Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.3% → 84.9%

Time: 21.1s

Alternatives: 14

Speedup: 3.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.3% 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: 84.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5 \cdot \left(M \cdot D\right)}{d}\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{t\_1}{\sqrt{-h}} \cdot \left(t\_1 \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot \left(1 - \frac{h \cdot t\_0}{\ell} \cdot \left(0.5 \cdot t\_0\right)\right)\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{-101}:\\ \;\;\;\;\frac{\left(1 - \frac{\left(\left(h \cdot 0.5\right) \cdot \frac{M \cdot D}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot 0.25\right)}{\ell \cdot d}\right) \cdot \frac{\left|d\right|}{\sqrt{h}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (* 0.5 (* M D)) d)) (t_1 (sqrt (- d))))
   (if (<= l -1e-310)
     (*
      (* (/ t_1 (sqrt (- h))) (* t_1 (sqrt (/ -1.0 l))))
      (- 1.0 (* (/ (* h t_0) l) (* 0.5 t_0))))
     (if (<= l 1.4e-101)
       (/
        (*
         (- 1.0 (/ (* (* (* h 0.5) (/ (* M D) d)) (* (* M D) 0.25)) (* l d)))
         (/ (fabs d) (sqrt h)))
        (sqrt l))
       (*
        (* (/ (sqrt d) (sqrt h)) (/ (sqrt d) (sqrt l)))
        (+
         1.0
         (* (/ h l) (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ -1.0 2.0)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (0.5 * (M * D)) / d;
	double t_1 = sqrt(-d);
	double tmp;
	if (l <= -1e-310) {
		tmp = ((t_1 / sqrt(-h)) * (t_1 * sqrt((-1.0 / l)))) * (1.0 - (((h * t_0) / l) * (0.5 * t_0)));
	} else if (l <= 1.4e-101) {
		tmp = ((1.0 - ((((h * 0.5) * ((M * D) / d)) * ((M * D) * 0.25)) / (l * d))) * (fabs(d) / sqrt(h))) / sqrt(l);
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * (sqrt(d) / sqrt(l))) * (1.0 + ((h / l) * (pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 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 = (0.5d0 * (m * d_1)) / d
    t_1 = sqrt(-d)
    if (l <= (-1d-310)) then
        tmp = ((t_1 / sqrt(-h)) * (t_1 * sqrt(((-1.0d0) / l)))) * (1.0d0 - (((h * t_0) / l) * (0.5d0 * t_0)))
    else if (l <= 1.4d-101) then
        tmp = ((1.0d0 - ((((h * 0.5d0) * ((m * d_1) / d)) * ((m * d_1) * 0.25d0)) / (l * d))) * (abs(d) / sqrt(h))) / sqrt(l)
    else
        tmp = ((sqrt(d) / sqrt(h)) * (sqrt(d) / sqrt(l))) * (1.0d0 + ((h / l) * ((((m * d_1) / (d * 2.0d0)) ** 2.0d0) * ((-1.0d0) / 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 = (0.5 * (M * D)) / d;
	double t_1 = Math.sqrt(-d);
	double tmp;
	if (l <= -1e-310) {
		tmp = ((t_1 / Math.sqrt(-h)) * (t_1 * Math.sqrt((-1.0 / l)))) * (1.0 - (((h * t_0) / l) * (0.5 * t_0)));
	} else if (l <= 1.4e-101) {
		tmp = ((1.0 - ((((h * 0.5) * ((M * D) / d)) * ((M * D) * 0.25)) / (l * d))) * (Math.abs(d) / Math.sqrt(h))) / Math.sqrt(l);
	} else {
		tmp = ((Math.sqrt(d) / Math.sqrt(h)) * (Math.sqrt(d) / Math.sqrt(l))) * (1.0 + ((h / l) * (Math.pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (0.5 * (M * D)) / d
	t_1 = math.sqrt(-d)
	tmp = 0
	if l <= -1e-310:
		tmp = ((t_1 / math.sqrt(-h)) * (t_1 * math.sqrt((-1.0 / l)))) * (1.0 - (((h * t_0) / l) * (0.5 * t_0)))
	elif l <= 1.4e-101:
		tmp = ((1.0 - ((((h * 0.5) * ((M * D) / d)) * ((M * D) * 0.25)) / (l * d))) * (math.fabs(d) / math.sqrt(h))) / math.sqrt(l)
	else:
		tmp = ((math.sqrt(d) / math.sqrt(h)) * (math.sqrt(d) / math.sqrt(l))) * (1.0 + ((h / l) * (math.pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0))))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(0.5 * Float64(M * D)) / d)
	t_1 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -1e-310)
		tmp = Float64(Float64(Float64(t_1 / sqrt(Float64(-h))) * Float64(t_1 * sqrt(Float64(-1.0 / l)))) * Float64(1.0 - Float64(Float64(Float64(h * t_0) / l) * Float64(0.5 * t_0))));
	elseif (l <= 1.4e-101)
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(Float64(Float64(h * 0.5) * Float64(Float64(M * D) / d)) * Float64(Float64(M * D) * 0.25)) / Float64(l * d))) * Float64(abs(d) / sqrt(h))) / sqrt(l));
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * Float64(sqrt(d) / sqrt(l))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (0.5 * (M * D)) / d;
	t_1 = sqrt(-d);
	tmp = 0.0;
	if (l <= -1e-310)
		tmp = ((t_1 / sqrt(-h)) * (t_1 * sqrt((-1.0 / l)))) * (1.0 - (((h * t_0) / l) * (0.5 * t_0)));
	elseif (l <= 1.4e-101)
		tmp = ((1.0 - ((((h * 0.5) * ((M * D) / d)) * ((M * D) * 0.25)) / (l * d))) * (abs(d) / sqrt(h))) / sqrt(l);
	else
		tmp = ((sqrt(d) / sqrt(h)) * (sqrt(d) / sqrt(l))) * (1.0 + ((h / l) * ((((M * D) / (d * 2.0)) ^ 2.0) * (-1.0 / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(0.5 * N[(M * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -1e-310], N[(N[(N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(h * t$95$0), $MachinePrecision] / l), $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.4e-101], N[(N[(N[(1.0 - N[(N[(N[(N[(h * 0.5), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.999999999999969e-311

   1. Initial program 64.2%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 68.1%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. neg-lowering-neg.f64 80.6

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

   6. Applied egg-rr 80.6%

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

      1. metadata-eval N/A

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

      2. pow1/2 N/A

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

      3. frac-2neg N/A

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

      4. div-inv N/A

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

      5. sqrt-prod N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. neg-lowering-neg.f64 N/A

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

      9. sqrt-lowering-sqrt.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. neg-lowering-neg.f64 86.8

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

   8. Applied egg-rr 86.8%

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

   if -9.999999999999969e-311 < l < 1.39999999999999995e-101

   1. Initial program 64.1%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-div N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 66.0

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

   4. Applied egg-rr 66.0%

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

   6. Applied egg-rr 72.4%

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

      1. associate-*l/ N/A

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

      2. sqrt-div N/A

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

      3. /-lowering-/.f64 N/A

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

      4. rem-sqrt-square N/A

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

      5. fabs-lowering-fabs.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 94.5

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

   8. Applied egg-rr 94.5%

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

   if 1.39999999999999995e-101 < l

   1. Initial program 72.3%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-div N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 77.9

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

   4. Applied egg-rr 77.9%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-div N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 86.2

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

   6. Applied egg-rr 86.2%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot \left(1 - \frac{h \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \left(0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{-101}:\\ \;\;\;\;\frac{\left(1 - \frac{\left(\left(h \cdot 0.5\right) \cdot \frac{M \cdot D}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot 0.25\right)}{\ell \cdot d}\right) \cdot \frac{\left|d\right|}{\sqrt{h}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 55.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{h} \cdot \left(\left(M \cdot \left(M \cdot D\right)\right) \cdot \left(D \cdot -0.125\right)\right)}{d \cdot \left(\ell \cdot \sqrt{\ell}\right)}\\ t_1 := \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-125}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (/ (* (sqrt h) (* (* M (* M D)) (* D -0.125))) (* d (* l (sqrt l)))))
        (t_1
         (*
          (+ 1.0 (* (/ h l) (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ -1.0 2.0))))
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))))))
   (if (<= t_1 -1e-125)
     t_0
     (if (<= t_1 0.0)
       (* (- d) (sqrt (/ 1.0 (* l h))))
       (if (<= t_1 INFINITY) (* (sqrt (/ d l)) (/ 1.0 (sqrt (/ h d)))) t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (sqrt(h) * ((M * (M * D)) * (D * -0.125))) / (d * (l * sqrt(l)));
	double t_1 = (1.0 + ((h / l) * (pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0)));
	double tmp;
	if (t_1 <= -1e-125) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = -d * sqrt((1.0 / (l * h)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt((d / l)) * (1.0 / sqrt((h / d)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.sqrt(h) * ((M * (M * D)) * (D * -0.125))) / (d * (l * Math.sqrt(l)));
	double t_1 = (1.0 + ((h / l) * (Math.pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0)));
	double tmp;
	if (t_1 <= -1e-125) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = -d * Math.sqrt((1.0 / (l * h)));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((d / l)) * (1.0 / Math.sqrt((h / d)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.sqrt(h) * ((M * (M * D)) * (D * -0.125))) / (d * (l * math.sqrt(l)))
	t_1 = (1.0 + ((h / l) * (math.pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0)))
	tmp = 0
	if t_1 <= -1e-125:
		tmp = t_0
	elif t_1 <= 0.0:
		tmp = -d * math.sqrt((1.0 / (l * h)))
	elif t_1 <= math.inf:
		tmp = math.sqrt((d / l)) * (1.0 / math.sqrt((h / d)))
	else:
		tmp = t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(sqrt(h) * Float64(Float64(M * Float64(M * D)) * Float64(D * -0.125))) / Float64(d * Float64(l * sqrt(l))))
	t_1 = Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))) * Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))))
	tmp = 0.0
	if (t_1 <= -1e-125)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(l * h))));
	elseif (t_1 <= Inf)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(1.0 / sqrt(Float64(h / d))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (sqrt(h) * ((M * (M * D)) * (D * -0.125))) / (d * (l * sqrt(l)));
	t_1 = (1.0 + ((h / l) * ((((M * D) / (d * 2.0)) ^ 2.0) * (-1.0 / 2.0)))) * (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0)));
	tmp = 0.0;
	if (t_1 <= -1e-125)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = -d * sqrt((1.0 / (l * h)));
	elseif (t_1 <= Inf)
		tmp = sqrt((d / l)) * (1.0 / sqrt((h / d)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Sqrt[h], $MachinePrecision] * N[(N[(M * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[(D * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(l * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-125], t$95$0, If[LessEqual[t$95$1, 0.0], N[((-d) * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.00000000000000001e-125 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 58.5%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-div N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 33.9

