]

Alternative 6: 73.1% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -2e-310)
   (*
    (* (/ (sqrt (- d)) (sqrt (- l))) (sqrt (/ d h)))
    (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D 2.0) (/ M d)) 2.0)))))
   (*
    (* (/ (sqrt d) (sqrt h)) (sqrt (/ d l)))
    (- 1.0 (* 0.5 (/ (* h (pow (* (/ D d) (* M 0.5)) 2.0)) l))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2e-310) {
		tmp = ((sqrt(-d) / sqrt(-l)) * sqrt((d / h))) * (1.0 - (0.5 * ((h / l) * pow(((D / 2.0) * (M / d)), 2.0))));
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * sqrt((d / l))) * (1.0 - (0.5 * ((h * pow(((D / d) * (M * 0.5)), 2.0)) / l)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (l <= (-2d-310)) then
        tmp = ((sqrt(-d) / sqrt(-l)) * sqrt((d / h))) * (1.0d0 - (0.5d0 * ((h / l) * (((d_1 / 2.0d0) * (m / d)) ** 2.0d0))))
    else
        tmp = ((sqrt(d) / sqrt(h)) * sqrt((d / l))) * (1.0d0 - (0.5d0 * ((h * (((d_1 / d) * (m * 0.5d0)) ** 2.0d0)) / l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2e-310) {
		tmp = ((Math.sqrt(-d) / Math.sqrt(-l)) * Math.sqrt((d / h))) * (1.0 - (0.5 * ((h / l) * Math.pow(((D / 2.0) * (M / d)), 2.0))));
	} else {
		tmp = ((Math.sqrt(d) / Math.sqrt(h)) * Math.sqrt((d / l))) * (1.0 - (0.5 * ((h * Math.pow(((D / d) * (M * 0.5)), 2.0)) / l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -2e-310:
		tmp = ((math.sqrt(-d) / math.sqrt(-l)) * math.sqrt((d / h))) * (1.0 - (0.5 * ((h / l) * math.pow(((D / 2.0) * (M / d)), 2.0))))
	else:
		tmp = ((math.sqrt(d) / math.sqrt(h)) * math.sqrt((d / l))) * (1.0 - (0.5 * ((h * math.pow(((D / d) * (M * 0.5)), 2.0)) / l)))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / 2.0) * Float64(M / d)) ^ 2.0)))));
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(D / d) * Float64(M * 0.5)) ^ 2.0)) / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -2e-310)
		tmp = ((sqrt(-d) / sqrt(-l)) * sqrt((d / h))) * (1.0 - (0.5 * ((h / l) * (((D / 2.0) * (M / d)) ^ 2.0))));
	else
		tmp = ((sqrt(d) / sqrt(h)) * sqrt((d / l))) * (1.0 - (0.5 * ((h * (((D / d) * (M * 0.5)) ^ 2.0)) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -2e-310], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / 2.0), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 71.0%

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

   3. Simplified 71.0%

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

      1. frac-2neg 78.9%

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

      2. sqrt-div 85.3%

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

   5. Applied egg-rr 76.8%

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

   if -1.999999999999994e-310 < l

   1. Initial program 64.4%

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

   3. Simplified 63.6%

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

      1. associate-*r/ 68.2%

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

      2. frac-times 69.0%

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

      3. *-commutative 69.0%

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

      4. frac-times 68.3%

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

      5. add-sqr-sqrt 46.1%

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

      6. add-sqr-sqrt 68.3%

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

      7. div-inv 68.3%

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

      8. metadata-eval 68.3%

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

   5. Applied egg-rr 68.3%

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

      1. sqrt-div 75.8%

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

      2. div-inv 75.8%

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

   7. Applied egg-rr 75.8%

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

      1. associate-*r/ 75.8%

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

      2. *-rgt-identity 75.8%

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

   9. Simplified 75.8%

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

4. Final simplification 76.3%

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

Alternative 7: 75.7% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (- 1.0 (* 0.5 (/ (* h (pow (* (/ D d) (* M 0.5)) 2.0)) l)))))
   (if (<= l -2e-310)
     (* (* (/ (sqrt (- d)) (sqrt (- l))) (sqrt (/ d h))) t_0)
     (* (* (/ (sqrt d) (sqrt h)) (sqrt (/ d l))) t_0))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 - (0.5 * ((h * pow(((D / d) * (M * 0.5)), 2.0)) / l));
	double tmp;
	if (l <= -2e-310) {
		tmp = ((sqrt(-d) / sqrt(-l)) * sqrt((d / h))) * t_0;
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * sqrt((d / l))) * t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (0.5d0 * ((h * (((d_1 / d) * (m * 0.5d0)) ** 2.0d0)) / l))
    if (l <= (-2d-310)) then
        tmp = ((sqrt(-d) / sqrt(-l)) * sqrt((d / h))) * t_0
    else
        tmp = ((sqrt(d) / sqrt(h)) * sqrt((d / l))) * t_0
    end if
    code = tmp
end function

Java

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

Python

def code(d, h, l, M, D):
	t_0 = 1.0 - (0.5 * ((h * math.pow(((D / d) * (M * 0.5)), 2.0)) / l))
	tmp = 0
	if l <= -2e-310:
		tmp = ((math.sqrt(-d) / math.sqrt(-l)) * math.sqrt((d / h))) * t_0
	else:
		tmp = ((math.sqrt(d) / math.sqrt(h)) * math.sqrt((d / l))) * t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(D / d) * Float64(M * 0.5)) ^ 2.0)) / l)))
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * sqrt(Float64(d / h))) * t_0);
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * sqrt(Float64(d / l))) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = 1.0 - (0.5 * ((h * (((D / d) * (M * 0.5)) ^ 2.0)) / l));
	tmp = 0.0;
	if (l <= -2e-310)
		tmp = ((sqrt(-d) / sqrt(-l)) * sqrt((d / h))) * t_0;
	else
		tmp = ((sqrt(d) / sqrt(h)) * sqrt((d / l))) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2e-310], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 71.0%

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

   3. Simplified 71.0%

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

      1. associate-*r/ 72.7%

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

      2. frac-times 72.7%

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

      3. *-commutative 72.7%

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

      4. frac-times 72.0%

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

      5. add-sqr-sqrt 44.6%

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

      6. add-sqr-sqrt 72.0%

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

      7. div-inv 72.0%

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

      8. metadata-eval 72.0%

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

   5. Applied egg-rr 72.0%

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

      1. frac-2neg 78.9%

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

      2. sqrt-div 85.3%

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

   7. Applied egg-rr 80.0%

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

   if -1.999999999999994e-310 < l

   1. Initial program 64.4%

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

   3. Simplified 63.6%

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

      1. associate-*r/ 68.2%

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

      2. frac-times 69.0%

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

      3. *-commutative 69.0%

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

      4. frac-times 68.3%

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

      5. add-sqr-sqrt 46.1%

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

      6. add-sqr-sqrt 68.3%

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

      7. div-inv 68.3%

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

      8. metadata-eval 68.3%

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

   5. Applied egg-rr 68.3%

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

      1. sqrt-div 75.8%

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

      2. div-inv 75.8%

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

   7. Applied egg-rr 75.8%

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

      1. associate-*r/ 75.8%

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

      2. *-rgt-identity 75.8%

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

   9. Simplified 75.8%

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

4. Final simplification 77.8%

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

Alternative 8: 75.7% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -2e-310)
   (*
    (* (/ (sqrt (- d)) (sqrt (- l))) (sqrt (/ d h)))
    (- 1.0 (* 0.5 (/ (* h (pow (* (/ D d) (* M 0.5)) 2.0)) l))))
   (*
    (* (/ (sqrt d) (sqrt h)) (sqrt (/ d l)))
    (+ 1.0 (* 0.5 (* (* h (pow (* M (* (/ D d) 0.5)) 2.0)) (/ -1.0 l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2e-310) {
		tmp = ((sqrt(-d) / sqrt(-l)) * sqrt((d / h))) * (1.0 - (0.5 * ((h * pow(((D / d) * (M * 0.5)), 2.0)) / l)));
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * sqrt((d / l))) * (1.0 + (0.5 * ((h * pow((M * ((D / d) * 0.5)), 2.0)) * (-1.0 / l))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (l <= (-2d-310)) then
        tmp = ((sqrt(-d) / sqrt(-l)) * sqrt((d / h))) * (1.0d0 - (0.5d0 * ((h * (((d_1 / d) * (m * 0.5d0)) ** 2.0d0)) / l)))
    else
        tmp = ((sqrt(d) / sqrt(h)) * sqrt((d / l))) * (1.0d0 + (0.5d0 * ((h * ((m * ((d_1 / d) * 0.5d0)) ** 2.0d0)) * ((-1.0d0) / l))))
    end if
    code = tmp
end function

Java

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

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -2e-310:
		tmp = ((math.sqrt(-d) / math.sqrt(-l)) * math.sqrt((d / h))) * (1.0 - (0.5 * ((h * math.pow(((D / d) * (M * 0.5)), 2.0)) / l)))
	else:
		tmp = ((math.sqrt(d) / math.sqrt(h)) * math.sqrt((d / l))) * (1.0 + (0.5 * ((h * math.pow((M * ((D / d) * 0.5)), 2.0)) * (-1.0 / l))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(D / d) * Float64(M * 0.5)) ^ 2.0)) / l))));
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * sqrt(Float64(d / l))) * Float64(1.0 + Float64(0.5 * Float64(Float64(h * (Float64(M * Float64(Float64(D / d) * 0.5)) ^ 2.0)) * Float64(-1.0 / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -2e-310)
		tmp = ((sqrt(-d) / sqrt(-l)) * sqrt((d / h))) * (1.0 - (0.5 * ((h * (((D / d) * (M * 0.5)) ^ 2.0)) / l)));
	else
		tmp = ((sqrt(d) / sqrt(h)) * sqrt((d / l))) * (1.0 + (0.5 * ((h * ((M * ((D / d) * 0.5)) ^ 2.0)) * (-1.0 / l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -2e-310], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(0.5 * N[(N[(h * N[Power[N[(M * N[(N[(D / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 71.0%