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

   4. Applied egg-rr 33.9%

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

   6. Applied egg-rr 33.7%

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

   7. Taylor expanded in h around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \]

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. *-lowering-*.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. *-lowering-*.f64 N/A

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

      18. *-lowering-*.f64 N/A

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

      19. unpow2 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \left(D \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \cdot \frac{\frac{-1}{8}}{d}\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \left(D \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \cdot \frac{\frac{-1}{8}}{d}\right) \]

      21. /-lowering-/.f64 31.6

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

   9. Simplified 31.6%

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

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{-1}{8}}{d}\right) \cdot \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}} \]

       2. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-1}{8}}{d}} \cdot \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \]

       3. sqrt-div N/A

          \[\leadsto \frac{\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-1}{8}}{d} \cdot \color{blue}{\frac{\sqrt{h}}{\sqrt{\ell \cdot \left(\ell \cdot \ell\right)}}} \]

       4. frac-times N/A

          \[\leadsto \color{blue}{\frac{\left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-1}{8}\right) \cdot \sqrt{h}}{d \cdot \sqrt{\ell \cdot \left(\ell \cdot \ell\right)}}} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-1}{8}\right) \cdot \sqrt{h}}{d \cdot \sqrt{\ell \cdot \left(\ell \cdot \ell\right)}}} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-1}{8}\right) \cdot \sqrt{h}}}{d \cdot \sqrt{\ell \cdot \left(\ell \cdot \ell\right)}} \]

       7. *-commutative N/A

          \[\leadsto \frac{\left(\color{blue}{\left(\left(D \cdot \left(M \cdot M\right)\right) \cdot D\right)} \cdot \frac{-1}{8}\right) \cdot \sqrt{h}}{d \cdot \sqrt{\ell \cdot \left(\ell \cdot \ell\right)}} \]

       8. *-commutative N/A

          \[\leadsto \frac{\left(\left(\color{blue}{\left(\left(M \cdot M\right) \cdot D\right)} \cdot D\right) \cdot \frac{-1}{8}\right) \cdot \sqrt{h}}{d \cdot \sqrt{\ell \cdot \left(\ell \cdot \ell\right)}} \]

       9. associate-*l* N/A

          \[\leadsto \frac{\color{blue}{\left(\left(\left(M \cdot M\right) \cdot D\right) \cdot \left(D \cdot \frac{-1}{8}\right)\right)} \cdot \sqrt{h}}{d \cdot \sqrt{\ell \cdot \left(\ell \cdot \ell\right)}} \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \frac{\color{blue}{\left(\left(\left(M \cdot M\right) \cdot D\right) \cdot \left(D \cdot \frac{-1}{8}\right)\right)} \cdot \sqrt{h}}{d \cdot \sqrt{\ell \cdot \left(\ell \cdot \ell\right)}} \]

       11. associate-*l* N/A

           \[\leadsto \frac{\left(\color{blue}{\left(M \cdot \left(M \cdot D\right)\right)} \cdot \left(D \cdot \frac{-1}{8}\right)\right) \cdot \sqrt{h}}{d \cdot \sqrt{\ell \cdot \left(\ell \cdot \ell\right)}} \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \frac{\left(\color{blue}{\left(M \cdot \left(M \cdot D\right)\right)} \cdot \left(D \cdot \frac{-1}{8}\right)\right) \cdot \sqrt{h}}{d \cdot \sqrt{\ell \cdot \left(\ell \cdot \ell\right)}} \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \frac{\left(\left(M \cdot \color{blue}{\left(M \cdot D\right)}\right) \cdot \left(D \cdot \frac{-1}{8}\right)\right) \cdot \sqrt{h}}{d \cdot \sqrt{\ell \cdot \left(\ell \cdot \ell\right)}} \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \frac{\left(\left(M \cdot \left(M \cdot D\right)\right) \cdot \color{blue}{\left(D \cdot \frac{-1}{8}\right)}\right) \cdot \sqrt{h}}{d \cdot \sqrt{\ell \cdot \left(\ell \cdot \ell\right)}} \]

       15. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto \frac{\left(\left(M \cdot \left(M \cdot D\right)\right) \cdot \left(D \cdot \frac{-1}{8}\right)\right) \cdot \color{blue}{\sqrt{h}}}{d \cdot \sqrt{\ell \cdot \left(\ell \cdot \ell\right)}} \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \frac{\left(\left(M \cdot \left(M \cdot D\right)\right) \cdot \left(D \cdot \frac{-1}{8}\right)\right) \cdot \sqrt{h}}{\color{blue}{d \cdot \sqrt{\ell \cdot \left(\ell \cdot \ell\right)}}} \]

       17. associate-*r* N/A

           \[\leadsto \frac{\left(\left(M \cdot \left(M \cdot D\right)\right) \cdot \left(D \cdot \frac{-1}{8}\right)\right) \cdot \sqrt{h}}{d \cdot \sqrt{\color{blue}{\left(\ell \cdot \ell\right) \cdot \ell}}} \]

       18. sqrt-prod N/A

           \[\leadsto \frac{\left(\left(M \cdot \left(M \cdot D\right)\right) \cdot \left(D \cdot \frac{-1}{8}\right)\right) \cdot \sqrt{h}}{d \cdot \color{blue}{\left(\sqrt{\ell \cdot \ell} \cdot \sqrt{\ell}\right)}} \]

   11. Applied egg-rr 36.8%

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

   if -1.00000000000000001e-125 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 47.4%

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

      1. unpow2 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. times-frac N/A

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

      9. associate-/r* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. /-lowering-/.f64 N/A

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

      17. *-lowering-*.f64 40.5

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

   4. Applied egg-rr 40.5%

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

   5. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-lowering-neg.f64 66.9

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

   7. Simplified 66.9%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 82.7%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 82.7%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. /-lowering-/.f64 82.7

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

   6. Applied egg-rr 82.7%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. /-lowering-/.f64 83.1

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

   8. Applied egg-rr 83.1%

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

   9. Taylor expanded in M around 0

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

   11. Simplified 83.1%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{-125}:\\ \;\;\;\;\frac{\sqrt{h} \cdot \left(\left(M \cdot \left(M \cdot D\right)\right) \cdot \left(D \cdot -0.125\right)\right)}{d \cdot \left(\ell \cdot \sqrt{\ell}\right)}\\ \mathbf{elif}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{elif}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq \infty:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{h} \cdot \left(\left(M \cdot \left(M \cdot D\right)\right) \cdot \left(D \cdot -0.125\right)\right)}{d \cdot \left(\ell \cdot \sqrt{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{d \cdot -8}}{\ell \cdot \sqrt{\frac{\ell}{h}}}\\ t_1 := \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (/ (* M (* D (* M D))) (* d -8.0)) (* l (sqrt (/ l h)))))
        (t_1
         (*
          (+ 1.0 (* (/ h l) (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ -1.0 2.0))))
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))))))
   (if (<= t_1 0.0)
     t_0
     (if (<= t_1 INFINITY) (* (sqrt (/ d l)) (/ 1.0 (sqrt (/ h d)))) t_0))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = ((M * (D * (M * D))) / (d * -8.0)) / (l * sqrt((l / h)));
	double t_1 = (1.0 + ((h / l) * (pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt((d / l)) * (1.0 / sqrt((h / d)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = ((M * (D * (M * D))) / (d * -8.0)) / (l * Math.sqrt((l / h)));
	double t_1 = (1.0 + ((h / l) * (Math.pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((d / l)) * (1.0 / Math.sqrt((h / d)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = ((M * (D * (M * D))) / (d * -8.0)) / (l * math.sqrt((l / h)))
	t_1 = (1.0 + ((h / l) * (math.pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0)))
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = math.sqrt((d / l)) * (1.0 / math.sqrt((h / d)))
	else:
		tmp = t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(M * Float64(D * Float64(M * D))) / Float64(d * -8.0)) / Float64(l * sqrt(Float64(l / h))))
	t_1 = Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))) * Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(1.0 / sqrt(Float64(h / d))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = ((M * (D * (M * D))) / (d * -8.0)) / (l * sqrt((l / h)));
	t_1 = (1.0 + ((h / l) * ((((M * D) / (d * 2.0)) ^ 2.0) * (-1.0 / 2.0)))) * (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0)));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = sqrt((d / l)) * (1.0 / sqrt((h / d)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(M * N[(D * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * -8.0), $MachinePrecision]), $MachinePrecision] / N[(l * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 57.1%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-div N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 33.4