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

   3. Simplified 71.0%

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

      1. associate-*r/ 72.7%

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

      2. frac-times 72.7%

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

      3. *-commutative 72.7%

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

      4. frac-times 72.0%

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

      5. add-sqr-sqrt 44.6%

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

      6. add-sqr-sqrt 72.0%

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

      7. div-inv 72.0%

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

      8. metadata-eval 72.0%

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

   5. Applied egg-rr 72.0%

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

      1. frac-2neg 78.9%

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

      2. sqrt-div 85.3%

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

   7. Applied egg-rr 80.0%

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

   if -1.999999999999994e-310 < l

   1. Initial program 64.4%

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

   3. Simplified 63.6%

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

      1. associate-*r/ 68.2%

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

      2. clear-num 68.2%

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

      3. frac-times 69.0%

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

      4. *-commutative 69.0%

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

      5. frac-times 68.3%

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

      6. add-sqr-sqrt 46.1%

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

      7. add-sqr-sqrt 68.3%

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

      8. div-inv 68.3%

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

      9. metadata-eval 68.3%

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

   5. Applied egg-rr 68.3%

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

      1. associate-/r/ 68.3%

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

      2. *-commutative 68.3%

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

      3. associate-*l* 68.3%

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

   7. Simplified 68.3%

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

      1. sqrt-div 75.8%

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

      2. div-inv 75.8%

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

   9. Applied egg-rr 75.8%

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

       1. associate-*r/ 75.8%

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

       2. *-rgt-identity 75.8%

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

   11. Simplified 75.8%

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

4. Final simplification 77.8%

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

Alternative 9: 75.7% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -2e-310)
   (*
    (* (/ (sqrt (- d)) (sqrt (- l))) (sqrt (/ d h)))
    (- 1.0 (* 0.5 (/ (* h (pow (* (/ D d) (* M 0.5)) 2.0)) l))))
   (*
    (* (/ (sqrt d) (sqrt h)) (sqrt (/ d l)))
    (- 1.0 (* 0.5 (* (/ 1.0 l) (* h (pow (* M (* D (/ 0.5 d))) 2.0))))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2e-310) {
		tmp = ((sqrt(-d) / sqrt(-l)) * sqrt((d / h))) * (1.0 - (0.5 * ((h * pow(((D / d) * (M * 0.5)), 2.0)) / l)));
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * sqrt((d / l))) * (1.0 - (0.5 * ((1.0 / l) * (h * pow((M * (D * (0.5 / d))), 2.0)))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (l <= (-2d-310)) then
        tmp = ((sqrt(-d) / sqrt(-l)) * sqrt((d / h))) * (1.0d0 - (0.5d0 * ((h * (((d_1 / d) * (m * 0.5d0)) ** 2.0d0)) / l)))
    else
        tmp = ((sqrt(d) / sqrt(h)) * sqrt((d / l))) * (1.0d0 - (0.5d0 * ((1.0d0 / l) * (h * ((m * (d_1 * (0.5d0 / d))) ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2e-310) {
		tmp = ((Math.sqrt(-d) / Math.sqrt(-l)) * Math.sqrt((d / h))) * (1.0 - (0.5 * ((h * Math.pow(((D / d) * (M * 0.5)), 2.0)) / l)));
	} else {
		tmp = ((Math.sqrt(d) / Math.sqrt(h)) * Math.sqrt((d / l))) * (1.0 - (0.5 * ((1.0 / l) * (h * Math.pow((M * (D * (0.5 / d))), 2.0)))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -2e-310:
		tmp = ((math.sqrt(-d) / math.sqrt(-l)) * math.sqrt((d / h))) * (1.0 - (0.5 * ((h * math.pow(((D / d) * (M * 0.5)), 2.0)) / l)))
	else:
		tmp = ((math.sqrt(d) / math.sqrt(h)) * math.sqrt((d / l))) * (1.0 - (0.5 * ((1.0 / l) * (h * math.pow((M * (D * (0.5 / d))), 2.0)))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(D / d) * Float64(M * 0.5)) ^ 2.0)) / l))));
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(1.0 / l) * Float64(h * (Float64(M * Float64(D * Float64(0.5 / d))) ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -2e-310)
		tmp = ((sqrt(-d) / sqrt(-l)) * sqrt((d / h))) * (1.0 - (0.5 * ((h * (((D / d) * (M * 0.5)) ^ 2.0)) / l)));
	else
		tmp = ((sqrt(d) / sqrt(h)) * sqrt((d / l))) * (1.0 - (0.5 * ((1.0 / l) * (h * ((M * (D * (0.5 / d))) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -2e-310], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(1.0 / l), $MachinePrecision] * N[(h * N[Power[N[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 71.0%

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

   3. Simplified 71.0%

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

      1. associate-*r/ 72.7%

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

      2. frac-times 72.7%

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

      3. *-commutative 72.7%

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

      4. frac-times 72.0%

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

      5. add-sqr-sqrt 44.6%

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

      6. add-sqr-sqrt 72.0%

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

      7. div-inv 72.0%

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

      8. metadata-eval 72.0%

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

   5. Applied egg-rr 72.0%

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

      1. frac-2neg 78.9%

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

      2. sqrt-div 85.3%

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

   7. Applied egg-rr 80.0%

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

   if -1.999999999999994e-310 < l

   1. Initial program 64.4%

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

   3. Simplified 63.6%

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

      1. associate-*r/ 68.2%

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

      2. clear-num 68.2%

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

      3. frac-times 69.0%

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

      4. *-commutative 69.0%

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

      5. frac-times 68.3%

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

      6. add-sqr-sqrt 46.1%

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

      7. add-sqr-sqrt 68.3%

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

      8. div-inv 68.3%

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

      9. metadata-eval 68.3%

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

   5. Applied egg-rr 68.3%

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

      1. associate-/r/ 68.3%

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

      2. *-commutative 68.3%

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

      3. associate-*l* 68.3%

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

   7. Simplified 68.3%

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

      1. *-commutative 68.3%

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

      2. associate-*r* 68.3%

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

      3. expm1-log1p-u 68.0%

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

      4. expm1-udef 65.4%

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

      5. associate-*r* 65.4%

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

      6. *-commutative 65.4%

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

      7. associate-*r/ 65.4%

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

   9. Applied egg-rr 65.4%

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

       1. expm1-def 68.0%

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

       2. expm1-log1p 68.3%

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

       3. associate-/l* 68.3%

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

       4. associate-/r/ 68.3%

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

   11. Simplified 68.3%

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

       1. sqrt-div 75.8%

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

       2. div-inv 75.8%

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

   13. Applied egg-rr 75.8%

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

       1. associate-*r/ 75.8%

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

       2. *-rgt-identity 75.8%

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

   15. Simplified 75.8%

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

4. Final simplification 77.9%

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

Alternative 10: 69.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{D}{d} \cdot \left(M \cdot 0.5\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq 1.7 \cdot 10^{+44}:\\ \;\;\;\;\left(1 - 0.5 \cdot {\left(t_0 \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \cdot \left(t_1 \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot t_1\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {t_0}^{2}}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ D d) (* M 0.5))) (t_1 (sqrt (/ d l))))
   (if (<= h 1.7e+44)
     (*
      (- 1.0 (* 0.5 (pow (* t_0 (sqrt (/ h l))) 2.0)))
      (* t_1 (sqrt (/ d h))))
     (*
      (* (/ (sqrt d) (sqrt h)) t_1)
      (- 1.0 (* 0.5 (/ (* h (pow t_0 2.0)) l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (D / d) * (M * 0.5);
	double t_1 = sqrt((d / l));
	double tmp;
	if (h <= 1.7e+44) {
		tmp = (1.0 - (0.5 * pow((t_0 * sqrt((h / l))), 2.0))) * (t_1 * sqrt((d / h)));
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * t_1) * (1.0 - (0.5 * ((h * pow(t_0, 2.0)) / l)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (d_1 / d) * (m * 0.5d0)
    t_1 = sqrt((d / l))
    if (h <= 1.7d+44) then
        tmp = (1.0d0 - (0.5d0 * ((t_0 * sqrt((h / l))) ** 2.0d0))) * (t_1 * sqrt((d / h)))
    else
        tmp = ((sqrt(d) / sqrt(h)) * t_1) * (1.0d0 - (0.5d0 * ((h * (t_0 ** 2.0d0)) / l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (D / d) * (M * 0.5);
	double t_1 = Math.sqrt((d / l));
	double tmp;
	if (h <= 1.7e+44) {
		tmp = (1.0 - (0.5 * Math.pow((t_0 * Math.sqrt((h / l))), 2.0))) * (t_1 * Math.sqrt((d / h)));
	} else {
		tmp = ((Math.sqrt(d) / Math.sqrt(h)) * t_1) * (1.0 - (0.5 * ((h * Math.pow(t_0, 2.0)) / l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (D / d) * (M * 0.5)
	t_1 = math.sqrt((d / l))
	tmp = 0
	if h <= 1.7e+44:
		tmp = (1.0 - (0.5 * math.pow((t_0 * math.sqrt((h / l))), 2.0))) * (t_1 * math.sqrt((d / h)))
	else:
		tmp = ((math.sqrt(d) / math.sqrt(h)) * t_1) * (1.0 - (0.5 * ((h * math.pow(t_0, 2.0)) / l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(D / d) * Float64(M * 0.5))
	t_1 = sqrt(Float64(d / l))
	tmp = 0.0
	if (h <= 1.7e+44)
		tmp = Float64(Float64(1.0 - Float64(0.5 * (Float64(t_0 * sqrt(Float64(h / l))) ^ 2.0))) * Float64(t_1 * sqrt(Float64(d / h))));
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * t_1) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (t_0 ^ 2.0)) / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (D / d) * (M * 0.5);
	t_1 = sqrt((d / l));
	tmp = 0.0;
	if (h <= 1.7e+44)
		tmp = (1.0 - (0.5 * ((t_0 * sqrt((h / l))) ^ 2.0))) * (t_1 * sqrt((d / h)));
	else
		tmp = ((sqrt(d) / sqrt(h)) * t_1) * (1.0 - (0.5 * ((h * (t_0 ^ 2.0)) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(D / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, 1.7e+44], N[(N[(1.0 - N[(0.5 * N[Power[N[(t$95$0 * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if h < 1.7e44

   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 71.1%

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

      1. add-sqr-sqrt 71.1%

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

      2. pow2 71.1%

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

      3. sqrt-prod 71.1%

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

      4. frac-times 71.1%

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

      5. *-commutative 71.1%

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

      6. frac-times 70.2%

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

      7. unpow2 70.2%

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

      8. sqrt-prod 47.2%

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

      9. add-sqr-sqrt 74.4%

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

      10. div-inv 74.4%

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

      11. metadata-eval 74.4%

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

   5. Applied egg-rr 74.4%

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

   if 1.7e44 < h

   1. Initial program 56.2%

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

   3. Simplified 54.4%

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

      1. associate-*r/ 64.5%

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

      2. frac-times 66.3%

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

      3. *-commutative 66.3%

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

      4. frac-times 66.1%

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

      5. add-sqr-sqrt 44.7%

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

      6. add-sqr-sqrt 66.1%

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

      7. div-inv 66.1%

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

      8. metadata-eval 66.1%

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

   5. Applied egg-rr 66.1%

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

      1. sqrt-div 78.0%

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

      2. div-inv 78.0%

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

   7. Applied egg-rr 77.9%

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

      1. associate-*r/ 78.0%

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

      2. *-rgt-identity 78.0%

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

   9. Simplified 77.9%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 1.7 \cdot 10^{+44}:\\ \;\;\;\;\left(1 - 0.5 \cdot {\left(\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\ell}\right)\\ \end{array} \]