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

   4. Applied egg-rr 33.4%

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

   6. Applied egg-rr 33.2%

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

   7. Taylor expanded in h around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \]

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. *-lowering-*.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. *-lowering-*.f64 N/A

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

      18. *-lowering-*.f64 N/A

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

      19. unpow2 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \left(D \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \cdot \frac{\frac{-1}{8}}{d}\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \left(D \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \cdot \frac{\frac{-1}{8}}{d}\right) \]

      21. /-lowering-/.f64 33.0

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

   9. Simplified 33.0%

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

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{-1}{8}}{d}\right) \cdot \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}} \]

       2. clear-num N/A

          \[\leadsto \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{-1}{8}}{d}\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}}} \]

       3. sqrt-div N/A

          \[\leadsto \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{-1}{8}}{d}\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}}} \]

       4. metadata-eval N/A

          \[\leadsto \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{-1}{8}}{d}\right) \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}} \]

       5. un-div-inv N/A

          \[\leadsto \color{blue}{\frac{\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{-1}{8}}{d}}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}}} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{-1}{8}}{d}}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}}} \]

   11. Applied egg-rr 66.5%

       \[\leadsto \color{blue}{\frac{\frac{M \cdot \left(\left(M \cdot D\right) \cdot D\right)}{d \cdot -8}}{\ell \cdot \sqrt{\frac{\ell}{h}}}} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 82.7%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 82.7%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. /-lowering-/.f64 82.7

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

   6. Applied egg-rr 82.7%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. /-lowering-/.f64 83.1

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

   8. Applied egg-rr 83.1%

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

   9. Taylor expanded in M around 0

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

   11. Simplified 83.1%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\frac{\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{d \cdot -8}}{\ell \cdot \sqrt{\frac{\ell}{h}}}\\ \mathbf{elif}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq \infty:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{d \cdot -8}}{\ell \cdot \sqrt{\frac{\ell}{h}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{\frac{h}{\ell}}}{\ell} \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-0.125}{d}\right)\\ t_1 := \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ (sqrt (/ h l)) l) (* (* D (* D (* M M))) (/ -0.125 d))))
        (t_1
         (*
          (+ 1.0 (* (/ h l) (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ -1.0 2.0))))
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))))))
   (if (<= t_1 0.0)
     t_0
     (if (<= t_1 INFINITY) (* (sqrt (/ d l)) (/ 1.0 (sqrt (/ h d)))) t_0))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (sqrt((h / l)) / l) * ((D * (D * (M * M))) * (-0.125 / d));
	double t_1 = (1.0 + ((h / l) * (pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt((d / l)) * (1.0 / sqrt((h / d)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.sqrt((h / l)) / l) * ((D * (D * (M * M))) * (-0.125 / d));
	double t_1 = (1.0 + ((h / l) * (Math.pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((d / l)) * (1.0 / Math.sqrt((h / d)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.sqrt((h / l)) / l) * ((D * (D * (M * M))) * (-0.125 / d))
	t_1 = (1.0 + ((h / l) * (math.pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0)))
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = math.sqrt((d / l)) * (1.0 / math.sqrt((h / d)))
	else:
		tmp = t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(sqrt(Float64(h / l)) / l) * Float64(Float64(D * Float64(D * Float64(M * M))) * Float64(-0.125 / d)))
	t_1 = Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))) * Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(1.0 / sqrt(Float64(h / d))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (sqrt((h / l)) / l) * ((D * (D * (M * M))) * (-0.125 / d));
	t_1 = (1.0 + ((h / l) * ((((M * D) / (d * 2.0)) ^ 2.0) * (-1.0 / 2.0)))) * (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0)));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = sqrt((d / l)) * (1.0 / sqrt((h / d)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] / l), $MachinePrecision] * N[(N[(D * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.125 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 57.1%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-div N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 33.4

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

   4. Applied egg-rr 33.4%

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

   6. Applied egg-rr 33.2%

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

   7. Taylor expanded in h around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right)} \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \]

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. *-lowering-*.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. *-lowering-*.f64 N/A

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

      18. *-lowering-*.f64 N/A

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

      19. unpow2 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \left(D \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \cdot \frac{\frac{-1}{8}}{d}\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \left(D \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \cdot \frac{\frac{-1}{8}}{d}\right) \]

      21. /-lowering-/.f64 33.0

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

   9. Simplified 33.0%

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

       1. associate-/r* N/A

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{h}{\ell}}{\ell \cdot \ell}}} \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{-1}{8}}{d}\right) \]

       2. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\frac{h}{\ell}}}{\sqrt{\ell \cdot \ell}}} \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{-1}{8}}{d}\right) \]

       3. pow2 N/A

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

       4. sqrt-pow1 N/A

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

       5. metadata-eval N/A

          \[\leadsto \frac{\sqrt{\frac{h}{\ell}}}{{\ell}^{\color{blue}{1}}} \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{-1}{8}}{d}\right) \]

       6. unpow1 N/A

          \[\leadsto \frac{\sqrt{\frac{h}{\ell}}}{\color{blue}{\ell}} \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{-1}{8}}{d}\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\frac{h}{\ell}}}{\ell}} \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{-1}{8}}{d}\right) \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\frac{h}{\ell}}}}{\ell} \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\frac{-1}{8}}{d}\right) \]

       9. /-lowering-/.f64 60.0

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

   11. Applied egg-rr 60.0%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 82.7%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 82.7%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. /-lowering-/.f64 82.7