Alternative 11: 73.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{-0.5}{\frac{\ell}{{\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 5e-307)
   (*
    (sqrt (/ d h))
    (*
     (sqrt (/ d l))
     (+ 1.0 (* h (/ -0.5 (/ l (pow (* D (/ M (* d 2.0))) 2.0)))))))
   (*
    (+ 1.0 (* (pow (* M (* (/ D d) 0.5)) 2.0) (* (/ h l) -0.5)))
    (/ d (* (sqrt h) (sqrt l))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 5e-307) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + (h * (-0.5 / (l / pow((D * (M / (d * 2.0))), 2.0))))));
	} else {
		tmp = (1.0 + (pow((M * ((D / d) * 0.5)), 2.0) * ((h / l) * -0.5))) * (d / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (l <= 5d-307) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 + (h * ((-0.5d0) / (l / ((d_1 * (m / (d * 2.0d0))) ** 2.0d0))))))
    else
        tmp = (1.0d0 + (((m * ((d_1 / d) * 0.5d0)) ** 2.0d0) * ((h / l) * (-0.5d0)))) * (d / (sqrt(h) * sqrt(l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 5e-307) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 + (h * (-0.5 / (l / Math.pow((D * (M / (d * 2.0))), 2.0))))));
	} else {
		tmp = (1.0 + (Math.pow((M * ((D / d) * 0.5)), 2.0) * ((h / l) * -0.5))) * (d / (Math.sqrt(h) * Math.sqrt(l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= 5e-307:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 + (h * (-0.5 / (l / math.pow((D * (M / (d * 2.0))), 2.0))))))
	else:
		tmp = (1.0 + (math.pow((M * ((D / d) * 0.5)), 2.0) * ((h / l) * -0.5))) * (d / (math.sqrt(h) * math.sqrt(l)))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 5e-307)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(h * Float64(-0.5 / Float64(l / (Float64(D * Float64(M / Float64(d * 2.0))) ^ 2.0)))))));
	else
		tmp = Float64(Float64(1.0 + Float64((Float64(M * Float64(Float64(D / d) * 0.5)) ^ 2.0) * Float64(Float64(h / l) * -0.5))) * Float64(d / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 5e-307)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + (h * (-0.5 / (l / ((D * (M / (d * 2.0))) ^ 2.0))))));
	else
		tmp = (1.0 + (((M * ((D / d) * 0.5)) ^ 2.0) * ((h / l) * -0.5))) * (d / (sqrt(h) * sqrt(l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 5e-307], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h * N[(-0.5 / N[(l / N[Power[N[(D * N[(M / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[Power[N[(M * N[(N[(D / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.00000000000000014e-307

   1. Initial program 70.5%

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

   3. Simplified 69.7%

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

      1. *-commutative 69.7%

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

      2. clear-num 69.7%

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

      3. un-div-inv 69.7%

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

      4. *-commutative 69.7%

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

      5. div-inv 69.7%

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

      6. metadata-eval 69.7%

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

   5. Applied egg-rr 69.7%

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

      1. associate-/r/ 72.4%

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

      2. *-commutative 72.4%

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

      3. associate-/l* 72.3%

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

      4. associate-*r/ 72.4%

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

      5. associate-/l* 72.3%

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

      6. associate-/l* 72.3%

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

      7. metadata-eval 72.3%

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

      8. associate-/l* 72.3%

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

      9. associate-*l/ 72.3%

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

      10. /-rgt-identity 72.3%

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

      11. associate-/r/ 72.4%

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

      12. associate-/r* 72.4%

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

      13. *-commutative 72.4%

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

      14. associate-/l/ 72.4%

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

   7. Simplified 72.4%

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

   if 5.00000000000000014e-307 < l

   1. Initial program 64.9%

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

   3. Simplified 64.1%

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

   5. Applied egg-rr 64.4%

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

      1. distribute-rgt1-in 73.6%

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

      2. +-commutative 73.6%

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

      3. associate-*l* 73.6%

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

   7. Simplified 73.6%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{-0.5}{\frac{\ell}{{\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 12: 66.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -8.8 \cdot 10^{-284}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \frac{{\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -8.8e-284)
   (*
    (sqrt (* (/ d l) (/ d h)))
    (+ 1.0 (* -0.5 (/ (pow (* M (/ (* D 0.5) d)) 2.0) (/ l h)))))
   (*
    (+ 1.0 (* (pow (* M (* (/ D d) 0.5)) 2.0) (* (/ h l) -0.5)))
    (/ d (* (sqrt h) (sqrt l))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -8.8e-284) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * (pow((M * ((D * 0.5) / d)), 2.0) / (l / h))));
	} else {
		tmp = (1.0 + (pow((M * ((D / d) * 0.5)), 2.0) * ((h / l) * -0.5))) * (d / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (d <= (-8.8d-284)) then
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 + ((-0.5d0) * (((m * ((d_1 * 0.5d0) / d)) ** 2.0d0) / (l / h))))
    else
        tmp = (1.0d0 + (((m * ((d_1 / d) * 0.5d0)) ** 2.0d0) * ((h / l) * (-0.5d0)))) * (d / (sqrt(h) * sqrt(l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -8.8e-284) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * (Math.pow((M * ((D * 0.5) / d)), 2.0) / (l / h))));
	} else {
		tmp = (1.0 + (Math.pow((M * ((D / d) * 0.5)), 2.0) * ((h / l) * -0.5))) * (d / (Math.sqrt(h) * Math.sqrt(l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -8.8e-284:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * (math.pow((M * ((D * 0.5) / d)), 2.0) / (l / h))))
	else:
		tmp = (1.0 + (math.pow((M * ((D / d) * 0.5)), 2.0) * ((h / l) * -0.5))) * (d / (math.sqrt(h) * math.sqrt(l)))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -8.8e-284)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 + Float64(-0.5 * Float64((Float64(M * Float64(Float64(D * 0.5) / d)) ^ 2.0) / Float64(l / h)))));
	else
		tmp = Float64(Float64(1.0 + Float64((Float64(M * Float64(Float64(D / d) * 0.5)) ^ 2.0) * Float64(Float64(h / l) * -0.5))) * Float64(d / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -8.8e-284)
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * (((M * ((D * 0.5) / d)) ^ 2.0) / (l / h))));
	else
		tmp = (1.0 + (((M * ((D / d) * 0.5)) ^ 2.0) * ((h / l) * -0.5))) * (d / (sqrt(h) * sqrt(l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -8.8e-284], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[Power[N[(M * N[(N[(D * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[Power[N[(M * N[(N[(D / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -8.8000000000000002e-284