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

   6. Applied egg-rr 82.7%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. /-lowering-/.f64 83.1

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

   8. Applied egg-rr 83.1%

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

   9. Taylor expanded in M around 0

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

   11. Simplified 83.1%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}}}{\ell} \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-0.125}{d}\right)\\ \mathbf{elif}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \leq \infty:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}}}{\ell} \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{-0.125}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5 \cdot \left(M \cdot D\right)}{d}\\ t_1 := 1 - \frac{h \cdot t\_0}{\ell} \cdot \left(0.5 \cdot t\_0\right)\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{t\_2}{\sqrt{-h}} \cdot \left(t\_2 \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot t\_1\\ \mathbf{elif}\;h \leq 7.2 \cdot 10^{-88}:\\ \;\;\;\;t\_1 \cdot \left(\left(\sqrt{d} \cdot \frac{1}{\sqrt{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{\left(\left(h \cdot 0.5\right) \cdot \frac{M \cdot D}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot 0.25\right)}{\ell \cdot d}\right) \cdot \frac{\left|d\right|}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (* 0.5 (* M D)) d))
        (t_1 (- 1.0 (* (/ (* h t_0) l) (* 0.5 t_0))))
        (t_2 (sqrt (- d))))
   (if (<= h -2e-310)
     (* (* (/ t_2 (sqrt (- h))) (* t_2 (sqrt (/ -1.0 l)))) t_1)
     (if (<= h 7.2e-88)
       (* t_1 (* (* (sqrt d) (/ 1.0 (sqrt h))) (sqrt (/ d l))))
       (/
        (*
         (- 1.0 (/ (* (* (* h 0.5) (/ (* M D) d)) (* (* M D) 0.25)) (* l d)))
         (/ (fabs d) (sqrt h)))
        (sqrt l))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (0.5 * (M * D)) / d;
	double t_1 = 1.0 - (((h * t_0) / l) * (0.5 * t_0));
	double t_2 = sqrt(-d);
	double tmp;
	if (h <= -2e-310) {
		tmp = ((t_2 / sqrt(-h)) * (t_2 * sqrt((-1.0 / l)))) * t_1;
	} else if (h <= 7.2e-88) {
		tmp = t_1 * ((sqrt(d) * (1.0 / sqrt(h))) * sqrt((d / l)));
	} else {
		tmp = ((1.0 - ((((h * 0.5) * ((M * D) / d)) * ((M * D) * 0.25)) / (l * d))) * (fabs(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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (0.5d0 * (m * d_1)) / d
    t_1 = 1.0d0 - (((h * t_0) / l) * (0.5d0 * t_0))
    t_2 = sqrt(-d)
    if (h <= (-2d-310)) then
        tmp = ((t_2 / sqrt(-h)) * (t_2 * sqrt(((-1.0d0) / l)))) * t_1
    else if (h <= 7.2d-88) then
        tmp = t_1 * ((sqrt(d) * (1.0d0 / sqrt(h))) * sqrt((d / l)))
    else
        tmp = ((1.0d0 - ((((h * 0.5d0) * ((m * d_1) / d)) * ((m * d_1) * 0.25d0)) / (l * d))) * (abs(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 t_0 = (0.5 * (M * D)) / d;
	double t_1 = 1.0 - (((h * t_0) / l) * (0.5 * t_0));
	double t_2 = Math.sqrt(-d);
	double tmp;
	if (h <= -2e-310) {
		tmp = ((t_2 / Math.sqrt(-h)) * (t_2 * Math.sqrt((-1.0 / l)))) * t_1;
	} else if (h <= 7.2e-88) {
		tmp = t_1 * ((Math.sqrt(d) * (1.0 / Math.sqrt(h))) * Math.sqrt((d / l)));
	} else {
		tmp = ((1.0 - ((((h * 0.5) * ((M * D) / d)) * ((M * D) * 0.25)) / (l * d))) * (Math.abs(d) / Math.sqrt(h))) / Math.sqrt(l);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (0.5 * (M * D)) / d
	t_1 = 1.0 - (((h * t_0) / l) * (0.5 * t_0))
	t_2 = math.sqrt(-d)
	tmp = 0
	if h <= -2e-310:
		tmp = ((t_2 / math.sqrt(-h)) * (t_2 * math.sqrt((-1.0 / l)))) * t_1
	elif h <= 7.2e-88:
		tmp = t_1 * ((math.sqrt(d) * (1.0 / math.sqrt(h))) * math.sqrt((d / l)))
	else:
		tmp = ((1.0 - ((((h * 0.5) * ((M * D) / d)) * ((M * D) * 0.25)) / (l * d))) * (math.fabs(d) / math.sqrt(h))) / math.sqrt(l)
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(0.5 * Float64(M * D)) / d)
	t_1 = Float64(1.0 - Float64(Float64(Float64(h * t_0) / l) * Float64(0.5 * t_0)))
	t_2 = sqrt(Float64(-d))
	tmp = 0.0
	if (h <= -2e-310)
		tmp = Float64(Float64(Float64(t_2 / sqrt(Float64(-h))) * Float64(t_2 * sqrt(Float64(-1.0 / l)))) * t_1);
	elseif (h <= 7.2e-88)
		tmp = Float64(t_1 * Float64(Float64(sqrt(d) * Float64(1.0 / sqrt(h))) * sqrt(Float64(d / l))));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(Float64(Float64(h * 0.5) * Float64(Float64(M * D) / d)) * Float64(Float64(M * D) * 0.25)) / Float64(l * d))) * Float64(abs(d) / sqrt(h))) / sqrt(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (0.5 * (M * D)) / d;
	t_1 = 1.0 - (((h * t_0) / l) * (0.5 * t_0));
	t_2 = sqrt(-d);
	tmp = 0.0;
	if (h <= -2e-310)
		tmp = ((t_2 / sqrt(-h)) * (t_2 * sqrt((-1.0 / l)))) * t_1;
	elseif (h <= 7.2e-88)
		tmp = t_1 * ((sqrt(d) * (1.0 / sqrt(h))) * sqrt((d / l)));
	else
		tmp = ((1.0 - ((((h * 0.5) * ((M * D) / d)) * ((M * D) * 0.25)) / (l * d))) * (abs(d) / sqrt(h))) / sqrt(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(0.5 * N[(M * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(N[(h * t$95$0), $MachinePrecision] / l), $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[h, -2e-310], N[(N[(N[(t$95$2 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[h, 7.2e-88], N[(t$95$1 * N[(N[(N[Sqrt[d], $MachinePrecision] * N[(1.0 / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(N[(N[(N[(h * 0.5), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.999999999999994e-310

   1. Initial program 64.2%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 68.1%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. neg-lowering-neg.f64 80.6

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

   6. Applied egg-rr 80.6%

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

      1. metadata-eval N/A

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

      2. pow1/2 N/A

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

      3. frac-2neg N/A

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

      4. div-inv N/A

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

      5. sqrt-prod N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. neg-lowering-neg.f64 N/A

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

      9. sqrt-lowering-sqrt.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. neg-lowering-neg.f64 86.8

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

   8. Applied egg-rr 86.8%

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

   if -1.999999999999994e-310 < h < 7.1999999999999999e-88

   1. Initial program 71.5%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 79.2%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. /-lowering-/.f64 79.2

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

   6. Applied egg-rr 79.2%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. unpow-prod-down N/A

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

      4. *-lowering-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow1/2 N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. sqrt-lowering-sqrt.f64 92.8

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

   8. Applied egg-rr 92.8%

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

   if 7.1999999999999999e-88 < h

   1. Initial program 68.5%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-div N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 74.3

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

   4. Applied egg-rr 74.3%

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

   6. Applied egg-rr 62.6%

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

      1. associate-*l/ N/A

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

      2. sqrt-div N/A

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

      3. /-lowering-/.f64 N/A

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

      4. rem-sqrt-square N/A

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

      5. fabs-lowering-fabs.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 84.3

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

   8. Applied egg-rr 84.3%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot \left(1 - \frac{h \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \left(0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)\right)\\ \mathbf{elif}\;h \leq 7.2 \cdot 10^{-88}:\\ \;\;\;\;\left(1 - \frac{h \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \left(0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)\right) \cdot \left(\left(\sqrt{d} \cdot \frac{1}{\sqrt{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{\left(\left(h \cdot 0.5\right) \cdot \frac{M \cdot D}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot 0.25\right)}{\ell \cdot d}\right) \cdot \frac{\left|d\right|}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \frac{0.5 \cdot \left(M \cdot D\right)}{d}\\ t_2 := 1 - \frac{h \cdot t\_1}{\ell} \cdot \left(0.5 \cdot t\_1\right)\\ \mathbf{if}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t\_2 \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t\_0\right)\\ \mathbf{elif}\;h \leq 8.8 \cdot 10^{-88}:\\ \;\;\;\;t\_2 \cdot \left(\left(\sqrt{d} \cdot \frac{1}{\sqrt{h}}\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{\left(\left(h \cdot 0.5\right) \cdot \frac{M \cdot D}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot 0.25\right)}{\ell \cdot d}\right) \cdot \frac{\left|d\right|}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1 (/ (* 0.5 (* M D)) d))
        (t_2 (- 1.0 (* (/ (* h t_1) l) (* 0.5 t_1)))))
   (if (<= h -2e-310)
     (* t_2 (* (/ (sqrt (- d)) (sqrt (- h))) t_0))
     (if (<= h 8.8e-88)
       (* t_2 (* (* (sqrt d) (/ 1.0 (sqrt h))) t_0))
       (/
        (*
         (- 1.0 (/ (* (* (* h 0.5) (/ (* M D) d)) (* (* M D) 0.25)) (* l d)))
         (/ (fabs d) (sqrt h)))
        (sqrt l))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double t_1 = (0.5 * (M * D)) / d;
	double t_2 = 1.0 - (((h * t_1) / l) * (0.5 * t_1));
	double tmp;
	if (h <= -2e-310) {
		tmp = t_2 * ((sqrt(-d) / sqrt(-h)) * t_0);
	} else if (h <= 8.8e-88) {
		tmp = t_2 * ((sqrt(d) * (1.0 / sqrt(h))) * t_0);
	} else {
		tmp = ((1.0 - ((((h * 0.5) * ((M * D) / d)) * ((M * D) * 0.25)) / (l * d))) * (fabs(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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = (0.5d0 * (m * d_1)) / d
    t_2 = 1.0d0 - (((h * t_1) / l) * (0.5d0 * t_1))
    if (h <= (-2d-310)) then
        tmp = t_2 * ((sqrt(-d) / sqrt(-h)) * t_0)
    else if (h <= 8.8d-88) then
        tmp = t_2 * ((sqrt(d) * (1.0d0 / sqrt(h))) * t_0)
    else
        tmp = ((1.0d0 - ((((h * 0.5d0) * ((m * d_1) / d)) * ((m * d_1) * 0.25d0)) / (l * d))) * (abs(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 t_0 = Math.sqrt((d / l));
	double t_1 = (0.5 * (M * D)) / d;
	double t_2 = 1.0 - (((h * t_1) / l) * (0.5 * t_1));
	double tmp;
	if (h <= -2e-310) {
		tmp = t_2 * ((Math.sqrt(-d) / Math.sqrt(-h)) * t_0);
	} else if (h <= 8.8e-88) {
		tmp = t_2 * ((Math.sqrt(d) * (1.0 / Math.sqrt(h))) * t_0);
	} else {
		tmp = ((1.0 - ((((h * 0.5) * ((M * D) / d)) * ((M * D) * 0.25)) / (l * d))) * (Math.abs(d) / Math.sqrt(h))) / Math.sqrt(l);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l))
	t_1 = (0.5 * (M * D)) / d
	t_2 = 1.0 - (((h * t_1) / l) * (0.5 * t_1))
	tmp = 0
	if h <= -2e-310:
		tmp = t_2 * ((math.sqrt(-d) / math.sqrt(-h)) * t_0)
	elif h <= 8.8e-88:
		tmp = t_2 * ((math.sqrt(d) * (1.0 / math.sqrt(h))) * t_0)
	else:
		tmp = ((1.0 - ((((h * 0.5) * ((M * D) / d)) * ((M * D) * 0.25)) / (l * d))) * (math.fabs(d) / math.sqrt(h))) / math.sqrt(l)
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(Float64(0.5 * Float64(M * D)) / d)
	t_2 = Float64(1.0 - Float64(Float64(Float64(h * t_1) / l) * Float64(0.5 * t_1)))
	tmp = 0.0
	if (h <= -2e-310)
		tmp = Float64(t_2 * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_0));
	elseif (h <= 8.8e-88)
		tmp = Float64(t_2 * Float64(Float64(sqrt(d) * Float64(1.0 / sqrt(h))) * t_0));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(Float64(Float64(h * 0.5) * Float64(Float64(M * D) / d)) * Float64(Float64(M * D) * 0.25)) / Float64(l * d))) * Float64(abs(d) / sqrt(h))) / sqrt(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l));
	t_1 = (0.5 * (M * D)) / d;
	t_2 = 1.0 - (((h * t_1) / l) * (0.5 * t_1));
	tmp = 0.0;
	if (h <= -2e-310)
		tmp = t_2 * ((sqrt(-d) / sqrt(-h)) * t_0);
	elseif (h <= 8.8e-88)
		tmp = t_2 * ((sqrt(d) * (1.0 / sqrt(h))) * t_0);
	else
		tmp = ((1.0 - ((((h * 0.5) * ((M * D) / d)) * ((M * D) * 0.25)) / (l * d))) * (abs(d) / sqrt(h))) / sqrt(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * N[(M * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(N[(N[(h * t$95$1), $MachinePrecision] / l), $MachinePrecision] * N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -2e-310], N[(t$95$2 * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 8.8e-88], N[(t$95$2 * N[(N[(N[Sqrt[d], $MachinePrecision] * N[(1.0 / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(N[(N[(N[(h * 0.5), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.999999999999994e-310

   1. Initial program 64.2%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 68.1%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. /-lowering-/.f64 68.1

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

   6. Applied egg-rr 68.1%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. neg-lowering-neg.f64 80.6

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

   8. Applied egg-rr 80.6%

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

   if -1.999999999999994e-310 < h < 8.8000000000000002e-88

   1. Initial program 71.5%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 79.2%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. /-lowering-/.f64 79.2

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

   6. Applied egg-rr 79.2%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. unpow-prod-down N/A

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

      4. *-lowering-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow1/2 N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. sqrt-lowering-sqrt.f64 92.8

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

   8. Applied egg-rr 92.8%

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

   if 8.8000000000000002e-88 < h

   1. Initial program 68.5%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-div N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 74.3

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

   4. Applied egg-rr 74.3%

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

   6. Applied egg-rr 62.6%

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

      1. associate-*l/ N/A

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

      2. sqrt-div N/A

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

      3. /-lowering-/.f64 N/A

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

      4. rem-sqrt-square N/A

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

      5. fabs-lowering-fabs.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 84.3

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

   8. Applied egg-rr 84.3%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \frac{h \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \left(0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;h \leq 8.8 \cdot 10^{-88}:\\ \;\;\;\;\left(1 - \frac{h \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \left(0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)\right) \cdot \left(\left(\sqrt{d} \cdot \frac{1}{\sqrt{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{\left(\left(h \cdot 0.5\right) \cdot \frac{M \cdot D}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot 0.25\right)}{\ell \cdot d}\right) \cdot \frac{\left|d\right|}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5 \cdot \left(M \cdot D\right)}{d}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \frac{h \cdot t\_0}{\ell} \cdot \left(0.5 \cdot t\_0\right)\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{\left(\left(h \cdot 0.5\right) \cdot \frac{M \cdot D}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot 0.25\right)}{\ell \cdot d}\right) \cdot \frac{\left|d\right|}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (* 0.5 (* M D)) d)))
   (if (<= l -1e-310)
     (*
      (- 1.0 (* (/ (* h t_0) l) (* 0.5 t_0)))
      (* (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ d l))))
     (/
      (*
       (- 1.0 (/ (* (* (* h 0.5) (/ (* M D) d)) (* (* M D) 0.25)) (* l d)))
       (/ (fabs d) (sqrt h)))
      (sqrt l)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (0.5 * (M * D)) / d;
	double tmp;
	if (l <= -1e-310) {
		tmp = (1.0 - (((h * t_0) / l) * (0.5 * t_0))) * ((sqrt(-d) / sqrt(-h)) * sqrt((d / l)));
	} else {
		tmp = ((1.0 - ((((h * 0.5) * ((M * D) / d)) * ((M * D) * 0.25)) / (l * d))) * (fabs(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) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 * (m * d_1)) / d
    if (l <= (-1d-310)) then
        tmp = (1.0d0 - (((h * t_0) / l) * (0.5d0 * t_0))) * ((sqrt(-d) / sqrt(-h)) * sqrt((d / l)))
    else
        tmp = ((1.0d0 - ((((h * 0.5d0) * ((m * d_1) / d)) * ((m * d_1) * 0.25d0)) / (l * d))) * (abs(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 t_0 = (0.5 * (M * D)) / d;
	double tmp;
	if (l <= -1e-310) {
		tmp = (1.0 - (((h * t_0) / l) * (0.5 * t_0))) * ((Math.sqrt(-d) / Math.sqrt(-h)) * Math.sqrt((d / l)));
	} else {
		tmp = ((1.0 - ((((h * 0.5) * ((M * D) / d)) * ((M * D) * 0.25)) / (l * d))) * (Math.abs(d) / Math.sqrt(h))) / Math.sqrt(l);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (0.5 * (M * D)) / d
	tmp = 0
	if l <= -1e-310:
		tmp = (1.0 - (((h * t_0) / l) * (0.5 * t_0))) * ((math.sqrt(-d) / math.sqrt(-h)) * math.sqrt((d / l)))
	else:
		tmp = ((1.0 - ((((h * 0.5) * ((M * D) / d)) * ((M * D) * 0.25)) / (l * d))) * (math.fabs(d) / math.sqrt(h))) / math.sqrt(l)
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(0.5 * Float64(M * D)) / d)
	tmp = 0.0
	if (l <= -1e-310)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(h * t_0) / l) * Float64(0.5 * t_0))) * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * sqrt(Float64(d / l))));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(Float64(Float64(h * 0.5) * Float64(Float64(M * D) / d)) * Float64(Float64(M * D) * 0.25)) / Float64(l * d))) * Float64(abs(d) / sqrt(h))) / sqrt(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (0.5 * (M * D)) / d;
	tmp = 0.0;
	if (l <= -1e-310)
		tmp = (1.0 - (((h * t_0) / l) * (0.5 * t_0))) * ((sqrt(-d) / sqrt(-h)) * sqrt((d / l)));
	else
		tmp = ((1.0 - ((((h * 0.5) * ((M * D) / d)) * ((M * D) * 0.25)) / (l * d))) * (abs(d) / sqrt(h))) / sqrt(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(0.5 * N[(M * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[l, -1e-310], N[(N[(1.0 - N[(N[(N[(h * t$95$0), $MachinePrecision] / l), $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(N[(N[(N[(h * 0.5), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -9.999999999999969e-311

   1. Initial program 64.2%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 68.1%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. /-lowering-/.f64 68.1

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

   6. Applied egg-rr 68.1%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. neg-lowering-neg.f64 80.6

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

   8. Applied egg-rr 80.6%

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

   if -9.999999999999969e-311 < l

   1. Initial program 69.6%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-div N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 74.0