   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%

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

      1. associate-*r/ 73.5%

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

      2. clear-num 73.5%

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

      3. frac-times 73.5%

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

      4. *-commutative 73.5%

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

      5. frac-times 72.7%

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

      6. add-sqr-sqrt 45.2%

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

      7. add-sqr-sqrt 72.7%

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

      8. div-inv 72.7%

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

      9. metadata-eval 72.7%

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

   5. Applied egg-rr 72.7%

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

      1. associate-/r/ 72.7%

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

      2. *-commutative 72.7%

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

      3. associate-*l* 72.7%

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

   7. Simplified 72.7%

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

   9. Applied egg-rr 18.2%

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

       1. expm1-def 31.2%

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

       2. expm1-log1p 61.5%

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

       3. *-commutative 61.5%

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

       4. *-commutative 61.5%

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

   11. Simplified 61.5%

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

   if -8.8000000000000002e-284 < d

   1. Initial program 64.0%

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

   3. Simplified 63.2%

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

   5. Applied egg-rr 62.0%

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

      1. distribute-rgt1-in 70.9%

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

      2. +-commutative 70.9%

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

      3. associate-*l* 70.9%

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

   7. Simplified 70.9%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.8 \cdot 10^{-284}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \frac{{\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 13: 54.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 1.35 \cdot 10^{-42} \lor \neg \left(d \leq 5.9 \cdot 10^{+80}\right) \land d \leq 1.35 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \frac{{\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (or (<= d 1.35e-42) (and (not (<= d 5.9e+80)) (<= d 1.35e+197)))
   (*
    (sqrt (* (/ d l) (/ d h)))
    (+ 1.0 (* -0.5 (/ (pow (* M (/ (* D 0.5) d)) 2.0) (/ l h)))))
   (* d (* (pow l -0.5) (pow h -0.5)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if ((d <= 1.35e-42) || (!(d <= 5.9e+80) && (d <= 1.35e+197))) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * (pow((M * ((D * 0.5) / d)), 2.0) / (l / h))));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((d <= 1.35d-42) .or. (.not. (d <= 5.9d+80)) .and. (d <= 1.35d+197)) then
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 + ((-0.5d0) * (((m * ((d_1 * 0.5d0) / d)) ** 2.0d0) / (l / h))))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if ((d <= 1.35e-42) || (!(d <= 5.9e+80) && (d <= 1.35e+197))) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * (Math.pow((M * ((D * 0.5) / d)), 2.0) / (l / h))));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if (d <= 1.35e-42) or (not (d <= 5.9e+80) and (d <= 1.35e+197)):
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * (math.pow((M * ((D * 0.5) / d)), 2.0) / (l / h))))
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if ((d <= 1.35e-42) || (!(d <= 5.9e+80) && (d <= 1.35e+197)))
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 + Float64(-0.5 * Float64((Float64(M * Float64(Float64(D * 0.5) / d)) ^ 2.0) / Float64(l / h)))));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if ((d <= 1.35e-42) || (~((d <= 5.9e+80)) && (d <= 1.35e+197)))
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * (((M * ((D * 0.5) / d)) ^ 2.0) / (l / h))));
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[Or[LessEqual[d, 1.35e-42], And[N[Not[LessEqual[d, 5.9e+80]], $MachinePrecision], LessEqual[d, 1.35e+197]]], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[Power[N[(M * N[(N[(D * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.35e-42 or 5.89999999999999973e80 < d < 1.35e197

   1. Initial program 67.4%

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

   3. Simplified 67.4%

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

      1. associate-*r/ 69.8%

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

      2. clear-num 69.8%

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

      3. frac-times 69.8%

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

      4. *-commutative 69.8%

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

      5. frac-times 68.9%

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

      6. add-sqr-sqrt 43.3%

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

      7. add-sqr-sqrt 68.9%

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

      8. div-inv 68.9%

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

      9. metadata-eval 68.9%

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

   5. Applied egg-rr 68.9%

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

      1. associate-/r/ 68.9%

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

      2. *-commutative 68.9%

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

      3. associate-*l* 68.9%

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

   7. Simplified 68.9%

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

   9. Applied egg-rr 15.5%

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

       1. expm1-def 26.1%

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

       2. expm1-log1p 54.5%

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

       3. *-commutative 54.5%

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

       4. *-commutative 54.5%

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

   11. Simplified 54.5%

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

   if 1.35e-42 < d < 5.89999999999999973e80 or 1.35e197 < d

   1. Initial program 68.6%

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

   3. Simplified 66.6%

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

      1. associate-*r/ 73.3%

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

      2. clear-num 73.3%

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

      3. frac-times 75.3%

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

      4. *-commutative 75.3%

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

      5. frac-times 75.4%

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

      6. add-sqr-sqrt 54.4%

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

      7. add-sqr-sqrt 75.4%

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

      8. div-inv 75.4%

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

      9. metadata-eval 75.4%

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

   5. Applied egg-rr 75.4%

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

      1. associate-/r/ 75.4%

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

      2. *-commutative 75.4%

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

      3. associate-*l* 75.4%

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

   7. Simplified 75.4%

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

   8. Taylor expanded in d around inf 67.3%

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

      1. unpow-1 67.3%

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

      2. metadata-eval 67.3%

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

      3. pow-sqr 67.3%

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

      4. rem-sqrt-square 67.3%

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

      5. rem-square-sqrt 67.1%

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

      6. fabs-sqr 67.1%

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

      7. rem-square-sqrt 67.3%

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

   10. Simplified 67.3%

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

       1. *-commutative 67.3%

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

       2. unpow-prod-down 76.9%

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

   12. Applied egg-rr 76.9%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.35 \cdot 10^{-42} \lor \neg \left(d \leq 5.9 \cdot 10^{+80}\right) \land d \leq 1.35 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \frac{{\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 14: 45.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.65 \cdot 10^{-135}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{-308}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\ell \cdot h\right)}^{-1.5}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -2.65e-135)
   (* (- d) (pow (* l h) -0.5))
   (if (<= l 6e-308)
     (* d (cbrt (pow (* l h) -1.5)))
     (* d (* (pow l -0.5) (pow h -0.5))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2.65e-135) {
		tmp = -d * pow((l * h), -0.5);
	} else if (l <= 6e-308) {
		tmp = d * cbrt(pow((l * h), -1.5));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2.65e-135) {
		tmp = -d * Math.pow((l * h), -0.5);
	} else if (l <= 6e-308) {
		tmp = d * Math.cbrt(Math.pow((l * h), -1.5));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -2.65e-135)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	elseif (l <= 6e-308)
		tmp = Float64(d * cbrt((Float64(l * h) ^ -1.5)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -2.65e-135], N[((-d) * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6e-308], N[(d * N[Power[N[Power[N[(l * h), $MachinePrecision], -1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.65e-135

   1. Initial program 72.0%

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

   3. Simplified 72.0%

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

      1. add-sqr-sqrt 72.0%

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

      2. pow2 72.0%

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

      3. sqrt-prod 72.0%

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

      4. frac-times 72.0%

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

      5. *-commutative 72.0%

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

      6. frac-times 70.8%

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

      7. unpow2 70.8%

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

      8. sqrt-prod 44.1%

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

      9. add-sqr-sqrt 74.5%

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

      10. div-inv 74.5%

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

      11. metadata-eval 74.5%

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

   5. Applied egg-rr 74.5%

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

      1. clear-num 73.2%

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

      2. sqrt-div 73.2%

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

      3. metadata-eval 73.2%

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

   7. Applied egg-rr 73.2%

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

   8. Taylor expanded in d around -inf 53.1%

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

      1. mul-1-neg 53.1%

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

      2. *-commutative 53.1%

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

      3. distribute-rgt-neg-in 53.1%

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

      4. unpow-1 53.1%

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

      5. metadata-eval 53.1%

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

      6. pow-sqr 53.2%

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

      7. rem-sqrt-square 53.2%

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

      8. rem-square-sqrt 53.0%

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

      9. fabs-sqr 53.0%

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

      10. rem-square-sqrt 53.2%

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

   10. Simplified 53.2%

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

   if -2.65e-135 < l < 6.00000000000000044e-308

   1. Initial program 69.2%

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

   3. Simplified 69.2%

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

      1. associate-*r/ 74.0%

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

      2. clear-num 74.0%

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

      3. frac-times 74.1%

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

      4. *-commutative 74.1%

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

      5. frac-times 74.1%

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

      6. add-sqr-sqrt 45.2%

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

      7. add-sqr-sqrt 74.1%

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

      8. div-inv 74.1%

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

      9. metadata-eval 74.1%

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

   5. Applied egg-rr 74.1%

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

      1. associate-/r/ 74.1%

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

      2. *-commutative 74.1%

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

      3. associate-*l* 74.1%

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

   7. Simplified 74.1%

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

   8. Taylor expanded in d around inf 24.4%

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

      1. unpow-1 24.4%

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

      2. metadata-eval 24.4%

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

      3. pow-sqr 24.4%

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

      4. rem-sqrt-square 22.3%

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

      5. rem-square-sqrt 22.3%

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

      6. fabs-sqr 22.3%

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

      7. rem-square-sqrt 22.3%

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

   10. Simplified 22.3%

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

       1. add-cbrt-cube 32.9%

          \[\leadsto d \cdot \color{blue}{\sqrt[3]{\left({\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

       2. pow3 32.9%

          \[\leadsto d \cdot \sqrt[3]{\color{blue}{{\left({\left(h \cdot \ell\right)}^{-0.5}\right)}^{3}}} \]

       3. pow-pow 32.9%

          \[\leadsto d \cdot \sqrt[3]{\color{blue}{{\left(h \cdot \ell\right)}^{\left(-0.5 \cdot 3\right)}}} \]

       4. metadata-eval 32.9%

          \[\leadsto d \cdot \sqrt[3]{{\left(h \cdot \ell\right)}^{\color{blue}{-1.5}}} \]

   12. Applied egg-rr 32.9%

       \[\leadsto d \cdot \color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{-1.5}}} \]

   if 6.00000000000000044e-308 < l

   1. Initial program 64.4%

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

   3. Simplified 63.6%

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

      1. associate-*r/ 68.2%

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

      2. clear-num 68.2%

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

      3. frac-times 69.0%

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

      4. *-commutative 69.0%

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

      5. frac-times 68.3%

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

      6. add-sqr-sqrt 46.1%

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

      7. add-sqr-sqrt 68.3%

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

      8. div-inv 68.3%

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

      9. metadata-eval 68.3%

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

   5. Applied egg-rr 68.3%

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

      1. associate-/r/ 68.3%

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

      2. *-commutative 68.3%

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

      3. associate-*l* 68.3%

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

   7. Simplified 68.3%

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

   8. Taylor expanded in d around inf 44.6%

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

      1. unpow-1 44.6%

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

      2. metadata-eval 44.6%

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

      3. pow-sqr 44.6%

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

      4. rem-sqrt-square 44.6%

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

      5. rem-square-sqrt 44.4%

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

      6. fabs-sqr 44.4%

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

      7. rem-square-sqrt 44.6%

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

   10. Simplified 44.6%

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

       1. *-commutative 44.6%

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

       2. unpow-prod-down 50.3%

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

   12. Applied egg-rr 50.3%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.65 \cdot 10^{-135}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{-308}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\ell \cdot h\right)}^{-1.5}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 15: 41.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{if}\;d \leq -1.9 \cdot 10^{-270}:\\ \;\;\;\;\left(-d\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (* l h) -0.5)))
   (if (<= d -1.9e-270) (* (- d) t_0) (* d t_0))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = pow((l * h), -0.5);
	double tmp;
	if (d <= -1.9e-270) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (l * h) ** (-0.5d0)
    if (d <= (-1.9d-270)) then
        tmp = -d * t_0
    else
        tmp = d * t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.pow((l * h), -0.5);
	double tmp;
	if (d <= -1.9e-270) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.pow((l * h), -0.5)
	tmp = 0
	if d <= -1.9e-270:
		tmp = -d * t_0
	else:
		tmp = d * t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(l * h) ^ -0.5
	tmp = 0.0
	if (d <= -1.9e-270)
		tmp = Float64(Float64(-d) * t_0);
	else
		tmp = Float64(d * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (l * h) ^ -0.5;
	tmp = 0.0;
	if (d <= -1.9e-270)
		tmp = -d * t_0;
	else
		tmp = d * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[d, -1.9e-270], N[((-d) * t$95$0), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -1.90000000000000021e-270