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

   4. Applied egg-rr 74.0%

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

   6. Applied egg-rr 63.0%

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

      1. associate-*l/ N/A

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

      2. sqrt-div N/A

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

      3. /-lowering-/.f64 N/A

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

      4. rem-sqrt-square N/A

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

      5. fabs-lowering-fabs.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 84.3

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

   8. Applied egg-rr 84.3%

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

4. Final simplification 82.5%

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

Alternative 8: 79.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := \frac{0.5 \cdot \left(M \cdot D\right)}{d}\\ \mathbf{if}\;h \leq -3.9 \cdot 10^{+151}:\\ \;\;\;\;\left(\frac{t\_0}{\sqrt{-h}} \cdot \frac{t\_0}{\sqrt{-\ell}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(\ell \cdot d\right)}\right)\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{h \cdot t\_1}{\ell} \cdot \left(0.5 \cdot t\_1\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{\left(\left(h \cdot 0.5\right) \cdot \frac{M \cdot D}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot 0.25\right)}{\ell \cdot d}\right) \cdot \frac{\left|d\right|}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (- d))) (t_1 (/ (* 0.5 (* M D)) d)))
   (if (<= h -3.9e+151)
     (*
      (* (/ t_0 (sqrt (- h))) (/ t_0 (sqrt (- l))))
      (- 1.0 (/ (* D (* D (* 0.125 (* h (* M M))))) (* d (* l d)))))
     (if (<= h -2e-310)
       (*
        (* d (sqrt (/ 1.0 (* l h))))
        (+ (* (/ (* h t_1) l) (* 0.5 t_1)) -1.0))
       (/
        (*
         (- 1.0 (/ (* (* (* h 0.5) (/ (* M D) d)) (* (* M D) 0.25)) (* l d)))
         (/ (fabs d) (sqrt h)))
        (sqrt l))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(-d);
	double t_1 = (0.5 * (M * D)) / d;
	double tmp;
	if (h <= -3.9e+151) {
		tmp = ((t_0 / sqrt(-h)) * (t_0 / sqrt(-l))) * (1.0 - ((D * (D * (0.125 * (h * (M * M))))) / (d * (l * d))));
	} else if (h <= -2e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * ((((h * t_1) / l) * (0.5 * t_1)) + -1.0);
	} else {
		tmp = ((1.0 - ((((h * 0.5) * ((M * D) / d)) * ((M * D) * 0.25)) / (l * d))) * (fabs(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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(-d)
    t_1 = (0.5d0 * (m * d_1)) / d
    if (h <= (-3.9d+151)) then
        tmp = ((t_0 / sqrt(-h)) * (t_0 / sqrt(-l))) * (1.0d0 - ((d_1 * (d_1 * (0.125d0 * (h * (m * m))))) / (d * (l * d))))
    else if (h <= (-2d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((((h * t_1) / l) * (0.5d0 * t_1)) + (-1.0d0))
    else
        tmp = ((1.0d0 - ((((h * 0.5d0) * ((m * d_1) / d)) * ((m * d_1) * 0.25d0)) / (l * d))) * (abs(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 t_0 = Math.sqrt(-d);
	double t_1 = (0.5 * (M * D)) / d;
	double tmp;
	if (h <= -3.9e+151) {
		tmp = ((t_0 / Math.sqrt(-h)) * (t_0 / Math.sqrt(-l))) * (1.0 - ((D * (D * (0.125 * (h * (M * M))))) / (d * (l * d))));
	} else if (h <= -2e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * ((((h * t_1) / l) * (0.5 * t_1)) + -1.0);
	} else {
		tmp = ((1.0 - ((((h * 0.5) * ((M * D) / d)) * ((M * D) * 0.25)) / (l * d))) * (Math.abs(d) / Math.sqrt(h))) / Math.sqrt(l);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(-d)
	t_1 = (0.5 * (M * D)) / d
	tmp = 0
	if h <= -3.9e+151:
		tmp = ((t_0 / math.sqrt(-h)) * (t_0 / math.sqrt(-l))) * (1.0 - ((D * (D * (0.125 * (h * (M * M))))) / (d * (l * d))))
	elif h <= -2e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * ((((h * t_1) / l) * (0.5 * t_1)) + -1.0)
	else:
		tmp = ((1.0 - ((((h * 0.5) * ((M * D) / d)) * ((M * D) * 0.25)) / (l * d))) * (math.fabs(d) / math.sqrt(h))) / math.sqrt(l)
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(-d))
	t_1 = Float64(Float64(0.5 * Float64(M * D)) / d)
	tmp = 0.0
	if (h <= -3.9e+151)
		tmp = Float64(Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(t_0 / sqrt(Float64(-l)))) * Float64(1.0 - Float64(Float64(D * Float64(D * Float64(0.125 * Float64(h * Float64(M * M))))) / Float64(d * Float64(l * d)))));
	elseif (h <= -2e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(Float64(Float64(Float64(h * t_1) / l) * Float64(0.5 * t_1)) + -1.0));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(Float64(Float64(h * 0.5) * Float64(Float64(M * D) / d)) * Float64(Float64(M * D) * 0.25)) / Float64(l * d))) * Float64(abs(d) / sqrt(h))) / sqrt(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(-d);
	t_1 = (0.5 * (M * D)) / d;
	tmp = 0.0;
	if (h <= -3.9e+151)
		tmp = ((t_0 / sqrt(-h)) * (t_0 / sqrt(-l))) * (1.0 - ((D * (D * (0.125 * (h * (M * M))))) / (d * (l * d))));
	elseif (h <= -2e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * ((((h * t_1) / l) * (0.5 * t_1)) + -1.0);
	else
		tmp = ((1.0 - ((((h * 0.5) * ((M * D) / d)) * ((M * D) * 0.25)) / (l * d))) * (abs(d) / sqrt(h))) / sqrt(l);
	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[(N[(0.5 * N[(M * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[h, -3.9e+151], N[(N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(D * N[(D * N[(0.125 * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -2e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(h * t$95$1), $MachinePrecision] / l), $MachinePrecision] * N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(N[(N[(N[(h * 0.5), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -3.89999999999999976e151

   1. Initial program 53.4%

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

   3. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{8} \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right) \cdot D}}{{d}^{2} \cdot \ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{D \cdot \left(\frac{1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot D\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot D\right)}}{{d}^{2} \cdot \ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \color{blue}{\left(D \cdot \left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{{d}^{2} \cdot \ell}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({M}^{2} \cdot h\right)\right)}\right)}{{d}^{2} \cdot \ell}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)}{{d}^{2} \cdot \ell}\right) \]

      17. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      18. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      20. *-lowering-*.f64 51.6

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

   5. Simplified 51.6%

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

      1. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      2. unpow1/2 N/A

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

      3. frac-2neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      4. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      7. neg-lowering-neg.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{\color{blue}{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{\mathsf{neg}\left(d\right)}}{\color{blue}{\sqrt{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      9. neg-lowering-neg.f64 51.6

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

   7. Applied egg-rr 51.6%

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

      1. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      2. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      3. frac-2neg N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(h\right)}}} \cdot \frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      4. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot \frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot \frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot \frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      7. neg-lowering-neg.f64 N/A

         \[\leadsto \left(\frac{\sqrt{\color{blue}{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot \frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot \frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(\frac{1}{8} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \]

      9. neg-lowering-neg.f64 68.6

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

   9. Applied egg-rr 68.6%

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

   if -3.89999999999999976e151 < h < -1.999999999999994e-310

   1. Initial program 68.8%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 73.2%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. /-lowering-/.f64 73.2

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

   6. Applied egg-rr 73.2%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. /-lowering-/.f64 72.7

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

   8. Applied egg-rr 72.7%

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

   9. Taylor expanded in d around -inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. sqrt-lowering-sqrt.f64 N/A

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

       5. /-lowering-/.f64 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. mul-1-neg N/A

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

       8. neg-lowering-neg.f64 85.7

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

   11. Simplified 85.7%

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

   if -1.999999999999994e-310 < h

   1. Initial program 69.6%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-div N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 74.0