   1. Initial program 71.5%

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

   3. Simplified 71.5%

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

      1. add-sqr-sqrt 71.4%

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

      2. pow2 71.4%

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

      3. sqrt-prod 71.4%

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

      4. frac-times 71.4%

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

      5. *-commutative 71.4%

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

      6. frac-times 70.7%

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

      7. unpow2 70.7%

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

      8. sqrt-prod 44.6%

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

      9. add-sqr-sqrt 73.2%

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

      10. div-inv 73.2%

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

      11. metadata-eval 73.2%

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

   5. Applied egg-rr 73.2%

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

      1. clear-num 72.3%

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

      2. sqrt-div 72.3%

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

      3. metadata-eval 72.3%

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

   7. Applied egg-rr 72.3%

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

   8. Taylor expanded in d around -inf 43.9%

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

      1. mul-1-neg 43.9%

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

      2. *-commutative 43.9%

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

      3. distribute-rgt-neg-in 43.9%

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

      4. unpow-1 43.9%

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

      5. metadata-eval 43.9%

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

      6. pow-sqr 43.9%

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

      7. rem-sqrt-square 43.9%

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

      8. rem-square-sqrt 43.8%

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

      9. fabs-sqr 43.8%

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

      10. rem-square-sqrt 43.9%

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

   10. Simplified 43.9%

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

   if -1.90000000000000021e-270 < d

   1. Initial program 64.2%

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

   3. Simplified 63.5%

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

      1. associate-*r/ 67.9%

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

      2. clear-num 67.9%

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

      3. frac-times 68.7%

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

      4. *-commutative 68.7%

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

      5. frac-times 68.0%

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

      6. add-sqr-sqrt 45.9%

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

      7. add-sqr-sqrt 68.0%

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

      8. div-inv 68.0%

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

      9. metadata-eval 68.0%

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

   5. Applied egg-rr 68.0%

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

      1. associate-/r/ 68.0%

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

      2. *-commutative 68.0%

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

      3. associate-*l* 68.0%

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

   7. Simplified 68.0%

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

   8. Taylor expanded in d around inf 43.7%

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

      1. unpow-1 43.7%

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

      2. metadata-eval 43.7%

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

      3. pow-sqr 43.8%

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

      4. rem-sqrt-square 43.8%

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

      5. rem-square-sqrt 43.6%

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

      6. fabs-sqr 43.6%

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

      7. rem-square-sqrt 43.8%

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

   10. Simplified 43.8%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.9 \cdot 10^{-270}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]

Alternative 16: 26.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ d \cdot \sqrt{\frac{1}{\ell \cdot h}} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (* d (sqrt (/ 1.0 (* l h)))))

C

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

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 * sqrt((1.0d0 / (l * h)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[d_, h_, l_, M_, D_] := N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.6%

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

3. Simplified 67.2%

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

4. Taylor expanded in d around inf 29.9%

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

5. Final simplification 29.9%

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

Alternative 17: 26.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (* d (sqrt (/ (/ 1.0 l) h))))

C

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

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 * sqrt(((1.0d0 / l) / h))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[d_, h_, l_, M_, D_] := N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.6%

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

3. Simplified 67.2%

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

4. Taylor expanded in d around inf 29.9%

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

   1. *-commutative 29.9%

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

   2. associate-/r* 29.9%

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

6. Simplified 29.9%

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

7. Final simplification 29.9%

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

Alternative 18: 25.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ d \cdot {\left(\ell \cdot h\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (* d (pow (* l h) -0.5)))

C

double code(double d, double h, double l, double M, double D) {
	return d * pow((l * h), -0.5);
}

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 * ((l * h) ** (-0.5d0))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d * Math.pow((l * h), -0.5);
}

Python

def code(d, h, l, M, D):
	return d * math.pow((l * h), -0.5)

Julia

function code(d, h, l, M, D)
	return Float64(d * (Float64(l * h) ^ -0.5))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d * ((l * h) ^ -0.5);
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.6%

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

3. Simplified 67.2%

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

   1. associate-*r/ 70.4%

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

   2. clear-num 70.4%

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

   3. frac-times 70.8%

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

   4. *-commutative 70.8%

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

   5. frac-times 70.1%

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

   6. add-sqr-sqrt 45.3%

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

   7. add-sqr-sqrt 70.1%

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

   8. div-inv 70.1%

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

   9. metadata-eval 70.1%

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

5. Applied egg-rr 70.1%

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

   1. associate-/r/ 70.1%

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

   2. *-commutative 70.1%

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

   3. associate-*l* 70.1%

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

7. Simplified 70.1%

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

8. Taylor expanded in d around inf 29.9%

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

   1. unpow-1 29.9%

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

   2. metadata-eval 29.9%

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

   3. pow-sqr 29.9%

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

   4. rem-sqrt-square 29.5%

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

   5. rem-square-sqrt 29.4%

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

   6. fabs-sqr 29.4%

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

   7. rem-square-sqrt 29.5%

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

10. Simplified 29.5%

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

11. Final simplification 29.5%

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

Reproduce

herbie shell --seed 2023322 
(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)))))