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

   4. Applied egg-rr 74.0%

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

   6. Applied egg-rr 63.0%

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

      1. associate-*l/ N/A

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

      2. sqrt-div N/A

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

      3. /-lowering-/.f64 N/A

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

      4. rem-sqrt-square N/A

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

      5. fabs-lowering-fabs.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 84.3

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

   8. Applied egg-rr 84.3%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -3.9 \cdot 10^{+151}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - \frac{D \cdot \left(D \cdot \left(0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(\ell \cdot d\right)}\right)\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{h \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \left(0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{\left(\left(h \cdot 0.5\right) \cdot \frac{M \cdot D}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot 0.25\right)}{\ell \cdot d}\right) \cdot \frac{\left|d\right|}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(M \cdot D\right)\\ t_1 := \frac{t\_0}{d}\\ t_2 := \frac{h \cdot t\_1}{\ell} \cdot \left(0.5 \cdot t\_1\right)\\ t_3 := d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ t_4 := \left(1 - t\_2\right) \cdot t\_3\\ \mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t\_3 \cdot \left(t\_2 + -1\right)\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-69}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;d \leq 4.7 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{\left|d\right|}{\sqrt{h}} \cdot \left(1 - \frac{\left(h \cdot t\_0\right) \cdot \left(M \cdot \left(D \cdot 0.25\right)\right)}{\ell \cdot \left(d \cdot d\right)}\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* 0.5 (* M D)))
        (t_1 (/ t_0 d))
        (t_2 (* (/ (* h t_1) l) (* 0.5 t_1)))
        (t_3 (* d (sqrt (/ 1.0 (* l h)))))
        (t_4 (* (- 1.0 t_2) t_3)))
   (if (<= d -2e-310)
     (* t_3 (+ t_2 -1.0))
     (if (<= d 4e-69)
       t_4
       (if (<= d 4.7e+148)
         (/
          (*
           (/ (fabs d) (sqrt h))
           (- 1.0 (/ (* (* h t_0) (* M (* D 0.25))) (* l (* d d)))))
          (sqrt l))
         t_4)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 0.5 * (M * D);
	double t_1 = t_0 / d;
	double t_2 = ((h * t_1) / l) * (0.5 * t_1);
	double t_3 = d * sqrt((1.0 / (l * h)));
	double t_4 = (1.0 - t_2) * t_3;
	double tmp;
	if (d <= -2e-310) {
		tmp = t_3 * (t_2 + -1.0);
	} else if (d <= 4e-69) {
		tmp = t_4;
	} else if (d <= 4.7e+148) {
		tmp = ((fabs(d) / sqrt(h)) * (1.0 - (((h * t_0) * (M * (D * 0.25))) / (l * (d * d))))) / sqrt(l);
	} else {
		tmp = t_4;
	}
	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) :: t_4
    real(8) :: tmp
    t_0 = 0.5d0 * (m * d_1)
    t_1 = t_0 / d
    t_2 = ((h * t_1) / l) * (0.5d0 * t_1)
    t_3 = d * sqrt((1.0d0 / (l * h)))
    t_4 = (1.0d0 - t_2) * t_3
    if (d <= (-2d-310)) then
        tmp = t_3 * (t_2 + (-1.0d0))
    else if (d <= 4d-69) then
        tmp = t_4
    else if (d <= 4.7d+148) then
        tmp = ((abs(d) / sqrt(h)) * (1.0d0 - (((h * t_0) * (m * (d_1 * 0.25d0))) / (l * (d * d))))) / sqrt(l)
    else
        tmp = t_4
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 0.5 * (M * D);
	double t_1 = t_0 / d;
	double t_2 = ((h * t_1) / l) * (0.5 * t_1);
	double t_3 = d * Math.sqrt((1.0 / (l * h)));
	double t_4 = (1.0 - t_2) * t_3;
	double tmp;
	if (d <= -2e-310) {
		tmp = t_3 * (t_2 + -1.0);
	} else if (d <= 4e-69) {
		tmp = t_4;
	} else if (d <= 4.7e+148) {
		tmp = ((Math.abs(d) / Math.sqrt(h)) * (1.0 - (((h * t_0) * (M * (D * 0.25))) / (l * (d * d))))) / Math.sqrt(l);
	} else {
		tmp = t_4;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = 0.5 * (M * D)
	t_1 = t_0 / d
	t_2 = ((h * t_1) / l) * (0.5 * t_1)
	t_3 = d * math.sqrt((1.0 / (l * h)))
	t_4 = (1.0 - t_2) * t_3
	tmp = 0
	if d <= -2e-310:
		tmp = t_3 * (t_2 + -1.0)
	elif d <= 4e-69:
		tmp = t_4
	elif d <= 4.7e+148:
		tmp = ((math.fabs(d) / math.sqrt(h)) * (1.0 - (((h * t_0) * (M * (D * 0.25))) / (l * (d * d))))) / math.sqrt(l)
	else:
		tmp = t_4
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(0.5 * Float64(M * D))
	t_1 = Float64(t_0 / d)
	t_2 = Float64(Float64(Float64(h * t_1) / l) * Float64(0.5 * t_1))
	t_3 = Float64(d * sqrt(Float64(1.0 / Float64(l * h))))
	t_4 = Float64(Float64(1.0 - t_2) * t_3)
	tmp = 0.0
	if (d <= -2e-310)
		tmp = Float64(t_3 * Float64(t_2 + -1.0));
	elseif (d <= 4e-69)
		tmp = t_4;
	elseif (d <= 4.7e+148)
		tmp = Float64(Float64(Float64(abs(d) / sqrt(h)) * Float64(1.0 - Float64(Float64(Float64(h * t_0) * Float64(M * Float64(D * 0.25))) / Float64(l * Float64(d * d))))) / sqrt(l));
	else
		tmp = t_4;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = 0.5 * (M * D);
	t_1 = t_0 / d;
	t_2 = ((h * t_1) / l) * (0.5 * t_1);
	t_3 = d * sqrt((1.0 / (l * h)));
	t_4 = (1.0 - t_2) * t_3;
	tmp = 0.0;
	if (d <= -2e-310)
		tmp = t_3 * (t_2 + -1.0);
	elseif (d <= 4e-69)
		tmp = t_4;
	elseif (d <= 4.7e+148)
		tmp = ((abs(d) / sqrt(h)) * (1.0 - (((h * t_0) * (M * (D * 0.25))) / (l * (d * d))))) / sqrt(l);
	else
		tmp = t_4;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(0.5 * N[(M * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / d), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(h * t$95$1), $MachinePrecision] / l), $MachinePrecision] * N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(1.0 - t$95$2), $MachinePrecision] * t$95$3), $MachinePrecision]}, If[LessEqual[d, -2e-310], N[(t$95$3 * N[(t$95$2 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4e-69], t$95$4, If[LessEqual[d, 4.7e+148], N[(N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(h * t$95$0), $MachinePrecision] * N[(M * N[(D * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.999999999999994e-310

   1. Initial program 64.2%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 68.1%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. /-lowering-/.f64 68.1

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

   6. Applied egg-rr 68.1%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. /-lowering-/.f64 67.8

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

   8. Applied egg-rr 67.8%

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

   9. Taylor expanded in d around -inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. sqrt-lowering-sqrt.f64 N/A

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

       5. /-lowering-/.f64 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. mul-1-neg N/A

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

       8. neg-lowering-neg.f64 77.1

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

   11. Simplified 77.1%

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

   if -1.999999999999994e-310 < d < 3.9999999999999999e-69 or 4.6999999999999997e148 < d

   1. Initial program 68.6%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 73.0%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. /-lowering-/.f64 73.0

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

   6. Applied egg-rr 73.0%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. /-lowering-/.f64 73.0

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

   8. Applied egg-rr 73.0%

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

   9. Taylor expanded in h around 0

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

       1. *-lowering-*.f64 N/A

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

       2. sqrt-lowering-sqrt.f64 N/A

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

       3. /-lowering-/.f64 N/A

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

       4. *-lowering-*.f64 79.2

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

   11. Simplified 79.2%

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

   if 3.9999999999999999e-69 < d < 4.6999999999999997e148

   1. Initial program 71.4%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-div N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 73.4

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

   4. Applied egg-rr 73.4%

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

   6. Applied egg-rr 71.2%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. sqrt-div N/A

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

      5. /-lowering-/.f64 N/A

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

      6. rem-sqrt-square N/A

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

      7. fabs-lowering-fabs.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. associate-/l* N/A

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

      11. associate-*r/ N/A

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

      12. frac-times N/A

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

      13. /-lowering-/.f64 N/A

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

   8. Applied egg-rr 86.8%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{h \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \left(0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}\right) + -1\right)\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-69}:\\ \;\;\;\;\left(1 - \frac{h \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \left(0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)\right) \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;d \leq 4.7 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{\left|d\right|}{\sqrt{h}} \cdot \left(1 - \frac{\left(h \cdot \left(0.5 \cdot \left(M \cdot D\right)\right)\right) \cdot \left(M \cdot \left(D \cdot 0.25\right)\right)}{\ell \cdot \left(d \cdot d\right)}\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{h \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}}{\ell} \cdot \left(0.5 \cdot \frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)\right) \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5 \cdot \left(M \cdot D\right)}{d}\\ \mathbf{if}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(\frac{h \cdot t\_0}{\ell} \cdot \left(0.5 \cdot t\_0\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{\left(\left(h \cdot 0.5\right) \cdot \frac{M \cdot D}{d}\right) \cdot \left(\left(M \cdot D\right) \cdot 0.25\right)}{\ell \cdot d}\right) \cdot \frac{\left|d\right|}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (* 0.5 (* M D)) d)))
   (if (<= h -2e-310)
     (* (* d (sqrt (/ 1.0 (* l h)))) (+ (* (/ (* h t_0) l) (* 0.5 t_0)) -1.0))
     (/
      (*
       (- 1.0 (/ (* (* (* h 0.5) (/ (* M D) d)) (* (* M D) 0.25)) (* l d)))
       (/ (fabs d) (sqrt h)))
      (sqrt l)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (0.5 * (M * D)) / d;
	double tmp;
	if (h <= -2e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * ((((h * t_0) / l) * (0.5 * t_0)) + -1.0);
	} else {
		tmp = ((1.0 - ((((h * 0.5) * ((M * D) / d)) * ((M * D) * 0.25)) / (l * d))) * (fabs(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) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 * (m * d_1)) / d
    if (h <= (-2d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((((h * t_0) / l) * (0.5d0 * t_0)) + (-1.0d0))
    else
        tmp = ((1.0d0 - ((((h * 0.5d0) * ((m * d_1) / d)) * ((m * d_1) * 0.25d0)) / (l * d))) * (abs(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 t_0 = (0.5 * (M * D)) / d;
	double tmp;
	if (h <= -2e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * ((((h * t_0) / l) * (0.5 * t_0)) + -1.0);
	} else {
		tmp = ((1.0 - ((((h * 0.5) * ((M * D) / d)) * ((M * D) * 0.25)) / (l * d))) * (Math.abs(d) / Math.sqrt(h))) / Math.sqrt(l);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (0.5 * (M * D)) / d
	tmp = 0
	if h <= -2e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * ((((h * t_0) / l) * (0.5 * t_0)) + -1.0)
	else:
		tmp = ((1.0 - ((((h * 0.5) * ((M * D) / d)) * ((M * D) * 0.25)) / (l * d))) * (math.fabs(d) / math.sqrt(h))) / math.sqrt(l)
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(0.5 * Float64(M * D)) / d)
	tmp = 0.0
	if (h <= -2e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(Float64(Float64(Float64(h * t_0) / l) * Float64(0.5 * t_0)) + -1.0));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(Float64(Float64(h * 0.5) * Float64(Float64(M * D) / d)) * Float64(Float64(M * D) * 0.25)) / Float64(l * d))) * Float64(abs(d) / sqrt(h))) / sqrt(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (0.5 * (M * D)) / d;
	tmp = 0.0;
	if (h <= -2e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * ((((h * t_0) / l) * (0.5 * t_0)) + -1.0);
	else
		tmp = ((1.0 - ((((h * 0.5) * ((M * D) / d)) * ((M * D) * 0.25)) / (l * d))) * (abs(d) / sqrt(h))) / sqrt(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(0.5 * N[(M * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[h, -2e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(h * t$95$0), $MachinePrecision] / l), $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(N[(N[(N[(h * 0.5), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < -1.999999999999994e-310

   1. Initial program 64.2%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 68.1%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. /-lowering-/.f64 68.1

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

   6. Applied egg-rr 68.1%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. /-lowering-/.f64 67.8

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

   8. Applied egg-rr 67.8%

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

   9. Taylor expanded in d around -inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. sqrt-lowering-sqrt.f64 N/A

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

       5. /-lowering-/.f64 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. mul-1-neg N/A

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

       8. neg-lowering-neg.f64 77.1

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

   11. Simplified 77.1%

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

   if -1.999999999999994e-310 < h

   1. Initial program 69.6%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-div N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 74.0

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

   4. Applied egg-rr 74.0%

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

   6. Applied egg-rr 63.0%

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

      1. associate-*l/ N/A

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

      2. sqrt-div N/A

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

      3. /-lowering-/.f64 N/A

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

      4. rem-sqrt-square N/A

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

      5. fabs-lowering-fabs.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 84.3

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

   8. Applied egg-rr 84.3%

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

4. Final simplification 80.8%

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

Alternative 11: 47.7% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -4.7 \cdot 10^{+140}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;h \leq 1.65 \cdot 10^{-308}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h -4.7e+140)
   (* (sqrt (/ d l)) (sqrt (/ d h)))
   (if (<= h 1.65e-308)
     (* (- d) (sqrt (/ 1.0 (* l h))))
     (/ d (* (sqrt h) (sqrt l))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -4.7e+140) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (h <= 1.65e-308) {
		tmp = -d * sqrt((1.0 / (l * h)));
	} else {
		tmp = 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 (h <= (-4.7d+140)) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else if (h <= 1.65d-308) then
        tmp = -d * sqrt((1.0d0 / (l * h)))
    else
        tmp = 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 (h <= -4.7e+140) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (h <= 1.65e-308) {
		tmp = -d * Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= -4.7e+140:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	elif h <= 1.65e-308:
		tmp = -d * math.sqrt((1.0 / (l * h)))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -4.7e+140)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (h <= 1.65e-308)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(l * h))));
	else
		tmp = 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 (h <= -4.7e+140)
		tmp = sqrt((d / l)) * sqrt((d / h));
	elseif (h <= 1.65e-308)
		tmp = -d * sqrt((1.0 / (l * h)));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, -4.7e+140], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 1.65e-308], N[((-d) * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -4.7000000000000003e140

   1. Initial program 56.6%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 59.1%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. /-lowering-/.f64 59.1

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

   6. Applied egg-rr 59.1%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. /-lowering-/.f64 59.1

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

   8. Applied egg-rr 59.1%

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

   9. Taylor expanded in M around 0

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

   11. Simplified 34.4%

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

   if -4.7000000000000003e140 < h < 1.6499999999999999e-308

   1. Initial program 67.9%

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

      1. unpow2 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. times-frac N/A

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

      9. associate-/r* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. /-lowering-/.f64 N/A

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

      17. *-lowering-*.f64 64.2

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

   4. Applied egg-rr 64.2%

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

   5. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-lowering-neg.f64 51.2

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

   7. Simplified 51.2%

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

   if 1.6499999999999999e-308 < h

   1. Initial program 69.6%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-div N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 74.0

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

   4. Applied egg-rr 74.0%

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

   5. Taylor expanded in d around inf

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 38.7

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

   7. Simplified 38.7%

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

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. *-lowering-*.f64 38.8

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

   9. Applied egg-rr 38.8%

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

       1. *-commutative N/A

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

       2. sqrt-prod N/A

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

       3. *-lowering-*.f64 N/A

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

       4. sqrt-lowering-sqrt.f64 N/A

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

       5. sqrt-lowering-sqrt.f64 48.7

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

   11. Applied egg-rr 48.7%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -4.7 \cdot 10^{+140}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;h \leq 1.65 \cdot 10^{-308}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.4% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{-221}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 2.5e-221)
   (* (- d) (sqrt (/ 1.0 (* l h))))
   (/ d (* (sqrt h) (sqrt l)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 2.5e-221) {
		tmp = -d * sqrt((1.0 / (l * h)));
	} else {
		tmp = 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 <= 2.5d-221) then
        tmp = -d * sqrt((1.0d0 / (l * h)))
    else
        tmp = 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 <= 2.5e-221) {
		tmp = -d * Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= 2.5e-221:
		tmp = -d * math.sqrt((1.0 / (l * h)))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 2.5e-221)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(l * h))));
	else
		tmp = 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 <= 2.5e-221)
		tmp = -d * sqrt((1.0 / (l * h)));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 2.5e-221], N[((-d) * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.49999999999999998e-221

   1. Initial program 65.7%

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

      1. unpow2 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. times-frac N/A

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

      9. associate-/r* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. /-lowering-/.f64 N/A

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

      17. *-lowering-*.f64 61.5

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

   4. Applied egg-rr 61.5%

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

   5. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-lowering-neg.f64 41.2

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

   7. Simplified 41.2%

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

   if 2.49999999999999998e-221 < l

   1. Initial program 68.7%

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

      1. metadata-eval N/A

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

      2. unpow1/2 N/A

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

      3. sqrt-div N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 73.9

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

   4. Applied egg-rr 73.9%

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

   5. Taylor expanded in d around inf

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 40.2

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

   7. Simplified 40.2%

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

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. *-lowering-*.f64 40.3

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

   9. Applied egg-rr 40.3%

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

       1. *-commutative N/A

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

       2. sqrt-prod N/A

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

       3. *-lowering-*.f64 N/A

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

       4. sqrt-lowering-sqrt.f64 N/A

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

       5. sqrt-lowering-sqrt.f64 51.2

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

   11. Applied egg-rr 51.2%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{-221}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 43.1% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7.6 \cdot 10^{-222}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 7.6e-222) (* (- d) (sqrt (/ 1.0 (* l h)))) (/ d (sqrt (* l h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 7.6e-222) {
		tmp = -d * sqrt((1.0 / (l * h)));
	} else {
		tmp = d / sqrt((l * h));
	}
	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 <= 7.6d-222) then
        tmp = -d * sqrt((1.0d0 / (l * h)))
    else
        tmp = d / sqrt((l * h))
    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 <= 7.6e-222) {
		tmp = -d * Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = d / Math.sqrt((l * h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= 7.6e-222:
		tmp = -d * math.sqrt((1.0 / (l * h)))
	else:
		tmp = d / math.sqrt((l * h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 7.6e-222)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(l * h))));
	else
		tmp = Float64(d / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 7.6e-222)
		tmp = -d * sqrt((1.0 / (l * h)));
	else
		tmp = d / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 7.6e-222], N[((-d) * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.59999999999999993e-222

   1. Initial program 65.7%

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

      1. unpow2 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. times-frac N/A

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

      9. associate-/r* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. /-lowering-/.f64 N/A

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

      17. *-lowering-*.f64 61.5

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

   4. Applied egg-rr 61.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{D}{d \cdot 2} \cdot \frac{\frac{M}{d}}{\frac{2}{M \cdot D}}\right)}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      3. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. neg-lowering-neg.f64 41.2

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   7. Simplified 41.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if 7.59999999999999993e-222 < l

   1. Initial program 68.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-lowering-sqrt.f64 73.9

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied egg-rr 73.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. *-lowering-*.f64 40.2

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   7. Simplified 40.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. sqrt-div N/A

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval N/A

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      6. *-lowering-*.f64 40.3

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   9. Applied egg-rr 40.3%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.6 \cdot 10^{-222}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 27.0% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (/ d (sqrt (* l h))))

C

double code(double d, double h, double l, double M, double D) {
	return d / sqrt((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((l * h))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d / Math.sqrt((l * h));
}

Python

def code(d, h, l, M, D):
	return d / math.sqrt((l * h))

Julia

function code(d, h, l, M, D)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d / sqrt((l * h));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.0%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. unpow1/2 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. sqrt-div N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. sqrt-lowering-sqrt.f64 38.2

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

4. Applied egg-rr 38.2%

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

5. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

   4. *-lowering-*.f64 25.1

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

7. Simplified 25.1%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
8. Step-by-step derivation

   1. sqrt-div N/A

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

   2. metadata-eval N/A

      \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

   3. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   6. *-lowering-*.f64 25.2

      \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

9. Applied egg-rr 25.2%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

10. Final simplification 25.2%

    \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(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)))))